This substantially revised and restructured second edition of an essential textbook presents dynamics and phase diagrams for advanced undergraduate and graduate courses in economic theory and quantitative economics. It offers an integrated analysis of dynamics that includes many more exercises and examples and a more comprehensive range of applications to economic theory. The user-friendly text is supported by a companion website offering a solutions manual and learning tools for teachers, students and researchers. First Edition Hb (1997): 0-521-47446-9 First Edition Pb (1997): 0-521-47973-8

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Economic Dynamics Phase Diagrams and Their Economic Application Second Edition

This is the substantially revised and restructured second edition of Ron Shone’s successful undergraduate and graduate textbook Economic Dynamics. The book provides detailed coverage of dynamics and phase diagrams including: quantitative and qualitative dynamic systems, continuous and discrete dynamics, linear and nonlinear systems and single equation and systems of equations. It illustrates dynamic systems using Mathematica, Maple and spreadsheets. It provides a thorough introduction to phase diagrams and their economic application and explains the nature of saddle path solutions. The second edition contains a new chapter on oligopoly and an extended treatment of stability of discrete dynamic systems and the solving of ﬁrst-order difference equations. Detailed routines on the use of Mathematica and Maple are now contained in the body of the text, which now also includes advice on the use of Excel and additional examples and exercises throughout. The supporting website contains a solutions manual and learning tools. ronald shone is Senior Lecturer in Economics at the University of Stirling. He is the author of eight books on economics covering the areas of microeconomics, macroeconomics and international economics at both undergraduate and postgraduate level. He has written a number of articles published in Oxford Economic Papers, the Economic Journal, Journal of Economic Surveys and Journal of Economic Studies.

Economic Dynamics Phase Diagrams and Their Economic Application Second Edition RONALD SHONE University of Stirling

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521816847 © Ronald Shone 2002 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2002 - -

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Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface to the second edition Preface to the ﬁrst edition

page xi xiii

PART I Dynamic modelling 1

Introduction 1.1 What this book is about 1.2 The rise in economic dynamics 1.3 Stocks, ﬂows and dimensionality 1.4 Nonlinearities, multiple equilibria and local stability 1.5 Nonlinearity and chaos 1.6 Computer software and economic dynamics 1.7 Mathematica and Maple 1.8 Structure and features Additional reading

3 3 5 8 12 15 17 20 24 25

2

Continuous dynamic systems 2.1 Some deﬁnitions 2.2 Solutions to ﬁrst-order linear differential equations 2.3 Compound interest 2.4 First-order equations and isoclines 2.5 Separable functions 2.6 Diffusion models 2.7 Phase portrait of a single variable 2.8 Second-order linear homogeneous equations 2.9 Second-order linear nonhomogeneous equations 2.10 Linear approximations to nonlinear differential equations 2.11 Solving differential equations with Mathematica 2.12 Solving differential equations with Maple Appendix 2.1 Plotting direction ﬁelds for a single equation with Mathematica Appendix 2.2 Plotting direction ﬁelds for a single equation with Maple Exercises Additional reading

26 26 37 39 41 45 53 54 59 64 66 70 73 77 79 80 84

vi

Contents 3

4

5

Discrete dynamic systems 3.1 Classifying discrete dynamic systems 3.2 The initial value problem 3.3 The cobweb model: an introduction 3.4 Equilibrium and stability of discrete dynamic systems 3.5 Solving ﬁrst-order difference equations 3.6 Compound interest 3.7 Discounting, present value and internal rates of return 3.8 Solving second-order difference equations 3.9 The logistic equation: discrete version 3.10 The multiplier–accelerator model 3.11 Linear approximation to discrete nonlinear difference equations 3.12 Solow growth model in discrete time 3.13 Solving recursive equations with Mathematica and Maple Appendix 3.1 Two-cycle logistic equation using Mathematica Appendix 3.2 Two-cycle logistic equation using Maple Exercises Additional reading

85 85 86 87 88 99 105 108 110 118 123

Systems of first-order differential equations 4.1 Deﬁnitions and autonomous systems 4.2 The phase plane, ﬁxed points and stability 4.3 Vectors of forces in the phase plane 4.4 Matrix speciﬁcation of autonomous systems 4.5 Solutions to the homogeneous differential equation system: real distinct roots 4.6 Solutions with repeating roots 4.7 Solutions with complex roots 4.8 Nodes, spirals and saddles 4.9 Stability/instability and its matrix speciﬁcation 4.10 Limit cycles 4.11 Euler’s approximation and differential equations on a spreadsheet 4.12 Solving systems of differential equations with Mathematica and Maple Appendix 4.1 Parametric plots in the phase plane: continuous variables Exercises Additional reading

142 142 145 149 156

Discrete systems of equations 5.1 Introduction 5.2 Basic matrices with Mathematica and Maple 5.3 Eigenvalues and eigenvectors

201 201 204 208

127 130 131 135 137 138 141

160 162 164 166 178 179 183 186 194 196 200

Contents

6

7

5.4 Mathematica and Maple for solving discrete systems 5.5 Graphing trajectories of discrete systems 5.6 The stability of discrete systems 5.7 The phase plane analysis of discrete systems 5.8 Internal and external balance 5.9 Nonlinear discrete systems Exercises Additional reading

214 220 223 235 239 245 247 250

Optimal control theory 6.1 The optimal control problem 6.2 The Pontryagin maximum principle: continuous model 6.3 The Pontryagin maximum principle: discrete model 6.4 Optimal control with discounting 6.5 The phase diagram approach to continuous time control models Exercises Additional reading

251 251 252 259 265

Chaos theory 7.1 Introduction 7.2 Bifurcations: single-variable case 7.3 The logistic equation, periodic-doubling bifurcations and chaos 7.4 Feigenbaum’s universal constant 7.5 Sarkovskii theorem 7.6 Van der Pol equation and Hopf bifurcations 7.7 Strange attractors 7.8 Rational choice and erratic behaviour 7.9 Inventory dynamics under rational expectations Exercises Additional reading

270 283 285 286 286 287 293 301 302 304 307 312 315 319 321

PART II Applied economic dynamics 8

Demand and supply models 8.1 Introduction 8.2 A simple demand and supply model in continuous time 8.3 The cobweb model 8.4 Cobwebs with Mathematica and Maple 8.5 Cobwebs in the phase plane 8.6 Cobwebs in two interrelated markets 8.7 Demand and supply with stocks 8.8 Stability of the competitive equilibrium 8.9 The housing market and demographic changes 8.10 Chaotic demand and supply

325 325 326 332 338 339 346 349 353 358 363

vii

viii

Contents Appendix 8.1 Obtaining cobwebs using Mathematica and Maple Exercises Additional reading 9

Dynamic theory of oligopoly 9.1 Static model of duopoly 9.2 Discrete oligopoly models with output adjusting completely and instantaneously 9.3 Discrete oligopoly models with output not adjusting completely and instantaneously 9.4 Continuous modelling of oligopoly 9.5 A nonlinear model of duopolistic competition (R&D) 9.6 Schumpeterian dynamics Exercises Additional reading

367 371 374 375 375 377 389 398 405 414 419 423

10

Closed economy dynamics 10.1 Goods market dynamics 10.2 Goods and money market dynamics 10.3 IS-LM continuous model: version 1 10.4 Trajectories with Mathematica, Maple and Excel 10.5 Some important propositions 10.6 IS-LM continuous model: version 2 10.7 Nonlinear IS-LM model 10.8 Tobin–Blanchard model 10.9 Conclusion Exercises Additional reading

424 425 429 431 437 442 447 453 455 465 467 469

11

The dynamics of inflation and unemployment 11.1 The Phillips curve 11.2 Two simple models of inﬂation 11.3 Deﬂationary ‘death spirals’ 11.4 A Lucas model with rational expectations 11.5 Policy rules 11.6 Money, growth and inﬂation 11.7 Cagan model of hyperinﬂation 11.8 Unemployment and job turnover 11.9 Wage determination models and the proﬁt function 11.10 Labour market dynamics Exercises Additional reading

470 470 472 484 490 493 494 500 506 509 513 516 518

12

Open economy dynamics: sticky price models 12.1 The dynamics of a simple expenditure model 12.2 The balance of payments and the money supply

519 519 524

Contents 12.3 Fiscal and monetary expansion under ﬁxed exchange rates 12.4 Fiscal and monetary expansion under ﬂexible exchange rates 12.5 Open economy dynamics under ﬁxed prices and ﬂoating Exercises Additional reading

532 539 545 551 552

13

Open economy dynamics: flexible price models 13.1 A simpliﬁed Dornbusch model 13.2 The Dornbusch model 13.3 The Dornbusch model: capital immobility 13.4 The Dornbusch model under perfect foresight 13.5 Announcement effects 13.6 Resource discovery and the exchange rate 13.7 The monetarist model Exercises Additional reading

553 554 559 564 567 573 581 586 589 592

14

Population models 14.1 Malthusian population growth 14.2 The logistic curve 14.3 An alternative interpretation 14.4 Multispecies population models: geometric analysis 14.5 Multispecies population models: mathematical analysis 14.6 Age classes and projection matrices Appendix 14.1 Computing a and b for the logistic equation using Mathematica Appendix 14.2 Using Maple to compute a and b for the logistic equation Appendix 14.3 Multispecies modelling with Mathematica and Maple Exercises Additional reading

593 593 596 601 603 619 626

The dynamics of fisheries 15.1 Biological growth curve of a ﬁshery 15.2 Harvesting function 15.3 Industry proﬁts and free access 15.4 The dynamics of open access ﬁshery 15.5 The dynamics of open access ﬁshery: a numerical example 15.6 The ﬁsheries control problem 15.7 Schooling ﬁshery 15.8 Harvesting and age classes

638 638 644 647 650

15

630 631 632 634 637

654 658 661 669

ix

x

Contents Exercises Additional reading

673 676

Answers to selected exercises Bibliography Author index Subject index

677 688 697 700

Preface to the second edition

I was very encouraged with the reception of the ﬁrst edition, from both staff and students. Correspondence eliminated a number of errors and helped me to improve clarity. Some of the new sections are in response to communications I received. The book has retained its basic structure, but there have been extensive revisions to the text. Part I, containing the mathematical background, has been considerably enhanced in all chapters. All chapters contain new material. This new material is largely in terms of the mathematical content, but there are some new economic examples to illustrate the mathematics. Chapter 1 contains a new section on dimensionality in economics, a much-neglected topic in my view. Chapter 3 on discrete systems has been extensively revised, with a more thorough discussion of the stability of discrete dynamical systems and an extended discussion of solving second-order difference equations. Chapter 5 also contains a more extensive discussion of discrete systems of equations, including a more thorough discussion of solving such systems. Direct solution methods using Mathematica and Maple are now provided in the main body of the text. Indirect solution methods using the Jordan form are new to this edition. There is also a more thorough treatment of the stability of discrete systems. The two topics covered in chapter 6 of the ﬁrst edition have now been given a chapter each. This has allowed topics to be covered in more depth. Chapter 6 on control theory now includes the use of Excel’s Solver for solving discrete control problems. Chapter 7 on chaos theory has also been extended, with a discussion of Sarkovskii’s theorem. It also contains a much more extended discussion of bifurcations and strange attractors. Changes to part II, although less extensive, are quite signiﬁcant. The mathematical treatment of cobwebs in chapter 8 has been extended and there is now a new section on stock models and another on chaotic demand and supply. Chapter 9 on dynamic oligopoly is totally new to this edition. It deals with both discrete and continuous dynamic oligopoly and goes beyond the typical duopoly model. There is also a discussion of an R&D dynamic model of duopoly and a brief introduction to Schumpeterian dynamics. Chapter 11 now includes a discussion of deﬂationary ‘death spirals’ which have been prominent in discussions of Japan’s downturn. Cagan’s model of hyperinﬂations is also a new introduction to this chapter. The open economy was covered quite extensively in the ﬁrst edition, so these chapters contain only minor changes. Population models now include a consideration of age classes and Leslie projection matrices. This material is employed

xii

Preface to the second edition in chapter 15 to discuss culling policy. The chapter on overlapping generations modelling has been dropped in this edition to make way for the new material. Part of the reason for this is that, as presented, it contained little by the way of dynamics. It had much more to say about nonlinearity. Two additional changes have been made throughout. Mathematica and Maple routines are now generally introduced into the main body of the text rather than as appendices. The purpose of doing this is to show that these programmes are ‘natural’ tools for the economist. Finally, there has been an increase in the number of questions attached to almost all chapters. As in the ﬁrst edition, the full solution to all these questions is provided on the Cambridge University website, which is attached to this book: one set of solutions provided in Mathematica notebooks and an alternative set of solutions provided in Maple worksheets. Writing a book of this nature, involving as it does a number of software packages, has become problematic with constant upgrades. This is especially true with Mathematica and Maple. Some of the routines provided in the ﬁrst edition no longer work in the upgrade versions. Even in the ﬁnal stages of preparing this edition, new upgrades were occurring. I had to make a decision, therefore, at which upgrade I would conclude. All routines and all solutions on the web site are carried out with Mathematica 4 and Maple 6. I would like to thank all those individuals who wrote or emailed me on material in the ﬁrst edition. I would especially like to thank Mary E. Edwards, Yee-Tien Fu, Christian Groth, Cars Hommes, Alkis Karabalis, Julio Lopez-Gallardo, Johannes Ludsteck and Yanghoon Song. I would also like to thank Simon Whitby for information and clariﬁcation on new material in chapter 9. I would like to thank Ashwin Rattan for his continued support of this project and Barbara Docherty for an excellent job of copy-editing, which not only eliminated a number of errors but improved the ﬁnal project considerably. The author and publishers wish to thank the following for permission to use copyright material: Springer-Verlag for the programme listing on p. 192 of A First Course in Discrete Dynamic Systems and the use of the Visual D Solve software package from Visual D Solve; Cambridge University Press for table 3 from British Economic Growth 1688–1959, p. 8. The publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate. March 2002

Preface to the ﬁrst edition

The conception of this book began in the autumn semester of 1990 when I undertook a course in Advanced Economic Theory for undergraduates at the University of Stirling. In this course we attempted to introduce students to dynamics and some of the more recent advances in economic theory. In looking at this material it was quite clear that phase diagrams, and what mathematicians would call qualitative differential equations, were becoming widespread in the economics literature. There is little doubt that in large part this was a result of the rational expectations revolution going on in economics. With a more explicit introduction of expectations into economic modelling, adjustment processes became the mainstay of many economic models. As such, there was a movement away from models just depicting comparative statics. The result was a more explicit statement of a model’s dynamics, along with its comparative statics. A model’s dynamics were explicitly spelled out, and in particular, vectors of forces indicating movements when the system was not in equilibrium. This led the way to solving dynamic systems by employing the theory of differential equations. Saddle paths soon entered many papers in economic theory. However, students found this material hard to follow, and it did not often use the type of mathematics they were taught in their quantitative courses. Furthermore, the material that was available was very scattered indeed. But there was another change taking place in Universities which has a bearing on the way the present book took shape. As the academic audit was about to be imposed on Universities, there was a strong incentive to make course work assessment quite different from examination assessment. Stirling has always had a long tradition of course work assessment. In the earlier period there was a tendency to make course work assessment the same as examination assessment: the only real difference being that examinations could set questions which required greater links between material since the course was by then complete. In undertaking this new course, I decided from the very outset that the course work assessment would be quite different from the examination assessment. In particular, I conceived the course work to be very ‘problem oriented’. It was my belief that students come to a better understanding of the economics, and its relation to mathematics, if they carry out problems which require them to explicitly solve models, and to go on to discuss the implications of their analysis. This provided me with a challenge. There was no material available of this type. Furthermore, many economics textbooks of an advanced nature, and certainly the

xiv

Preface to the ﬁrst edition published articles, involved setting up models in general form and carrying out very tedious algebraic manipulations. This is quite understandable. But such algebraic manipulation does not give students the same insight it may provide the research academic. A compromise is to set out models with speciﬁc numerical coefﬁcients. This has at least four advantages. It allowed the models to be solved explicitly. This means that students can get to grips with the models themselves fairly quickly and easily. Generalisation can always be achieved by replacing the numerical coefﬁcients by unspeciﬁed parameters. Or alternatively, the models can be solved for different values, and students can be alerted to the fact that a model’s solution is quite dependent on the value (sign) of a particular parameter. The dynamic nature of the models can more readily be illustrated. Accordingly concentration can be centred on the economics and not on the mathematics. Explicit solutions to saddle paths can be obtained and so students can explicitly graph these solutions. Since it was the nature of saddle paths which gave students the greatest conceptual difﬁculty, this approach soon provided students with the insight into their nature that was lacking from a much more formal approach. Furthermore, they acquired this insight by explicitly dealing with an economic model. I was much encouraged by the students’ attitude to this ‘problem oriented’ approach. The course work assignments that I set were far too long and required far more preparation than could possibly be available under examination conditions. However, the students approached them with vigour during their course work period. Furthermore, it led to greater exchanges between students and a positive externality resulted. This book is an attempt to bring this material together, to extend it, and make it more widely available. It is suitable for core courses in economic theory, and reading for students undertaking postgraduate courses and to researchers who require to acquaint themselves with the phase diagram technique. In addition, it can also be part of courses in quantitative economics. Outside of economics, it is also applicable to courses in mathematical modelling. Finally, I would like to thank Cambridge University Press and the department of economics at Stirling for supplying the two mathematical software programmes; the copy editor, Anne Rix, for an excellent job on a complex manuscript; and my wife, Anne Thomson, for her tolerance in bringing this book about. January 1997

PART I

Dynamic modelling

CHAPTER 1

Introduction

1.1 What this book is about This is not a book on mathematics, nor is it a book on economics. It is true that the over-riding emphasis is on the economics, but the economics under review is speciﬁed very much in mathematical form. Our main concern is with dynamics and, most especially with phase diagrams, which have entered the economics literature in a major way since 1990. By their very nature, phase diagrams are a feature of dynamic systems. But why have phase diagrams so dominated modern economics? Quite clearly it is because more emphasis is now placed on dynamics than in the past. Comparative statics dominated economics for a long time, and much of the teaching is still concerned with comparative statics. But the breakdown of many economies, especially under the pressure of high inﬂation, and the major inﬂuence of inﬂationary expectations, has directed attention to dynamics. By its very nature, dynamics involves time derivatives, dx/dt, where x is a continuous function of time, or difference equations, xt − xt−1 where time is considered in discrete units. This does not imply that these have not been considered or developed in the past. What has been the case is that they have been given only cursory treatment. The most distinguishing feature today is that dynamics is now taking a more central position. In order to reveal this emphasis and to bring the material within the bounds of undergraduate (and postgraduate) courses, it has been necessary to consider dynamic modelling, in both its continuous and discrete forms. But in doing this the over-riding concern has been with the economic applications. It is easy to write a text on the formal mathematics, but what has always been demonstrated in teaching economics is the difﬁculty students have in relating the mathematics to the economics. This is as true at the postgraduate level as it is at the undergraduate level. This linking of the two disciplines is an art rather than a science. In addition, many books on dynamics are mathematical texts that often choose simple and brief examples from economics. Most often than not, these reduce down to a single differential equation or a single difference equation. Emphasis is on the mathematics. We do this too in part I. Even so, the concentration is on the mathematical concepts that have the widest use in the study of dynamic economics. In part II this emphasis is reversed. The mathematics is chosen in order to enhance the economics. The mathematics is applied to the economic problem rather than the

4

Economic Dynamics (simple) economic problem being applied to the mathematics. We take a number of major economic areas and consider various aspects of their dynamics. Because this book is intended to be self-contained, then it has been necessary to provide the mathematical background. By ‘background’ we, of course, mean that this must be mastered before the economic problem is reviewed. Accordingly, part I supplies this mathematical background. However, in order not to make part I totally mathematical we have discussed a number of economic applications. These are set out in part I for the ﬁrst time, but the emphasis here is in illustrating the type of mathematics they involve so that we know what mathematical techniques are required in order to investigate them. Thus, the Malthusian population growth model is shown to be just a particular differential equation, if population growth is assumed to vary continuously over time. But equally, population growth can be considered in terms of a discrete time-period model. Hence, part I covers not only differential equations but also difference equations. Mathematical speciﬁcation can indicate that topics such as A, B and C should be covered. However, A, B and C are not always relevant to the economic problem under review. Our choice of material to include in part I, and the emphasis of this material, has been dictated by what mathematics is required to understand certain features of dynamic economic systems. It is quite clear when considering mathematical models of differential equations that the emphasis has been, and still is, with models from the physical sciences. This is not surprising given the development of science. In this text, however, we shall concentrate on economics as the raison d’ˆetre of the mathematics. In a nutshell, we have taken a number of economic dynamic models and asked: ‘What mathematics is necessary to understand these?’ This is the emphasis of part I. The content of part I has been dictated by the models developed in part II. Of course, if more economic models are considered then the mathematical background will inevitably expand. What we are attempting in this text is dynamic modelling that should be within the compass of an undergraduate with appropriate training in both economics and quantitative economics. Not all dynamic questions are dealt with in this book. The over-riding concern has been to explain phase diagrams. Such phase diagrams have entered many academic research papers over the past decade, and the number is likely to increase. Azariades (1993) has gone as far as saying that Dynamical systems have spread so widely into macroeconomics that vector ﬁelds and phase diagrams are on the verge of displacing the familiar supply–demand schedules and Hicksian crosses of static macroeconomics. (p. xii)

The emphasis is therefore justiﬁed. Courses in quantitative economics generally provide inadequate training to master this material. They provide the basics in differentiation, integration and optimisation. But dynamic considerations get less emphasis – most usually because of a resource constraint. But this is a most unfortunate deﬁciency in undergraduate teaching that simply does not equip students to understand the articles dealing with dynamic systems. The present book is one attempt to bridge this gap. I have assumed some basic knowledge of differentiation and integration, along with some basic knowledge of difference equations. However, I have made great

Introduction pains to spell out the modelling speciﬁcations and procedures. This should enable a student to follow how the mathematics and economics interrelate. Such knowledge can be imparted only by demonstration. I have always been disheartened by the idea that you can teach the mathematics and statistics in quantitative courses, and you can teach the economics in economics courses, and by some unspeciﬁed osmosis the two areas are supposed to fuse together in the minds of the student. For some, this is true. But I suspect that for the bulk of students this is simply not true. Students require knowledge and experience in how to relate the mathematics and the economics. As I said earlier, this is more of an art than a science. But more importantly, it shows how a problem excites the economist, how to then specify the problem in a formal (usually mathematical) way, and how to solve it. At each stage ingenuity is required. Economics at the moment is very much in the mould of problem solving. It appears that the procedure the investigator goes through1 is: (1) (2) (3) (4)

Specify the problem Mathematise the problem See if the problem’s solution conforms to standard mathematical solutions Investigate the properties of the solution.

It is not always possible to mathematise a problem and so steps (2)–(4) cannot be undertaken. However, in many such cases a verbal discussion is carried out in which a ‘story’ is told about the situation. This is no more than a heuristic model, but a model just the same. In such models the dynamics are part of the ‘story’ – about how adjustment takes place over time. It has long been argued by some economists that only those problems that can be mathematised get investigated. There are advantages to formal modelling, of going beyond heuristics. In this book we concentrate only on the formal modelling process.

1.2 The rise in economic dynamics Economic dynamics has recently become more prominent in mainstream economics. This inﬂuence has been quite pervasive and has inﬂuenced both microeconomics and macroeconomics. Its inﬂuence in macroeconomics, however, has been much greater. In this section we outline some of the main areas where economic dynamics has become more prominent and the possible reasons for this rise in the subject. 1.2.1

Macroeconomic dynamics

Economists have always known that the world is a dynamic one, and yet a scan of the books and articles over the past twenty years or so would make one wonder if they really believed it. With a few exceptions, dynamics has been notably absent from published works. This began to change in the 1970s. The 1970s became a watershed in both economic analysis and economic policy. It was a turbulent time. 1

For an extended discussion of the modelling process, see Mooney and Swift (1999, chapter 0).

5

6

Economic Dynamics Economic relationships broke down, stagﬂation became typical of many Western economies, and Conservative policies became prominent. Theories, especially macroeconomic theories, were breaking down, or at best becoming poor predictors of economic changes. The most conspicuous change was the rapid (and accelerating) rise in inﬂation that occurred with rising unemployment. This became a feature of most Western economies. Individuals began to expect price rises and to build this into their decision-making. If such behaviour was to be modelled, and it was essential to do so, then it inevitably involved a dynamic model of the macroeconomy. More and more, therefore, articles postulated dynamic models that often involved inﬂationary expectations. Inﬂation, however, was not the only issue. As inﬂation increased, as OPEC changed its oil price and as countries discovered major resource deposits, so there were major changes to countries’ balance of payments situations. Macroeconomists had for a long time considered their models in the context of a closed economy. But with such changes, the ﬁxed exchange rate system that operated from 1945 until 1973 had to give way to ﬂoating. Generalised ﬂoating began in 1973. This would not have been a problem if economies had been substantially closed. But trade in goods and services was growing for most countries. Even more signiﬁcant was the increase in capital ﬂows between countries. Earlier trade theories concentrated on the current account. But with the growth of capital ﬂows, such models became quite unrealistic. The combination of major structural changes and the increased ﬂows of capital meant that exchange rates had substantial impacts on many economies. It was no longer possible to model the macroeconomy as a closed economy. But with the advent of generalised ﬂoating changes in the exchange rate needed to be modelled. Also, like inﬂation, market participants began to formulate expectations about exchange rate movements and act accordingly. It became essential, then, to model exchange rate expectations. This modelling was inevitably dynamic. More and more articles considered dynamic models, and are still doing so. One feature of signiﬁcance that grew out of both the closed economy modelling and the open economy modelling was the stock-ﬂow aspects of the models. Keynesian economics had emphasised a ﬂow theory. This was because Keynes himself was very much interested in the short run – as he aptly put it: ‘In the long run we are all dead.’ Even growth theories allowed investment to take place (a ﬂow) but assumed the stock of capital constant, even though such investment added to the capital stock! If considering only one or two periods, this may be a reasonable approximation. However, economists were being asked to predict over a period of ﬁve or more years. More importantly, the change in the bond issue (a ﬂow) altered the National Debt (a stock), and also the interest payment on this debt. It is one thing to consider a change in government spending and the impact this has on the budget balance; but the budget, or more signiﬁcantly the National Debt, gives a stock dimension to the long-run forces. Governments are not unconcerned with the size of the National Debt. The same was true of the open economy. The balance of payments is a ﬂow. The early models, especially those ignoring the capital account, were concerned only with the impact of the difference between the exports and imports of goods and

Introduction

7

services. In other words, the inﬂow and outﬂow of goods and services to and from an economy. This was the emphasis of modelling under ﬁxed exchange rates. But a deﬁcit leads to a reduction in the level of a country’s stock of reserves. A surplus does the opposite. Repeated deﬁcits lead to a repeated decline in a country’s level of reserves and to the money stock. Printing more money could, of course, offset the latter (sterilisation), but this simply complicates the adjustment process. At best it delays the adjustment that is necessary. Even so, the adjustment requires both a change in the ﬂows and a change in stocks. What has all this to do with dynamics? Flows usually (although not always) take place in the same time period, say over a year. Stocks are at points in time. To change stock levels, however, to some desired amount would often take a number of periods to achieve. There would be stock-adjustment ﬂows. These are inherently dynamic. Such stock-adjustment ﬂows became highly signiﬁcant in the 1970s and needed to be included in the modelling process. Models had to become more dynamic if they were to become more realistic or better predictors. These general remarks about why economists need to consider dynamics, however, hide an important distinction in the way dynamics enters economics. It enters in two quite different and fundamental ways (Farmer 1999). The ﬁrst, which has its counterpart in the natural sciences, is from the fact that the present depends upon the past. Such models typically are of the form yt = f ( yt−1 )

(1.1)

where we consider just a one-period lag. The second way dynamics enters macroeconomics, which has no counterpart in the natural sciences, arises from the fact that economic agents in the present have expectations (or beliefs) about the future. Again taking a one-period analysis, and denoting the present expectation about the variable y one period from now by Eyt+1 , then yt = g(Eyt+1 ) Let us refer to the ﬁrst lag as a past lag and the second a future lag. There is certainly no reason to suppose modelling past lags is the same as modelling future lags. Furthermore, a given model can incorporate both past lags and future lags. The natural sciences provide the mathematics for handling past lags but has nothing to say about how to handle future lags. It is the future lag that gained most attention in the 1970s, most especially with the rise in rational expectations. Once a future lag enters a model it becomes absolutely essential to model expectations, and at the moment there is no generally accepted way of doing this. This does not mean that we should not model expectations, rather it means that at the present time there are a variety of ways of modelling expectations, each with its strengths and weaknesses. This is an area for future research. 1.2.2

Environmental issues

Another change was taking place in the 1970s. Environmental issues were becoming, and are becoming, more prominent. Environmental economics as a subject began to have a clear delineation from other areas of economics. It is true that

(1.2)

8

Economic Dynamics environmental economics already had a body of literature. What happened in the 1970s and 1980s was that it became a recognised sub-discipline. Economists who had considered questions in the area had largely conﬁned themselves to the static questions, most especially the questions of welfare and cost–beneﬁt analysis. But environmental issues are about resources. Resources have a stock and there is a rate of depletion and replenishment. In other words, there is the inevitable stock-ﬂow dimension to the issue. Environmentalists have always known this, but economists have only recently considered such issues. Why? Because the issues are dynamic. Biological species, such as ﬁsh, grow and decline, and decline most especially when harvested by humans. Forests decline and take a long time to replace. Fossil fuels simply get used up. These aspects have led to a number of dynamic models – some discrete and some continuous. Such modelling has been inﬂuenced most particularly by control theory. We shall brieﬂy cover some of this material in chapters 6 and 15. 1.2.3

The implication for economics

All the changes highlighted have meant a signiﬁcant move towards economic dynamics. But the quantitative courses have in large part not kept abreast of these developments. The bulk of the mathematical analysis is still concerned with equilibrium and comparative statics. Little consideration is given to dynamics – with the exception of the cobweb in microeconomics and the multiplier–accelerator model in macroeconomics. Now that more attention has been paid to economic dynamics, more and more articles are highlighting the problems that arise from nonlinearity which typify many of the dynamic models we shall be considering in this book. It is the presence of nonlinearity that often leads to more than one equilibrium; and given more than one equilibrium then only local stability properties can be considered. We discuss these issues brieﬂy in section 1.4.

1.3 Stocks, ﬂows and dimensionality Nearly all variables and parameters – whether they occur in physics, biology, sociology or economics – have units in which they are deﬁned and measured. Typical units in physics are weight and length. Weight can be measured in pounds or kilograms, while length can be measured in inches or centimetres. We can add together length and we can add together weight, but what we cannot do is add length to weight. This makes no sense. Put simply, we can add only things that have the same dimension. DEFINITION Any set of additive quantities is a dimension. A primary dimension is not expressible in terms of any other dimension; a secondary dimension is deﬁned in terms of primary dimensions.2 2

An elementary discussion of dimensionality in economics can be found in Neal and Shone (1976, chapter 3). The deﬁnitive source remains De Jong (1967).

Introduction To clarify these ideas, and other to follow, we list the following set of primary dimensions used in economics: (1) (2) (3) (4)

Money [M ] Resources or quantity [Q] Time [T ] Utility or satisfaction [S]

Apples has, say, dimension [Q1] and bananas [Q2]. We cannot add an apple to a banana (we can of course add the number of objects, but that is not the same thing). The value of an apple has dimension [M ] and the value of a banana has dimension [M], so we can add the value of an apple to the value of a banana. They have the same dimension. Our reference to [Q1] and [Q2] immediately highlights a problem, especially for macroeconomics. Since we cannot add apples and bananas, it is sometimes assumed in macroeconomics that there is a single aggregate good, which then involves dimension [Q]. For any set of primary dimensions, and we shall use money [M ] and time [T ] to illustrate, we have the following three propositions: (1) (2) (3)

If a ∈ [M] and b ∈ [M] then a ± b ∈ [M] If a ∈ [M] and b ∈ [T] then ab ∈ [MT] and a/b ∈ [MT −1 ] If y = f (x) and y ∈ [M] then f (x) ∈ [M].

Proposition (1) says that we can add or subtract only things that have the same dimension. Proposition (2) illustrates what is meant by secondary dimensions, e.g., [MT −1 ] is a secondary or derived dimension. Proposition (3) refers to equations and states that an equation must be dimensionally consistent. Not only must the two sides of an equation have the same value, but it must also have the same dimension, i.e., the equation must be dimensionally homogeneous. The use of time as a primary dimension helps us to clarify most particularly the difference between stocks and ﬂows. A stock is something that occurs at a point in time. Thus, the money supply, Ms, has a certain value on 31 December 2001. Ms is a stock with dimension [M], i.e., Ms ∈ [M]. A stock variable is independent of the dimension [T]. A ﬂow, on the other hand, is something that occurs over a period of time. A ﬂow variable must involve the dimension [T −1 ]. In demand and supply analysis we usually consider demand and supply per period of time. Thus, qd and qs are the quantities demanded and supplied per period of time. More speciﬁcally, qd ∈ [QT −1 ] and qs ∈ [QT −1 ]. In fact, all ﬂow variables involve dimension [T −1 ]. The nominal rate of interest, i, for example, is a per cent per period, so i ∈ [T −1 ] and is a ﬂow variable. Inﬂation, π, is the percentage change in prices per period, say a year. Thus, π ∈ [T −1 ]. The real rate of interest, deﬁned as r = i − π, is dimensionally consistent since r ∈ [T −1 ], being the difference of two variables each with dimension [T −1 ]. Continuous variables, such as x(t), can be a stock or a ﬂow but are still deﬁned for a point in time. In dealing with discrete variables we need to be a little more careful. Let xt denote a stock variable. We deﬁne this as the value at the end of period t.3 Figure 1.1 uses three time periods to clarify our discussion: t − 1, t and 3

We use this convention throughout this book.

9

10

Economic Dynamics

Figure 1.1.

period t + 1. Thus xt−1 is the stock at the end of period t − 1 and xt is the stock at the end of period t. Now let zt be a ﬂow variable over period t, and involving dimension [T −1 ]. Of course, there is also zt−1 and zt+1 . Now return to variable x. It is possible to consider the change in x over period t, which we write as xt = xt − xt−1 This immediately shows up a problem. Let xt have dimension [Q], then by proposition (1) so would xt . But this cannot be correct! xt is the change over period t and must involve dimension [T −1 ]. So how can this be? The correct formulation is, in fact, (1.3)

xt − xt−1 xt = ∈ [QT −1 ] t t − (t − 1) Implicit is that t = 1 and so xt = xt − xt−1 . But this ‘hides’ the dimension [T −1 ]. This is because t ∈ [T], even though it has a value of unity, xt /t ∈ [QT −1 ]. Keeping with the convention xt = xt − xt−1 , then xt ∈ [QT −1 ] is referred to as a stock-ﬂow variable. xt must be kept quite distinct from zt . The variable zt is a ﬂow variable and has no stock dimension. xt , on the other hand, is a difference of two stocks deﬁned over period t. Example 1.1 Consider the quantity equation MV = Py. M is the stock of money, with dimension [M]. The variable y is the level of real output. To make dimensional sense of this equation, we need to assume a single-good economy. It is usual to consider y as real GDP over a period of time, say one year. So, with a single-good economy with goods having dimension [Q], then y ∈ [QT −1 ]. If we have a single-good economy, then P is the money per unit of the good and has dimension [MQ−1 ]. V is the income velocity of circulation of money, and indicates the average number of times a unit of money circulates over a period of time. Hence V ∈ [T −1 ]. Having considered the dimensions of the variables separately, do we have dimensional consistency? MV ∈ [M][T −1 ] = [MT −1 ] Py ∈ [MQ−1 ][QT −1 ] = [MT −1 ] and so we do have dimensional consistency. Notice in saying this that we have utilised the feature that dimensions ‘act like algebra’ and so dimensions cancel, as with [QQ−1 ]. Thus Py ∈ [MQ−1 ][QT −1 ] = [MQ−1 QT −1 ] = [MT −1 ]

Introduction

11

Example 1.2 Consider again the nominal rate of interest, denoted i. This can more accurately be deﬁned as the amount of money received over some interval of time divided by the capital outlay. Hence, i∈

[MT −1 ] = [T −1 ] [M]

Example 1.3 Consider the linear static model of demand and supply, given by the following equations. qd = a − bp qs = c + dp q d = qs = q

a, b > 0 d>0

(1.4)

with equilibrium price and quantity a−c , b+d and with dimensions p∗ =

q∗ =

qd , qs ∈ [QT −1 ],

ad + bc b+d

p ∈ [MQ−1 ]

The model is a ﬂow model since qd and qs are deﬁned as quantities per period of time.4 It is still, however, a static model because all variables refer to time period t. Because of this we conventionally do not include a time subscript. Now turn to the parameters of the model. If the demand and supply equations are to be dimensionally consistent, then a, c ∈ [QT −1 ]

and

b, d ∈ [Q2 T −1 M −1 ]

Then a − c ∈ [QT −1 ] b + d ∈ [Q2 T −1 M −1 ] p∗ ∈

[QT −1 ] [Q2 T −1 M −1 ]

= [MQ−1 ]

Also ad ∈ [QT −1 ][Q2 T −1 M −1 ] = [Q3 T −2 M −1 ] bc ∈ [Q2 T −1 M −1 ][QT −1 ] = [Q3 T −2 M −1 ] q∗ ∈

[Q3 T −2 M −1 ] = [QT −1 ] [Q2 T −1 M −1 ]

Where a problem sometimes occurs in writing formulas is when parameters have values of unity. Consider just the demand equation and suppose it takes the 4

We could have considered a stock demand and supply model, in which case qd and qs would have dimension [Q]. Such a model would apply to a particular point in time.

12

Economic Dynamics form qd = a − p. On the face of it this is dimensionally inconsistent. a ∈ [QT −1 ] and p ∈ [MQ−1 ] and so cannot be subtracted! The point is that the coefﬁcient of p is unity with dimension [Q2 T −1 M −1 ], and this dimension gets ‘hidden’. Example 1.4 A typically dynamic version of example 1.3 is the cobweb model qdt = a − bpt

(1.5)

qst qdt

= c + dpt−1

a, b > 0 d>0

= qst = qt

Here we do subscript the variables since now two time periods are involved. Although qdt and qst are quantities per period to time with dimension [QT −1 ], they both refer to period t. However, p ∈ [MQ−1 ] is for period t in demand but period t − 1 for supply. A model that is speciﬁed over more than one time period is a dynamic model. We have laboured dimensionality because it is still a much-neglected topic in economics. Yet much confusion can be avoided with a proper understanding of this topic. Furthermore, it lies at the foundations of economic dynamics.

1.4 Nonlinearities, multiple equilibria and local stability Nonlinearities, multiple equilibria and local stability/instability are all interlinked. Consider the following simple nonlinear difference equation (1.6)

xt = f (xt−1 ) An equilibrium (a ﬁxed point) exists, as we shall investigate fully later in the book, if x∗ = f (x∗ ). Suppose the situation is that indicated in ﬁgure 1.2(a), then an equilibrium point is where f (xt−1 ) cuts the 45◦ -line. But in this example three such ﬁxed points satisfy this condition: x1∗ , x2∗ and x3∗ . A linear system, by contrast, can cross the 45◦ -line at only one point (we exclude here the function coinciding with the 45◦ -line), as illustrated in ﬁgures 1.2(b) and 1.2(c). It is the presence of the nonlinearity that leads to multiple equilibria. If we consider a sequence of points {xt } beginning at x0 , and if for a small neighbourhood of a ﬁxed point x∗ the sequence {xt } converges on x∗ , then x∗ is said to be locally asymptotically stable. We shall explain this in more detail later in the book. Now consider the sequence in the neighbourhood of each ﬁxed point in ﬁgure 1.2(a). We do this for each point in terms of ﬁgure 1.3. In the case of x1∗ , for any initial point x0 (or x0 ) in the neighbourhood of x1∗ , the sequence {xt } will converge on x1∗ . This is also true for the ﬁxed point x3∗ . However, it is not true for the ﬁxed point x2∗ , represented by point b. The ﬁxed point x2∗ is locally asymptotically unstable. On the other hand both x1∗ and x3∗ are locally asymptotically stable. Suppose we approximate the nonlinear system in the neighbourhood of each of the ﬁxed points. This can be done by means of a Taylor expansion about the appropriate ﬁxed point. These are shown by each of the dotted lines in ﬁgure 1.3.

Introduction

13 Figure 1.2.

xt=(f(xt−1))

Observation of these lines indicates that for equilibrium points x1∗ and x3∗ the linear approximation has a slope less than unity. On the other hand, the linear approximation about x2∗ has a slope greater than unity. It is this feature that allows us to deal with the dynamics of a nonlinear system – so long as we keep within a small neighbourhood of a ﬁxed point.

14

Economic Dynamics

Figure 1.3.

Although a great deal of attention has been given to linear difference and differential equations, far less attention has been given to nonlinear relationships. This is now changing. Some of the most recent researches in economics are considering nonlinearities. Since, however, there is likely to be no general solutions for nonlinear relationships, both mathematicians and economists have, with minor

Introduction exceptions, been content to investigate the local stability of the ﬁxed points to a nonlinear system. The fact that a linear approximation can be taken in the neighbourhood of a ﬁxed point in no way removes the fact that there can be more than one ﬁxed point, more than one equilibrium point. Even where we conﬁne ourselves only to stable equilibria, there is likely to be more than one. This leads to some new and interesting policy implications. In simple terms, and using ﬁgure 1.2(a) for illustrative purposes, the welfare attached to point x1∗ will be different from that attached to x3∗ . If this is so, then it is possible for governments to choose between the two equilibrium points. Or, it may be that after investigation one of the stable equilibria is found to be always superior. With linear systems in which only one equilibrium exists, such questions are meaningless. Multiple equilibria of this nature create a problem for models involving perfect foresight. If, as such models predict, agents act knowing the system will converge on equilibrium, will agents assume the system converges on the same equilibrium? Or, even with perfect foresight, can agents switch from one (stable) equilibrium to another (stable) equilibrium? As we shall investigate in this book, many of the rational expectations solutions involve saddle paths. In other words, the path to equilibrium will arise only if the system ‘jumps’ to the saddle path and then traverses this path to equilibrium. There is something unsatisfactory about this modelling process and its justiﬁcation largely rests on the view that the world is inherently stable. Since points off the saddle path tend the system ever further away from equilibrium, then the only possible (rational) solution is that on the saddle path. Even if we accept this argument, it does not help in analysing systems with multiple equilibrium in which more than one stable saddle path exits. Given some initial point off the saddle path, to which saddle path will the system ‘jump’? Economists are only just beginning to investigate these difﬁcult questions.

1.5 Nonlinearity and chaos Aperiodic behaviour had usually been considered to be the result of either exogenous shocks or complex systems. However, nonlinear systems that are simple and deterministic can give rise to aperiodic, or chaotic, behaviour. The crucial element leading to this behaviour is the fact that the system is nonlinear. For a linear system a small change in a parameter value does not affect the qualitative nature of the system. For nonlinear systems this is far from true. For some small change (even very small) both the quantitative and qualitative behaviour of the system can dramatically change. Strangely, nonlinearity is the norm. But in both the physical sciences and economics linearity has been the dominant mode of study for over 300 years. Nonlinearity is the most commonly found characteristic of systems and it is therefore necessary for the scientist, including the social scientist, to take note of this. The fact that nonlinear systems can lead to aperiodic or chaotic behaviour has meant a new branch of study has arisen – chaos theory. It may be useful to point out that in studying any deterministic system three characteristics of the system must be known (Hilborn 1994, p. 7):

15

16

Economic Dynamics (1) (2) (3)

the time-evolution values, the parameter values, and the initial conditions.

A system for which all three are known is said to be deterministic. If such a deterministic system exhibits chaos, then it is very sensitive to initial conditions. Given very small differences in initial conditions, then the system will after time behave very differently. But this essentially means that the system is unpredictable since there is always some imprecision in specifying initial conditions,5 and therefore the future path of the system cannot be known in advance. In this instance the future path of the system is said to be indeterminable even though the system itself is deterministic. The presence of chaos raises the question of whether economic ﬂuctuations are generated by the ‘endogenous propagation mechanism’ (Brock and Malliaris 1989, p. 305) or from exogenous shocks to the system. The authors go on, Theories that support the existence of endogenous propagation mechanisms typically suggest strong government stabilization policies. Theories that argue that business cycles are, in the main, caused by exogenous shocks suggest that government stabilization policies are, at best, an exercise in futility and, at worst, harmful. (pp. 306–7)

This is important. New classical economics assumes that the macroeconomy is asymptotically stable so long as there are no exogenous shocks. If chaos is present then this is not true. On the other hand, new Keynesian economics assumes that the economic system is inherently unstable. What is not clear, however, is whether this instability arises from random shocks or from the presence of chaos. As Day and Shafer (1992) illustrate, in the presence of nonlinearity a simple Keynesian model can exhibit chaos. In the presence of chaos, prediction is either hazardous or possibly useless – and this is more true the longer the prediction period. Nonlinearity and chaos is quite pervasive in economics. Azariadis (1993) has argued that much of macroeconomics is (presently) concerned with three relationships: the Solow growth model, optimal growth, and overlapping generations models. The three models can be captured in the following discrete versions: (i) (1.7)

(ii) (iii)

(1 − δ)kt + sf (kt ) 1+n kt+1 = f (kt ) + (1 − δ)kt − ct u (ct ) = ρu (ct+1 )[ f (kt+1 ) + (1 − δ)] kt+1 =

(1 + n)kt+1 = z[ f (kt+1 ) + (1 − δ), w(kt )]

The explanation of these equations will occur later in the book. Sufﬁce it to say here that Azariadis considers that the business of mainstream macroeconomics amounts to ‘complicating’ one of [these] dynamical systems . . . and exploring what happens as new features are added. (p. 5) 5

As we shall see in chapter 7, even a change in only the third or fourth decimal place can lead to very different time paths. Given the poor quality of economic data, not to mention knowledge of the system, this will always be present. The literature refers to this as the butterﬂy effect.

Introduction

17

All these major concerns involve dynamical systems that require investigation. Some have found to involve chaotic behaviour while others involve multiple equilibria. All three involve nonlinear equations. How do we represent these systems? How do we solve these systems? Why do multiple equilibria arise? How can we handle the analysis in the presence of nonlinearity? These and many more questions have been addressed in the literature and will be discussed in this book. They all involve an understanding of dynamical systems, both in continuous time and in discrete time. The present book considers these issues, but also considers dynamic issues relevant to microeconomics. The present book also tries to make the point that even in the area of macroeconomics, these three systems do not constitute the whole of the subject matter. As one moves into the realms of policy questions, open economy issues begin to dominate. For this reason, the present book covers much more of the open economy when discussing macroeconomic issues. Of importance here is the differential speeds of adjustment in the various sectors of the economy. Such asymmetry, however, is also relevant to closed economy models, as we shall see.

1.6 Computer software and economic dynamics Economic dynamics has not been investigated for a long time because of the mathematical and computational requirements. But with the development of computers, especially ready-made software packages, economists can now fairly easily handle complex dynamic systems. Each software package has its comparative advantage. This is not surprising. But for this reason I would not use one package to do everything. Spreadsheets – whether Excel, QuattroPro, Lotus 1-2-3, etc. – are all good at manipulating data and are particularly good at displaying sequential data. For this reason they are especially useful at computing and displaying difference equations. This should not be surprising. Difference equations involve recursive formulae, but recursion is the basis of the copy command in spreadsheets, where entries in the cells being copied have relative (and possibly absolute) cell addresses. If we have a difference equation of the form xt = f (xt−1 ), then so long as we have a starting value x0 , it is possible to compute the next cell down as f (x0 ). If we copy down n−1 times, then xn is no more than f (xn−1 ). Equally important is the fact that f (xt−1 ) need not be linear. There is inherently no more difﬁculty in copy2 or f (xt−1 ) = ing f (xt−1 ) = a + bxt−1 than in copying f (xt−1 ) = a + bxt−1 + cxt−1 a + b sin(xt−1 ). The results may be dramatically different, but the principle is the same. Nonlinear equations are becoming more important in economics, as we indicated in the previous section, and nonlinear difference equations have been at the heart of chaos. The most famous is the logistic recursive equation xt = f (xt−1 , λ) = λxt−1 (1 − xt−1 ) It is very easy to place the value of λ in a cell that can then be referred to using an absolute address reference. In the data column all one does is specify x0 and then x1 is computed from f (x0 , λ), which refers to the relative address of x0 and

(1.8)

18

Economic Dynamics

Figure 1.4.

the absolute address of λ. This is then copied down as many times as one likes, as illustrated in ﬁgure 1.4.6 This procedure allows two things to be investigated: (1) (2)

different values for λ different initial values (different values for x0 ).

Equally important, xt can be plotted against t and the implications of changing λ and/or x0 can immediately be observed. This is one of the real beneﬁts of the Windows spreadsheets. There is no substitute for interactive learning. In writing this 6

In this edition, all spreadsheets are created in Microsoft Excel.

Introduction

19

book there were a number of occasions when I set up spreadsheets and investigated the property of some system and was quite surprised by the plot of the data. Sometimes this led me to reinvestigate the theory to establish why I saw what I did. The whole process, sometimes frustrating, was a most satisfying learning experience. The scope of using spreadsheets for investigating recursive equations cannot be emphasised enough. But they can also be used to investigate recursive systems. Often this is no more difﬁcult than a single equation, it just means copying down more than one column. For example, suppose we have the system xt = axt−1 + byt−1 yt = cxt−1 + dyt−1

(1.9)

Then on a spreadsheet all that needs to be speciﬁed is the values for a, b, c and d and the initial values for x and y, i.e., x0 and y0 . Then x1 and y1 can be computed with relative addresses to x0 and y0 and absolute addresses to a, b, c and d. Given these solutions then all that needs to be done is to copy the cells down. Using this procedure it is possible to investigate some sophisticated systems. It is also possible to plot trajectories. The above system is autonomous (it does not involve t explicitly) and so {x(t), y(t)} can be plotted using the spreadsheet’s x-y plot. Doing this allows the display of some intriguing trajectories – and all without any intricate mathematical knowledge.7 Having said this, I would not use a spreadsheet to do econometrics, nor would I use Mathematica or Maple to do so – not even regression. Economists have many econometrics packages that specialise in regression and related techniques. They are largely (although not wholly) for parameter estimation and diagnostic testing. Mathematica and Maple (see the next section) can be used for statistical work, and each comes with a statistical package that accompanies the main programme, but they are inefﬁcient and unsuitable for the economist. But the choice is not always obvious. Consider, for example, the logistic equation xt = f (xt−1 ) = 3.5xt−1 (1 − xt−1 ) It is possible to compute a sequence {xt } beginning at x0 = 0.1 and to print the 10th through to the 20th iteration using the following commands in Mathematica8 clear[f] f[x-]:=3.5x(1-x); StartingValue:.1; FirstIteration=10: LastIteration=20; i=0; y=N[StartingValue]; While[i=FirstIteration, Print[i, `` ``, N[y,8] ] ]; y = f[y]; i =i+1]

7 8

See Shone (2001) for an introductory treatment of economic dynamics using spreadsheets. Taken from Holmgren (1994, appendix A1).

(1.10)

20

Economic Dynamics which would undoubtedly appeal to a mathematician or computer programmer. The same result, however, can be achieved much simpler by means of a spreadsheet by inputting 0.1 in the ﬁrst cell and then obtaining 3.5x0 (1 − x0 ) in the second cell and copying down the next 18 cells. Nothing more is required than knowing how to enter a formula and copying down.9 There are advantages, however, to each approach. The spreadsheet approach is simple and requires no knowledge of Mathematica or programming. However, there is not the same control over precision (it is just as acceptable to write N[y,99] for precision to 99 signiﬁcant digits in the above instructions). Also what about the iteration from the 1000th through to 1020th? Use of the spreadsheet means accepting its precision; while establishing the iterations from 1000 onwards still requires copying down the ﬁrst 998 entries! For the economist who just wants to see the dynamic path of a sequence {xt }, then a spreadsheet may be all that is required. Not only can the sequence be derived, but also it can readily be graphed. Furthermore, if the formula is entered as f (x) = rx(1 − x), then the value of r can be given by an absolute address and then changed.10 Similarly, it is a simple matter of changing x0 to some value other than 0.1. Doing such manipulations immediately shows the implications on a plot of {xt }, most especially its convergence or divergence. Such interactive learning is quick, simple and very rewarding. The message is a simple one. Know your tools and use the most suitable. A hammer can put a nail in a plank of wood. It is possible to use a pair of pliers and hit the nail, but no tradesman would do this. Use the tool designed for the task. I will not be dealing with econometrics in this book, but the message is general across software: use the software for which it is ‘best’ suited. This does beg the question of what a particular software package is best suited to handle. In this book we intend to answer this by illustration. Sometimes we employ one software package rather than another. But even here there are classes of packages. It is this that we concentrate on. Which package in any particular class is often less important: they are close substitutes. Thus, we have four basic classes of software: (1) (2) (3) (4)

Spreadsheets Excel, QuattroPro, Lotus 1-2-3, etc. Mathematics Mathematica, Maple, MatLab, MathCad, DERIVE, etc. Statistical SPSS, Systat, Statgraphics, etc. Econometrics Shazam, TSP, Microﬁt, etc.

1.7 Mathematica and Maple An important feature of the present book is the ready use of both Mathematica and Maple.11 These packages for mathematics are much more than gloriﬁed calculators because each of them can also be applied to symbolic manipulation: they can expand the expression (x + y)2 into x2 + y2 + 2xy, they can carry out differentiation and integration and they can solve standard differential equations – and much 9 10 11

Occam’s razor would suggest the use of the spreadsheet in this instance. We use r rather than λ to avoid Greek symbols in the spreadsheet. There are other similar software packages on the market, such as DERIVE and MathCad, but these are either more specialised or not as extensive as Mathematica or Maple.

Introduction

21 Figure 1.5.

more. Of course, computer algebra requires some getting used to. But so did the calculator (and the slide rule even more so!). But the gains are extensive. Once the basic syntax is mastered and a core set of commands, much can be accomplished. Furthermore, it is not necessary to learn everything in these software packages. They are meant to be tools for a variety of disciplines. The present book illustrates the type of tools they provide which are useful for the economist. By allowing computer software to carry out the tedious manipulations – whether algebraic or numeric – allows concentration to be directed towards the problem in hand. Both Mathematica and Maple have the same basic structure. They are composed of three parts: (1) (2) (3)

a kernel, which does all the computational work, a front end, which displays the input/output and interacts with the user, and a set of libraries of specialist routines.

This basic structure is illustrated in ﬁgure 1.5. What each programme can do depends very much on which version of the programme that is being used. Both programmes have gone through many upgrades. In this second edition we use Mathematica for Windows version 4 and Maple 6 (upgrade 6.01).12 Each programme is provided for a different platform. The three basic platforms are DOS, Windows and UNIX. In the case of each programme, the kernel, which is the heart of the programme, is identical for the different platforms. It is the front end that differs across the three platforms. In this book it is the Windows platform that is being referred to in the case of both programmes. The front end of Maple is more user friendly to that of Mathematica, but Mathematica’s kernel is far more comprehensive than that of Maple.13 Both have extensive specialist library packages. For the economist, it is probably ease of use 12

13

Mathematica for Windows has been frequently upgraded, with a major change occurring with Mathematica 3. Maple was Maple V up to release 5, and then become Maple 6. Both packages now provide student editions. Mathematica’s palettes are far more extensive than those of Maple (see Shone 2001).

22

Economic Dynamics that matters most, and Maple’s front end is far more user friendly and far more intuitive than that of Mathematica. Having said this, each has its strengths and in this book we shall highlight these in the light of applicability to economics. The choice is not always obvious. For instance, although the front end of Maple is more user friendly, I found Mathematica’s way of handling differential equations easier and more intuitive, and with greater control over the graphical output. Certainly both are comprehensive and will handle all the types of mathematics encountered in economics. Accordingly, the choice between the two packages will reduce to cost and ease of use. Having mentioned the front end, what do these look like for the two packages? Figure 1.6 illustrates the front end for a very simple function, namely y = x3, where each programme is simply required to plot the function over the interval −3 < x < 3 and differentiate it. Both programmes now contain the graphical output in the same window.14 In Mathematica (ﬁgure 1.6a) a postscript rendering of the graph is displayed in the body of the page. This can be resized and copied to the clipboard. It can also be saved as an Encapsulated Postscript (EPS), Bitmap (BMP), Enhanced Metaﬁle (EMF) and a Windows Metaﬁle. However, many more graphical formats are available using the Export command of Mathematica. To use this the graphic needs to have a name. For instance, the plot shown in ﬁgure 1.6 could be called plot16, i.e., the input line would now be plot16=Plot [(x^3,{x,-3,3}]

Suppose we wish to export this with a ﬁle name Fig01 06. Furthermore, we wish to export it as an Encapsulated Postscript File (EPS), then the next instruction would be Export[``Fig01-06.eps’’,plot16, ``EPS’’]

In the case of Maple (ﬁgure 1.6b) the plot can be copied to the clipboard and pasted or can be exported as an Encapsulated Postscript (EPS), Graphics Interchange Format (GIF), JPEG Interchange Format (JPG), Windows Bitmap (BMP) and Windows Metaﬁle (WMF). For instance, to export the Maple plot in ﬁgure 1.6, simply right click the plot, choose ‘Export As’, then choose ‘Encapsulated Postscript (EPS) . . .’ and then simply give it a name, e.g., Fig01 06. The ‘eps’ ﬁle extension is automatically added. Moving plots into other programmes can be problematic. This would be necessary, for example, if a certain degree of annotation is required to the diagram. This is certainly the case in many of the phase diagrams constructed in this book. In many instances, diagrams were transported into CorelDraw for annotation.15 When importing postscript ﬁles it is necessary to use CorelDraw’s ‘.eps,*.ps (interpreted)’ import ﬁlter. In this book we often provide detailed instructions on deriving solutions, especially graphical solutions, to a number of problems. Sometimes these are provided in the appendices. Since the reader is likely to be using either Mathematica or Maple, then instructions for each of these programmes are given in full in the body 14 15

This was not always the case with Maple. In earlier versions, the graphical output was placed in separate windows. CorelDraw has also gone through a number of incarnations. This book uses CorelDraw 9.0.

Introduction

23 Figure 1.6.

of the text for the most important features useful to the economist. This allows the reader to choose whichever programme they wish without having to follow instructions on the use of the alternative one, with which they are probably not familiar. Although this does involve some repeat of the text, it seems the most sensible approach to take. The routines contained here may not always be the most efﬁcient – at

24

Economic Dynamics least in the eyes of a computer programmer – but they are straightforward and can readily be reproduced without any knowledge of computer programming. Furthermore, they have been written in such a way that they can easily be adapted for any similar investigation by the reader.

1.8 Structure and features This book takes a problem solving, learning by doing approach to economic dynamics. Chapters 2–5 set out the basic mathematics for continuous and discrete dynamical systems with some references to economics. Chapter 2 covers continuous single-equation dynamics, while chapter 3 deals with discrete single-equation dynamics. Chapter 4 covers continuous dynamical systems of equations and chapter 5 deals with discrete dynamical systems of equations. Chapters 6 and 7 cover two quite distinct dynamical topics that do not ﬁt into the continuous/discrete categorisation so neatly. Chapter 6 deals with control theory and chapter 7 with chaos theory. Both these topics are more advanced, but can be taken up at any stage. Each deals with both continuous and discrete modelling. Chapters 1–7 constitute part I and set out the mathematical foundation for the economic topics covered in part II. Part II contains chapters 8–15, and deals with problems and problem solving. Each subject intermingles continuous and discrete modelling according to the problem being discussed and the approach taken to solving it. We begin with demand and supply in chapter 8. Chapter 9 also deals with a topic in microeconomics, namely the dynamics of oligopoly. This chapter is new to this edition. We then introduce the basic modelling of macroeconomics in terms of closed economy dynamics, emphasising the underlying dynamics of the IS–LM model and extending this to the Tobin–Blanchard model. Next we consider the important topics of inﬂation and unemployment. Here we are more restrictive, considering just certain dynamic aspects of these interrelated topics. Chapters 12 and 13 deal with open economy dynamics, a much-neglected topic in macroeconomics until recently. Chapter 12 deals with the open economy under the assumption of a ﬁxed price level, while Chapter 13 deals with open economy dynamics under the assumption of ﬂexible prices. It will be seen that the modelling approach between these two differs quite considerably. In chapter 14 we consider population models, which can be considered a microeconomic topic. Not only does it deal with single populations, but it also considers the interaction between two populations. Finally, chapter 15 on ﬁsheries economics also deals with a microeconomic topic that is a central model in the theory of environmental economics. All the topics covered in part II are contained in core courses in economic theory. The main difference here is the concentration on the dynamics of these topics and the techniques necessary to investigate them. All chapters, with the exception of this one, contain exercises. These not only enhance the understanding of the material in the chapter, but also extend the analysis. Many of these questions, especially in part II, are problem solving type exercises. They require the use of computer software to carry them out. Sometimes this is no more than using a spreadsheet. However, for some problems the power of a mathematical programme is required. It is in carrying out the exercises that one learns

Introduction by doing. In a number of the exercises the answers are provided in the question. When this is not the case, answers to a number of the questions are supplied at the end of the book. The present book has a number of features. The coverage is both up-to-date and deals with discrete as well as continuous models. The book is fairly self-contained, with part I supplying all the mathematical background for discussing dynamic economic models, which is the content of part II. Many recent books on dynamic economics deal largely with macroeconomics only. In this book we have attempted a more balanced coverage between microeconomics and macroeconomics. Part I in large part treats continuous models and discrete models separately. In part II, however, the economics dictates to a large extent whether a particular model is discrete or continuous – or even both. A feature of both part I and part II is a discussion of the phase diagram for analysing dynamic models. A major emphasis is problem solving, and to this end we supply copious solved problems in the text. These range from simple undergraduate economic models to more sophisticated ones. In accomplishing this task ready use has been made of three software packages: Mathematica, Maple and Excel. The text has detailed instructions on using both Mathematica and Maple, allowing the reader to duplicate the models in the text and then to go beyond these. In order to reinforce the learning process, the book contains copious exercises. Detailed solutions using both Mathematica and Maple are provided on the Cambridge University website. Additional reading Additional material on the economic content of this chapter can be found in Azariades (1993), Brock and Malliaris (1989), Bullard and Butler (1993), Day and Shafer (1992), De Jong (1967), Farmer (1999), Mizrach (1992), Mooney and Swift (1999), Mullineux and Peng (1993), Neal and Shone (1976) and Scheinkman (1990). Additional material on Mathematica can be found in, Abell and Braselton (1992, 1997a, 1997b), Blachman (1992), Brown, Porta and Uhl (1991), Burbulla and Dodson (1992), Coombes et al. (1998), Crandall (1991), Don (2001), Gray and Glynn (1991), Huang and Crooke (1997), Ruskeepaa (1999), Schwalbe and Wagon (1996), Shaw and Tigg (1994), Shone (2001), Skeel and Keiper (1993), Varian et al. (1993), Wagon (1991) and Wolfram (1999). Additional material on Maple can be found in Abell and Braselton (1994a, 1994b, 1999), Devitt (1993), Ellis et al. (1992), Gander and Hrebicek (1991), Heck (1993), Koﬂer (1997), Kreyszig and Norminton (1994) and Nicolaides and Walkington (1996).

25

CHAPTER 2

Continuous dynamic systems

2.1 Some deﬁnitions A differential equation is an equation relating: (a) (b) (c) (d)

the derivatives of an unknown function, the function itself, the variables in terms of which the function is deﬁned, and constants.

More brieﬂy, a differential equation is an equation that relates an unknown function and any of its derivatives. Thus dy + 3xy = ex dx is a differential equation. In general dy = f (x, y) dx is a general form of a differential equation. In this chapter we shall consider continuous dynamic systems of a single variable. In other words, we assume a variable x is a continuous function of time, t. A differential equation for a dynamic equation is a relationship between a function of time and its derivatives. One typical general form of a differential equation is dx = f (t, x) dt

(2.1)

Examples of differential equations are: (i)

dx + 3x = 4 + e−t dt

(ii)

d2 x dx + 4t − 3(1 − t2 )x = 0 dt2 dt

(iii)

dx = kx dt

(iv)

∂u ∂v + + 4u = 0 ∂t dt

Continuous dynamic systems

27

In each of the ﬁrst three examples there is only one variable other than time, namely x. They are therefore referred to as ordinary differential equations. When functions of several variables are involved, such as u and v in example (iv), such equations are referred to as partial differential equations. In this book we shall be concerned only with ordinary differential equations. Ordinary differential equations are classiﬁed according to their order. The order of a differential equation is the order of the highest derivative to appear in the equation. In the examples above (i) and (iii) are ﬁrst-order differential equations, while (ii) is a second-order differential equation. Of particular interest is the linear differential equation, whose general form is a0 (t)

dn x dn−1 x + a1 (t) n−1 + . . . + an (t)x = g(t) n dt dt

(2.2)

If a0 (t), a1 (t), . . . , an (t) are absolute constants, and so independent of t, then equation (2.2) is a constant-coefﬁcient nth-order differential equation. Any differential equation not conforming to equation (2.2) is referred to as a nonlinear differential equation. The nth-order differential equation (2.2) is said to be homogeneous if g(t) ≡ 0 and nonhomogeneous if g(t) is not identically equal to zero. Employing these categories, the examples given above are as follows: (i) (ii) (iii)

a linear constant-coefﬁcient differential equation with nonhomogeneous term g(t) = 4 + e−t a second-order linear homogeneous differential equation a linear constant-coefﬁcient homogeneous differential equation.

In the present book particular attention will be directed to ﬁrst-order linear differential equations which can be expressed in the general form h(t)

dx + k(t)x = g(t) dt

by dividing throughout by h(t) we have the simpler form dx + a(t)x = b(t) dt The problem is to ﬁnd all functions x(t) which satisfy equation (2.3). However, in general equation (2.3) is hard to solve. In only a few cases can equation (2.1) or (2.3) be solved explicitly. One category that is sometimes capable of solution is autonomous or time-invariant differential equations, especially if they are linear. Equation (2.1) would be autonomous if ∂f/∂t = 0 and nonautonomous if ∂f/∂t = 0. In the examples of ordinary differential equations given above only (iii) is an autonomous differential equation. A solution to a nth-order differential equation is an n-times differential function x = φ(t) which when substituted into the equation satisﬁes it exactly in some interval a < t < b.

(2.3)

28

Economic Dynamics Example 2.1 Consider (iii) above. This is an autonomous ﬁrst-order homogeneous differential equation. Rearranging the equation we have dx 1 =k dt x Integrating both sides with respect to t yields dx 1 dt = k dt dt x ln x(t) = kt + c0 where c0 is the constant of integration. Taking exponentials of both sides yields x(t) = cekt where c = ec0 . It is readily veriﬁed that this is indeed a solution by differentiating it and substituting. Thus kcekt = kx = kcekt which holds identically for any a < t < b. Example 2.2 To check whether x(t) = 1 + t + cet is a solution of dx/dt = x − t, we can differentiate x with respect to t and check whether the differential equation holds exactly. Thus dx = 1 + cet dt ... 1 + cet = 1 + t + cet − t Hence x(t) = 1 + t + cet is indeed a solution. Example 2.3 Check whether p(t) =

ap0 bp0 + (a − bp0 )e−at

is a solution to the differential equation dp = p(a − bp) dt Differentiating the solution function with respect to t we obtain dp = −ap0 [bp0 + (a − bp0 )e−at ]−2 (−a(a − bp0 )e−at ) dt a2 p0 (a − bp0 )e−at = [bp0 + (a − bp0 )e−at ]2

Continuous dynamic systems

29

while substituting for p we obtain 2 ap0 a2 p0 −b ap − bp = bp0 + (a − bp0 )e−at bp0 + (a − bp0 )e−at 2

=

a2 p0 (a − bp0 )e−at [bp0 + (a − bp0 )e−at ]2

which is identically true for all values of t. Equation x(t) = cekt is an explicit solution to example (iii) because we can solve directly x(t) as a function of t. On occasions it is not possible to solve x(t) directly in terms of t, and solutions arise in the implicit form F(x, t) = 0

(2.4)

Solutions of this type are referred to as implicit solutions. A graphical solution to a ﬁrst-order differential equation is a curve whose slope at any point is the value of the derivative at that point as given by the differential equation. The graph may be known precisely, in which case it is a quantitative graphical representation. On the other hand, the graph may be imprecise, as far as the numerical values are concerned; yet we have some knowledge of the solution curve’s general shape and features. This is a graph giving a qualitative solution. The graph of a solution, whether quantitative or qualitative, can supply considerable information about the nature of the solution. For example, maxima and minima or other turning points, when the solution is zero, when the solution is increasing and when decreasing, etc. Consider, for example, dx/dt = t2 whose solution is x(t) =

t3 +c 3

where c is the constant of integration. There are a whole series of solution curves depending on the value of c. Four such curves are illustrated in ﬁgure 2.1, with solutions x(t) =

t3 + 8, 3

x(t) =

t3 + 2, 3

x(t) =

t3 , 3

x(t) =

t3 −3 3 Figure 2.1.

30

Economic Dynamics A general solution to a differential equation is a solution, whether expressed explicitly or implicitly, which contains all possible solutions over an open interval. In the present example, all solutions are involved for all possible values of c. A particular solution involves no arbitrary constants. Thus, if c = 2 then x(t) = (t3/3) + 2 represents a particular solution. It is apparent that a second-order differential equation would involve integrating twice and so would involve two arbitrary constants of integration. In general the solution to an nth-order differential equation will involve n arbitrary constants. It follows from this discussion that general solutions are graphically represented by families of solution curves, while a particular solution is just one solution curve. Consider further the general solution in the above example. If we require that x = 0 when t = 0, then this is the same as specifying c = 0. Similarly if x = 2 when t = 0, then this is the same as specifying that c = 2. It is clear, then, that a particular solution curve to a ﬁrst-order differential equation is equivalent to specifying a point (x0 , t0 ) through which the solution curve must pass (where t0 need not be zero). In other words, we wish to ﬁnd a solution x = x(t) satisfying x(t0 ) = x0 . The condition x(t0 ) = x0 is called the initial condition of a ﬁrst-order differential equation. A ﬁrst-order differential equation, together with an initial condition, is called a ﬁrst-order initial value problem. In many applications we ﬁnd that we need to impose an initial condition on the solution. Consider the following ﬁrst-order initial value problem

(2.5)

dx = kx x(t0 ) = x0 dt Rearranging and integrating over the interval t0 to t1 we obtain t1 t1 dx 1 dt = k dt t0 dt x t0 [ln x]tt0 = [k t]tt0 x(t) ln = k(t − t0 ) x0 x(t) = x0 ek(t−t0 ) This is a particular solution that satisﬁes the initial condition. We shall conclude this section with some applications taken from economics and some noneconomic examples. At this stage our aim is simply to set out the problem so as to highlight the type of ordinary differential equations that are involved, the general or speciﬁc nature of the solution and whether the solution satisﬁes some initial value. Example 2.4 A simple continuous price-adjustment demand and supply model takes the form:

(2.6)

qd = a + bp

b0

dp = α(qd − qs ) α > 0 dt

Continuous dynamic systems

31 Figure 2.2.

where quantities, qd and qs and price, p, are assumed to be continuous functions of time. Substituting the demand and supply equations into the price adjustment equation we derive the following dp − α(b − d) = α(a − c) dt which is a ﬁrst-order linear nonhomogeneous differential equation. Using a typical software programme for solving differential equations, the solution path is readily found to be c−a c−a p(t) = + p0 − e−α(d−b)t b−d b−d which satisﬁes the initial condition. For d – b > 0 the solution path for different initial prices is illustrated in ﬁgure 2.2 Example 2.5 Suppose we have the same basic demand and supply model as in example 2.4 but now assume that demand responds not only to the price of the good but also to the change in the price of the good. In other words, we assume that if the price of the good is changing, then this shifts the demand curve. We shall leave open the question at this stage of whether the demand curve shifts to the right or the left as a result of the price change. The model now takes the form qd = a + bp + f

dp dt

b < 0, f = 0

qs = c + dp

d>0

dp = α(qd − qs ) dt

α>0

(2.7)

32

Economic Dynamics This is effectively a stock-adjustment model. Stocks (inventories) change according to the difference between supply and demand, and price adjusts according to the accumulation–decumulation of stocks. Thus, if i(t) denotes the inventory holding of stocks at time t, then di = qs − q d dt t

i = i0 +

(qs − qd )dt

and prices adjust according to di dp = −α = −α(qs − qd ) dt dt = α(qd − qs ) α > 0 which is the third equation in the model. Substituting the demand and supply equations into the price-adjustment equation results in the following ﬁrst-order linear nonhomogeneous differential equation α(b − d) α(a − c) dp − p= dt 1 − αf 1 − αf with solution p(t) =

c−a b−d

+ p0 −

c−a b−d

e

−α(d−b)t 1−αf

which satisﬁes the initial condition p(0) = p0 . For this model there are far more varieties of solution paths, depending on the values of the various parameters. Some typical solution paths are illustrated in ﬁgure 2.3. We shall discuss the stability of such systems later. Figure 2.3.

Continuous dynamic systems

33 Figure 2.4.

Example 2.6 Assume population, p, grows at a constant rate k, where we assume that p is a continuous function of time, t. This means that the percentage change in the population is a constant k. Hence dp 1 =k dt p which immediately gives the ﬁrst-order linear homogeneous differential equation dp − kp = 0 dt with solution p(t) = p0 ekt which satisﬁes the initial condition p(0) = p0 . Typical solution paths for this Malthusian population growth are illustrated in ﬁgure 2.4. Example 2.7 In many scientiﬁc problems use is made of radioactive decay. Certain radioactive elements are unstable and within a certain period the atoms degenerate to form another element. However, in a speciﬁed time period the decay is quite speciﬁc. In the early twentieth century the famous physicist Ernest Rutherford showed that the radioactivity of a substance is directly proportional to the number of atoms

(2.8)

34

Economic Dynamics present at time t. If dn/dt denotes the number of atoms that degenerate per unit of time, then according to Rutherford

(2.9)

dn = −λn λ > 0 dt where λ is the decay constant of the substance concerned and n is a continuous function of time. This is a ﬁrst-order linear homogeneous differential equation and is identical in form to the exponential population growth speciﬁed in example 2.6 above. We shall return to this example later when we consider its solution and how the solution is used for calculating the half-life of a radioactive substance and how this is used to authenticate paintings and such items as the Turin shroud. Example 2.8 In this example we consider a continuous form of the Harrod–Domar growth model. In this model savings, S, is assumed to be proportional to income, Y; investment, I, i.e., the change in the capital stock, is proportional to the change in income over time; and in equilibrium investment is equal to savings. If s denotes the average (here equal to the marginal) propensity to save, and v the coefﬁcient for the investment relationship, then the model can be captured by the following set of equations S = sY I = K˙ = vY˙

(2.10)

I=S where a dot above a variable denotes the ﬁrst-time derivate, i.e., dx/dt. Substituting, we immediately derive the following homogeneous differential equation vY˙ = sY s Y=0 Y˙ − v with initial condition I0 = S0 = sY0 It also follows from the homogeneous equation that the rate of growth of income is equal to s/v, which Harrod called the ‘warranted rate of growth’. The solution path satisfying the initial condition is readily established to be Y(t) = Y0 e(s/v)t Example 2.91 It is well known that the Solow growth model reduces down to a simple autonomous differential equation. We begin with a continuous production function 1

We develop this model in detail here because it has once again become of interest and is the basis of new classical growth models and real business cycle models. A discrete version of the model is developed in chapter 3.

Continuous dynamic systems

35

Y = F(K, L), which is twice differentiable and homogeneous of degree one (i.e. constant returns to scale). Let k = K/L denote the capital/labour ratio and y = Y/L the output/labour ratio. Then K F(K, L) Y = =F , 1 = F(k, 1) = f (k) L L L i.e. y = f (k) with f (0) = 0, f (k) > 0, f (k) < 0, k > 0 We make two further assumptions: 1.

The labour force grows at a constant rate n, and is independent of any economic variables in the system. Hence L˙ = nL

2.

L(0) = L0

Savings is undertaken as a constant fraction of output (S = sY ) and savings equal investment, which is simply the change in the capital stock plus replacement investment, hence I = K˙ + δK S = sY K˙ + δK = sY K(0) = K0

Now differentiate the variable k with respect to time, i.e., derive dk/dt, dK dL L −K dk = k˙ = dt 2 dt dt L K 1 dK 1 dL − k˙ = L dt L L dt K 1 dK K 1 dL = − L K dt L L dt ˙ ˙ K L − =k K L But

sY − δK sY L K˙ sf (k) = = −δ −δ = K K L K k

and nL L˙ = =n L L Hence k˙ = sf (k) − δk − nk = sf (k) − (n + δ)k

(2.11)

36

Economic Dynamics with initial conditions K0 = k0 k(0) = L0 We cannot solve equation (2.11) because the production function is not explicitly deﬁned. Suppose we assume that the production function F(K, L) conforms to a Cobb–Douglas, i.e., we assume Y = aK α L1−α α K Y =a L L

00

41

42

Economic Dynamics

Figure 2.6.

Hence, f (x) reaches a minimum at x = x∗ where f (x) cuts the line y = ax/b. It must follow, then, that for x > x∗ , f (x) is positively sloped. This can be veriﬁed immediately f (x) = ax − bf (x) x > x∗

implying

... f (x) > 0

ax > f (x) or ax > bf (x) b

All the analysis so far allows us to graph the properties, as shown in ﬁgure 2.6. The curve f (x) cuts the y-axis at a/b, declines and reaches a minimum where f (x) cuts the line y = ax/b, and then turns up. Although we cannot identify f (x) or the solution value of x∗ , we do know that ∗ x is nonzero. But can we obtain additional information about the shape of f (x)? Yes – if we consider isoclines. Isoclines and direction ﬁelds Given dy = ax − by dx then for every (x,y)-combination this equation speciﬁes the slope at that point. A plot of all such slopes gives the direction ﬁeld for the differential equation, and gives the ‘ﬂow of solutions’. (The slopes at given points can be considered as small lines, like iron ﬁlings, and if many of these are drawn the direction ﬁeld is revealed – just like iron ﬁlings reveal magnetic forces.) However, it is

Continuous dynamic systems

43 Figure 2.7.

not possible to consider all points in the (x,y)-plane. One procedure is to consider the points in the (x,y)-plane associated with a ﬁxed slope. If m denotes a ﬁxed slope, then f (x, y) = m denotes all combinations of x and y for which the slope is equal to m. f (x, y) = m is referred to as an isocline. The purpose of constructing these isoclines is so that a more accurate sketch of f (x) can be obtained. For dy/dx = ax − by = m the isoclines are the curves (lines) ax − by = m ax m − or y = b b These are shown in ﬁgure 2.7. Of course, the slope of f(x, y) at each point along an isocline is simply the value of m. Thus, along y = ax/b the slope is zero or inclination arctan0 = 0◦ . Along y = (ax/b) − (1/b) the slope is unity or inclination arc tan1 = 45◦ ; while along y = (ax/b) − (2/b) the slope is 2 or inclination arc tan2 = 63◦ . Hence, for values of m rising the slope rises towards inﬁnity (but never reaching it). We have already established that along y = ax/b the slope is zero and so there are turning points all along this isocline. For m negative and increasing, the slope becomes greater in absolute terms. Consider ﬁnally m = a/b. Then the isocline is a a 1 ax ax − − y= = b b b b b2 with intercept −a/b2 . Then along this isocline the slope of the directional ﬁeld is identical to the slope of the isocline. Hence, the direction ﬁelds look quite different either side of this isocline. Above it the solution approaches this isocline asymptotically from above. Hence, the function f (x) takes the shape of the heavy curve in ﬁgure 2.7. In general we do not know the intercept or the turning point. In this instance we consider the approximate integral curves, which are the continuous lines drawn in ﬁgure 2.7. Such integral curves can take a variety of shapes.

44

Economic Dynamics We can summarise the method of isoclines as follows: (1)

From the differential equation dy = φ(x, y) dx determine the family of isoclines φ(x, y) = m

(2)

and construct several members of this family. Consider a particular isocline φ(x, y) = m0 . All points (x, y) on this isocline have the same slope m0 . Obtain the inclination α0 = arctan m0

(3) (4)

0 ≤ α0 ≤ 180◦

Along the isocline φ(x, y) = m0 construct line elements with inclination α0 . (This establishes part of the direction ﬁeld.) Repeat step 2 for each isocline. Draw smooth curves to represent the approximate integral curves indicated by the line elements of step 3.

It is apparent that this is a very tedious procedure. Luckily, a number of mathematical software packages now compute direction ﬁelds and can be used to construct isoclines (see appendices 2.1 and 2.2).

Example 2.12 dy = 2x − y dx In sections 2.11 and 2.12 we give the instructions on using software packages to solve this differential equation explicitly, and in appendices 2.1 and 2.2 we provide Mathematica and Maple instructions, respectively, for plotting solution curves along with the direction ﬁeld. The result is shown in ﬁgure 2.8 Throughout this book we shall provide a number of direction ﬁeld diagrams of differential equation systems. In some cases we can readily obtain the solution explicitly, as shown in ﬁgure 2.9(a) for the Malthusian population and ﬁgure 2.9(b) for the logistic growth curve,4 which features prominently in the present text. In the previous section we derived a differential equation for the Solow growth model under the assumption that production conformed to a Cobb–Douglas production function. Although we explicitly solved this using the Bernoulli equation, its solution was not at all obvious. In such cases we can obtain considerable insight into the solution paths by considering the direction ﬁeld. Thus, in ﬁgure 2.10 we illustrate this feature of the Solow growth model for three initial values of k, the 4

See example 2.15 in section 2.5.

Continuous dynamic systems

45 Figure 2.8.

capital/labour ratio: one below the equilibrium level, another equal to the equilibrium level and a third above the equilibrium level. It is quite clear from the solution paths and the direction ﬁeld that the equilibrium k∗ is locally stable (see exercise 14). Direction ﬁelds can usefully be employed for two further areas of study. First, when considering nonlinear differential equations whose solution may not be available. In this case the qualitative features of the solution can be observed from the direction ﬁeld. Second, in the case of simultaneous equation systems, the examples given so far refer to only one variable along with time. But suppose we are investigating a system of two variables, say x and y, both of which are related to time. In these cases we can observe much about the solution trajectories from considering the direction ﬁeld in the plane of x and y – which later we shall refer to as the phase plane. We shall investigate such differential equation systems in detail in chapter 4.

2.5 Separable functions Earlier we solved for the ﬁrst-order linear homogeneous differential equation dx − kx = 0 dt for the initial condition x(0) = x0 (see equation (2.5)). We did this by ﬁrst re-writing equation (2.18) in the form dx 1 =k dt x

(2.18)

46

Economic Dynamics

Figure 2.9.

Hence integrating both sides with respect to t gives dx 1 dt = k dt + c0 dt x ln x = kt + c0 x(t) = cekt which gives the solution x(t) = x0 ekt In other words, we could solve x(t) explicitly in terms of t.

Continuous dynamic systems

47 Figure 2.10.

But equation (2.18) is just a particular example of a more general differential equation dx g(t) = dt f (x) Any differential equation which can be written in terms of two distinct functions f (x) and g(t) is said to be a separable differential equation. Some examples are the following: (i) (ii) (iii)

1 dx = 2 dt x dx = x(2 − x) dt dx 1 = dt 2xt

Our interest in these particular differential equations is because they are often possible to solve fairly readily since we can write one side in terms of x and the other in terms of t. Thus, writing (2.19) in the form f (x)

dx = g(t) dt

we can then integrate both sides with respect to t dx f (x) dt = g(t)dt + c0 dt or F[x(t)] = g(t)dt + c0

(2.19)

48

Economic Dynamics where

F[x(t)] =

f (x)dx

Using this equation we can solve for x = x(t) which gives the general solution to equation (2.19). Example 2.13 Radioactive decay and half-life Equation (2.9) speciﬁed the differential equation that represented the radioactive decay of atomic particles. We can employ the feature of separability to solve this equation. Thus, if dn = −λn λ > 0 dt then we can re-write this equation dn = −λ dt n Integrating both sides, and letting c0 denote the coefﬁcient of integration, then dn = − λ dt + c0 n ln n = −λt + c0 n = e−λt+c0 = ce−λt

c = ec0

At t = t0 , n = n0 . From this initial condition we can establish the value of c n0 = ce−λt0 c = n0 eλt0 n = n0 e−λt eλt0 = n0 e−λ(t−t0 ) The half-life of a radioactive substance is the time necessary for the number of nuclei to reduce to half the original level. Since n0 denotes the original level then half this number is n0 /2. The point in time when this occurs we denote t1/2 . Hence n0 = n0 e−λ(t1/2 −t0 ) 2 1 = e−λ(t1/2 −t0 ) 2 − ln 2 = −λ(t1/2 − t0 ) ln 2 0.693 = t0 + ... t1/2 = t0 + λ λ Usually, t0 = 0 and so 0.693 λ These results are illustrated in ﬁgure 2.11. t1/2 =

Continuous dynamic systems

49 Figure 2.11.

Example 2.14 Testing for art forgeries5 All paintings contain small amounts of the radioactive element lead-210 and a smaller amount of radium-226. These elements are contained in white lead which is a pigment used by artists. Because of the smelting process from which the pigment comes, lead-210 gets transferred to the pigment. On the other hand, over 90 per cent of the radium is removed. The result of the smelting process is that lead-210 loses its radioactivity very rapidly, having a half-life of about 22 years; radium-226 on the other hand has a half-life of 1,600 years (see example 2.7). For most practical purposes we can treat radium-226 emissions as constant. Let l(t) denote the amount of lead-210 per gram of white lead at time t, and l0 the amount present at the time of manufacture, which we take to be t0 . The disintegration of radium-226 we assume constant at r. If λ is the decay constant of lead-210, then dl = −λl + r l(t0 ) = l0 dt with solution r

l(t) = 1 − e−λ(t−t0 ) + l0 e−λ(t−t0 ) λ Although l(t) and r can readily be measured, this is not true of l0 , and therefore we cannot determine t − t0 . We can, however, approach the problem from a different perspective. Assume that the painting of interest, if authentic, is 300 years old and if new is at the present time t. Then t − t0 = 300. If we substitute this into the previous result and simplify we obtain λl0 = λl(t)e300λ − r(e300λ − 1) 5

This is based on the analysis presented in Braun (1983, pp. 11–17).

50

Economic Dynamics It is possible to estimate λl0 for an authentic painting. It is also possible to estimate λl0 for the lead in the painting under investigation. If the latter is absurdly large relative to the former, then we can conclude that it is a forgery. A very conservative estimate would indicate that any value for λl0 in excess of 30,000 disintegrations per minute per gram of white lead is absurd for an authentic painting aged 300 years. Using 22 years for the half-life of lead-210, then the value of λ is (ln2/22) and e300λ = e(300/22) ln 2 = 2(150/11) To estimate the present disintegration rate of lead-210 the disintegration rate of polonium-210 is used instead because it has the same disintegration rate as lead210 and because it is easier to measure. In order, then, to authenticate the ‘Disciples at Emmaus’, purported to be a Vermeer, it is established that the disintegration rate of polonium-210 per minute per gram of white lead in this particular painting is 8.5 and that of radium-226 is 0.8. Using all this information then we can estimate the value of λl0 for the‘Disciples at Emmaus’ as follows: λl0 = (8.5)2150/11 − 0.8(2150/11 − 1) = 98,050 which is considerably in excess of 30,000. We, therefore, conclude that the ‘Disciples at Emmaus’ is not an authentic Vermeer. Example 2.15 The logistic curve In this example we shall consider the logistic equation in some detail. Not only does this illustrate a separable differential equation, but also it is an equation that occurs in a number of areas of economics. It occurs in population growth models, which we shall consider in part II, and in product diffusion models. It is the characteristic equation to represent learning, and hence occurs in a number of learning models. We shall justify the speciﬁcation of the equation in part II; here we are concerned only with solving the following growth equation for the variable x dx = kx(a − x) dt

(2.20)

The differential equation is ﬁrst separated dx = k dt (a − x)x Integrating both sides, and including the constant of integration, denoted c0 dx = k dt + c0 (a − x)x However

1 1 1 1 = + (a − x)x a x a−x

Continuous dynamic systems Hence 1 a

dx + x

51

dx = k dt + c0 a−x

1 [ln x − ln |a − x|] = kt + c0 a 1 x ln = kt + c0 a a − x x = akt + ac0 ln a − x Taking anti-logs, we have x = eakt+ac0 = eac0 eakt = ceakt a−x where c = eac0 . Substituting for the initial condition, i.e., t = t0 then x = x0 , we can solve for the constant c, as follows x0 = ceakt0 a − x0 x0 c= e−akt0 a − x0 Substituting, then x0 e−akt0 eakt a − x0 x0 = eak(t−t0 ) a − x0

x = a−x

Solving for x

x0 eak(t−t0 ) a − x0 x= x0 1+ eak(t−t0 ) a − x0 a

Which can be further expressed6 x=

ax0 (a − x0 )e−ak(t−t0 ) + x0

From the logistic equation (2.21) we can readily establish the following results, assuming that x0 is less than a: 1. 2.

6

For t = t0 then x = x0 As t → ∞ then x → a

The logistic growth equation is a particular example of the Bernoulli function and can be solved in a totally different way using a simple transformation. See n. 2 and exercise 6.

(2.21)

52

Economic Dynamics

Figure 2.12.

3.

An inﬂexion occurs at the point a − x0 1 ln t = t0 + ak x0 a x= 2 The logistic curve is shown in ﬁgure 2.12. Example 2.16 Constant elasticity of demand Let a commodity x be related to price p with a constant elasticity of demand ε, then dx p = −ε ε > 0 dp x We can rearrange this as dx dp = −ε x p

Integrating both sides and adding a constant of integration, then dp dx = −ε x p ln x = −ε ln p + c0 = −ε ln p + ln c

where c0 = ln c

−ε

= ln cp Therefore x = cp−ε

which is the general expression for a demand curve with constant elasticity of demand.

Continuous dynamic systems

53

2.6 Diffusion models In recent years we have seen the widespread use of desktop computers, and more recently the increased use of the mobile phone. The process by which such innovations are communicated through society and the rate at which they are taken up is called diffusion. Innovations need not be products. They can just as easily be an idea or some contagious disease. Although a variety of models have been discussed in the literature (e.g. Davies 1979; Mahajan and Peterson 1985), the time path of the diffusion process most typically takes the form of the S-shaped (sigmoid) curve. Considering the mobile phone, we would expect only a few adoptions in the early stages, possibly business people. The adoption begins to accelerate, diffusing to the public at large and even to youngsters. But then it begins to tail off as saturation of the market becomes closer. At the upper limit the market is saturated. Although this is a verbal description of the diffusion process, and suggests an S-shaped mathematical formulation of the process, it supplies no exact information about the functional form. In particular, the slope, which indicates the speed of the diffusion; or the asymptote, which indicates the level of saturation. Furthermore, such diffusion processes may differ between products. The typical diffusion model can be expressed dN(t) = g(t)(m − N(t)) dt where N(t) is the cumulative number of adopters at time t, m is the maximum number of potential adopters and g(t) is the coefﬁcient of diffusion. dN(t)/dt then represents the rate of diffusion at time t. Although we refer to the number of adopters, the model is assumed to hold for continuous time t. It is possible to think of g(t) as the probability of adoption at time t, and so g(t)(m − N(t)) is the expected number of adopters at time t. Although a number of speciﬁcations of g(t) have been suggested, most are a special case of

(2.22)

g(t) = a + bN(t) So the diffusion equation generally used is dN(t) = (a/m + bN(t))(m − N(t)) dt ˙ = N(t)/m, ˙ If we divide (2.23) throughout by m and deﬁne F(t) = N(t)/m, with F(t) then dF(t) = (a + bF(t))(1 − F(t)) dt This is still a logistic equation that is separable, and we can re-arrange and integrate by parts (see example 2.15) to solve for F(t) F(t) =

1 − e−(a+b)t 1 + (b/a)e−(a+b)t

This speciﬁcation, however, is not the only possibility. The Gompertz function also exhibits the typical S-shaped curve (see exercise 2), and using this we can

(2.23)

(2.24)

(2.25)

54

Economic Dynamics express the diffusion process as dN(t) = bN(t)(ln m − ln N(t)) dt

(2.26)

or dF(t) = bF(t)(− ln F(t)) dt Suppressing the time variable for convenience, then the two models are F˙ = (a + bF)(1 − F) and F˙ = bF(− ln F) Pursuing the logistic equation, we can graph F˙ against F. When F = 0 then ˙ F = a and when F˙ = 0 then (a + bF)(1 − F) = 0 with solutions F1 = −b/a

and

F2 = 1

Since F˙ denotes the rate of diffusion, then the diffusion rate is at a maximum (penetration is at its maximum rate) when F¨ = 0, i.e., when d2 F/dt2 = 0. Differentiating and solving for F, which we denote Fp (for maximum penetration rate), we obtain 1 a a 1 m am implying Np = m · Fp = m − Fp = − = − 2 2b 2 2b 2 2b In order to ﬁnd the time tp when F(tp) is at a maximum penetration rate, we must ﬁrst solve for F(t). This we indicated above. Since we need to ﬁnd the value of t satisfying F(t) = Fp, then we need to solve 1 − e−(a+b)t a 1 = − 1 + (b/a)e−(a+b)t 2 2b

(2.27)

for t, which we can do using a software package. This gives the time for the maximum penetration of b ln a tp = a+b Since d2 F/dt2 = 0 at Fp, then this must denote the inﬂexion point of F(t). The stylised information is shown in ﬁgure 2.13. Notice that the time for the maximum penetration is the same for both F(t) and N(t). Also note that F(t) involves only the two parameters a and b; while N(t) involves the three parameters a, b and m.

2.7 Phase portrait of a single variable This book is particularly concerned with phase diagrams. These diagrams help to convey the dynamic properties of differential and difference equations – either single equations or simultaneous equations. To introduce this topic and to lay down some terminology, we shall consider here just a single variable. Let x denote

Continuous dynamic systems

55 Figure 2.13.

a variable which is a continuous function of time, t. Let x (t) denote an autonomous differential equation, so that x (t) is just a function of x and independent of t. Assume that we can solve for x (t) for any point in time t. Then at any point in time we have a value for x (t). The path of solutions as t varies is called a trajectory, path or orbit. The x-axis containing the trajectory is called the phase line. If x (t) = 0 then the system is at rest. This must occur at some particular point in time, say t0 . The solution value would then be x(t0 ) = x∗ . The point x∗ is referred to variedly as a rest point, ﬁxed point, critical point, equilibrium point or steady-state solution. For the Malthusian population equation p (t) = kp, there is

56

Economic Dynamics

Figure 2.14.

only one ﬁxed point, namely p∗ = 0. In the case of the logistic growth equation x (t) = kx(a − x) there are two ﬁxed points, one at x1∗ = 0 and the other at x2∗ = a. In example 2.4 on demand and supply the ﬁxed point, the equilibrium point, is given by p∗ =

c−a b−d

which is also the ﬁxed point for example 2.5. For the Harrod–Domar growth model (example 2.8) there is only one stationary point, only one equilibrium point, and that is Y ∗ = 0. For the Solow growth model, in which the production function conforms to a Cobb–Douglas (example 2.9), there are two stationary values, one at k1∗ = 0 and the other at 1 sa −( α−1 ) ∗ k2 = n+δ Whether a system is moving towards a ﬁxed point or away from a ﬁxed point is of major importance. A trajectory is said to approach a ﬁxed point if x(t) → x∗ as t → ∞, in this case the ﬁxed point is said to be an attractor. On the other hand, if x(t) moves away from x∗ as t increases, then x∗ is said to be a repellor. Fixed points, attractors and repellors are illustrated in ﬁgure 2.14. Also illustrated in ﬁgure 2.14 is the intermediate case where the trajectory moves ﬁrst towards the ﬁxed point and then away from the ﬁxed point. Since this can occur from two different directions, they are illustrated separately, but both appear as a shunting motion, and the ﬁxed point is accordingly referred to as a shunt. Consider once again the logistic growth equation x (t) = kx(a − x), as illustrated in ﬁgure 2.15. Figure 2.15(a) illustrates the differential equation, ﬁgure 2.15(b) illustrates the phase line7 and ﬁgure 2.15(c) denotes the path of x(t) against time. The stationary points on the phase line are enclosed in small circles to identify them. The arrows marked on the phase line, as in ﬁgure 2.15(b), indicate the direction of change in x(t) as t increases. In general, x∗ = 0 is uninteresting, and for any initial value of x not equal to zero, the system moves towards x∗ = a, as illustrated in ﬁgure 2.15(c). Even if x initially begins above the level x∗ = a, the system moves over time towards x∗ = a. In other words, x∗ = a is an attractor. 7

Some textbooks in economics confusingly refer to ﬁgure 2.15(b) as a phase diagram.

Continuous dynamic systems

57 Figure 2.15.

If any trajectory starting ‘close to’ a ﬁxed point8 stays close to it for all future time, then the ﬁxed point is said to be stable. A ﬁxed point is asymptotically stable if it is stable as just deﬁned, and also if any trajectory that starts close to the ﬁxed point approaches the ﬁxed point as t → ∞. Considering the logistic equation as shown in ﬁgure 2.15, it is clear that x∗ = a is an asymptotically stable rest point. Figure 2.15 also illustrates another feature of the characteristics of a ﬁxed point. The origin, x∗ = 0, is a repellor while x∗ = a is an attractor. In the neighbourhood of the origin, the differential equation has a positive slope. In the neighbourhood 8

We shall be more explicit about the meaning of ‘close to’ in section 4.2.

58

Economic Dynamics of the attractor, the differential equation has a negative slope. In fact, this is a typical feature of instability/stability. A ﬁxed point is unstable if the slope of the differential equation in the neighbourhood of this point is positive; it is stable if the slope of the differential equation in the neighbourhood of this point is negative. If there is only one ﬁxed point in a dynamic system, then such a ﬁxed point is either globally stable or globally unstable. In the case of a globally stable system, for any initial value not equal to the ﬁxed point, then the system will converge on the ﬁxed point. For a globally unstable system, for any initial value not equal to the ﬁxed point, then the system will move away from it. Consider example 2.4, a simple continuous price-adjustment demand and supply model with the differential equation dp = α(a − c) + α(b − d)p dt

α>0

For a solution (a ﬁxed point, an equilibrium point) to exist in the positive quadrant then a > c and so the intercept is positive. With conventional shaped demand and supply curves, then b < 0 and d > 0, respectively, so that the slope of the differential equation is negative. The situation is illustrated in ﬁgure 2.16(a).

Figure 2.16.

Continuous dynamic systems

59

Given linear demand and supply then there is only one ﬁxed point. The system is either globally stable or globally unstable. It is apparent from ﬁgure 2.16 that the ﬁxed point is an attractor, as illustrated in ﬁgure 2.16(b). Furthermore, the differential equation is negatively sloped for all values of p. In other words, whenever the price is different from the equilibrium price (whether above or below), it will converge on the ﬁxed point (the equilibrium price) over time. The same qualitative characteristics hold for example 2.5, although other possibilities are possible depending on the value/sign of the parameter f . Example 2.6 on population growth, and example 2.7 on radioactive decay, also exhibit linear differential equations and are globally stable/unstable only for p = 0 and n = 0, respectively. Whether they are globally stable or globally unstable depends on the sign of critical parameters. For example, in the case of Malthusian population, if the population is growing, k > 0, then for any initial positive population will mean continuously increased population over time. If k < 0, then for any initial positive population will mean continuously declining population over time. In the case of radioactive decay, λ is positive, and so there will be a continuous decrease in the radioactivity of a substance over time. The Harrod–Domar growth model, example 2.8, is qualitatively similar to the Malthusian population growth model, with the ‘knife-edge’ simply indicating the unstable nature of the ﬁxed point. The Solow growth model, example 2.9, on the other hand, exhibits multiple equilibria. There cannot be global stability or instability because such statements have meaning only with reference to a single ﬁxed point system. In the case of multiple ﬁxed points, statements about stability or instability must be made in relation to a particular ﬁxed point. Hence, with systems containing multiple equilibria we refer to local stability or local instability, i.e., reference is made only to the characteristics of the system in the neighbourhood of a ﬁxed point. For instance, for the Solow growth model with a Cobb–Douglas production function homogeneous of degree one there are two ﬁxed points k1∗

=0

and

k2∗

=

1 sa ( 1−α ) n+δ

The ﬁrst is locally unstable while the second is locally stable, as we observed in ﬁgure 2.10. The ﬁrst ﬁxed point is a repellor while the second ﬁxed point is an attractor. The slope of the differential equation in the neighbourhood of the origin has a positive slope, which is characteristic of a repellor; while the slope of the differential equation in the neighbourhood of the second ﬁxed point is negative, which is characteristic of an attractor. These characteristics of the slope of the differential equation in the neighbourhood of a system’s ﬁxed points and the features of the phase line are illustrated in ﬁgure 2.17.

2.8 Second-order linear homogeneous equations A general second-order linear homogeneous differential equation with constant coefﬁcients is a

d2 y dy + b + cy = 0 dt2 dt

(2.28)

60

Economic Dynamics

Figure 2.17.

Or ay (t) + by (t) + cy(t) = 0 If we can ﬁnd two linearly independent solutions9 y1 and y2 then the general solution is of the form y = c1 y1 + c2 y2 where c1 and c2 are arbitrary constants. Suppose y = e xt . Substituting we obtain ax2 e xt + bxe xt + ce xt = 0 e xt (ax 2 + bx + c) = 0 Hence, y = e xt is a solution if and only if ax2 + bx + c = 0 9

See exercises 9 and 10 for a discussion of linear dependence and independence.

Continuous dynamic systems which is referred to as the auxiliary equation of the homogeneous equation. The quadratic has two solutions √ √ −b − b2 − 4ac −b + b2 − 4ac , s= r= 2a 2a If b2 > 4ac the roots r and s are real and distinct; if b2 = 4ac the roots are real and equal; while if b2 < 4ac the roots are complex conjugate. There are, therefore, three types of solutions. Here we shall summarise them. 2.8.1

Real and distinct (b2 > 4ac)

If the auxiliary equation has distinct real roots r and s, then ert and est are linearly independent solutions to the second-order linear homogeneous equation. The general solution is y(t) = c1 ert + c2 est where c1 and c2 are arbitrary constants. If y(0) and y (0) are the initial conditions when t = 0, then we can solve for c1 and c2 y(0) = c1 er(0) + c2 es(0) = c1 + c2 y (t) = rc1 ert + sc2 est y (0) = rc1 er(0) + sc2 es(0) = rc1 + sc2 Hence c1 =

y (0) − sy(0) , r−s

c2 =

y (0) − ry(0) s−r

and the particular solution is y (0) − ry(0) st y (0) − sy(0) rt e + e y(t) = r−s s−r which satisﬁes the initial conditions y(0) and y (0). Example 2.17 Suppose dy d2 y + 4 − 5y = 0 2 dt dt Then the auxiliary equation is x2 + 4x − 5 = 0 (x + 5)(x − 1) = 0 Hence, r = −5 and s = 1, with the general solution y(t) = c1 e−5t + c2 et

61

62

Economic Dynamics If y(0) = 0 and y (0) = 1, then 1 1 =− −5 − 1 6 1 1 = c2 = 1 − (−5) 6

c1 =

So the particular solution is 1 −5t 1 t y(t) = − e + e 6 6 2.8.2

Real and equal roots (b2 = 4ac)

If r is a repeated real root to the differential equation ay (t) + by (t) + c = 0 then a general solution is y(t) = c1 ert + c2 tert where c1 and c2 are arbitrary constants (see exercise 9). If y(0) and y (0) are the two initial conditions, then y(0) = c1 + c2 (0) = c1 y (t) = rc1 ert + rc2 tert + c2 ert y (0) = rc1 + c2 Hence c1 = y(0),

c2 = y (0) − ry(0)

So the particular solution is y(t) = y(0)ert + [y (0) − ry(0)]tert Example 2.18 y (t) + 4y (t) + 4y(t) = 0 Then the auxiliary equation is x2 + 4x + 4 = 0 (x + 2)2 = 0 Hence, r = −2 and the general solution is y(t) = c1 e−2t + c2 te−2t If y(0) = 3 and y (0) = 7, then c1 = y(0) = 3 c2 = y (0) − ry(0) = 7 − (−2)(3) = 13

Continuous dynamic systems so the particular solution is y(t) = 3e−2t + 13te−2t = (3 + 13t)e−2t 2.8.3

Complex conjugate (b2 < 4ac)

If the auxiliary equation has complex conjugate roots r and s where r = α + iβ and s = α − iβ then eαt cos(βt)

eαt sin(βt)

and

are linearly independent solutions to the second-order homogeneous equation (see exercise 10). The general solution is y(t) = c1 eαt cos(βt) + c2 eαt sin(βt) where c1 and c2 are arbitrary constants. If y(0) and y (0) are the initial conditions when t = 0, then we can solve for c1 and c2 y(0) = c1 cos(0) + c2 sin(0) = c1 y (t) = (αc1 + βc2 )eαt cos(βt) + (αc2 − βc1 )eαt sin(βt) y (0) = (αc1 + βc2 )e0 cos(0) + (αc2 − βc1 )e0 sin(0) = αc1 + βc2 i.e. c1 = y(0)

and

c2 =

Hence, the particular solution is αt

y(t) = y(0)e cos(βt) +

y (0) − αy(0) β

y (0) − αy(0) αt e sin(βt) β

Example 2.19 y (t) + 2y (t) + 2y(t) = 0,

y(0) = 2

The auxiliary equation is x2 + 2x + 2 = 0 with complex conjugate roots √ −2 + 4 − 4(2) = −1 + i r= √2 −2 − 4 − 4(2) s= = −1 − i 2 The general solution is y(t) = c1 e−t cos(t) + c2 e−t sin(t)

and

y (0) = 1

63

64

Economic Dynamics The coefﬁcients are c1 = y(0) = 2 c2 =

y (0) − αy(0) =3 β

Hence the particular solution is y(t) = 2e−t cos(t) + 3e−t sin(t)

2.9 Second-order linear nonhomogeneous equations A second-order linear nonhomogeneous equation with constant coefﬁcients takes the form a

(2.29)

dy d2 y + b + cy = g(t) dt2 dt

or ay (t) + by (t) + cy(t) = g(t) Let L(y) = ay (t) + by (t) + cy(t) then equation (2.29) can be expressed as L( y) = g(t). The solution to equation (2.29) can be thought of in two parts. First, there is the homogeneous component, L(y) = 0. As we demonstrated in the previous section, if the roots are real and distinct then yc = c1 ert + c2 est The reason for denoting this solution as yc will become clear in a moment. Second, it is possible to come up with a particular solution, denoted yp , which satisﬁes L(yp ) = g(t). yc is referred to as the complementary solution satisfying L(y) = 0, while yp is the particular solution satisfying L(yp ) = g(t). If both yc and yp are solutions, then so is their sum, y = yc + yp , which is referred to as the general solution to a linear nonhomogeneous differential equation. Hence, the general solution to equation (2.29) if the roots are real and distinct takes the form y(t) = yc + yp = c1 ert + c2 est + yp The general solution y(t) = yc + yp holds even when the roots are not real or distinct. The point is that the complementary solution arises from the solution to L(y) = 0. As in the previous section there are three possible cases: (1)

Real and distinct roots yc = c1 ert + c2 est

(2)

Real and equal roots yc = c1 ert + c2 tert

Continuous dynamic systems (3)

Complex conjugate roots yc = c1 eαt cos(βt) + c2 eαt sin(βt)

In ﬁnding a solution to a linear nonhomogeneous equation, four steps need to be followed: Step 1 Find the complementary solution yc . Step 2 Find the general solution yh by solving the higher-order equation Lh ( yh ) = 0 where yh is determined from L(y) and g(t). Step 3 Obtain yq = yh − yc . Step 4 Determine the unknown constant, the undetermined coefﬁcients, in the solution yq by requiring L( yq ) = g(t) and substituting these into yq, giving the particular solution yp .

Example 2.20 Suppose y (t) + y (t) = t Step 1 This has the complementary solution yc , which is the solution to the auxiliary equation x2 + x = 0 x(x + 1) = 0 with solutions r = 0 and s = −1 and yc = c1 e0t + c2 e−t = c1 + c2 e−t Step 2 The differential equation needs to be differentiated twice to obtain Lh (yh ) = 0. Thus, differentiating twice y(4) (t) + y(3) (t) = 0 with auxiliary equation x4 + x3 = 0 with roots 0, −1, 0, 0. Hence10 yh = c1 e0t + c2 e−t + c3 te0t + c4 t2 e0t = c1 + c2 e−t + c3 t + c4 t2 10

We have here used the property that ert , tert and t2 ert are linearly independent and need to be combined with a root repeating itself three times (see exercise 9(ii)).

65

66

Economic Dynamics Step 3 Obtain yq = yh − yc . Thus yq = (c1 + c2 e−t + c3 t + c4 t2 ) − (c1 + c2 e−t ) = c3 t + c4 t2 Step 4 To ﬁnd c3 and c4 , the undetermined coefﬁcients, we need L(yq ) = t. Hence yq (t) + yq (t) = t But from step 3 we can derive yq = c3 + 2c4 t yq = 2c4 Hence 2c4 + c3 + 2c4 t = t Since the solution must satisfy the differential equation identically for all t, then the result just derived must be an identity for all t and so the coefﬁcients of like terms must be equal. Hence, we have the two simultaneous equations 2c4 + c3 = 0 2c4 = 1 with solutions c4 = 1/2 and c3 = −1. Thus yp = −t + 12 t2 and the solution is y(t) = c1 + c2 e−t − t + 12 t2 It is also possible to solve for c1 and c2 if we know y(0) and y (0). Although we have presented the method of solution, many software packages have routines built into them, and will readily supply solutions if they exist. The economist can use such programmes to solve the mathematics and so concentrate on model formulation and model features. This we shall do in part II.

2.10 Linear approximations to nonlinear differential equations Consider the differential equation x˙ = f (x) here f is nonlinear and continuously differentiable. In general we cannot solve such equations explicitly. We may be able to establish the ﬁxed points of the system by solving the equation f (x) = 0, since a ﬁxed point is characterised by x˙ = 0. Depending on the nonlinearity there may be more than one ﬁxed point.

Continuous dynamic systems

67

If f is continuously differentiable in an open interval containing x = x∗ , then we approximate f using the Taylor expansion f (x) = f (x∗ ) + f (x∗ )(x − x∗ ) f n (x∗ )(x − x∗ ) f (x∗ )(x − x∗ ) + ... + + Rn (x, x∗ ) 2! n! where Rn (x, x∗ ) is the remainder. In particular, a ﬁrst-order approximation takes the form +

f (x) = f (x∗ ) + f (x∗ )(x − x∗ ) + R2 (x, x∗ ) If the initial point x0 is sufﬁciently close to x∗ , then R2 (x, x∗ ) 0. Furthermore, if we choose x∗ as being a ﬁxed point, then f (x∗ ) = 0. Hence we can approximate f (x) about a ﬁxed point x∗ with f (x) = f (x∗ )(x − x∗ )

(2.30)

Example 2.21 Although we could solve the Solow growth model explicitly if the production function was a Cobb–Douglas by using a transformation suggested by Bernoulli, it provides a good example of a typical nonlinear differential equation problem. Our equation is k˙ = f (k) = sakα − (n + δ)k This function has two ﬁxed points obtained from solving k[sakα−1 − (n + δ)] = 0 namely k1∗ = 0

and

k2∗ =

sa n+δ

1 −( α−1 )

Taking a ﬁrst-order Taylor expansion about point k∗, we have f (k) = f (k∗ ) + f (k∗ )(k − k∗ ) where f (k∗ ) = αsa(k∗ )α−1 − (n + δ) f (k∗ ) = 0

and

Consider ﬁrst k∗ = k1∗ = 0, then f (k1∗ ) = lim f (k) = lim [αsakα−1 − (n + δ)] = ∞ k→0

Next consider k = f

(k2∗ )

=

k2∗

k→0

> 0, then f (k2∗ ) = 0 and

αsa(k2∗ )α−1

− (n + δ) = αsa

= α(n + δ) − (n + δ) = −(n + δ)(1 − α)

sa n+δ

1 −( α−1 ) α−1

− (n + δ)

68

Economic Dynamics

Figure 2.18.

Hence f (k) = −(n + δ)(1 − α)(k − k∗ ) Since 0 < α < 1 and n and δ are both positive, then this has a negative slope about k2∗ and hence k2∗ is a locally stable equilibrium. The situation is shown in ﬁgure 2.18. The ﬁrst-order linear approximation about the non-zero equilibrium is then k˙ = f (k) = −(n + δ)(1 − α)(k − k∗ ) with the linear approximate solution k(t) = k2∗ + (k(0) − k2∗ )e−(n+δ)(1−α)t As t → ∞ then k(t) → k2∗ . What we are invoking here is the following theorem attributed to Liapunov THEOREM 2.1 If x˙ = f (x) is a nonlinear equation with a linear approximation f (x) = f (x∗ ) + f (x∗ )(x − x∗ ) about the equilibrium point x∗ , and if x∗ is (globally) stable for the linear approximation, then x∗ is asymptotically stable for the original nonlinear equation. Care must be exercised in using this theorem. The converse of the theorem is generally not true. In other words, it is possible for x∗ to be stable for the nonlinear system but asymptotically unstable for its linear approximation.

Continuous dynamic systems

69 Figure 2.19.

Example 2.22 Consider x˙ = f (x) = a(x − x∗ )3

− ∞ < x < ∞,

a>0

There is a unique equilibrium at x = x∗ = 0 which is globally stable. This is readily seen in terms of ﬁgure 2.19, which also displays the phase line. Now consider its linear approximation at x = x∗ f (x) = 3a(x − x∗ )2 f (x∗ ) = 0 and so x˙ = f (x) = f (x∗ ) + f (x∗ )(x − x∗ ) = 0 which does not exhibit global stability. This is because for any x0 = x∗ then x = x0 for all t since x˙ = 0. Consequently, x0 does not approach x∗ in the limit, and so x∗ = 0 cannot be asymptotically stable.

70

Economic Dynamics We shall return to linear approximations in chapter 3 when considering difference equations, and then again in chapters 4 and 5 when we deal with nonlinear systems of differential and difference equations. These investigations will allow us to use linear approximation methods when we consider economic models in part II.

2.11 Solving differential equations with Mathematica 2.11.1

First-order equations

Mathematica has two built in commands for dealing with differential equations, which are the DSolve command and the NDSolve command. The ﬁrst is used to ﬁnd a symbolic solution to a differential equation; the second ﬁnds a numerical approximation. Consider the following ﬁrst-order differential equation dy = f (y, t) dt In particular, we are assuming that y is a function of t, y(t). Then we employ the DSolve command by using DSolve[y’[t]==f[y[t],t],y[t],t]

Note a number of aspects of this instruction: (1) (2) (3) (4)

The equation utilises the single apostrophe, so y (t) denotes dy/dt The function f (y(t), t) may or may not be independent of t y(t) is written in the equation rather than simply y The second term, y(t), is indicating what is being solved for, and t denotes the independent variable.

It is possible to ﬁrst deﬁne the differential equation and use the designation in the DSolve command. Thus Eq = y’[t]==f[y[t],t] DSolve[Eq,y[t],t]

If Mathematica can solve the differential equation then this is provided in the output. Sometimes warnings are provided, especially if inverse functions are being used. If Mathematica can ﬁnd no solution, then the programme simply repeats the input. The user does not need to know what algorithm is being used to solve the differential equation. What matters is whether a solution can be found. What is important to understand, however, is that a ﬁrst-order differential equation (as we are discussing here) involves one unknown constant of integration. The output will, therefore, involve an unknown constant, which is denoted C[1]. Consider the examples of ﬁrst-order differential equations used in various places throughout this chapter shown in table 2.1. Mathematica has no difﬁculty solving all these problems, but it does provide a warning with the last stating: ‘The equations appear to involve transcendental functions of the variables in an essentially non-algebraic way.’ What is also illustrated by these solutions is that the output may not, and usually is not, provided in

Continuous dynamic systems Table 2.1 First-order differential equations with Mathematica Problem

Input instructions

(i)

dx = kx dt

DSolve[x’[t]==kx[t],x[t],t]

(ii)

dx = 1 + cet dt

DSolve[x’[t]==1+cExp[t],x[t],t]

(iii)

dp − α(b − d)p = α(a − c) dt

DSolve[p’[t]-α (b-d)p[t]==α(a-c),p[t],t]

(iv)

dx = kx(a − x) dt k˙ = sakα −(n + δ)k

(v)

DSolve[x’[t]==kx[t](a-x[t]),x[t],t] DSolve[k’[t]==sak[t]α -(n+δ)k[t],k[t],t]

Table 2.2 Mathematica input instructions for initial value problems Problem

Input instructions

(i)

dp = p(a − bp), p(0) = p0 dt

(ii)

dn = −λn, n(0) = n0 dt

DSolve[{n’[t]==-λn[t],n[0]==n0},n[t],t]

(iii)

dy = x2 − 2x + 1, y(0) = 1 dx

DSolve[{y’[x]==x2 -2x+1,y[0]==1},y[x],x]

DSolve[{p’[t]==p[t](a-bp[t]),p[0]==p0}, p[t],t]

a way useful for economic interpretation. So some manipulation of the output is often necessary. It will be noted that none of the above examples involve initial conditions, which is why all outputs involve the unknown constant C[1]. Initial value problems are treated in a similar manner. If we have the initial value problem, dy = f ( y, t) dt

y(0) = y0

then the input instruction is DSolve[{y’[t]==f[y[t],t],y[0]==y0},y[t],t]

For example, look at table 2.2.

2.11.2

Second-order equations

Second-order differential equations are treated in fundamentally the same way. If we have the homogeneous second-order differential equation a

dy d2 y + b + cy = 0 dt2 dt

71

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Economic Dynamics Table 2.3 Mathematica input instructions for second-order differential equations Problem

Input Instructions

(i)

dy d2 y + 4 − 5y = 0 dt2 dt

DSolve[y’’[t]+4y’[t]-5y[t]==0,y[t],t]

(ii) (iii) (iv)

y (t) + 4y (t) + 4y(t) = 0 y (t) + 2y (t) + 2y(t) = 0 y (t) + y (t) = t

DSolve[y’’[t]+4y’[t]+4y[t]==0,y[t],t] DSolve[y’’[t]+2y’[t]+2y[t]==0,y[t],t] DSolve[y’’[t]+y’[t]==t,y[t],t]

Table 2.4 Mathematica input instructions for initial value problems Problem (i)

(ii)

Input instructions d2 y dy + 4 − 5y = 0, dt2 dt y(0) = 0, y (0) = 1 y (t) + 4y (t) + 4y(t) = 0, y(0) = 3, y (0) = 7

DSolve[{y’’[t]+4y’[t]-5y[t]==0,y[0]==0, y’[0]==1},y[t],t] DSolve[{y’’[t]+4y’[t]+4y[t]==0,y[0]==3, y’[0]==7},y[t],t]

then the input instruction is11 DSolve[ay’’[t]+by’[t]+cy[t]==0,y[t],t]

If we have the nonhomogeneous second-order differential equation d2 y dy + b + cy = g(t) dt2 dt then the input instruction is a

DSolve[ay’’[t]+by’[t]+cy[t]==g[t],y[t],t]

Of course, the solutions are far more complex because they can involve real and distinct roots, real and equal roots and complex conjugate roots. But the solution algorithms that are built into Mathematica handle all these. Furthermore, secondorder differential equations involve two unknowns, which are denoted C[1] and C[2] in Mathematica’s output. The Mathematica input instructions for some examples used in this chapter are shown in table 2.3. Initial value problems follow the same structure as before (table 2.4). 2.11.3

NDSolve

Many differential equations, especially nonlinear and nonautonomous differential equations, cannot be solved by any of the known solution methods. In such cases a numerical approximation can be provided using the NDSolve command. In using NDSolve it is necessary, however, to provide initial conditions and the range for 11

Do not use the double quotes in these equations; rather input the single quote twice.

Continuous dynamic systems the independent variable. Given the following initial value problem dy = f ( y(t), t) dt the input instruction is

y(0) = y0

NDSolve[{y’[t]==f[y[t],t],y[0]==y0}, y[t],{t,tmin,tmax}]

Mathematica provides output in the form of an InterpolatingFunction that represents an approximate function obtained using interpolation. This InterpolatingFunction can then be plotted. Since it is usual to plot an InterpolatingFunction, then it is useful to give the output a name. For example, given the problem dy = sin(3t − y) dt the instruction is

y(0) = 0.5,

t ∈ [0, 10]

sol=NDSolve[{y’[t]==Sin[3t-y[t]],y[0]==0.5}, y[t],{t,0,10}]

Although the output is named ‘sol’, the solution is still for the variable y(t). So the plot would involve the input Plot[y[t] /. sol, {t,0,10}]

Note that the range for t in the plot is identical to the range given in the NDSolve command. Higher-order ordinary differential equations are treated in the same way. For example, given the initial value problem dy d2 y + 0.5 + sin(y) = 0, dt2 dt the input instruction is

y(0) = −1, y (0) = 0,

t ∈ [0, 15]

sol=NDSolve[{y’’[t]+0.5y’[t]+Sin[y[t]]==0, y[0]==-1,y’[0]==0}, y[t],{t,0,15}]

with plot Plot[y[t] /. sol, {t,0,15}]

2.12 Solving differential equations with Maple 2.12.1

First-order equations

Maple has a built in command for dealing with differential equations, which is the dsolve command. This command is used to ﬁnd a symbolic solution to a differential equation. The command dsolve(. . . , numeric) ﬁnds a numerical approximation. Consider the following ﬁrst-order differential equation dy = f ( y, t) dt

73

74

Economic Dynamics In particular, we are assuming that y is a function of t, y(t). Then we employ the dsolve command by using dsolve(diff(y(t),t)=f(y(t),t),y(t));

Note a number of aspects of this instruction: (1) (2) (3) (4)

The equation utilises diff( y(t), t) to denote dy/dt The function f (y(t), t) may or may not be independent of t y(t) is written in the equation rather than simply y The second term, y(t), is indicating what is being solved for and that t is the independent variable.

It is possible to ﬁrst deﬁne the differential equation and use the designation in the dsolve command. Thus Eq:=diff(y(t),t); dsolve(Eq,y(t));

If Maple can solve the differential equation then this is provided in the output. If Maple can ﬁnd no solution, then the programme simply gives a blank output. The user does not need to know what algorithm is being used to solve the differential equation. What matters is whether a solution can be found. What is important to understand, however, is that a ﬁrst-order differential equation (as we are discussing here) involves one unknown constant of integration. The output will, therefore, involve an unknown constant, which is denoted C1. Consider the following examples of ﬁrst-order differential equations used in various places throughout this chapter (table 2.5). Maple has no difﬁculty solving all these problems. What is illustrated by these solutions is that the output may not, and usually is not, provided in a way useful for economic interpretation. So some manipulation of the output is often necessary. It will be noted that none of the above examples involves initial conditions, which is why all outputs involve the unknown constant C1. Initial value problems Table 2.5 Maple input instructions for ﬁrst-order differential equations Problem (i) (ii) (iii)

dx = kx dt dx = 1 + cet dt dp − α(b − d)p = α(a − c) dt

(iv)

dx = kx(a − x) dt

(v)

k˙ = sak α − (n + δ)k

Input instructions dsolve(diff(x(t),t)=k*x(t),x(t)); dsolve(diff(x(t),t)=1+c*exp(t),x(t)); dsolve(diff(p(t),t)-alpha*(b-d)*p(t)= alpha*(a-c),p(t)); dsolve(diff(x(t),t)= k*x(t)*(a-x(t)),x(t)); dsolve(diff(k(t),t)=s*a*k(t)^alpha(n+delta)*k(t),k(t));

Continuous dynamic systems Table 2.6 Maple input instructions for ﬁrst-order initial value problems Problem

Input instructions

(i)

dp = p(a − bp), p(0) = p0 dt

(ii)

dn = −λn, n(0) = n0 dt

(iii)

dy = x2 − 2x + 1, y(0) = 1 dx

dsolve({diff(p(t),t)= p(t)*(a-b*p(t)),p(0)=p0},p(t)); dsolve({diff(n(t),t)= -lambda*n(t),n(0)=n0},n(t)); dsolve({diff(y(x),x)= x^2-2*x+1,y(0)=1},y(x));

are treated in a similar manner. If we have the initial value problem, dy = f ( y, t) dt

y(0) = y0

then the input instruction is dsolve({diff(y(t),t)=f(y(t),t),y(0)=y0},y(t));

For example, look at table 2.6. 2.12.2

Second-order equations

Second-order differential equations are treated in fundamentally the same way. If we have the homogeneous second-order differential equation a

dy d2 y + b + cy = 0 2 dt dt

then the input instruction is dsolve(a*diff(y(t),t$2)+b*diff(y(t),t)+c*y(t)=0,y(t));

If we have the nonhomogeneous second-order differential equation a

d2 y dy + b + cy = g(t) 2 dt dt

then the input instruction is dsolve(a*diff(y(t),t$2)+b*diff(y(t),t)+c*y(t) =g(t),y(t));

Of course, the solutions are far more complex because they can involve real and distinct roots, real and equal roots and complex conjugate roots. But the solution algorithms that are built into Maple handle all these. Furthermore, second-order differential equations involve two unknowns, which are denoted C1 and C2 in Maple’s output. The input instructions for some examples used in this chapter are shown in table 2.7. Initial value problems follow the same structure as before (table 2.8).

75

76

Economic Dynamics Table 2.7 Maple input instructions for second-order differential equations Problem

Input instructions

(i)

dy d2 y + 4 − 5y = 0 dt2 dt

(ii)

y (t) + 4y (t) + 4y(t) = 0

(iii)

y (t) + 2y (t) + 2y(t) = 0

(iv)

y (t) + y (t) = t

dsolve(diff(y(t),t$2)+4*diff(y(t),t)5*y(t)=0,y(t)); dsolve(diff(y(t),t$2)+4*diff(y(t),t)+ 4*y(t)=0,y(t)); dsolve(diff(y(t),t$2)+2*diff(y(t),t)+ 2*y(t)=0,y(t)); dsolve(diff(y(t),t$2)+diff(y(t),t)= t,y(t));

Table 2.8 Maple input instructions for second-order initial value problems Problem

Input instructions

(i)

dy d2 y + 4 − 5y = 0, dt2 dt y(0) = 0, y (0) = 1

(ii)

y (t) + 4y (t) + 4y(t) = 0, y(0) = 3, y (0) = 7

2.12.3

dsolve({diff(y(t),t$2)+4*diff(y(t),t)5*y(t)=0,y(0)=0,D(y)(0)=1},y(t)); dsolve({diff(y(t),t$2)+4*diff(y(t),t)+ 4*y(t)=0,y(0)=3,D(y)(0)=7},y(t));

dsolve(. . . , numeric)

Many differential equations, especially nonlinear and nonautonomous differential equations, cannot be solved by any of the known solution methods. In such cases a numerical approximation can be provided using the dsolve(. . . , numeric) command. In using the numerical version of the dsolve command, it is necessary to provide also the initial condition. Given the following initial value problem, dy = f ( y(t), t) dt the input instruction is

y(0) = y0

dsolve({diff(y(t),t)=f(y(t),t),y(0)=y0},y(t),numeric);

Maple provides output in the form of a proc function (i.e. a procedural function) that represents an approximate function obtained using interpolation. This procedure can then be plotted. Since it is usual to plot such a procedural function, it is useful to give the output a name. Furthermore, since the plot is of a procedural function, it is necessary to use the odeplot rather than simply the plot command. In order to do this, however, it is ﬁrst necessary to load the plots subroutine with the following instruction. with(plots):

For example, given the problem dy = sin(3t − y) dt

y(0) = 0.5,

t ∈ [0,10]

Continuous dynamic systems the instruction for solving this is Sol1=dsolve({diff(y(t),t)=sin(3*t-y(t)),y(0)=0.5}, y(t),numeric);

Although the output is named ‘Sol’, the solution is still for the variable y(t). So the plot would involve the input odeplot(Sol1,[t,y(t)],0..10);

Note that the range for t is given only in the odeplot instruction. Higher-order ordinary differential equations are treated in the same way. For example, given the initial value problem, dy d2 y + 0.5 + sin(y) = 0, 2 dt dt the input instruction is

y(0) = −1, y (0) = 0,

t ∈ [0,15]

Sol2=dsolve({diff(y(t),t$2)+0.5*diff(y(t), t)+sin(y(t))=0, y(0)=-1,D(y)(0)=0},y(t),numeric);

with plot odeplot(Sol2,[t,y(t)],0..15);

Appendix 2.1 Plotting direction ﬁelds for a single equation with Mathematica Figure 2.8 (p. 45) Given the differential equation dy = 2x − y dx the direction ﬁeld and isoclines can be obtained using Mathematica as follows: Step 1 Load the PlotField subroutine with the instruction a,{a,-2,2,.5}]

Note the following: (a) the solution to the differential equation is evaluated by letting C[1], the constant of integration, take the value of a. This is accomplished by adding the term ‘/. C[1]->a’ (b) a is then given values between −2 and 2 in increments of 0.5. Step 6 Plot the trajectories using the Plot and Evaluate commands plottraj=Plot[ Evaluate[trajectories], {x,-2,2} ]

Note that it is important to give the domain for x the same as in the direction ﬁeld plot. Step 7 Combine the direction ﬁeld plot and the trajectories plot using the Show command (not available prior to version 2.0) Show[arrows,plottraj]

This ﬁnal result is shown in ﬁgure 2.8.

Figure 2.9(a) (p. 46) This follows similar steps as for ﬁgure 2.8, and so here we shall simply list the input lines, followed by a few notes. (1) (2) (3) (4) (5) (6)

Input Input Input Input Input Input

(7)

Input

{"t","p"} ]

Input (2) and (3) are simply to check the initial population size and the ﬁnal population size. Input (6) has {1, kp} (with k = 0.01) as the ﬁrst element in the

Continuous dynamic systems PlotVectorField. Input (7) indicates some options that can be used with the Show command. These too could be employed in (a) above.

Figure 2.9(b) (p. 46) Before we can plot the logistic function we need to solve it. In this example we shall employ the ﬁgures for a and b we derive in chapter 14 for the UK population over the period 1781–1931. a = 0.02

and

b = 0.000436

and with p0 = 13. (1) (2)

Input Input

(3) (4) (5)

Input Input Input

(6)

Input

{"t","p"} ]

Note again that the PlotVectorField has the ﬁrst element in the form {1, (a − bp)p} (with a = 0.02 and b = 0.000436).

Appendix 2.2 Plotting direction ﬁelds for a single equation with Maple Figure 2.8 (p. 45) Given the differential equation dy = 2x − y dx the direction ﬁeld and isoclines can be obtained using Maple as follows: Step 1 Load the DEtools subroutine with the instruction with(DEtools):

Note the colon after the instruction. Step 2 Deﬁne the differential equation and a set of points for the isoclines. Eq:= diff(y(x),x)=2*x-y Points:={[-2,2],[-1,1],[-1,0.5],[-0.5,-2], [0,-2],[0.5,-1.5],[0.5,-1],[1,-1],[1.5,-0.5]};

Step 3 Obtain the direction ﬁeld and the integral curves with the instruction DEplot(Eq,y(x),x=-2..2,Points,y=-2..2, arrows=slim, linecolour=blue);

79

80

Economic Dynamics Note that the direction ﬁeld has six elements: (i) (ii) (iii) (iv) (v) (vi)

the differential equation y(x) indicates that x is the independent variable and y the dependent variable the range for the x-axis the initial points the range for the y-axis a set of options; here we have two options: (a) arrows are to be drawn slim (the default is thin) (b) the colour of the lines is to be blue (the default is yellow).

Figure 2.9(a) (p. 46) This follows similar steps as ﬁgure 2.8 and so we shall be brief. We assume a new session. Input the following: (1) (2) (3) (4) (5) (6)

with(DEtools): equ:=p0*exp(k*t); newequ:=subs(p0=13,k=0.01,equ); inisol:=evalf(subs(t=0,newequ)); finsol:=evalf(subs(t=150,newequ)); DEplot(diff(p(t),t)=0.01*p,p(t),t=0..150,{[0,13]}, p=0..60, arrows=slim,linecolour=blue);

Instructions (2), (3), (4) and (5) input the equation and evaluate it for the initial point (time t = 0) and at t = 150. The remaining instruction plots the direction ﬁeld and one integral curve through the point (0, 13).

Figure 2.9(b) (p. 46) The logistic equation uses the values a = 0.02 and b = 0.000436 and p0 = 13. The input instructions are the following, where again we assume a new session: (1) (2)

with(DEtools): DEplot(diff(p(t),t)=(0.02-0.000436*p)*p,p(t), t=0..150,{[0,13]},p=0..50,arrows=slim, linecolour=blue);

Exercises 1.

Show the following are solutions to their respective differential equations (i)

dy = ky dx

(ii)

−x dy = dx y

(iii)

dy −2y = dx x

y = cekx y = x 2 + y2 = c y=

a x2

Continuous dynamic systems 2.

Analyse the qualitative and quantitative properties of the Gompertz equation for population growth p˙ = p (t) = kp(a − ln p)

3.

4.

5.

6.

Solve the following separable differential equations (i)

dy = x(1 − y2 ) dx

(ii)

dy = 1 − 2y + y2 dx

(iii)

y2 dy = 2 dx x

−10

(3.5)

Substituting, we obtain a − bpt = c + dpt−1 or

pt =

a−c b

d − pt−1 b

which is a ﬁrst-order nonhomogeneous dynamic system. It is also an autonomous dynamic system since it does not depend explicitly on t. This model is illustrated in ﬁgure 3.1. The demand and supply curves are indicated by D and S, respectively. Because we have a ﬁrst-order system, we need one initial starting price. Suppose this is p0 . This gives a quantity supplied in the next period of q1 , read off the supply curve, and indicated by point a. But since demand equals supply in any one period, this gives a demand of also q1 , while this demand implies a price of p1 in period 1. This in turn means that supply in period 2 is q2 . And so the sequence continues. We shall refer to this model frequently in this chapter.

(3.6)

88

Economic Dynamics

Figure 3.1.

3.4 Equilibrium and stability of discrete dynamic systems If yt+1 = f (yt ) is a discrete dynamic system, then y∗ is a ﬁxed point or equilibrium point of the system if f ( yt ) = y∗ for all t

(3.7)

A useful implication of this deﬁnition is that y∗ is an equilibrium value of the system yt+1 = f (yt ) if and only if y∗ = f (y∗ ) For example, in the cobweb model (3.6) we have, d ∗ a−c ∗ − p p = b b Hence p∗ =

a−c b+d

where p∗ ≥ 0 if a ≥ c

With linear demand and supply curves, therefore, there is only one ﬁxed point, one equilibrium point. However, such a ﬁxed point makes economic sense (i.e. for price to be nonnegative) only if the additional condition a ≥ c is also satisﬁed. As with ﬁxed points in continuous dynamic systems, a particularly important consideration is the stability/instability of a ﬁxed point. Let y∗ denote a ﬁxed point for the discrete dynamic system yt+1 = f ( yt ). Then (Elaydi 1996, p. 11)

Discrete dynamic systems (i)

(ii)

(iii)

The equilibrium point y∗ is stable if given ε > 0 there exists δ > 0 such that n f ( y0 ) − y∗ < ε y0 − y∗ < δ implies for all n > 0. If y∗ is not stable then it is unstable. The equilibrium point y∗ is a repelling ﬁxed point if there exists ε > 0 such that f (y0 ) − y∗ > y0 − y∗ implies 0 < y0 − y∗ < ε The point y∗ is an asymptotically stable (attracting) equilibrium point2 if it is stable and there exists η > 0 such that y0 − y∗ < η implies lim yt = y∗ t→∞

∗

If η = ∞ then y is globally asymptotically stable. All these are illustrated in ﬁgure 3.2(a)–(e). In utilising these concepts we employ the following theorem (Elaydi 1996, section 1.4). THEOREM 3.1 Let y∗ be an equilibrium point of the dynamical system yt+1 = f (yt ) where f is continuously differentiable at y∗ . Then (i) if f ( y∗ ) < 1 then y∗ is an asymptotically stable (attracting) ﬁxed point (ii) if f ( y∗ ) > 1 then y∗ is unstable and is a repelling ﬁxed point (iii) if f ( y∗ ) = 1 and

(iv)

(a) if f ( y∗ ) = 0, then y∗ is unstable (b) if f ( y∗ ) = 0 and f ( y∗ ) > 0, then y∗ is unstable (c) if f ( y∗ ) = 0 and f ( y∗ ) < 0, then y∗ is asymptotically stable ∗ if f ( y ) = −1 and (a) if −2f ( y∗ ) − 3[ f (y∗ )]2 < 0, then y∗ is asymptotically stable (b) if −2f ( y∗ ) − 3[ f (y∗ )]2 > 0 , then y∗ is unstable.

The attraction and repulsion of a ﬁxed point can readily be illustrated for a ﬁrst-order system. Suppose f ( yt ) is linear for the ﬁrst-order system yt+1 = f ( yt ). This is represented by the lines denoted L in ﬁgures 3.3(a) and (b), where yt+1 is marked on the vertical axis and yt on the horizontal axis. The equilibrium condition requires yt+1 = yt for all t, hence this denotes a 45◦ -line, denoted by E in ﬁgures 3.3(a) and (b). The ﬁxed point in each case, therefore, is y∗ . 2

Sometimes an asymptotically stable (attracting) equilibrium point is called a sink.

89

90

Economic Dynamics

Figure 3.2.

Consider ﬁrst ﬁgure 3.3(a). We require an initial value for y to start the sequence, which is denoted y0 . Given y0 in period 0, then we have y1 in period 1, as read off from the line L. In terms of the horizontal axis, this gives a value of y1 as read off the 45◦ -line (i.e. the horizontal movement across). But this means that in period 2 the value of y is y2 , once again read off from the line L. In terms of the horizontal axis this also gives a value y2 , read horizontally across. Regardless of the initial value y0 , the sequence converges on y∗ , and this is true whether y0 is below y∗ , as in the ﬁgure, or is above y∗ . Using the same analysis, it is clear that in ﬁgure 3.3(b), starting from an initial value of y of y0 , the sequence diverges from y∗ . If y0 is below y∗ then the system creates smaller values of y and moves away from y∗ in the negative direction. On the other hand, if y0 is above y∗ , then the sequence diverges from y∗ with the sequence diverging in the positive direction. Only if

Discrete dynamic systems

91 Figure 3.3.

y0 = y∗ will the system remain at rest. Hence, y∗ in ﬁgure 3.3(a) is an attractor while y∗ in ﬁgure 3.3(b) is a repellor. It is apparent from ﬁgure 3.2 that the essential difference between the two situations is that the line in ﬁgure 3.3(a) has a (positive) slope less than 45◦ , while in ﬁgure 3.3(b) the slope is greater than 45◦ . Another feature can be illustrated in a similar diagram. Consider the following simple linear dynamic system yt+1 = −yt + k

92

Economic Dynamics Given this system, the ﬁrst few terms in the sequence are readily found to be: yt+1 = −yt + k yt+2 = −yt+1 + k = −(−yt + k) + k = yt yt+3 = −yt+2 + k = −yt + k yt+4 = −yt+3 + k = −(−yt + k) + k = yt It is apparent that this is a repeating pattern. If y0 denotes the initial value, then we have y0 = y2 = y4 = . . .

and

y1 = y3 = y5 = . . .

We have here an example of a two-cycle system that oscillates between −y0 + k and y0 . There is still a ﬁxed point to the system, namely y∗ = −y∗ + k k y∗ = 2 but it is neither an attractor nor a repellor. The situation is illustrated in ﬁgure 3.4, where again the line L denotes the difference equation and the line E gives the equilibrium condition. The two-cycle situation is readily revealed by the fact that the system cycles around a rectangle. Return to the linear cobweb model given above, equation (3.5). Suppose the slope of the (linear) demand curve is the same as the slope of the (linear) supply

Figure 3.4.

Discrete dynamic systems curve but with opposite sign. Then b = d and b a−c pt = − pt−1 d d a−c = − pt−1 d or pt+1 = −pt + k where k =

a−c d

which is identical to the situation shown in ﬁgure 3.4, and must produce a two-cycle result. In general, a solution yn is periodic if yn+m = yn for some ﬁxed integer m and all n. The smallest integer for m is called the period of the solution. For example, given the linear cobweb system qdt = 10 − 2pt qst = 4 + 2pt−1 qdt = qst it is readily established that the price cycles between p0 and 3–p0 , while the quantity cycles between 4 + 2p0 and 10 − 2p0 (see exercise 12). In other words p0 = p2 = p4 = . . .

and

p1 = p3 = p5 = . . .

so that yn+2 = yn for all n and hence we have a two-cycle solution. More formally: DEFINITION If a sequence {yt } has (say) two repeating values y1 and y2 , then y1 and y2 are called period points, and the set {y1 ,y2 } is called a periodic orbit. Geometrically, a k-periodic point for the discrete system yt+1 = f (yt ) is the y-coordinate of the point where the graph of f k ( y) meets the diagonal line yt+1 = yt . Thus, a three-period cycle is where f 3 (y) meets the line yt+1 = yt . In establishing the stability/instability of period points we utilise the following theorem. THEOREM 3.2 Let b be a k-period point of f. Then b is (i) (ii) (iii)

stable if it is a stable ﬁxed point of f k asymptotically stable (attracting) if it is an attracting ﬁxed point of f k repelling if it is a repelling ﬁxed point of f k .

93

94

Economic Dynamics In deriving the stability of a periodic point we require, then, to compute [ f k ( y)] , and to do this we utilise the chain rule [ f k (y)] = f ( y∗1 )f ( y∗2 ) . . . f ( y∗n ) where y∗1 , y∗2 , . . . , y∗k are the k-periodic points. For example, if y∗1 and y∗2 are two periodic points of f 2 ( y), then 2 [ f ( y)] = f ( y∗ )f ( y∗ ) 1

2

and is asymptotically stable if ∗ ∗ f (y )f ( y ) < 1 1 2 All other stability theorems hold in a similar fashion. Although it is fairly easy to determine the stability/instability of linear dynamic systems, this is not true for nonlinear systems. In particular, such systems can create complex cycle phenomena. To illustrate, and no more than illustrate, the more complex nature of systems that arise from nonlinearity, consider the following quadratic equation yt+1 = ayt − by2t First we need to establish any ﬁxed points. It is readily established that two ﬁxed points arise since by∗ ∗ ∗ ∗2 ∗ y = ay − by = ay 1 − a which gives two ﬁxed points a−1 b The situation is illustrated in ﬁgure 3.5, where the quadratic is denoted by the graph G, and the line E as before denotes the equilibrium condition. The two equilibrium points, the two ﬁxed points of the system, are where the graph G intersects the line E. Depending on the values for a and b, it is of course possible for the graph G to be totally below the line E, in which case only one equilibrium point exists, namely y∗ = 0. Whether one or more equilibria exist, the question of interest is whether such a ﬁxed point is stable or unstable. Suppose we attempt to establish which by means of a numerical example y∗ = 0

and

y∗ =

yt+1 = 2yt − y2t The situation is illustrated in ﬁgure 3.6, where G denotes the graph of the difference equation, and the line E the equilibrium condition. The two equilibrium values are readily found to be y∗ = 0 and y∗ = 1. As in the linear system, we need to consider a starting value, which we denote y0 , then y1 = 2y0 − y20 . But this is no more than the value as read off the graph G. In terms of the horizontal axis, this value is read off by moving horizontally across to the E-line, as shown more clearly in ﬁgure 3.7. Given y1 then y2 = 2y1 − y21 as read off the graph G, which gives y2 on the horizontal axis when read horizontally

Discrete dynamic systems

95 Figure 3.5.

Figure 3.6.

off the E-line. And so on. It would appear, therefore, that y∗ = 1 is an attractor. Even if y0 is above y∗ = 1, the system appears to converge on y∗ = 1. Similarly, y∗ = 0 appears to be a repellor. It is useful to use a spreadsheet not only to establish the sequence {yn }, but also to graph the situation. A spreadsheet is ideal for recursive equations because the relation gives the next element in the sequence, and for given initial values, the sequence is simply copied to all future cells. A typical spreadsheet for the present example is illustrated in ﬁgure 3.8, where we have identiﬁed the formulas in the initial cells.

96 Figure 3.7.

Figure 3.8.

Economic Dynamics

Discrete dynamic systems Given such a spreadsheet, it is possible to change the initial value y0 and see the result in the sequence and on the various graphs that can be constructed.3 For instance, considering yt+1 = 3.2yt − 0.8y2t readily establishes that the equilibrium value is y∗ = 2.75, but that this is not reached for any initial value not equal to it. For any initial value not equal to the equilibrium value, then the system will tend towards a two-cycle with values 2.05 and 3.20, as can readily be established by means of a spreadsheet. It is also easy to establish that for any value slightly above or slightly below 2.75, i.e., in the neighbourhood of the equilibrium point, then the system diverges further from the equilibrium. In other words, the equilibrium is locally unstable. What is not apparent, however, is why the system will tend towards a two-cycle result. We shall explain why in section 3.7. Nor should it be assumed that only a two-cycle result can arise from the logistic equation. For instance, the logistic equation yt+1 = 3.84yt (1 − yt ) has a three-cycle (see exercise 13). We can approach stability/instability from a slightly different perspective. Consider the ﬁrst-order difference equation yt+1 = f ( yt ) with ﬁxed points satisfying a = f (a). Let y denote yt+1 and x denote yt , then the difference equation is of the form y = f (x). Expanding this equation around an equilibrium point (a, a) we have y − a = f (a)(x − a) or y = a[1 − f (a)] + f (a)x which is simply a linear equation with slope f (a). The situation is illustrated in ﬁgure 3.9. This procedure reduces the problem of stability down to that of our linear model. There we noted that if the absolute slope of f (x) was less than the 45◦ -line, as in ﬁgure 3.9, then the situation was stable, otherwise it was unstable. To summarise, If f (a) < 1 then a is an attractor or stable If f (a) > 1 then a is a repellor or unstable If f (a) = 1 then the situation is inconclusive.4 We can use such a condition for each ﬁxed point. 3

4

Many spreadsheets now allow graphics to be displayed within the spreadsheet, as shown here – especially those using the Windows environment. Hence, any change in initial values or parameter values results in an immediate change in the displayed graph. This is a very interactive experimentation. However, it is possible to utilise higher derivatives to obtain more information about the ﬁxed point a, as pointed out in theorem 3.1 (p. 89).

97

98

Economic Dynamics

Figure 3.9.

Example 3.1 yt+1 = 2yt − y2t The ﬁxed points can be found from a = 2a − a2 a2 − a = 0 a(a − 1) = 0 a=0

and

a=1

To establish stability, let y = f (x) = 2x − x2 then f (x) = 2 − 2x f (0) = 2

and

f (1) = 0

Since f (0) > 1 then a = 0 is unstable Since f (1) < 1 then a = 1 is stable Example 3.2 yt+1 = 3.2yt − 0.8y2t

Discrete dynamic systems

99

The ﬁxed points can be found from a = 3.2a − 0.8a2 0.8a2 − 2.2a = 0 a(0.8a − 2.2) = 0 a=0

a = 2.75

and

To establish stability let y = f (x) = 3.2x − 0.8x2 then f (x) = 3.2 − 1.6x f (0) = 3.2 Since

Since

and

f (2.75) = −1.2

f (0) > 1 then a = 0 is unstable f (2.75) > 1 then a = 2.75 is unstable.

Although a = 2.75 is unstable, knowledge about f (x) does not give sufﬁcient information to determine what is happening to the sequence {yn } around the point a = 2.75.

3.5 Solving ﬁrst-order difference equations For some relatively simple difference equations it is possible to ﬁnd analytical solutions. The simplest difference equation is a ﬁrst-order linear homogeneous equation of the form yt+1 = ayt

(3.8)

If we consider the recursive nature of this system, beginning with the initial value y0 , we have y1 = ay0 y2 = ay1 = a(ay0 ) = a2 y0 y3 = ay2 = a(a2 y0 ) = a3 y0 .. . yn = an y0 The analytical solution is, therefore, yn = an y0 satisfying the initial value y0 . The properties of this system depend only on the value and sign of the parameter a. There is only one ﬁxed point to such a system, y∗ = 0. For positive y0 , if a exceeds unity, then the series gets larger and larger over time, tending to inﬁnity in the limit. If 0 < a < 1, then the series gets smaller

(3.9)

100

Economic Dynamics

Figure 3.10.

and smaller over time, tending to zero in the limit. If a is negative, then the series will alternate between positive and negative numbers. However, if −1 < a < 0 the values of the alternating series becomes smaller and smaller, tending to zero in the limit. While if a < −1, then the series alternates but tends to explode over time. The various solution paths are plotted in ﬁgure 3.10. Example 3.3 A number of systems satisfy this general form. Consider the Malthusian population discussed in chapter 2, but now speciﬁed in discrete form. Between time t and t + 1 the change in the population is proportional to the population size. If pt denotes the population size in period t, then pt+1 = pt+1 − pt is proportional to pt . If k denotes the proportionality factor, then pt+1 = kpt Or pt+1 = (1 + k)pt which has the analytical solution pt = (1 + k)t p0

Discrete dynamic systems

101

where p0 is the initial population size. If population is growing at all, k > 0, then this population will grow over time becoming ever larger. We shall discuss population more fully in chapter 14. Example 3.4 As a second example, consider the Harrod–Domar growth model in discrete time St = sYt It = v(Yt − Yt−1 ) St = It This gives a ﬁrst-order homogeneous difference equation of the form v Yt−1 Yt = v−s with solution Yt =

v v−s

t Y0

If v > 0 and v > s then v/(v − s) > 1 and the solution is explosive and nonoscillatory. On the other hand, even if v > 0 if s > v then the solution oscillates, being damped if s < 2v, explosive if s > 2v or constant if s = 2v. The analytical solution to the ﬁrst-order linear homogeneous equation is useful because it also helps to solve ﬁrst-order linear nonhomogeneous equations. Consider the following general ﬁrst-order linear nonhomogeneous equation yt+1 = ayt + c A simple way to solve such equations, and one particularly useful for the economist, is to transform the system into deviations from its ﬁxed point, deviations from equilibrium. Let y∗ denote the ﬁxed point of the system, then y∗ = ay∗ + c c y∗ = 1−a Subtracting the equilibrium equation from the recursive equation gives yt+1 − y∗ = a(yt − y∗ ) Letting xt+1 = yt+1 − y∗ and xt = yt − y∗ then this is no more than a simple homogeneous difference equation in x xt+1 = axt with solution xt = a t x 0 Hence, yt − y∗ = at ( y0 − y∗ )

(3.10)

102

Economic Dynamics or

(3.11)

c c t + a y0 − yt = 1−a 1−a

which clearly satisﬁes the initial condition. Example 3.5 Consider, for example, the cobweb model we developed earlier in the chapter, equation (3.5), with the resulting recursive equation d a−c − pt−1 pt = b b and with equilibrium p∗ =

a−c b+d

Taking deviations from the equilibrium, we have d pt − p∗ = − ( pt−1 − p∗ ) b which is a ﬁrst-order linear homogeneous difference equation, with solution d t pt − p∗ = − ( p0 − p∗ ) b or (3.12)

pt =

a−c b+d

d t a−c p0 − + − b b+d

With the usual shaped demand and supply curves, i.e., b > 0 and d > 0, then d/b > 0, hence (−d/b)t will alternate in sign, being positive for even numbers of t and negative for odd numbers of t. Furthermore, if 0 < |−d/b| < 1 then the series will become damped, and in the limit tend towards the equilibrium price. On the other hand, if |−d/b| > 1 then the system will diverge from the equilibrium price. These results are veriﬁed by means of a simple numerical example and solved by means of a spreadsheet, as shown in ﬁgure 3.11. The examples we have just discussed can be considered as special cases of the following recursive equation: (3.13)

yn+1 = an yn

y0 at n = 0

The solution to this more general case can be derived as follows: y1 = a0 y0 y2 = a1 y1 = a1 a0 y0 y3 = a2 y2 = a2 a1 a0 y0 .. . yn = an−1 an−2 . . . a1 a0 y0

Discrete dynamic systems

103 Figure 3.11.

or yn =

n−1

ak y 0

(3.14)

k=0

Hence, if ak = a for all k, then

n−1 ak = an and

yn = an y0

k=0

Consider an even more general case: that of the nonhomogeneous ﬁrst-order equation given by yn+1 = an yn + gn

a0 , g0 , y0 at n = 0

(3.15)

Then y1 = a0 y0 + g0 y2 = a1 y1 + g1 = a1 (a0 y0 + g0 ) + g1 = a1 a0 y0 + a1 g0 + g1 y3 = a2 y2 + g2 = a2 (a1 a0 y0 + a1 g0 + g1 ) + g2 = a2 a1 a0 y0 + a2 a1 g0 + a2 g1 + g2 .. . with solution for yn of

n−1 n−1 n−1 yn = ak y 0 + ak g i k=0

i=0

k=i+1

(3.16)

104

Economic Dynamics We can consider two special cases: Case A : ak = a for all k Case B : ak = a

gk = b for all k

and

Case A ak = a for all k In this case we have yn+1 = ayn + gn

g0, y0 at n = 0

Using the general result above, then n−1

ak = an

n−1

and

k=0

ak = an−i−1

k=i+1

Hence, yn = an y0 +

(3.17)

n−1

an−i−1 gi

i=0

Case B ak = a and gk = b for all k In this case we have yn+1 = ayn + b

y0 at n = 0

We already know that if ak = a for all k then n−1

ak = an

and

k=0

n−1

ak = an−i−1

k=i+1

and so yn = an y0 + b

n−1

an−i−1

i=0

This case itself, however, can be divided into two sub-categories: (i) where a = 1 and (ii) where a = 1. Case (i) a = 1 If a = 1 then n−1

an−i−1 = n

i=0

and so yn = y0 + bn Case (ii) a = 1 Let S=

n−1 i=0

an−i−1

Discrete dynamic systems

105

then aS =

n−1

an−i

i=0

S − aS = (1 − a)S = 1 − an 1 − an S= 1−a and

yn = an y0 + b

1 − an 1−a

Combining these two we can summarise case B as follows y0 + bn a = 1 yn = 1 − an an y0 + b a = 1 1−a These particular formulas are useful in dealing with recursive equations in the area of ﬁnance. We take these up in the exercises. These special cases can be derived immediately using either Mathematica or Maple with the following input instructions5 : Mathematica RSolve[{y[n+1]==y[n]+b, y[0]==y0},y[n],n] RSolve[{y[n+1]==a y[n]+b, y[0]==y0},y[n],n]

Maple rsolve({y(n+1)=y(n)+b, y(0)=y0},y(n)); rsolve({y(n+1)=a*y(n)+b, y(0)=y0},y(n));

3.6 Compound interest If an amount A is compounded annually at a market interest rate of r for a given number of years, t, then the payment received at time t, Pt , is given by Pt = A(1 + r)t On the other hand, if it is compounded m times each year, then the payment received is r mt Pt = A 1 + m If compounding is done more frequently over the year, then the amount received is larger. The actual interest rate being paid, once allowance is made for the compounding, is called the effective interest rate, which we denote re. The relationship between re and (r, m) is developed as follows r m A(1 + re) = A 1 + m r m i.e. re = 1 + −1 m It follows that re ≥ r. 5

See section 3.13 on solving recursive equations with Mathematica and Maple.

(3.18)

106

Economic Dynamics

Figure 3.12.

Example 3.6 A bank is offering a savings account paying 7% interest per annum, compounded quarterly. What is the effective interest rate? 0.07 4 re = 1 + − 1 = 0.072 4 or 7.2%. If we assume that m is a continuous variable, then given an interest rate of say, 7%, we can graph the relationship between re and m. A higher market interest rate leads to a curve wholly above that of the lower interest rate, as shown in ﬁgure 3.12. Returning to the compounding result, if an amount is compounded at an annual interest rate r, then at time t we have the relationship Yt = (1 + r)Yt−1 . If we generalise this further and assume an additional deposit (or withdrawal) in each period, at , then the resulting recursive equation is Yt = (1 + r)Yt−1 + at−1 Or more generally, we have the recursive equation Yt = at−1 + bYt−1 Many problems reduce to this kind of relationship. For example, population of a species at time t may be proportional to its size in the previous period, but predation may take place each period. Or, human populations may grow proportionally but immigration and emigration occurs in each period. Solving the recursive equation can be achieved by iteration. Let the initial values be Y0 and a0 , respectively, then Y1 = a0 + bY0 Y2 = a1 + bY1 = a1 + b(a0 + bY0 ) = a1 + ba0 + b2 Y0 Y3 = a2 + bY2 = a2 + b(a1 + ba0 + b2 Y0 ) = a2 + ba1 + b2 a0 + b3 Y0 Y4 = a3 + bY3 = a3 + b(a2 + ba1 + b2 a0 + b3 Y0 ) = a3 + ba2 + b2 a1 + b3 a0 + b4 Y0

Discrete dynamic systems and so on. The general result emerging is Yt = at−1 + bat−2 + b2 at−3 + · · · + bt−1 a0 + bt Y0 or Yt =

t−1

bt−1−k ak + bt Y0

k=0

Having derived the general result two cases are of interest. The ﬁrst is where ak = a for all k; the second is where ak = a and b = 1 for all k. Case (i) ak = a for all k In this case we have Yt = a + bYt−1 with the general result Yt = a

t−1

bt−1−k + bt Y0

k=0

or

Yt = a

1 − bt 1−b

+ bt Y0

It is useful for the economist to see this result from a different perspective. In equilibrium Yt = Y for all t. So Y = a + bY a Y= 1−b Re-arranging the result for case (i), we have abt a − + bt Y0 1−b 1−b a a t Yt = b Y0 − + 1−b 1−b

Yt =

It is clear from this result that if |b| < 1 then the series converges on the equilibrium. If 0 < b < 1, there is steady convergence; while if −1 < b < 0, the convergence oscillates. If |b| > 1, the system is unstable. Case (ii) ak = a and b = 1 for all k In this case Yt = a + Yt−1 with result Yt = a

t−1

(1)t−1−k + Y0

k=0

i.e. Yt = at + Y0

107

108

Economic Dynamics Example 3.7 An investor makes an initial deposit of £10,000 and an additional £250 each year. The market interest rate is 5% per annum. What are his accumulated savings after ﬁve years? For this problem, Y0 = £10,000, ak = £250 for all k and b = (1 + r) = 1.05. Hence 1 − (1.05)5 + (1.05)5 (10000) = £14,144.20 Y5 = 250 1 − 1.05

3.7 Discounting, present value and internal rates of return Since the future payment when interest is compounded is Pt = P0 (1 + r)t , then it follows that the present value, PV, of an amount Pt received in the future is PV =

Pt (1 + r)t

and r is now referred to as the discount rate. Consider an annuity. An annuity consists of a series of payments of an amount A made at constant intervals of time for n periods. Each payment receives interest from the date it is made until the end of the nth-period. The last payment receives no interest. The future value, FV, is then FV = A(1 + r)n−1 + A(1 + r)n−2 + · · · + (1 + r)A + A Utilising a software package, the solution is readily found to be (1 + r)n − 1 FV = A r On the other hand, the present value of an annuity requires each future payment to be discounted by the appropriate discount factor. Thus the payment A received at the end of the ﬁrst period is worth A/(1 + r) today, while a payment A at the end of the second period is worth A/(1 + r)2 today. So the present value of the annuity is PV = with solution

A A A A + + ··· + + 2 n−1 (1 + r) (1 + r) (1 + r) (1 + r)n

1 − (1 + r)−n PV = A r

Example 3.8 £1,000 is deposited at the end of each year in a savings account that earns 6.5% interest compounded annually. (a) (b)

At the end of ten years, how much is the account worth? What is the present value of the payments stream?

Discrete dynamic systems (1 + 0.065)10 − 1 (1 + r)n − 1 = 1000 = £13494.40 r 0.065 1 − (1 + 0.065)−10 1 − (1 + r)−n = 1000 = £7188.83 PV = A r 0.065

(a) (b)

FV = A

Discounting is readily used in investment appraisal and cost–beneﬁt analysis. Suppose Bt and Ct denote the beneﬁts and costs, respectively, at time t. Then the present value of such ﬂows are Bt /(1 + r)t and Ct /(1 + r)t , respectively. It follows, then, that the net present value, NPV, of a project with ﬁnancial ﬂows over n-periods is NPV =

n t=0

n n Bt Ct Bt − Ct − = t t (1 + r) (1 + r) (1 + r)t t=0 t=0

Notice that for t = 0 the beneﬁts B0 and the costs C0 involve no discounting. In many projects no beneﬁts accrue in early years only costs. If NPV > 0 then a project (or investment) should be undertaken. Example 3.9 Bramwell plc is considering buying a new welding machine to increase its output. The machine would cost £40,000 but would lead to increased revenue of £7,500 each year for the next ten years. Half way through the machine’s lifespan, in year 5, there is a one-off maintenance expense of £5,000. Bramwell plc consider that the appropriate discount rate is 8%. Should they buy the machine? NPV = −40000 +

10 7500 5000 − t (1 + r) (1 + r)5 t=1

The second term is simply the present value of an annuity of £7,500 received for ten years and discounted at 8%. The present value of this is 1 − (1 + r)−n 1 − (1 + 0.08)−10 PV = A = 7500 r 0.08 Hence

NPV = −40000 + 7500

1 − (1.08)−10 5000 = £6922.69 − 0.08 (1.08)5

Since NPV > 0, then Bramwell plc should go ahead with the investment. Net present value is just one method for determining projects. One difﬁculty, as the above example illustrates, is that it is necessary to make an assumption about the appropriate discount rate. Since there is often uncertainty about this, computations are often carried out for different discount rates. An alternative is to use the internal rate of return (IRR). The internal rate of return is the discount rate that leads to a zero net present value. Thus, the internal rate of return is the value of r satisfying n Bt − Ct =0 (1 + r)t t=0

109

110

Economic Dynamics Although software programmes can readily solve for the internal rate of return, there is a problem in the choice of r. n Bt − Ct (1 + r)t t=0

is a polynomial with the highest power of n, and so theoretically there are n possible roots to this equation. Of course, we can rule out negative values and complex values. For example, the choice problem for Bramwell plc involves r10 as the highest term and so there are ten possible solutions to the equation 5000 1 − (1 + r)−10 − =0 −40000 + 7500 r (1 + r)5 Eight solutions, however, are complex and another is negative. This leaves only one positive real-valued solution, namely r = 0.1172 or r = 11.72%. Since such a return is well above the typical market interest rate, then the investment should be undertaken. The point is, however, that multiple positive real-valued solutions are possible.

3.8 Solving second-order difference equations 3.8.1

Homogeneous

Consider the following general second-order linear homogeneous equation (3.19)

yn+2 = ayn+1 + byn Similar to the solution for a ﬁrst-order linear homogeneous equation, we can suppose the solution takes the form yn = c1 r n + c2 sn for some constants r and s and where c1 and c2 depend on the initial conditions y0 and y1 . If this indeed is correct, then c1 rn+2 + c2 sn+2 = a(c1 rn+1 + c2 sn+1 ) + b(c1 r n + c2 sn ) Re-arranging and factorising, we obtain c1 r n (r2 − ar − b) + c2 sn (s2 − as − b) = 0 So long as r and s are chosen to be the solution values to the general quadratic equation x2 − ax − b = 0 i.e. x = r and x = s, where r = s, then yn = c1 rn + c2 sn is a solution to the dynamic system. This quadratic equation is referred to as the characteristic equation of the dynamical system. If r > s, then we call y1 = c1 r n the dominant solution and r the dominant characteristic root. Furthermore, given we have obtained the solution values r and s, and given the initial conditions, y0 and y1 , then we can solve for the two unknown coefﬁcients,

Discrete dynamic systems c1 and c2 . Since y0 = c1 r0 + c2 s0 = c1 + c2 y1 = c1 r + c2 s then c1 =

y1 − sy0 r−s

and

c2 =

y1 − ry0 s−r

The solution values r and s to the characteristic equation of the dynamic system are the solutions to a quadratic. As in all quadratics, three possibilities can occur: (i) (ii) (iii)

distinct real roots identical real roots complex conjugate roots

Since the solution values to the quadratic equation are √ −a ± a2 + 4b r, s = 2 then we have distinct real roots if a2 > −4b, identical roots if a2 = −4b, and complex conjugate roots if a2 < −4b. Example 3.10 (real distinct roots) Suppose yn+2 = yn+1 + 2yn The characteristic equation is given by x2 − x − 2 = 0 i.e. (x − 2)(x + 1) = 0 Hence, we have two real distinct roots, x = 2 and x = −1, and the general solution is yn = c1 (2)n + c2 (−1)n If we know y0 = 5 and y1 = 4, then c1 =

4 − (−1)(5) y1 − sy0 = =3 r−s 2 − (−1)

c2 =

4 − (2)(5) y1 − ry0 = =2 s−r (−1) − 2

Hence, the particular solution satisfying these initial conditions is given by yn = 3(2)n + 2(−1)n As ﬁgure 3.13 makes clear, this is an explosive system that tends to inﬁnity over time. The limiting behaviour of the general solution yn = c1 r n + c2 sn is determined by the behaviour of the dominant solution. If, for example, r is the dominant

111

112

Economic Dynamics

Figure 3.13.

characteristic root and |r| > |s|, then s n yn = r n c1 + c2 r Since |s/r| < 1, then (s/r)n → 0 as n → ∞. Therefore, lim yn = lim c1 r n

n→∞

n→∞

There are six different situations that can arise depending on the value of r. (1) (2) (3) (4) (5) (6)

r > 1, then the sequence {c1 r n } diverges to inﬁnity and the system is unstable r = 1, then the sequence {c1 r n } is a constant sequence 0 ≤ r < 1, then the sequence {c1 r n } is monotonically decreasing to zero and the system is stable −1 < r ≤ 0, then the sequence {c1 r n } is oscillating around zero and converging on zero, so the system is stable r = −1, then the sequence {c1 r n } is oscillating between two values r < −1, then the sequence {c1 r n } is oscillating but increasing in magnitude. Identical real roots

If the roots are real and equal, i.e., r = s, then the solution becomes yn = (c1 + c2 )r n = c3 r n But if c3 rn is a solution, then so is c4 nrn (see Chiang 1992, p. 580 or Goldberg 1961, p. 136 and exercise 14), hence the general solution when the roots are equal is given by yn = c3 r n + c4 nr n We can now solve for c3 and c4 given the two initial conditions y0 and y1 y0 = c3 r0 + c4 (0)r0 = c3 y1 = c3 r + c4 (1)r = (c3 + c4 )r

Discrete dynamic systems

113

Hence c3 = y 0 y1 y1 − ry0 − c3 = c4 = r r Therefore, the general solution satisfying the two initial conditions, is y1 − ry0 n yn = y0 r + nr n r Example 3.11 (equal real roots) Let yn+2 = 4yn+1 − 4yn This has the characteristic equation x2 − 4x + 4 = (x − 2)2 = 0 Hence, r = 2. yn = c3 (2)n + c4 n(2)n Suppose y0 = 6 and y1 = 4, then c3 = y0 = 6 4 − (2)(6) y1 − ry0 = = −4 c4 = r 2 Hence, the particular solution is yn = 6(2)n − 4n(2)n which tends to minus inﬁnity as n increases, as shown in ﬁgure 3.14. In the case of the general solution yn = (c3 + c4 n)r n (1) (2) (3)

If |r| ≥ 1, then yn diverges monotonically If r ≤ −1, then the solution oscillates If |r| < 1, then the solution converges to zero Figure 3.14.

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Economic Dynamics Complex conjugate roots6 If the roots are complex conjugate then r = α + βi and s = α − βi and Rn cos(βt)

and

Rn sin(βt)

are solutions and the general solution is yn = c1 Rn cos(θn) + c2 Rn sin(θn) where R=

α2 + β 2, β α and sin θ = cos θ = R R β sin θ = or tan θ = cos θ α

Example 3.12 (complex conjugate) Consider yn+2 − 4yn+1 + 16yn = 0 The characteristic equation is x2 − 4x + 16 = 0 with roots r, s =

4±

√

16 − 64 =2± 2

√

48 i 2

i.e. r =2+ s=2−

√

1 48i 2 √ 1 48i 2

α=2 β=

1 2

√

48

and polar coordinates √ √ R = 22 + ( 12 48)2 = 4 + 12 = 4 2 1 α cos θ = = = R 4 2

and

β sin θ = = R

1 2

√ √ 48 3 = 4 2

Implying θ = π/3. Hence nπ nπ yn = c1 4n cos + c2 4n sin 3 3 6

In this section the complex roots are expressed in polar coordinate form (see Allen 1965 or Chiang 1984).

Discrete dynamic systems

115

Given y0 and y1 , it is possible to solve for c1 and c2 . Speciﬁcally c1 = y0

π y1 − y0 4 cos π 3 c2 = 4 sin 3 If r and s are complex conjugate, then yn oscillates because the cosine function oscillates. There are, however, three different types of oscillation: (1)

(2) (3)

R > 1. In this instance the characteristic roots r and s lie outside the unit circle, shown in ﬁgure 3.15(a). Hence yn is oscillating, but increasing in magnitude. The system is unstable. R = 1. In this instance the characteristic roots r and s lie on the unit circle, and the system oscillates with a constant magnitude, ﬁgure 3.15(b). R < 1. In this instance the characteristic roots r and s lie inside the unit circle and the system oscillates but converges to zero as n → ∞, ﬁgure 3.15(c). The system is stable.

Figure 3.15.

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Economic Dynamics 3.8.2

Nonhomogeneous

A constant coefﬁcient nonhomogeneous second-order difference equation takes the general form yn+2 + ayn+1 + byn = g(n)

(3.20)

If g(n) = c, a constant, then yn+2 + ayn+1 + byn = c which is the form we shall consider here. As with second-order differential equations considered in chapter 2, we can break the solution down into a complementary component, yc , and a particular component, yp , i.e., the general solution yn , can be expressed yn = yc + yp The complementary component is the solution to the homogeneous part of the recursive equation, i.e., yc is the solution to yn+2 + ayn+1 + byn = 0 which we have already outlined in the previous section. Since yn = y∗ is a ﬁxed point for all n, then this will satisfy the particular solution. Thus y∗ + ay∗ + by∗ = c c y∗ = 1+a+b so long as 1 + a + b = 0. Example 3.13 yn+2 − 4yn+1 + 16yn = 26 Then y∗ − 4y∗ + 16y∗ = 26 y∗ = 2 Hence, yp = 2. The general solution is, then πn πn + c2 4n +2 yn = c1 4n cos 3 3 Example 3.14 yn+2 − 5yn+1 + 4yn = 4

Discrete dynamic systems In this example, 1 + a + b = 0 and so it is not possible to use y∗ as a solution. In this instance we try a moving ﬁxed point, ny∗ . Thus (n + 2)y∗ − 5(n + 1)y∗ + 4ny∗ = 4 −3y∗ = 4 y∗ = −

4 3

−4n ... ny∗ = 3 For the complementary component we need to solve the homogeneous equation yn+2 − 5yn+1 + 4yn = 0 whose characteristic equation is x2 − 5x + 4 = 0 with solutions

√

5±3 25 − 16 = 2 2 i.e. r = 4 and s = 1. Hence r, s =

5±

yn = c1 4n + c2 1n − (4n/3) = c1 4n + c2 − (4n/3) Example 3.15 yn+2 + yn+1 − 2yn = 12

y0 = 4

and

y1 = 5

The particular solution cannot be solved for y∗ (since 1 + a + b = 0) and so we employ ny∗ (n + 2)y∗ + (n + 1)y∗ − 2ny∗ = 12 (n + 2 + n + 1 − 2n)y∗ = 12 12 =4 y∗ = 3 Hence, ny∗ = 4n = yp . The complementary component is derived by solving the characteristic equation x2 + x − 2 = 0 (x + 2)(x − 1) = 0 giving r = 1 and s = −2. Giving the complementary component of yc = c1 r n + c2 sn = c1 (1)n + c2 (−2)n = c1 + c2 (−2)n Hence, the general solution is yn = yc + yp = c1 + c2 (−2)n + 4n

117

118

Economic Dynamics Given y0 = 4 and y1 = 5, then y0 = c1 + c2 = 4 y1 = c1 − 2c2 + 4 = 5 with solutions c1 = 3

and c2 = 1

Hence, the general solution satisfying the given conditions is yn = 3 + (−2)n + 4n For the nonhomogeneous second-order linear difference equation yn+2 + ayn+1 + byn = c yn → y∗ ,where y∗ is the ﬁxed point, if and only if the complementary solution, yc , tends to zero as n tends to inﬁnity; while yn will oscillate about y∗ if and only if the complementary solution oscillates about zero. Since the complementary solution is the solution to the homogeneous part, we have already indicated the stability of these in section 3.8.1. In the case of the second-order linear difference equations, both homogeneous and nonhomogeneous, it is possible to have explicit criteria on the parameters a and b for stability. These are contained in the following theorem (Elaydi 1996, pp. 87–8). THEOREM 3.3 The conditions 1 + a + b > 0,

1 − a + b > 0,

1−b>0

are necessary and sufﬁcient for the equilibrium point of both homogeneous and nonhomogeneous second-order difference equations to be asymptotically stable.

3.9 The logistic equation: discrete version Suppose yt+1 = ayt − by2t

(3.21)

where b is the competition coefﬁcient.7 Then yt+1 = (1 + a)yt − by2t This is a nonlinear recursive equation and cannot be solved analytically as it stands. However, with a slight change we can solve the model.8 Let y2t yt yt+1 7 8

We shall discuss this coefﬁcient more fully in chapter 14. This approximate solution is taken from Grifﬁths and Oldknow (1993, p. 16).

Discrete dynamic systems

119

then yt+1 = (1 + a)yt − byt yt+1 Solving we obtain yt+1 =

(1 + a)yt 1 + byt

This can be transformed by dividing both sides by yt+1 yt 1 1 1+a = yt yt+1 1 + byt i.e. 1 1 1 + byt 1 b = = + yt+1 (1 + a)yt (1 + a) yt 1+a Let xt = 1/yt , then 1 b xt + xt+1 = 1+a 1+a In equilibrium xt+1 = xt = . . . = x∗ , hence 1 b ∗ x∗ + x = 1+a 1+a Solving for x∗ we obtain the ﬁxed point b a Subtracting the equilibrium equation from the recursive equation we obtain x∗ =

1 (xt − x∗ ) 1+a which has the general solution t 1 ∗ (x0 − x∗ ) xt − x = 1+a or b b xt = + (1 + a)−t x0 − a a xt+1 − x∗ =

Substituting back xt = 1/yt for all t 1 1 b b −t = + (1 + a) − yt a y0 a Hence, 1 yt = b 1 b −t − + (1 + a) a y0 a or yt =

ay0 by0 + (1 + a)−t (a − by0 )

(3.22)

120

Economic Dynamics

Figure 3.16.

It is readily established that a lim yt = t→∞ b Three typical plots are shown in ﬁgure 3.16, for y0 < a/b, y0 = a/b and y0 > a/b. Return to the original formulation yt+1 − yt = ayt − by2t i.e. yt+1 = (1 + a)yt − by2t It is not possible to solve this nonlinear equation, although our approximation is quite good (see exercise 6). But the equation has been much investigated by mathematicians because of its possible chaotic behaviour.9 In carrying out this investigation it is normal to respecify the equation in its generic form xt+1 = λxt (1 − xt )

(3.23)

It is this simple recursive formulation that is often employed for investigation because it involves only a single parameter, λ. The reader is encouraged to set up this equation on a spreadsheet, which is very straightforward. If λ = 3.2 it is readily established that the series will, after a sufﬁcient time period, oscillate between two values: a1 = 0.799455 and a2 = 0.513045. This two-cycle is typical of the logistic equation for a certain range of λ. To establish the range of λ is straightforward but algebraically tedious. Here we shall give the gist of the solution, and leave appendices 3.1 and 3.2 to illustrate how Mathematica and Maple, respectively, can be employed to solve the tedious algebra. Let f (x) = λx(1 − x) then a two-cycle result will occur if a = f ( f (a)) 9

We shall investigate chaos in chapter 7.

Discrete dynamic systems where a is a ﬁxed point. Hence a = f [λa(1 − a)] = λ[λa(1 − a)][1 − λa(1 − a)] = λ2 a(1 − a)[1 − λa(1 − a)] It is at this point where Mathematica or Maple is used to solve this equation. The range for a stable two-cycle is established by solving10 −1 < f (a1 )f (a2 ) < 1 where a1 and a2 are the two relevant solutions. Since f (x) = λ(1 − x) − λx then we can compute f (a1 ) f (a2 ), which is a surprisingly simple equation of the form 4 + 2λ − λ2 Hence, we have a stable two-cycle if −1 < 4 + 2λ − λ2 < 1 Discarding negative values for λ, we establish the range to be 3 < λ < 3.449. Given we have already a1 and a2 solved for any particular value of λ, then we can ﬁnd these two stable solutions for any λ in the range just established. Thus, for λ = 3.2 it is readily established using Mathematica or Maple, that a1 = 0.799455 and a2 = 0.513045, which are the same results as those established using a spreadsheet. For λ < 3 we have a single ﬁxed point which is stable, which again can readily be established by means of the same spreadsheet. Finally, if λ = 3.84 the system converges on a three-cycle result with a1 = 0.149407, a2 = 0.488044 and a3 = 0.959447 (see exercise 13). Example 3.16 As an application of the logistic equation, different from its normal application in population models (see chapter 14), we turn to the issue of productivity growth discussed by Baumol and Wolff (1991). Let qt denote the rate of growth of productivity outside of the research development industries; yt the activity level of the information producing industry (the R&D industries); and pt the price of information. The authors now assume three relationships: (1)

Information contributes to productivity growth according to: (i) yt+1 = a + byt

(2)

10

The price of information grows in proportion to productivity in the sector outside of the R&D industries, so: pt+1 − pt = νqt+1 (ii) pt

See theorem 3.2, p. 93 and Sandefur (1990, chapter 4).

121

122

Economic Dynamics (3)

Information demand has a constant elasticity, so: yt+1 − yt pt+1 − pt (iii) = −ε yt pt

Substituting (i) into (ii) and the result into (iii), we obtain yt+1 − yt = −εν(a + byt ) yt Assume εν = k > 0 then yt+1 − yt = −k(a + byt ) yt i.e. yt+1 = (1 − ak)yt − kby2t which is no more than a logistic equation. In equilibrium yt = y∗ for all t, hence y∗ = (1 − ak)y∗ − kby∗2 y∗ (ak + kby∗ ) = 0 and a b It is possible to consider the stability in the locality of the equilibrium. Since y∗ = 0 or y∗ = −

yt+1 = (1 − ak)yt − kby2t let yt+1 = y and yt = x, then y = (1 − ak)x − kbx2 = f (x) f (x) = (1 − ak) − 2kbx and

dyt+1 dy = dx y∗ =−a/b dyt y∗ =−a/b = (1 − ak) − 2kb(−a/b) = 1 + ak

Hence the stability is very dependent on the sign/value of ak. Letting yt+1 = Ayt − By2t , A = (1 − ak), B = kb then using our earlier approximation (equation (3.22)) we have yt =

Ay0 By0 + (1 + A)−t (A − By0 )

yt =

(1 − ak)y0 kby0 + (2 − ak)−t (1 − ak − kby0 )

i.e.

Various paths for this solution are possible depending on the values of v and ε. For instance, if v = 1 and ε ≤ 2, then k = εν ≤ 2, and if a < 1 then ak < 2, which

Discrete dynamic systems is highly probable. Even with ak < 1, two possibilities arise: (i) (ii)

if a < 0 then ak < 2 if 0 < a < 1 then 0 < ak < 2

with various paths for yt . This should not be surprising because we have already established that the discrete logistic equation has a variety of paths and possible cycles.

3.10 The multiplier–accelerator model A good example that illustrates the use of recursive equations, and the variety of solution paths for income in an economy, is that of the multiplier–accelerator model ﬁrst outlined by Samuelson (1939). Consumption is related to lagged income while investment at time t is related to the difference between income at time t − 1 and income at time t − 2.11 In our formulation we shall treat government spending as constant, and equal to G in all periods. The model is then Ct = a + bYt−1 It = v(Yt−1 − Yt−2 ) Gt = G for all t E t = C t + I t + Gt Yt = E t which on straight substitution gives rise to the second-order nonhomogeneous recursive equation Yt − (b + v)Yt−1 + vYt−2 = a + G The particular solution is found by letting Yt = Y ∗ for all t. Hence Y ∗ − (b + v)Y ∗ + vY ∗ = a + G a+G i.e. Y ∗ = 1−b In other words, in equilibrium, income equals the simple multiplier result. The complementary result, Yc , is obtained by solving the homogeneous component Yt − (b + v)Yt−1 + vYt−2 = 0 which has the characteristic equation x2 − (b + v)x + v = 0 with solutions r, s =

11

(b + v) ±

(b + v)2 − 4v 2

Samuelson originally related investment to lagged consumption rather than lagged income.

123

124

Economic Dynamics Example 3.17 Determine the path of income for the equations Ct = 50 + 0.75Yt−1 It = 4(Yt−1 − Yt−2 ) G = 100 The equilibrium is readily found to be Y ∗ = 600, which is the particular solution. The complementary solution is found by solving the quadratic x2 − (19/4)x + 4 = 0 i.e. r = 3.6559

and

s = 1.0941

Since r and s are real and distinct, then the solution is Yt = c1 (3.6559)t + c2 (1.0941)t + 600 and c1 and c2 can be obtained if we know Y0 and Y1 . Of more interest is the fact that the model can give rise to a whole variety of paths for Yt depending on the various parameter values for b and v. It is to this issue that we now turn. From the roots of the characteristic equation given above we have three possible outcomes: (i) (ii) (iii)

real distinct roots real equal roots complex roots

(b + v)2 > 4v (b + v)2 = 4v (b + v)2 < 4v

In determining the implications of these possible outcomes we use the two properties of roots r+s=b+v rs = v It also follows using these two results that (1 − r)(1 − s) = 1 − (r + s) + rs = 1 − (b + v) + v = 1−b and since 0 < b < 1, then 0 < (1 − r)(1 − s) < 1. With both roots real and distinct, the general solution is Yt = c1 rt + c2 st + Y ∗ where r is the larger of the two roots. The path of Yt is determined by the largest root, r > s. Since b > 0 and v > 0, then rs = v > 0 and so the roots must have the same sign. Furthermore, since r + s = b + v > 0, then both r and s must be positive. The path of income cannot oscillate. However, it will be damped if the largest root lies between zero and unity. Thus, a damped path occurs if 0 < s < r < 1, which arises if 0 < b < 1 and v < 1. Similarly, the path is explosive if the largest root exceeds unity, i.e., if r > s > 1, which implies 0 < b < 1 and rs = v > 1.

Discrete dynamic systems

125

With only one real root, r, the same conditions hold. Hence, in the case of real roots with 0 < b < 1, the path of income is damped for 0 < v < 1 and explosive for v > 1. If the solution is complex conjugate then r = α + βi and s = α − βi and the general solution Yt = c1 Rt cos(tθ) + c2 Rt sin(θt) + Y ∗ exhibits oscillations, whose damped or explosive nature depends on the amplitude, R. From our earlier analysis we know R = α 2 + β 2 . But + 4v − (b + v)2 b+v and β= α= 2 2 Hence √ b + v 2 4v − (b + v)2 = v R= + 2 4 For damped oscillations, R < 1, i.e., v < 1; while for explosive oscillations, R > 1, i.e., v > 1. All cases are drawn in ﬁgure 3.17. The dividing line between real and complex roots is the curve (b + v)2 = 4v, which was drawn using Mathematica’s ImplicitPlot command and annotated in CorelDraw. A similar result can be derived using Maple. The instructions for each software are: Mathematica 1 then x∗ is a repellor or unstable if f (x∗ ) = 1 then the stability of x∗ is inconclusive. Example 3.18 Consider xt =

4xt−1 − 3

This has two equilibria found by solving x2 − 4x + 3 = 0, i.e., x1∗ = 1 and x2∗ = 3, and shown by the points a and b in ﬁgure 3.19. The linear approximation is xt = f (x∗ ) + f (x∗ )(xt−1 − x∗ ) Figure 3.19.

Discrete dynamic systems Take ﬁrst x1∗ = 1, then f (x1∗ ) = 1 f (x1∗ ) = 2(4x1∗ − 3)−1/2 = 2 Hence xt = 1 + 2(xt−1 − 1) = −1 + 2xt−1 which is unstable since f (x1∗ ) = 2 > 1. Next consider x2∗ = 3 f (x2∗ ) = 3 f (x2∗ ) = 2(4x2∗ − 3)−1/2 = Hence

2 3

2 xt = 3 + (xt−1 − 3) 3 2 = 1 + xt−1 3

which is stable since f (x2∗ ) = 2/3 < 1. Example 3.19 yt+1 = f (yt ) = 3.2yt − 0.8y2t Letting yt = y∗ for all t we can readily establish two equilibria: y∗1 = 0 and y∗2 = 2.75. Considering the nonzero equilibrium, then f ( y∗2 ) = 2.75 f ( y∗2 ) = 3.2 − 1.6y∗2 = −1.2 Hence, the linear approximation is yt+1 = 2.75 − 1.2( yt − 2.75) = 6.05 − 1.2yt The situation is shown in ﬁgure 3.20. The solution to this model is yt+1 = 2.75 + (−1.2)t (y0 − 2.75) which is oscillatory and explosive. Although the linear approximation leads to an explosive oscillatory equilibrium, the system in its nonlinear form exhibits a two-cycle with values 2.0522 and 3.1978.12 What the linear approximation reveals is the movement away from y∗ = 2.75. What it cannot show is that it will converge on a two-cycle. This example, therefore, illustrates the care required in interpreting the stability of nonlinear difference equations using their linear approximations. 12

This can be established quite readily with a spreadsheet or as explained in appendices 3.1 and 3.2.

129

130

Economic Dynamics

Figure 3.20.

3.12 Solow growth model in discrete time We have already established in chapter 2, example 2.9, that a homogeneous of degree one production function can be written y = f (k), where y is the output/labour ratio and k is the capital/labour ratio. In discrete time we have13 yt = f (kt−1 ) where yt = Yt /Lt−1 and kt−1 = Kt−1 /Lt−1 . Given the same assumptions as example 2.9, savings is given by St = sYt and investment as It = Kt − Kt−1 + δKt−1 , where δ is the rate of depreciation. Assuming saving is equal to investment in period t, then sYt = Kt − Kt−1 + δKt−1 = Kt − (1 − δ)Kt−1 Dividing both sides by Lt−1 , then sYt Kt (1 − δ)Kt−1 = − Lt−1 Lt−1 Lt−1 Lt Kt Kt−1 = − (1 − δ) Lt Lt−1 Lt−1 But if population is growing at a constant rate n, as is assumed in this model, then Lt − Lt−1 =n Lt−1 Lt i.e. =1+n Lt−1 13

A little care is required in discrete models in terms of stocks and ﬂows (see section 1.3). Capital and labour are stocks and are deﬁned at the end of the period. Hence, Kt and Lt are capital and labour at the end of period t. Flows, such as income, investment and savings are ﬂows over a period of time. Thus, Yt , It and St are ﬂows over period t.

Discrete dynamic systems Hence syt = kt (1 + n) − (1 − δ)kt−1 or (1 + n)kt − (1 − δ)kt−1 = sf (kt−1 ) which can be expressed (1 − δ)kt−1 + sf (kt−1 ) 1+n i.e. kt = h(kt−1 ) kt =

With constant returns to scale and assuming a Cobb–Douglas production function, then α yt = f (kt−1 ) = akt−1

a > 0, 0 < α < 1

Example 3.20 This can be investigated by means of a spreadsheet, where we assume a = 5,

α = 0.25,

s = 0.1,

n = 0.02,

δ=0

and let k0 = 20. Alternatively, using a Taylor expansion about k∗ > 0, then (1 − δ)(kt−1 − k∗ ) + αsa(k∗ )α−1 (kt−1 − k∗ ) 1+n (1 − δ) + αsa(k∗ )α−1 = h(k∗ ) + (kt−1 − k∗ ) 1+n (1 − δ) + αsa(k∗ )α−1 = k∗ + (kt−1 − k∗ ) 1+n

kt = h(k∗ ) +

The situation is illustrated in ﬁgure 3.21.

3.13 Solving recursive equations with Mathematica and Maple Both Mathematica and Maple come with a solver for solving recursive equations. RSolve in Mathematica and rsolve in Maple. They both operate in fundamentally the same way, and both can solve only linear recursive equations. While rsolve is built into the main kernel of Maple, the RSolve command of Mathematica is contained in the DiscreteMath package, and so must ﬁrst be loaded with the following command. Needs[``DiscreteMath`RSolve`’’]

(Note the back single-quote on RSolve.) These solvers are particularly useful for solving many difference equations. There are, however, some differences in the two

131

132

Economic Dynamics

Figure 3.21.

solvers. One difference is shown immediately by attempting to solve the recursive equation xt = axt−1 . The input and output from each programme is as follows. Mathematica RSolve[x[t]==ax[t-1],x[t],t] {{x[t]->a1+t C[1]

Maple rsolve(x(t)=a*x(t-1),x(t)); x(0)at

While Maple’s output looks quite familiar, Mathematica’s looks decidedly odd. The reason for this is that Mathematica is solving for a ‘future’ variable. If the input had been RSolve[x[t+1]== ax[t],x[t],t]

Then the solution would be {{x[t]->at C[1]}}

which is what we would expect. Note also that while Mathematica leaves unsolved the unknown constant, which it labels C[1], Maple assumes the initial condition is x(0) for t = 0. If attempting to solve yt = ayt−1 + byy−2 for example, then when using Mathematica, this should be thought of as yt+2 = ayt+1 + byt when solving for yt . With this caveat in mind, we can explore the RSolve and rsolve commands in more detail.

Discrete dynamic systems When an initial condition is supplied the caveat just alluded to is of no consequence. Thus, if we wish to solve xt = axt−1

x0 = 2

then the instructions are: Mathematica RSolve[{x[t]==ax[t-1],x[0]==2},x[t],t]

with result {{x[t]->2at }}

Maple rsolve({x(t)=a*x(t-1),x(0)=2},x(t));

with result 2at

So no difference arises when initial conditions are supplied. Using either the RSolve command of Mathematica or the rsolve command of Maple, we can readily check the following equations used in this chapter: (i)

yt+1 = ayt

(ii)

pt+1 = (1 + k)pt v Yt−1 Yt = v+s yt+1 = ayt + c a−c d pt = − pt−1 b b

(iii) (iv) (v)

The following observations, however, should be borne in mind. (1)

(2) (3)

(4)

(5)

When using both Mathematica and Maple to solve the Harrod–Domar model, problem (iii), the recursive equation should be thought of as Yt+1 = (v/(v + s))Yt and solved accordingly. On some occasions it is necessary to use additional commands, especially the Simplify command (Mathematica) or the simplify command (Maple). Mathematica sometimes supplies ‘If’ conditions in the solutions, usually to do with t ≥ −1 for example. This partly arises from the caveat mentioned above. Many of these can be avoided by writing the equations in terms of future lags, as in the case of the Harrod–Domar model. A number of solutions involve complex output that is not always meaningful. This is especially true of general algebraic problems, such as solving yt+2 = ayt+1 + byt . Even when results have been simpliﬁed, it is not always possible to interpret the results in an economically meaningful way. For instance, in problem (v), it is impossible for a computer software package to ‘know’ that (a − c)/(b + d) is the equilibrium price and that it is more economically meaningful to take the difference ( p0 − (a − c)/(b + d)). Economic insight is still a vital element.

133

134

Economic Dynamics Problems (i)–(v) are all recursive equations of the ﬁrst-order. The same basic form is used to solve higher-order recursive equations. Given the recursive equation yt+2 = ayt+1 + byt then this can be solved with the instructions: Mathematica RSolve[y[t+2]==ay[t+1]+by[t],y[t],t]

Maple rsolve( y(t+2)=a*y(t+1)+b*y(t),y(t));

But because this is a general recursive equation the output in each case is quite involved. Mathematica’s output even more so, since it involves Binomial equations! What is revealed by the output is the need to know two initial conditions to solve such second-order recursive equations: Solving yt+2 = yt+1 + 2yt with initial conditions y(0) = 5 and y(1) = 4, we have Mathematica RSolve[{y[t+2]==y[t+1]+2y[t],y[0]==5,y[1]==4},y[t],t]

with output {{y[t]->2(-1)t + 3 2t }}

Maple rsolve({r(t+2)=y(t+1)+2*y(t),y(0)=5,y(1)=4},y(t));

with output 2(-1)t + 3 2t

Furthermore, there is no difﬁculty with repeated roots, which occur in solving yt+2 = 4yt+1 − 4yt . For initial conditions y(0) = 6 and y(1) = 4, we have solutions Mathematica : {{y[t]->-21+t (-3 + 2t)}} Maple : (-4t-4)2t + 10 2t Here we see that output in the two packages need not look the same, and often does not, yet both are identical; and identical to 6(2)t – 4t(2)t which we derived in the text. Complex roots, on the other hand, are solved by giving solutions in their complex form rather than in trigonometric form. The RSolve and rsolve commands, therefore, allow a check of the following equations in this chapter. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi)

yt+2 = ayt+1 + byt yt+2 = yt+1 + 2yt y(0) = 5, y(1) = 4 yt+2 = 4yt+1 − 4yt yt+2 = 4yt+1 − 4yt y(0) = 6, y(1) = 4 yt+2 = 4yt+1 − 16yt yt+2 = ayt+1 − byt + c yt+2 = 4yt+1 − 16yt + 26 yt+2 = 5yt+1 − 4yt + 4 yt+2 = −yt+1 + 2yt + 12 y(0) = 4, y(1) = 5 Yt = (b + v)Yt−1 − vYt−2 + (a + G) Yt = 4.75Yt−1 + 4Yt−2 + 150

Discrete dynamic systems Neither Mathematica nor Maple, however, can solve directly the logistic equation yt+1 =

(1 + a)yt 1 + byt

This is readily accomplished using the substitution provided in section 3.9. It is worth pointing out that in the case of numerical examples, if all that is required is a plot of the sequence of points, then there is no need to solve the recursive (or difference) equation. We conclude this section, therefore, with simple instructions for doing this.14 The equation we use as an example is pt = 5.6 − 0.4pt−1

p0 = 1

Mathematica Clear[p]; p[0]=1; p[t-]:=p[t]=5.6-0.4p[t-1]; data=Table[{t,p[t]},{t,0,20}]; ListPlot[data,PlotJoined->True,PlotRange->All];

Maple t:=’t’: p:=’p’: p:=proc(t)option remember; 5.6-0.4*p(t-1)end: p(0):=1: data:=seq([t,p(t)],t=0..20)]; plot(data,colour=black,thickness=2);

Notice that the instructions in Maple require a ‘small’ procedural function. It is important in using this to include the option remember, which allows the programme to remember values already computed. Higher-order recursive equations and nonlinear recursive equations are dealt with in exactly the same way. With discrete dynamic models, however, it is often easier and quicker to set the model up on a spreadsheet (see Shone 2001).

Appendix 3.1 Two-cycle logistic equation using Mathematica THEOREM The number a satisﬁes the equation a = f ( f (a)) if a is either a ﬁxed point or is part of a two-cycle for the dynamical system xn+1 = f (xn ) 14

I am grateful to Johannes Ludsteck, Centre for European Economic Research (ZEW), for the method of computing tables from recursive equations in Mathematica, which is more efﬁcient than the one I provided in the ﬁrst edition.

135

136

Economic Dynamics Example (the generic logistic equation) xn+1 = rxn (1 − xn ) In[1]:= f[x-]=rx(1-x) Out[1]= r (1-x) x In[2]:= eq1=f[f[x]] Out[2]= r2 (1-x)

x (1-r (1-x)x)

In[3]:= soll=Solve[eq1==x,x] Out[3]=

r+r2 -r √-3-2r+r2 }, {{x --> 0}, {x --> -1+r 2 r },{x --> 2r

2 √ {x --> r+r +r -3-2r+r }} 2 2

2r In[4]:= a1=so11[[3, 1, 2]] 2 2 √ Out[4]= r+r -r -3-2r+r 2 2r In[5]:= a2=soll[[4, 1, 2]] 2 2 √ Out[5]= r+r +r -3-2r+r 2r2 In[6]:= g[x-]=∂ x f[x]

Out[6]= r(1-x)-rx In[7]:= eq2=Simplify[g[a1]g[a2]] Out[7]= 4+2r-r2 In[8]:= sol2=Nsolve[eq2==0,r] Out[8]= {{r --> -1.23607}, {r --> 3.23607}} In[9]:= rstar=sol2[[2, 1, 2]] Out[9]= 3.23607 In[10]:= a1/.r -> rstar Out[10]= 0.5 In[11]:= a2/.r -> rstar Out[11]= 0.809017 In[12]:= a1/.r -> 3.2 Out[12]= 0.513045 In[13]:= a2/.r -> 3.2 Out[13]= 0.799455 In[14]:= Nsolve[-1==4+2r-r2 ,r] Out[14]= {{r --> -1.44949}, {r --> 3.44949}} In[15]:= Nsolve[4+2r-r2 ==1, r] Out[15]= {{r --> -1.}, {r --> 3.}}

Considering only positive roots, we have: r=3

and r = 3.44949

Discrete dynamic systems

Appendix 3.2 Two-cycle logistic equation using Maple > f:=x->r* x* (1-x); f := x → rx(1 − x) > > eq1:=f(f(x)); eq1 := r2 x(1 − x)(1 − rx(1 − x)) > sol1:=solve(eq1=x,x); 1 1 1 r + + −3 − 2r + r2 −1 + r 2 2 2 , , Sol1 := 0, r r 1 1 1 r+ − −3 − 2r + r2 2 2 2 r > a1:=sol1[3]; 1 1 1 r+ + −3 − 2r + r2 2 2 a1 := 2 r > a2:=sol1[4]; 1 1 1 r+ − −3 − 2r + r2 2 2 2 a2 := r > g:=diff(f(x),x); g := r(1 − x) − rx > eq2:=expand(subs(x=a1,g)*subs(x=a2,g)); eq2 := 4 + 2r − r2 > sol2:=solve(eq2=0,r); √ √ sol2 := 1 − 5, 1 + 5 > rstar:=sol2[2]; √ rstar := 1 + 5 > evalf(subs(r=rstar,a1)); .8090169946 > evalf(subs(r=rstar,a2)); .4999999997 > evalf(subs(r=3.2,a1)); .7994554906 > evalf(subs(r=3.2,a2)); .5130445094

137

138

Economic Dynamics > solve(eq2=-1,r); √ √ 1 − 6, 1 + 6 > evalf(%); −1.449489743, 3.449489743 > solve(eq2=1,r); −1, 3

Exercises 1.

2.

3. 4.

Classify the following difference equations: (i) yt+2 = yt+1 − 0.5yt + 1 (ii) yt+2 = 2yt + 3 yt+1 − yt =4 (iii) yt (iv) yt+2 − 2yt+1 + 3yt = t Suppose you borrow an amount P0 , the principal, but you repay a ﬁxed amount, R, each period. Formulate the general amount, Pt+1 , owing in period t + 1, with interest payment r%. Solve for Pn . In question 2, suppose the repayment is also variable, with amount repaid in period t of Rt . Derive the solution Pn . Establish whether the following are stable or unstable and which are cyclical. (i) yt+1 = −0.5yt + 3 (ii) 2yt+1 = −3yt + 4 (iii) yt+1 = −yt + 6 (iv) yt+1 = 0.5yt + 3

5.

6.

(v) 4yt+2 + 4yt+1 − 2 = 0 Consider yt+1 = y3t − y2t + 1 (i) Show that a = 1 is a ﬁxed point of this system. (ii) Illustrate that a = 1 is a shunt by considering points either side of unity for y0 . Use a spreadsheet to compare yt+1 = (1 + a)yt − by2t and yt+1 =

(1 + a)yt 1 + byt

using (i) a = 1.5 (ii) a = 1.5 (iii) a = 2.2 (iv) a = 2.2 (v) a = 1.8

b = 0.1 b = 0.1 b = 0.1 b = 0.1 b = 0.15

y0 = 1 y0 = 22 y0 = 1 y0 = 25 y0 = 11.5

Discrete dynamic systems 7.

Derive the cobweb system for the price in each of the following demand and supply systems, and establish whether the equilibrium price is (i) stable, (ii) unstable, or (iii) oscillatory. (i) qdt = 10 − 3 pt qst = 2 + pt−1 qdt = qst

8.

(ii) qdt = 25 − 4 pt qst = 3 + 4 pt−1 qdt = qst

(iii) qdt = 45 − 2.5 pt qst = 5 + 7.5 pt−1 qdt = qst

Suppose we have the macroeconomic model Ct = a + bYt−1 Et = Ct + It + Gt Yt = Et

9.

10.

where C and Y are endogenous and I and G are exogenous. Derive the general solution for Yn . Under what conditions is the equilibrium of income, Y ∗ , stable? Verify your results of question 8 by using a spreadsheet and letting I = 10, G = 20, a = 50, Y0 = 20, and b = 0.8 and 1.2, respectively. For what period does the system converge on Y ∗ − Y0 within 1% deviation from equilibrium? For the same initial value Y0 , is the period longer or shorter in approaching equilibrium the higher the value of b? Given qdt = a − bpt qst = c + dpet pet = pt−1 − e(pt−1 − pt−2 ) (i) Show that if in each period demand equals supply, then the model exhibits a second-order nonhomogeneous difference equation for pt . (ii) Use a spreadsheet to investigate the path of price and quantity for the parameter values a = 10 b=3

11.

12.

13.

c = 2 e = 0.5 d=1

In the linear cobweb model of demand and supply, demonstrate that the steeper the demand curve relative to the supply curve, the more damped the oscillations and the more rapidly equilibrium is reached. Using a spreadsheet, verify for the linear cobweb model of demand and supply that whenever the absolute slope of the demand curve is equal to the absolute slope of the supply curve, both price and quantity have a two-period cycle. Given the following logistic model yt+1 = 3.84yt (1 − yt ) set this up on a spreadsheet. Set y0 = 0.1 and calculate yn for the ﬁrst 100 elements in the series. Use the 100th element as the starting value and then re-compute the next 100 elements in the series. Do the same again, and verify that this system tends to a three-cycle with a1 = 0.149407 a2 = 0.488044

a3 = 0.959447

139

140

Economic Dynamics 14.

f1 (x) and f2 (x) are linearly dependent if and only if there exist constants b1 and b2 , not all zero, such that b1 f1 (x) + b2 f2 (x) = 0 for every x. If the set of functions is not linearly dependent, then f1 (x) and f2 (x) are linearly independent. Show that y1 = Y ∗

15.

and

y2 = tY ∗

are linearly independent. Given the following version of the Solow model with labour augmenting technical progress Yt = F(Kt , At Lt ) Kt+1 = Kt + δKt St = sYt I t = St Lt+1 − Lt =n Lt t At = γ A0 (i) show that (1 − δ)kˆ t + sf (kˆ t ) kˆ t+1 = γ (1 + n) where kˆ is the capital/labour ratio measured in efﬁciency units, i.e., kˆ = (K/AL).

16.

(ii) Approximate this result around kˆ ∗ > 0. Given the model qdt = a − bpt b > 0 qst = c + dpet d > 0 pet = pet−1 − λ( pt−1 − pet−1 )

0 0 y< 3 Hence, for any point at which x lies below the x-line, then x is rising. Two are shown in ﬁgure 4.6 by the horizontal arrows that are pointing to the right. By the same reasoning, to the left of the x-line we have x y> implying x˙ < 0 3 Hence, for any point at which x lies above the x-line, then x is falling. Two are shown in ﬁgure 4.6 by the horizontal arrows pointing to the left. By the same reasoning we can establish to the right of the y-line y < 2x

implying

y˙ < 0, hence y is falling

implying

y˙ > 0, hence y is rising

while to its left y > 2x

Again these are shown by the vertical arrows pointing down and up respectively in ﬁgure 4.6. It is clear from ﬁgure 4.6 that we have four quadrants, which we have labelled I, II, III and IV, and that the general direction of force in each quadrant is shown by the arrow between the vertical and the horizontal. It can be seen from ﬁgure 4.6 that in quadrants I and III forces are directing the system towards the origin, towards the ﬁxed point. In quadrants II and IV, however,

152

Economic Dynamics the forces are directing the system away from the ﬁxed point. We can immediately conclude, therefore, that the ﬁxed point cannot be a stable point. Can we conclude that for any initial value of x and y, positioning the system in quadrants I or III, that the trajectory will tend over time to the ﬁxed point? No, we cannot make any such deduction! For instance, if the system began in quadrant I, and began to move towards the ﬁxed point, it could over time pass into quadrant IV, and once in quadrant IV would move away from the ﬁxed point. In fact, this is precisely the trajectory shown in ﬁgure 4.5. Although the trajectory shown in ﬁgure 4.5 moves from quadrant I into quadrant IV, this need not be true of all initial points beginning in quadrant I. Depending on the initial value for x and y, it is quite possible for the system to move from quadrant I into quadrant II, ﬁrst moving towards the ﬁxed point and then away from it once quadrant II has been entered. This would be the situation, for example, if the initial point was (x0 , y0 ) = (4, 2) (see exercise 2). This complex nature of the solution paths can be observed by considering the direction ﬁeld for the differential equation system. The direction ﬁeld, along with the equilibrium lines are shown in ﬁgure 4.7. Why the dynamic forces seem to operate in this way we shall investigate later in this chapter. Example 4.5 The following system of linear differential equations x˙ = −3x + y y˙ = x − 3y with initial condition x0 = 4 and y0 = 5 has solution equations x(t) =

9e2t − 1 2e4t

and

y(t) =

1 + 9e2t 2e4t

which gives rise to a trajectory which approaches the ﬁxed point (x∗ , y∗ ) = (0, 0), as shown in ﬁgure 4.8. The path of x(t) and y(t), represented by the phase line, as t increases is shown by the direction of the arrows. Figure 4.7.

Systems of ﬁrst-order differential equations

153 Figure 4.8.

No matter what the initial point, it will be found that each trajectory approaches the ﬁxed point (x∗ , y∗ ) = (0, 0). In other words, the ﬁxed point (the equilibrium point) is globally stable. Considering the vector of forces for this system captures this feature. The equilibrium solution lines are y = 3x for x˙ = 0 x y = for y˙ = 0 3 with a ﬁxed point at the origin. To the right of the x-line we have y < 3x or −3x + y < 0

implying

x˙ < 0 so x is falling

While to the left of the x-line we have y > 3x or −3x + y > 0

implying

x˙ > 0 so x is rising

Similarly, to the right of the y-line we have y<

x or 0 < −3y + x 3

implying

y˙ > 0 so y is rising

While to the left of the y-line we have y>

x or 0 > −3y + x 3

implying

y˙ < 0 so y is falling

All this information, including the vectors of force implied by the above results, is illustrated in ﬁgure 4.9. It is clear that no matter in which of the four quadrants the system begins, all the forces push the system towards the ﬁxed point. This means that even if the trajectory crosses from one quadrant into another, it is still being directed towards the ﬁxed point. The ﬁxed point must be globally stable. If the initial point is (x0 , y0 ) = (4, 5), then the trajectory remains in quadrant I and tends to the ﬁxed point (x∗ , y∗ ) = (0, 0) over time. However, ﬁgure 4.9 reveals much more. If the system should pass from one quadrant into an adjacent quadrant, then the trajectory is still being directed towards the ﬁxed point, but the movement

154

Economic Dynamics

Figure 4.9.

Figure 4.10.

of the system is clockwise.3 This clockwise motion is shown most explicitly by including the direction ﬁeld on the equilibrium lines, as shown in ﬁgure 4.10. Example 4.6 The two examples discussed so far both have ﬁxed points at the origin. However, this need not always be the case. Consider the following system of linear 3

The nature of the trajectory can be established by noting that (9e2t − 1)2 (1 + 9e2t )2 + 4e8t 4e8t 81e4t + 1 = 2e8t 2 2 i.e. x + y = φ(t) x2 + y2 =

This is a circle at any moment in time whose radius is governed by φ(t). But over time the limit of φ(t) is zero. Consequently, as time passes the radius must diminish. This means that the trajectory conforms to a spiralling path of ever-decreasing radius. Furthermore, the vector forces indicate that the spiral moves in a clockwise direction regardless of the initial value.

Systems of ﬁrst-order differential equations

155 Figure 4.11.

Figure 4.12.

nonhomogeneous autonomous differential equations x˙ = −2x − y + 9 y˙ = −y + x + 3 The equilibrium lines in the phase plane can readily be found by setting x˙ = 0 and y˙ = 0. Thus x˙ = 0 y˙ = 0

implying implying

y = −2x + 9 y=x+3

which can be solved to give a ﬁxed point, an equilibrium point, namely (x∗ , y∗ ) = (2, 5). The solution equations for this system for initial condition, x0 = 2 and y0 = 2 are √ √ x(t) = 2 + 2 3 sin( 3t/2)e−(3t/2) √ √ √ y(t) = 5 − (3 cos( 3t/2)) − 3 sin( 3t/2)e−(3t/2) The equilibrium lines along with the trajectory are illustrated in ﬁgure 4.11. The analysis of this example is the same as for examples 4.4 and 4.5. In this case the ﬁxed point is at (x∗ , y∗ ) = (2, 5). The vectors of force are illustrated in ﬁgure 4.12 by the arrows. What is apparent from this ﬁgure is that the system is globally stable, and the dynamic forces are sending the system towards the ﬁxed point in a counter-clockwise motion. As we illustrated in ﬁgure 4.11, if the initial point is (x0 , y0 ) = (2, 2), then the system begins in quadrant III and tends to the ﬁxed point over time in a counter-clockwise direction, passing ﬁrst into quadrant

156

Economic Dynamics

Figure 4.13.

II and then into I as it moves towards the ﬁxed point. A similar behaviour occurs if the initial point is (x0 , y0 ) = (3, 1) beginning in quadrant III (see exercise 4). Once again the vector forces can be seen in terms of the direction ﬁeld, which we show in ﬁgure 4.13, along with the equilibrium lines.

4.4 Matrix speciﬁcation of autonomous systems

(4.9)

(4.10)

(4.11)

(4.12)

The examples so far discussed illustrate that even with simple linear autonomous systems, the type of dynamic behaviour is quite varied. In order to pursue the stability/instability aspects of systems of autonomous equations it is more convenient to specify the models in terms of matrices and vectors. Example 4.4 can be written x˙ x 1 −3 = −2 1 y˙ y example 4.5 as x˙ −3 = 1 y˙

1 −3

x y

while example 4.6 can be written x˙ x 9 −2 −1 = + 1 −1 y˙ y 3 In fact, we can readily generalise such linear autonomous systems with constant coefﬁcients. Although we shall talk here of only two variables, it can readily be generalised to n. Deﬁne the following vectors and matrices a11 a12 x1 b x˙1 , A= , x= , b= 1 x˙ = x˙2 a21 a22 x2 b2

Systems of ﬁrst-order differential equations

157

Then systems 4.4 and 4.5 are simply speciﬁc examples of the homogeneous linear system x˙ = Ax

(4.13)

while system 4.6 is a speciﬁc example of the nonhomogeneous linear system x˙ = Ax + b

(4.14)

For linear homogeneous systems, if the determinant of A is not zero, then the only solution, the only ﬁxed point, is x∗ = 0, i.e., (x1∗ = 0 and x2∗ = 0). On the other hand, for nonhomogeneous linear systems, the equilibrium can be found, so long as A is nonsingular, from 0 = Ax∗ + b x∗ = −A−1 b

(4.15)

When considering the issue of stability/instability it is useful to note that linear nonhomogeneous systems can always be reduced to linear homogeneous systems in terms of deviations from equilibrium if an equilibrium exists. For x˙ = Ax + b 0 = Ax∗ + b subtracting we immediately have in deviation form x˙ = A(x − x∗ )

(4.16) ∗

(x1∗ , x2∗ ).

which is homogeneous in terms of deviations from the ﬁxed point x = There will be no loss of generality, therefore, if we concentrate on linear homogeneous systems. The matrix A is of particular importance in dealing with stability and instability. Two important properties of such a square matrix are its trace, denoted tr(A), and its determinant, denoted det(A), where4 tr(A) = a11 + a22 a11 a12 = a11 a22 − a12 a21 det(A) = a21 a22

(4.17)

It should be noted that both the trace and the determinant are scalars. The matrix A is nonsingular if det(A) = 0. There is another property of the matrix A that arises for special linear systems. Consider the following general linear system y = Ax This can be viewed as a transformation of the vector x into the vector y. But suppose that x is transformed into a multiple of itself, i.e., y = λx, where λ is a scalar of proportionality. Then Ax = λx or (A − λI)x = 0 4

See Chiang (1984) or any book on linear algebra.

(4.18)

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Economic Dynamics But equation (4.18) will have a nonzero solution if and only if λ is chosen such that φ(λ) = det(A − λI) = 0

(4.19)

The values of λ which satisfy equation (4.19) are called the eigenvalues of the matrix A, and the solution to the system that are obtained using these values are called the eigenvectors corresponding to that eigenvalue. Let us clarify these concepts with a simple example. Example 4.7 Let

1 A= −2 then

1 4

1 − λ 1 det(A − λI) = = λ2 − 5λ + 6 = 0 −2 4 − λ

Let the two roots, the two eigenvalues, of this quadratic be denoted r and s, respectively. Then r = 3 and s = 2. In this example we have two distinct real roots. To determine the eigenvectors, we must substitute for a particular value of λ in the equation (A − λi I)vi = 0

(i = r, s)

With λ = r = 3 then 1 1 3 0 −2 A − 3I = − = −2 4 0 3 −2

1 1

whose determinant value is zero as required. Hence −2 1 vr1 0 = −2 1 vr2 0 which has the single condition −2vr1 + vr2 = 0, and vr2 is determined in terms of vr1 . Thus, if vr1 = c, then vr2 = 2c. Accordingly, the eigenvector, denoted vr is 1 vr = c 2 Since c is an arbitrary constant this is usually normalised to unity, and so the eigenvector is simply denoted 1 vr = 2 Of course, this eigenvector was derived from the root r = 3, which is why we have labelled it vr . We also have a second eigenvector associated with the root λ = s = 2. Following through exactly the same procedure we ﬁnd that

Systems of ﬁrst-order differential equations

159 Figure 4.14.

−vs1 + vs2 = 0 and hence 1 s v = 1 Figure 4.14 illustrates the two eigenvectors associated with the two eigenvalues. Although this illustration had two distinct roots, it is quite clear that for any system of two dimensions, there will be two roots that correspond to one of the following possibilities: (1) (2) (3)

real and distinct real and equal complex conjugate.

It is possible to relate these three possibilities to conditions imposed on the trace and determinant of the matrix A. To see this let a b A= c d Then

a − λ b det(A − λI) = = λ2 − (a + d)λ + (ad − bc) = 0 c d − λ

But a + d = tr(A) and ad − bc = det(A). Hence the characteristic equation can be expressed λ2 − tr(A)λ + det(A) = 0 with solutions r= s=

tr(A) + tr(A) −

tr(A)2 − 4det(A) 2

tr(A)2 − 4det(A) 2

(4.20)

160

Economic Dynamics It follows immediately that the roots are: (1) (2) (3)

real and distinct if tr(A)2 > 4det(A) real and equal if tr(A)2 = 4det(A) complex conjugate if tr(A)2 < 4det(A).

4.5 Solutions to the homogeneous differential equation system: real distinct roots Suppose we have an n-dimensional dynamic system x˙ = Ax

(4.21)

where

x˙1 x˙2 x˙ = . , .. x˙n

a11 .. A= . an1

a12 .. .

··· .. .

a1n .. , .

an2

···

ann

x1 x2 x=. .. xn

and suppose u1 , u2 , . . . , un are n linearly independent solutions, then a linear combination of these solutions is also a solution. We can therefore express the general solution as the linear combination x = c1 u1 + c2 u2 + . . . + cn un where c1 , c2 , . . . , cn are arbitrary constants. In the case of just two variables, we are after the general solution x = c1 u1 + c2 u2 In chapter 2, where we considered a single variable, we had a solution x = cert This would suggest that we try the solution5 x = eλt v where λ is an unknown constant and v is an unknown vector of constants. If we do this and substitute into the differential equation system we have λeλt v = Aeλt v eliminating the term eλt we have λv = Av i.e. (A − λI)v = 0 For a nontrivial solution we require that det(A − λI) = 0 5

Here u = ert v.

Systems of ﬁrst-order differential equations

161

We investigated this problem in the last section. What we wish to ﬁnd is the eigenvalues of A and the associated eigenvectors. Return to the situation with only two variables, and let the two roots (the two eigenvalues) be real and distinct, which we shall again label as r and s. Let vr be the eigenvector associated with the root r and vs be the eigenvector associated with the root s. Then so long as r = s u1 = ert vr

and

u2 = est vs

are independent solutions, while x = c1 ert vr + c2 est vs is a general solution. Example 4.8 Find the general solution to the dynamic system x˙ = x + y y˙ = −2x + 4y We can write this in matrix form x˙ 1 1 x = y˙ −2 4 y The matrix A of this system has already been investigated in terms of example 4.7. Note, however, that det(A) > 0. In example 4.7 we found that the two eigenvalues were r = 3 and s = 2 and the associated eigenvectors were 1 1 r s , v = v = 2 1 Then the general solution is 3t 1 2t 1 + c2 e x = c1 e 2 1 or, in terms of x and y x(t) = c1 e3t + c2 e2t y(t) = 2c1 e3t + c2 e2t Given an initial condition, it is possible to solve for c1 and c2 . For example, if x(0) = 1 and y(0) = 3, then 1 = c1 + c2 3 = 2c1 + c2 which gives c1 = 2 and c2 = −1. Leading to our ﬁnal result x(t) = 2e3t − e2t y(t) = 4e3t − e2t

(4.22)

162

Economic Dynamics

4.6 Solutions with repeating roots In chapter 2 we used ceλt

cteλt

and

for a repeated root. If λ = r which is a repeated root, then either there are two independent eigenvectors v1 and v2 which will lead to the general solution x = c1 ert v1 + c2 ert v2 or else there is only one associated eigenvector, say v. In this latter case we use the result x = c1 ert v1 + c2 (ert tv + ert v2 )

(4.23)

In this latter case the second solution satisﬁes ert tv + ert v2 and is combined with the solution ert v1 to obtain the general solution (see Boyce and DiPrima 1997, pp. 390–6). We shall consider two examples, the ﬁrst with a repeating root, but with two linearly independent eigenvectors, and a second with a repeating root but only one associated eigenvector. Example 4.9 Consider x˙ = x y˙ = y Then

x˙ 1 = y˙ 0

where A=

1 0

0 1

x y

0 , 1

det(A) = 1,

A − λI =

1−λ 0

0 1−λ

Hence, det(A − λI) = (1 − λ)2 = 0, with root λ = r = 1 (twice). Using this value of λ then 0 0 (A − I) = 0 0 Since (A − rI)v = 0 (r = 1) is satisﬁed for any vector v, then we can choose any arbitrary set of linearly independent vectors for eigenvectors. Let these be 1 0 v1 = and v2 = 0 1 then the general solution is rt 1 rt 0 x = c1 e + c2 e 0 1

Systems of ﬁrst-order differential equations or x(t) = c1 ert y(t) = c2 ert Example 4.10 Let x˙ = x − y y˙ = x + 3y Then

with

x˙ 1 = y˙ 1

1 A= 1

−1 3

x y

−1 , 3

det(A) = 4,

1 − λ −1 A − λI = 1 3−λ

Hence, det(A − λI) = λ2 − 4λ + 4 = (λ − 2)2 , with root λ = r = 2 (twice). Using λ = r = 2 −1 −1 A − rI = 1 1 and

x −1 (A − rI) = y 1

−1 1

x y

which implies −x − y = 0. Given we normalise x to x = 1, then y = −1. The ﬁrst solution is then 1 e2t −1 To obtain the second solution we might think of proceeding as in the single variable case, but this is not valid (see Boyce and DiPrima 1997, pp. 390–6). What we need to use is e2t tv + e2t v2 where we need to ﬁnd the elements of v2 . Since we know v, then the second solution u2 = (x2 , y2 ) is x2 v 1 = e2t t + e2t 1 y2 v2 −1 i.e. x2 = e2t (t + v1 ) y2 = e2t (−t + v2 ) Hence x˙ = 2e2t (t + v1 ) + e2t = e2t (2t + 2v1 + 1) y˙ = 2e2t (−t + v2 ) + e2t = e2t (−2t + 2v2 + 1)

163

164

Economic Dynamics Substituting all these results into the differential equation system we have e2t (2t + 2v1 + 1) = e2t (t + v1 ) − e2t (−t + v2 ) e2t (−2t + 2v2 − 1) = e2t (t + v1 ) + 3e2t (−t + v2 ) Eliminating e2t and simplifying, we obtain v1 + v2 = −1 v1 + v2 = −1 which is a dependent system. Since we require only one solution, set v2 = 0, giving v1 = −1. This means solution x2 is 1 2 2t 2t −1 x =e t +e −1 0 Hence, the general solution is 1 1 2t 2t 2t −1 x = c1 e + c2 e t +e −1 0 −1 or x = c1 e2t + c2 (t − 1)e2t y = −c1 e2t − c2 te2t

4.7 Solutions with complex roots For the system x˙ = Ax with characteristic equation det(A−λI) = 0, if tr(A)2 < 4det(A), then we have complex conjugate roots. Return to our situation of just two roots, λ = r and λ = s. Then r = α + βi s = α − βi

(4.24)

But this implies that the eigenvectors vr and vs associated with r and s, respectively, are also complex conjugate. Example 4.11 Consider x˙ = −3x + 4y y˙ = −2x + y Then

with

−3 x˙ = y˙ −2

−3 A= −2

4 1 4 , 1

x y

det(A) = 5,

−3 − λ 4 A − λI = −2 1−λ

Systems of ﬁrst-order differential equations and det(A − λI) = λ2 + 2λ + 5, which leads to the roots √ √ −2 − 4 − 20 −2 + 4 − 20 = −1 + 2i and s = = −1 − 2i r= 2 2 The associated eigenvectors are −2 − 2i 4 0 r r (A − λI)v = v = −2 2 − 2i 0 i.e. −(2 + 2i)v1r + 4vr2 = 0 −2vr1 + (2 − 2i)vr2 = 0 Let vr1 = 2, then vr2 = 2(2 + 2i)/4 = 1 + i. Thus 2 u1 = e(−1+2i)t 1+i Turning to the second root. With λ = s = −1 − 2i then s −2 + 2i 4 v1 s (A − λI)v = −2 2 + 2i vs2 i.e. (−2 + 2i)vs1 + 4vs2 = 0 −2vs1 + (2 + 2i)vs2 = 0 Choose vs1 = 2, then vs2 = −(−2 + 2i)(2)/4 = 1 − i. Hence the second solution is 2 u2 = e−(1+2i)t 1−i i.e. vs is the complex conjugate of vr . Hence the general solution is 2 2 + c2 e−(1+2i)t x = c1 e(−1+2i)t 1+i 1−i These are, however, imaginary solutions. To convert them to real solutions we employ two results. One is Euler’s identity (see exercise 10 of chapter 2), i.e. eiθ = cos θ + i sin θ The other employs the real elements of vr (or vs ). Let vr generally be written u1 + w1 i r v = u2 + w2 i and deﬁne

u b1 = 1 u2

and

w b2 = − 1 w2

then the two solutions can be written in the form6 u1 = eαt (b1 cos βt + b2 sin βt) u2 = eαt (b2 cos βt − b1 sin βt) 6

See Giordano and Weir (1991, pp. 180–1).

165

166

Economic Dynamics with the general solution x = c1 u1 + c2 u2

(4.25)

Continuing our example, where λ = −1 + 2i, i.e., α = −1 and β = 2, then 2 2 0 r , i.e. b1 = and b2 = v = 1+i 1 −1 Hence

2 0 cos 2t + sin 2t 1 −1 0 2 u2 = e−t cos 2t − sin 2t −1 1 u1 = e−t

and x = c1 u1 + c2 u2 or x(t) = c1 e−t 2 cos 2t − 2c2 e−t sin 2t = 2e−t (c1 cos 2t − c2 sin 2t) y(t) = c1 e−t cos 2t − c1 e−t sin 2t − c2 e−t cos 2t − c2 e−t sin 2t = e−t [(c1 − c2 ) cos 2t − (c1 + c2 ) sin 2t)]

4.8 Nodes, spirals and saddles Here we shall consider only a two-variable system of the general form x˙ = Ax which to have solutions of the form x = eλt v must satisfy (A − λI)v = 0 and λ must be the eigenvalue and v the eigenvector associated with the matrix A. We shall denote the two eigenvalues as λ = r and λ = s and the two associated eigenvectors vr and vs, respectively. We have already discussed the general solution of the form x = c1 ert vr + c2 est vs In this section we shall extract some geometric properties from the various possible solutions. First such a system will have a critical point, denoted x∗ , if Ax = 0. If A is nonsingular, or det(A) = 0, then the only solution is x∗ = 0. The only critical point is at the origin. The solution function x = φ(t) satisﬁes the differential equations, and this shows the solution path in the phase plane. In terms of vectors, the situation is illustrated in ﬁgure 4.15. The (x,y)-plane denotes the phase plane and the origin is a critical point, ﬁxed point or equilibrium point. At time t = 0 we have x(0) = x0 and y(0) = y0 . At time t there is a vector with coordinates (x(t), y(t)) and the movement of the system as time increases is indicated by the arrows along the solution path.

Systems of ﬁrst-order differential equations

167 Figure 4.15.

Case 1 (Real distinct roots of the same sign) Here we are considering the general solution x = c1 ert vr + c2 est vs where r and s are real and distinct and are either both positive or both negative. We shall assume that r is the larger root in absolute value |r| > |s|. Suppose both roots are negative, then r < s < 0. Further, suppose the associated eigenvectors vr and vs are as shown in ﬁgure 4.16 by the heavy arrows. Thus it is quite clear that as t → ∞, ert → 0 and est → 0, and so x → 0 regardless of the value of c1 and c2 . Of particular signiﬁcance is that if the initial point lies on vr , then c2 = 0 and the system moves down the line through vr and approaches the origin over time. Similarly, if the initial point lies on vs , then c1 = 0 and the system moves down the line vs , approaching the origin in the limit. The critical point is called a node.7 In the present case we have a stable node. If r and s are both positive, then the system will move away from the ﬁxed point over time. This is because both x and y grow exponentially. In this case we have an unstable node. Example 4.12 Let x˙ = −2x + y y˙ = x − 2y with

A=

7

−2 1

1 , −2

Sometimes called an improper node.

det(A) = 3,

A − λI =

−2 − λ 1 1 −2 − λ

168

Economic Dynamics

Figure 4.16.

Hence det(A − λI) = λ2 + 4λ + 3 = (λ + 3)(λ + 1) = 0, which leads to roots λ = r = −3 and λ = s = −1. Using these values for the eigenvalues, the eigenvectors are 1 1 and vs = vr = −1 1 which gives the general solution 1 1 x = c1 e−3t + c2 e−t −1 1 or x(t) = c1 e−3t + c2 e−t y(t) = −c1 e−3t + c2 e−t The solution is illustrated in ﬁgure 4.17,8 where the solution paths are revealed by the direction ﬁeld, indicating quite clearly that the origin is a stable node. Case 2 (Real distinct roots of opposite sign) Consider again x = c1 ert vr + c2 est vs where r and s are both real but of opposite sign. Let r > 0 and s < 0. Suppose the eigenvectors are those as shown in ﬁgure 4.18. If a solution starts on the line 8

Notice that the solution paths tend towards the eigenvector vs .

Systems of ﬁrst-order differential equations

169 Figure 4.17.

Figure 4.18.

through vr then c2 = 0. The solution will therefore remain on vr . Since r is positive, then over time the solution moves away from the origin, away from the ﬁxed point. On the other hand, if the system starts on the line through vs , then c1 = 0, and since s < 0, then as t → ∞ the system tends towards the ﬁxed point. For initial points off the lines through the eigenvectors, then the positive root will dominate the system. Hence for points above vr and vs , the solution path will veer towards the line through vr . The same is true for any initial point below vr and above vs . On the other hand, an initial point below the line through vs will be dominated by the larger root and the system will veer towards minus inﬁnity. In this case the node is called a saddle point. The line through vr is called the unstable arm, while the line through vs is called the stable arm. Saddle path equilibria are common in economics and one should look out for them in terms of real distinct roots of opposite sign and the fact that det(A) is negative. It will also be important to establish the stable and unstable arms of

170

Economic Dynamics the saddle point, which are derived from the eigenvectors associated with the characteristic roots. Because of the importance of saddle points in economics, we shall consider two examples here. Example 4.13 Let x˙ = x + y y˙ = 4x + y then

with

x˙ 1 = y˙ 4

1 A= 4

1 1

x y

1 , 1

det(A) = −3,

1−λ A − λI = 4

1 1−λ

giving det(A − λI) = λ2 − 2λ − 3 = (λ − 3)(λ + 1) = 0. Hence, λ = r = 3 and λ = s = −1. For λ = r = 3 then −2 1 vr = 0 (A − λI)vr = 4 −2 i.e. −2vr1 + vr2 = 0 4vr1 − 2vr2 = 0 Let vr1 = 1, then vr2 = 2. Hence, one solution is 1 1 u1 = ert and vr = 2 2 For λ = s = −1, then (A − λI)vs =

2 1 s v =0 4 2

i.e. 2vs1 + vs2 = 0 4vs1 + 2vs2 = 0 Let vs1 = 1, then vs2 = −2. Hence, a second solution is 1 1 2 st s and v = u =e −2 −2 The situation is illustrated in ﬁgure 4.19. The solution paths are revealed by the direction ﬁeld. The ﬁgure quite clearly shows that the unstable arm of the saddle is the line through the eigenvector vr , while the stable arm of the saddle is the line through the eigenvector vs .

Systems of ﬁrst-order differential equations

171 Figure 4.19.

Example 4.14 Let x˙ = 3x − 2y y˙ = 2x − 2y then

x˙ 3 = y˙ 2

with A=

3 2

−2 −2 −2 , −2

x y

det(A) = −2,

A − λI =

3−λ −2 2 −2 − λ

giving det(A − λI) = λ2 − λ − 2 = (λ − 2)(λ + 1) = 0. Hence, λ = r = 2 and λ = s = −1. For λ = r = 2 then 1 −2 r r v =0 (A − λI)v = 2 −4 i.e. vr1 − 2vr2 = 0 2vr1 − 4vr2 = 0 Let vr1 = 2, then vr2 = 1. Hence, one solution is 2 1 rt 2 r u =e and v = 1 1 For λ = s = −1, then

4 −2 s (A − λI)v = v =0 2 −1 s

172

Economic Dynamics

Figure 4.20.

i.e. 4vs1 − 2vs2 = 0 2vs1 − vs2 = 0 Let vs1 = 1, then vs2 = 2. Hence, a second solution is 1 1 and vs = u2 = est 2 2 The situation is illustrated in ﬁgure 4.20. The solution paths are revealed by the direction ﬁeld. The unstable arm of the saddle is the line through the eigenvector vr , while the stable arm of the saddle is the line through the eigenvector vs . Case 3 (Real equal roots) In this case λ = r = s. Throughout assume the repeated root is negative. (If it is positive then the argument is identical but the movement of the system is reversed.) There are two sub-cases to consider in line with our earlier analysis: (a) independent eigenvectors, and (b) one independent eigenvector. The two situations were found to be: (a) (b)

x = c1 ert v1 + c2 ert v2 x = c1 ert v + c2 [ert tv + ert v2 ]

Consider each case in turn. In example 4.9 we found for two independent eigenvectors x(t) = c1 ert y(t) = c2 ert Hence, x/y = c1 /c2 is independent of t and depends only on the components of vr and vs and the arbitrary constants c1 and c2 . This is a general result and so all solutions lie on straight lines through the origin, as shown in ﬁgure 4.21. In this case the origin is a proper node that is stable. Had the repeated root been positive, then we would have an unstable proper node. It is this situation we gave an example

Systems of ﬁrst-order differential equations

173 Figure 4.21.

Figure 4.22.

of at the beginning of section 4.6. The direction ﬁeld, along with the independent vectors is shown in ﬁgure 4.22 for this example. For the second sub-case, where again r < 0, for large t the dominant term must be c2 ert tv, and hence as t → ∞ every trajectory must approach the origin and in such a manner that it is tangent to the line through the eigenvector v. Certainly, if c2 = 0 then the solution must lie on the line through the eigenvector v, and approaches the origin along this line, as shown in ﬁgure 4.23. (Had r > 0, then every trajectory would have moved away from the origin.) The approach of the trajectories to the origin depends on the eigenvectors v and v2 . One possibility is illustrated in ﬁgure 4.23. To see what is happening, express

174

Economic Dynamics

Figure 4.23.

the general solution as x = [c1 ert v + c2 ert v2 + c2 ert tv] = [(c1 v + c2 v2 ) + c2 tv]ert = uert Then u = (c1 v + c2 v2 ) + c2 tv which is a vector equation of a straight line which passes through the point c1 v + c2 v2 and is parallel to v. Two such points are illustrated in ﬁgure 4.23, one at point a (c2 > 0) and one at point b (c2 < 0). We shall not go further into the mathematics of such a node here. What we can do, however, is highlight the variety of solution paths by means of two numerical examples. The ﬁrst, in ﬁgure 4.24, has the orientation of the trajectories as illustrated in ﬁgure 4.23, while ﬁgure 4.25 has the reverse orientation. Whatever the orientation, the critical point is again an improper node that is stable. Had r > 0, then the critical point would be an improper node that is unstable. Case 4 (Complex roots, α = 0 and β > 0) In this case we assume the roots λ = r and λ = s are complex conjugate and with r = α + βi and s = α − βi, and α = 0 and β > 0. Systems having such complex roots can be expressed x˙ = αx + βy y˙ = −βx + αy

Systems of ﬁrst-order differential equations

175 Figure 4.24.

Figure 4.25.

or

x˙ α = y˙ −β

β α

x y

Now express the system in terms of polar coordinates with R and θ, where y and tan θ = R 2 = x 2 + y2 x and R˙ = αR which results in R = ceαt

where c is a constant

Similarly θ˙ = −β giving θ = −βt + θ0

where θ(0) = θ0

What we have here are parametric equations R = ceαt θ = −βt + θ0

176

Economic Dynamics

Figure 4.26.

in polar coordinates of the original system. Since β > 0 then θ decreases over time, and so the motion is clockwise. Furthermore, as t → ∞ then either R → 0 if α < 0 or R → ∞ if α > 0. Consequently, the trajectories spiral either towards the origin or away from the origin depending on the value of α. The two possibilities are illustrated in ﬁgure 4.26. The critical point in such situations is called a spiral point. Case 5 (Complex roots, α = 0 and β > 0) In this case we assume the roots λ = r and λ = s are complex conjugate with r = βi and s = −βi (i.e. α = 0). In line with the analysis in case 4, this means x˙ 0 β x = y˙ −β 0 y resulting in R˙ = 0 and θ˙ = −β, giving R = c and θ = −βt + θ0 , where c and θ 0 are constants. This means that the trajectories are closed curves (circles or ellipses) with centre at the origin. If β > 0 the movement is clockwise while if β < 0 the movement is anticlockwise. A complete circuit around the origin denotes the phase

Systems of ﬁrst-order differential equations

177 Figure 4.27.

of the cycle, which is 2π/β. The critical point is called the centre. These situations are illustrated in ﬁgure 4.27.

Summary From the ﬁve cases discussed we arrive at a number of observations. 1.

2. 3.

After a sufﬁcient time interval, the trajectory of the system tends towards three types of behaviour: (i) the trajectory approaches inﬁnity (ii) the trajectory approaches the critical point (iii) the trajectory traverses a closed curve surrounding the critical point. Through each point (x0 , y0 ) in the phase plane there is only one trajectory. Considering the set of all trajectories, then three possibilities arise: (i) All trajectories approach the critical point. This occurs when (a) tr(A)2 > 4det(A), r < s < 0 (b) tr(A)2 < 4det(A), r = α + βi, s = α − βi and α < 0.

178

Economic Dynamics (ii) All trajectories remain bounded but do not approach the critical point as t → ∞. This occurs when tr(A)2 < 4det(A) and r = βi and s = −βi(α = 0). (iii) At least one of the trajectories tends to inﬁnity as t → ∞. This occurs when (a) tr(A)2 > 4det(A), r > 0 and s > 0 or r < 0 and s > 0 (b) tr(A)2 < 4det(A), r = α + βi, s = α − βi and α > 0.

4.9 Stability/instability and its matrix speciﬁcation Having outlined the methods of solution for linear systems of homogeneous autonomous equations, it is quite clear that the characteristic roots play an important part in these. Here we shall continue to pursue just the two-variable cases. For the system x˙ = ax + by y˙ = cx + dy where

A=

(4.26)

a c

b d

and

A − λI =

a−λ c

b d−λ

we have already shown that a unique critical point exists if A is nonsingular, i.e., det(A) = 0 and that tr(A) ± tr(A)2 − 4det(A) r, s = 2 Furthermore, if: (i) (ii) (iii)

tr(A)2 > 4det(A) the roots are real and distinct tr(A)2 = 4det(A) the roots are real and equal tr(A)2 < 4det(A) the roots are complex conjugate.

This leads to our ﬁrst distinction. To illustrate the variety of solutions we plot the tr(A) on the horizontal axis and the det(A) on the vertical, which is valid because these are scalars. The plane is then divided by plotting the curve tr(A)2 = 4det(A) (i.e. x2 = 4y), which is a parabola with minimum at the origin, as shown in ﬁgure 4.28. Below the curve tr(A)2 > 4det(A) and so the roots are real and distinct; above the curve the roots are complex conjugate; while along the curve the roots are real and equal. We can further sub-divide the situations according to the sign/value of the two roots. Take ﬁrst real distinct roots that lie strictly below the curve. If both roots are negative then the tr(A) must be negative, and since det(A) is positive, then we are in the region below the curve and above the x-axis, labelled region I in ﬁgure 4.28. In this region the critical point is asymptotically stable. In region II, which is also below the curve and above the x-axis, both roots are positive and the system is unstable.

Systems of ﬁrst-order differential equations

179 Figure 4.28.

If both roots are opposite in sign, we have found that the det(A) is negative and the critical point is a saddle. Hence, below the x-axis, marked region III, the critical point is an unstable saddle point. Notice that this applies whether the trace is positive or negative. The complex region is sub-divided into three categories. In region IV the sign of α in the complex conjugate roots α ± βi is strictly negative and the spiral trajectory tends towards the critical point in the limit. In region V α is strictly positive and the critical point is an unstable one with the trajectory spiralling away from it. Finally in region VI, which is the y-axis above zero, α = 0 and the critical point has a centre with a closed curve as a trajectory. It is apparent that the variety of possibilities can be described according to the tr(A) and det(A) along with the characteristic roots of A. The list with various nomenclature is given in table 4.1.

4.10 Limit cycles9 A limit cycle is an isolated closed integral curve, which is also called an orbit. A limit cycle is asymptotically stable if all the nearby cycles tend to the closed orbit from both sides. It is unstable if the nearby cycles move away from the closed orbit on either side. It is semi-stable if the nearby cycles move towards the closed orbit on one side and away from it on the other. Since the limiting trajectory is a periodic orbit rather than a ﬁxed point, then the stability or instability is called an orbital stability or instability. There is yet another case, common 9

This section utilises the VisualDSolve package provided by Schwalbe and Wagon (1996). It can be loaded into Mathematica with the Needs command. This package provides considerable visual control over the display of phase portraits.

180

Economic Dynamics Table 4.1 Stability properties of linear systems Matrix and eigenvalues

Type of point

Type of stability

tr(A) < 0, det(A) > 0, tr(A)2 > 4det(A) r 0, tr(A)2 > 4det(A) r>s>0 det(A) < 0 r > 0, s < 0 or r < 0, s > 0 tr(A) < 0, det(A) > 0, tr(A)2 = 4det(A) r=s 0, det(A) > 0, tr(A)2 = 4det(A) r=s>0 tr(A) < 0, det(A) > 0, tr(A)2 < 4det(A) r = α + βi, s = α − βi, α < 0 tr(A) > 0, det(A) > 0, tr(A)2 < 4det(A) r = α + βi, s = α − βi, α > 0 tr(A) = 0, det(A) > 0 r = βi, s = −βi

Improper node

Asymptotically stable

Improper node

Unstable

Saddle point

Unstable saddle

Star node or proper node

Stable

Star node or proper node

Unstable

Spiral node

Asymptotically stable

Spiral node

Unstable

Centre

Stable

in predatory–prey population models. If a system has closed orbits that other trajectories neither approach nor diverge from, then the closed orbits are said to be stable. Geometrically, we have a series of concentric orbits, each one denoting a closed trajectory. In answering the question: ‘When do limit cycles occur?’ we draw on the Poincar´e–Bendixson theorem. This theorem is concerned with a bounded region, which we shall call R, in which the long-term motion of a two-dimensional system is limited to it. If for region R, any trajectory starting within R stays within R for all time, then two possibilities arise: (1) (2)

the trajectory approaches a ﬁxed point of the system as t → ∞; or the trajectory approaches a limit cycle as t → ∞.

When trajectories that start in R remain in R for all time, then the region R is said to be the invariant set for the system. Trajectories cannot escape such a set. The following points about limit cycles are worth noting. (1) (2)

(3) (4) (5) (6)

Limit cycles are periodic motions and so the system must involve complex roots. For a stable limit cycle, the interior nearby paths must diverge from the singular point (the ﬁxed point). This occurs if the trace of the Jacobian of the system is positive. For a stable limit cycle, the outer nearby paths must converge on the closed orbit, which requires a negative trace. Points (2) and (3) mean that for a stable limit cycle the trace must change sign in the region where the limit cycle occurs. The Poincar´e–Bendixson theorem holds only for two-dimensional spaces. If the Poincar´e–Bendixson theorem is satisﬁed, then it can be shown that if there is more than one limit cycle they alternate between being stable and unstable. Furthermore, the outermost one and the innermost one must

Systems of ﬁrst-order differential equations

181

be stable. This means that if there is only one limit cycle satisfying the theorem, it must be stable. Example 4.15 The following well-known example has a limit cycle composed of the unit circle (see Boyce and DiPrima 1997, pp. 523–7): x = y + x − x(x2 + y2 ) y = −x + y − y(x2 + y2 ) Utilising the VisualDSolve package within Mathematica, we can show the limit cycle and two trajectories: one starting at point (0.5,0.5) and the other at point (1.5,1.5). The input instructions are: In[2]:= PhasePlot [{x’ [t] == y[t] + x[t] - x[t] (x[t]ˆ2 + y[t]ˆ2), y’ [t] == -x[t] + y[t] - y[t] (x[t]ˆ2 + y[t]ˆ2)}, {x[t], y[t]}, {t, 0, 10}, {x, -2, 2}, {y, -2, 2}, InitialValues -> {{0.5, 0.5}, {1.5, 1.5}}, ShowInitialValues -> True, FlowField -> False, FieldLength -> 1.5, FieldMeshSize -> 25, WindowShade -> White, FieldColor -> Black, Nullclines -> True, PlotStyle -> AbsoluteThickness [1.2], InitialPointStyle -> AbsolutePointSize [3], ShowEquilibria -> True, DirectionArrow -> True, AspectRatio -> 1, AxesLabel -> {x, y}, PlotLabel -> ‘‘Unit Limit Cycle”];

which produces ﬁgure 4.29 showing a unit limit cycle. Example 4.16 (Van der Pol equation) The Van der Pol equation is a good example illustrating an asymptotically stable limit cycle. It also illustrates that a second-order differential equation can be reduced to a system of ﬁrst-order differential equations that are more convenient for Figure 4.29.

182

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Figure 4.30.

solving. The Van der Pol equation takes the form: (4.27)

x¨ − µ(1 − x2 )x˙ + x = 0 ˙ then y˙ = x, ¨ so we have the two equations, Let y = x,

(4.28)

x˙ = y y˙ = µ(1 − x2 )y − x To illustrate the limit cycle, let µ = 1. The phase portrait that results is shown in ﬁgure 4.30. Here we take two initial points: (a) point (0.5,0.5), which starts inside the limit cycle; and (b) point (1.5,4), which begins outside the limit cycle. Example 4.17 Walrasian price and quantity adjustment and limit cycles The presence of limit cycles is illustrated by a Walrasian model which includes both price and quantity adjustments (see Flaschel et al. 1997 and Mas-Colell 1986). Let Y denote output of a one good economy and L labour input. Y = f (L) is a production function which is twice differentiable and invertible with L = f −1 (Y) = φ(Y) and φ (Y) > 0. In equilibrium the price, p, is equal to marginal wage cost, where marginal wage cost is also given by φ (Y). Thus, p∗ = φ (Y). For simplicity we assume that the marginal wage cost is a linear function of Y, with φ (Y) = c1 + c2 Y. Aggregate demand takes the form D(p, L) and in equilibrium is equal to supply, i.e., D[p∗ , φ(Y ∗ )] = Y ∗ . Finally, we have both a price and a quantity adjustment:

(4.29)

p˙ = α[D(p, φ(Y)) − Y] Y˙ = β[p − φ (Y)]

α>0 β>0

These establish two differential equations in p and Y. Consider the following numerical example. Let φ (Y) = 0.87 + 0.5Y D( p) = −0.02p3 + 0.8p2 − 9p + 50 then (p∗ , Y ∗ ) = (13, 24.26) with isoclines: p˙ = 0 Y˙ = 0

Y = −0.02p3 + 0.8p2 − 9p + 50 p = 0.87 + 0.5Y

or

Y = 1.74 + 2p

Figure 4.31 reproduces the ﬁgures derived in Flaschel et al. 1997 using Mathematica, for α = 1 and different values of the parameter β. Not only do the ﬁgures

Systems of ﬁrst-order differential equations

183 Figure 4.31.

illustrate a stable limit cycle, but they also illustrate that the limit cycle shrinks as β increases.

4.11 Euler’s approximation and differential equations on a spreadsheet10 Although differential equations are for continuous time, if our main interest is the trajectory of a system over time, sometimes it is convenient to use a spreadsheet to do this. To accomplish this task we employ Euler’s approximation. For a single variable the situation is shown in ﬁgure 4.32. We have the differential equation dx = f (x, t) x(t0 ) = x0 dt Let x = φ(t) denote the unknown solution curve. At time t0 we know x0 = φ(t0 ). We also know dx/dt at t0 , which is simply f (x0 , t0 ). If we knew x = φ(t), then the value at time t1 would be φ(t1 ). But if we do not have an explicit form for x = φ(t), we can still plot φ(t) by noting that at time t0 the slope at point P is f (x0 , t0 ), which is given by the differential equation. The value of x1 at time t1 (point R) is given by x1 = x0 + f (x0 , t0 )t

(4.30)

t = t1 − t0

This process can be repeated for as many steps as one wishes. If f is autonomous, so dx/dt = f (x), then xn = xn−1 + f (xn−1 )t It is clear from ﬁgure 4.32 that point R will deviate from its ‘true’ value at point Q, the larger the step size, given by t. If the step size is reduced, then the approximation is better. 10

See Shone (2001) for a treatment of differential equations with spreadsheets.

(4.31)

184

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Figure 4.32.

The procedure generalises quite readily to systems of equations. Let

(4.32)

(4.33)

dx = f (x, y) dt dy = g(x, y) dt denote a system of autonomous differential equations and let the initial value of the system be x(t0 ) = x0 and y(t0 ) = y0 . Then x1 = x0 + f (x0 , y0 )t y1 = y0 + g(x0 , y0 )t which can be repeated for further (approximate) values on the solution curve. For an autonomous system in which y = φ(x), then such a procedure allows us to plot the trajectory in the phase plane. Example 4.18 Consider example 4.6 given by the differential equations: dx = f (x, y) = −2x − y + 9 dt dy = g(x, y) = −y + x + 3 dt with x(t0 ) = 2 and y(t0 ) = 2. Given these values, and letting t = 0.01, then f (x0 , y0 ) = −2(2) − 2 + 9 = 3 g(x0 , y0 ) = −2 + 2 + 3 = 3

Systems of ﬁrst-order differential equations

185

Hence, x1 = x0 + f (x0 , y0 )t = 2 + 3(0.01) = 2.03 y1 = y0 + g(x0 , y0 )t = 2 + 3(0.01) = 2.03 and f (x1 , y1 ) = −2(2.03) − 2.03 + 9 = 2.91 g(x1 , y1 ) = −2.03 + 2.03 + 3 = 3 giving x2 = x1 + f (x1 , y1 )t = 2.03 + 2.91(0.01) = 2.0591 y2 = y1 + g(x1 , y1 )t = 2.03 + 3(0.01) = 2.06 This process is repeated. But all this can readily be set out on a spreadsheet, as shown in ﬁgure 4.33. The ﬁrst two columns are simply the differential equations. Columns (3) and (4) employ the Euler approximation using relative addresses and the absolute address for t. The x-y plot gives the trajectory of the system in the phase plane, with initial value (x0 , y0 ) = (2, 2). As can be seen from the embedded graph in the spreadsheet, this trajectory is the same as that shown in ﬁgure 4.11 (p. 155) The advantage of using Euler’s approximation, along with a spreadsheet, is that no explicit solution need be obtained – assuming that one exists. By reducing the step size a smoother trajectory results. It is also easy to increase the number of steps.

Figure 4.33.

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Figure 4.34.

Example 4.19 (The Lorenz curve) The Lorenz equations are given by:

(4.34)

dx = σ (y − x) dt dy = rx − y − xz dt dz = xy − bz dt with parameter values σ = 10, r = 28, b = 8/3 and we take a step size of t = 0.01. In this example we take 2,000 steps, however ﬁgure 4.34 only shows the ﬁrst few steps. In ﬁgure 4.35 we have three generated plots, (i) (x, y), (ii) (x, z) and (iii) (y, z). These diagrams illustrate what is referred to as strange attractors, a topic we shall return to when we discuss chaos theory.

4.12 Solving systems of differential equations with Mathematica and Maple Chapter 2 sections 2.11 and 2.12 outlined how to utilise Mathematica and Maple to solve single differential equations. The method for solving systems of such equations is fundamentally the same. Consider the system,

(4.35)

dx = f (x, y, t) dt dy = g(x, y, t) dt

Systems of ﬁrst-order differential equations

187 Figure 4.35.

Then the solution method in each case is: Mathematica DSolve[{x’[t]==f[x[t],y[t],t],y’[t]==g[x[t], y[t],t]}, {x[t],y[t]},t]

Maple dsolve({diff(x(t),t)=f(x(t),y(t),t),diff(y(t),t)=g(x(t), y(t),t)}, {x(t),y(t)});

If initial conditions x(0) = x0 and y(0) = y0 are provided, then the input instructions are:

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Economic Dynamics Mathematica DSolve[{x’[t]==f[x[t],y[t],t],y’[t]==g[x[t],y[t],t], x(0)==x0,y(0)==y0},{x[t],y[t]},t]

Maple dsolve(({diff(x(t),t)=f(x(t),y(t),t),diff(y(t),t) =g(x(t),y(t),t), x(0)=x0,y(0)=y0},{x(t),y(t)});

It is often easier to deﬁne the equations, variables and initial conditions ﬁrst. Not only is it easier to see, but much easier in correcting any mistakes. For instance in Mathematica deﬁne eq:={x’[t]==f[x[t],y[t],t],y’[t]==g[x[t],y[t],t], x[0]==x0,y[0]==y0} var:={x[t],y[t]}

and then solve using DSolve[eq,var,t]

In Maple deﬁne: eq:=diff(x(t),t)=f(x(t),y(t),t),diff(y(t),t)= g(x(t),y(t),t); init:=x(0)=x0,y(0)=y0; var:={x(t),y(t)};

and then solve using dsolve({eq,init},var);

Example 4.4 in the text can then be solved in each package as follows Mathematica eq:={x’[t]==x[t]-3y[t], y’[t]==-2x[t]+y[t], x[0]==4,y[0]==5} var:={x[t],y[t]} DSolve[eq,var,t]

Maple eq:=diff(x(t),t)=x(t)-3*y(t), diff(y(t),t)= -2*x(t)+y(t); init:=x(0)=4,y(0)=5; var:={x(t),y(t)}; dsolve({eq,init},var);

Although the output looks different in the two cases, they are equivalent and identical to that provided in the text.

Systems of ﬁrst-order differential equations

189

So long as solutions exist, then the packages will solve the system of equations. Thus, the system of three equations with initial values: x (t) = x(t) y (t) = x(t) + 3y(t) − z(t) z (t) = 2y(t) + 3x(t) x(0) = 1, y(0) = 1, z(0) = 2

(4.36)

can be solved in a similar manner with no difﬁculty. In the case of nonlinear systems of differential equations, or where no explicit solution can be found, then it is possible to use the NDSolve command in Mathematica and the dsolve(. . . , numeric) command in Maple to obtain numerical approximations to the solutions. These can then be plotted. But often more information can be obtained from direction ﬁeld diagrams and phase portraits. A direction ﬁeld shows a series of small arrows that are tangent vectors to solutions of the system of differential equations. These highlight possible ﬁxed points and most especially the ﬂow of the system over the plane. A phase portrait, on the other hand, is a sample of trajectories (solution curves) for a given system. Figure 4.36(a) shows a direction ﬁeld and ﬁgure 4.36(b) a phase portrait. In many instances direction ﬁelds and phase portraits are combined on the one diagram – as we have done in many diagrams in this chapter. The phase portrait can be derived by solving a system of differential equations, if a solution exists. Where no known solution exists, trajectories can be obtained by using numerical Figure 4.36.

190

Economic Dynamics solutions. These are invariably employed for systems of nonlinear differential equation systems.

4.12.1

Direction ﬁelds and phase portraits with Mathematica

Direction ﬁelds in Mathematica are obtained using the PlotVectorField command. In order to use this command it is ﬁrst necessary to load the PlotField package. There is some skill required in getting the best display of direction ﬁelds using the PlotVectorField command, and the reader should consult the references supplied on using Mathematica in chapter 1. Given the system of differential equations (4.35), then a direction ﬁeld can be obtained with the instructions Needs[``Graphics`PlotField`’’] dfield=PlotVectorField[{f(x,y,t),g(x,y,t)}, {x,xmin,xmax}, {y,ymin,ymax}, DisplayFunction->Identity] Show[dfield, DisplayFunction->$DisplayFunction]

To obtain a ‘good’ display it is often necessary to adjust scaling, change the arrow lengths and change the aspect ratio. All these, and other reﬁnements, are accomplished by optional instructions. Thus, ﬁgure 4.36(a) can be obtained from the following input Needs[``Graphics`PlotField`’’] dfield=PlotVectorField[{1-y,x2 +y2 }, {x,-2,2},{y,-1,3}, Frame->True, PlotPoints->20, DisplayFunction->Identity] Show[dfield, DisplayFunction->$DisplayFunction]

The phase portrait is not straightforward in Mathematica and requires solving the differential equations, either with DSolve command, if an explicit solution can be found, or the NDSolve command for a numerical approximation. If an explicit solution can be found with the DSolve command, then phase portraits can be obtained with the ParametricPlot command on supplying different values for the constants of integration. On the other hand, if a numerical approximation is required, as is often the case with nonlinear systems, then it is necessary to obtain a series of solution curves for different initial conditions. In doing this quite a few other commands of Mathematica are needed. Consider the Van der Pol model, equation (4.28), a simple set of instructions to produce a diagram similar to that of ﬁgure 4.30 is eq1:= {x’[t]==y[t],y’[t]==(1-x[t]^2)y[t]-x[t], x[0]==0.5,y[0]==0.5} eq2:= {x’[t]==y[t],y’[t]==(1-x[t]^2)y[t]-x[t], x[0]==0.5,y[0]==4} var:={x,y} trange:={t,0,20}

Systems of ﬁrst-order differential equations sol1=NDSolve[eq1,var,trange] sol2=NDSolve[eq2,var,trange] graph1=ParametricPlot[Evaluate[{x[t],y[t]} /.sol1], {t,0,20},PlotPoints->500, DisplayFunction->Identity]; graph2=ParametricPlot[Evaluate[{x[t],y[t]} /.sol2], {t,0,20},PlotPoints->500, DisplayFunction->Identity]; Show[{graph1,graph2},AxesLabel->{``x’’,``y’’}, DisplayFunction->$DisplayFunction];

The more trajectories that are required the more cumbersome these instructions become. It is then that available packages, such as the one provided by Schwalbe and Wagon (1996), become useful. For instance, ﬁgure 4.36(b) can be produced using the programme provided by Schwalbe and Wagon with the following set of instructions: PhasePlot[{x’[t]==1-y[t],y’[t]==x[t]^2+y[t],^2}, {x[t],y[t]},{t,0,3},{x,-2,2},{y,-1,3}, InitialValues->{{-2,-1},{-1.75,-1},{-1.5,-1}, {-1,0},{-1,-1},{-0.5,-1},{0,-1},{-1.25,0}, {0.5,-1},{1,-1}}, PlotPoints->500, ShowInitialValues->False, DirectionArrows->False, AspectRatio->1, AxesLabel->{x,y}]

When considering just one trajectory in the phase plane, the simple instructions given above can sufﬁce. For instance, consider the Lorenz curve, given in equation (4.34), with parametric values σ = 10, r = 28, and b = 8/3. We can construct a three-dimensional trajectory from the initial point (x0, y0, z0) = (5, 0, 0) using the following input instructions: eqs:={x’[t]==10(y[t]-x[t]),y’[t]==28x[t]-y[t]-x[t]z[t], z’[t]==x[t]y[t]-(8/3)z[t], x[0]==5,y[0]==0,z[0]==0} var:={x,y,z} lorenzsol=NDSolve[eqs,var,{t,0,30},MaxSteps->3000] lorenzgraph=ParametricPlot3D[ Evaluate[x[t],y[t],z[t]} /.lorenzsol], {t,0,30},PlotPoints->2000,PlotRange->All];

The resulting phase line is shown in ﬁgure 4.37. This goes beyond the possibilities of a spreadsheet, and ﬁgure 4.37 should be compared with the three twodimensional plots given in ﬁgure 4.35.

191

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Economic Dynamics

Figure 4.37.

4.12.2

Direction ﬁelds and phase portraits with Maple

Direction ﬁelds and phase portraits are more straightforward in Maple and use the same basic input commands. Given the system of differential equations (4.35), then a direction ﬁeld can be obtained with the instructions with(DEtools): with(plots): Dfield:=dfieldplot( [diff(f(x(t),t)=f(x(t),y(t),t), diff(y(t),t)=g(x(t),y(t),t)], [x(t),y(t)], t=tmin..tmax, x=xmin..xmax, y=ymin..ymax); display(Dfield);

Notice that the instruction ‘with(plots):’ is required for use of the display command. To obtain a ‘good’ display it is often necessary to add options with respect to arrows. For example, a Maple version of ﬁgure 4.36(a) can be achieved with the following input with(DEtools): with(plots): Dfield:=dfieldplot( [diff(x(t),t)=1-y(t),diff(y(t),t)=x(t)^2 +y(t)^2], [x(t),y(t)], t=0..1, x=-2..2, y=-1..3, arrows=SLIM): display(Dfield);

The phase portrait, not surprisingly, uses the phaseportrait command of Maple. This particular command plots solution curves by means of numerical methods.

Systems of ﬁrst-order differential equations In a two-equation system, the programme will produce a direction ﬁeld plot by default if the system is a set of autonomous equations. Since we require only the solution curves, then we include an option that indicates no arrows. To illustrate the points just made, consider the Van der Pol model, equation (4.28), a simple set of instructions to produce a Maple plot similar to ﬁgure 4.30 is phaseportrait( [D(x)(t)=y(t), D(y)(t)=(1-x(t)^2)*y(t)-x(t)], [x(t),y(t)], t=0..10, [ [x(0)=0.5,y(0)=0.5],[x(0)=0.5,y(0)=4] ], stepsize=.05 linecolour=blue, arrows=none, thickness=1);

Producing more solution curves in Maple is just a simple case of specifying more initial conditions. For instance, a Maple version of ﬁgure 4.36(b) can be produced with the following instructions: with(DEtools): phaseportrait( [D(x)(t)=1-y(t),D(y)(t)=x(t)^2+y(t)^2], [x(t),y(t)], t=0..3, [[x(0)=-2,y(0)=-1],[x(0)=-1.75,y(0)=-1], [x(0)=1.5,y(0)=-1],[x(0)=-1,y(0)=0], [x(0)=-1,y(0)=-1], [x(0)=-0.5,y(0)=-1], [x(0)=0,y(0)=-1], [x(0)=-1.25,y(0)=0], [x(0)=0.5,y(0)=-1, [x(0)=1,y(0)=-1]], x=-2..2, y=-1..3, stepsize=.05, linecolour=blue, arrows=none, thickness=1);

Trajectories for three-dimensional plots are also possible with Maple. Consider once again the Lorenz curve, given in equation (4.34), with parameter values σ = 10, r = 28 and b = 8/3. We can construct a three-dimensional trajectory from the initial point (x0, y0, z0) = (5, 0, 0) using the following input instructions: with(DEtools): DEplot3d( [diff(x(t),t)=10*(y(t)-x(t)), diff(y(t),t)=28*x(t)-y(t)-x(t)*z(t), diff(z(t),t)=x(t)*y(t)-(8/3)*z(t)], [x(t),y(t),z(t)], t=0..30, [[x(0)=5,y(0)=0,z(0)=0]], stepsize=.01, linecolour=BLACK, thickness=1);

193

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Figure 4.38.

The resulting phase line is shown in ﬁgure 4.38. This goes beyond the possibilities of a spreadsheet, and ﬁgure 4.38 should be compared with the three twodimensional plots given in ﬁgure 4.35. It is worth noting that ﬁgure 4.38 is the default plot and the orientation can readily be changed by clicking on the ﬁgure and revolving.

Appendix 4.1 Parametric plots in the phase plane: continuous variables A trajectory or orbit is the path of points {x(t), y(t)} in 2-dimensional space and {x(t), y(t), z(t)} in 3-dimensional space as t varies. Such plots are simply parametric plots as far as computer programmes are concerned. There are two methods for deriving the points (x(t), y(t)) or (x(t), y(t), z(t)): (1) (2)

Solve for these values Derive numerical values by numerical means: (a) by solving numerically, or (b) deriving by recursion.

Method 2(a) is used particularly in the case of differential equations, while method 2(b) is used for difference (or recursive) equations. In each of these cases initial conditions must be supplied.

4A.1 Two-variable case Consider the solution values for x and y in example 4.1, which are x(t) = 2e2t

and

y(t) = 3et

Both x and y are expressed in terms of a common parameter, t, so that when t varies we can establish how x and y vary. More speciﬁcally, if t denotes time, then (x(t), y(t)) denotes a point at time t in the (x,y)-plane, i.e., a Cartesian representation of the parametric point at time t. If the differential equation system which generated x(t) and y(t) is autonomous, then there is only one solution curve, and we can express this in the form y = φ(x), where y0 = φ(x0 ) and (x0 , y0 ) is some initial point, i.e., x(0) = x0 and y(0) = y0 at t = 0. In the present example this is readily

Systems of ﬁrst-order differential equations found since 2e2t 2e2t 2 x = = = 2 t 2 2t y (3e ) 9e 9 Hence

y=

9x 2

Whether or not it is possible to readily ﬁnd a Cartesian representation of the parametric curve, it is a simple matter to plot the parametric curve itself using software packages. Example 4.1 with Mathematica The two commands used in this set of instructions, DSolve and ParametricPlot are now both contained in the main package:11 Clear[x,y] sol=DSolve[{x’[t]==2x[t],y’[t]==y[t],x[0]==2, y[0]==3}, {x[t],y[t]},t] solx=sol[[1,1,2]] soly=sol[[1,2,2]] x[t-]:=solx y[t-]:=soly traj=ParametricPlot[{x[t],y[t]},{t,0,1}]

If the equations for x(t) and y(t) are already known, then only the last instruction need be given. For example, if it is known that x(t) = 2e2t and y(t) = 3et then all that is required is traj=ParametricPlot[{2e2t ,3et },{t,0,1}]

Example 4.1 with Maple To use Maple’s routine for plotting parametric equations that are solutions to differential equations it is necessary to load the plots package ﬁrst. The following input instructions will produce the trajectory for example 4.1: restart; with(plots): sys:={diff(x(t),t)=2*x(t),diff(y(t),t)=y(t), x(0)=2,y(0)=3} vars:={x(t),y(t)}: sol:=dsolve(sys,vars,numeric); odeplot(sol,[x(t),y(t)],0..1,labels=[x,y]);

11

In earlier versions, DSolve and ParametricPlot needed to be loaded ﬁrst since these were contained in the additional packages. This is no longer necessary, since both are contained in the basic built in functions.

195

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Economic Dynamics Notice that we have placed a semi-colon after the ‘sol’ instruction so that you can observe that Maple produces a procedural output, which is then used in the odeplot. If the equations for x(t) and y(t) are already known, then the plot command can be used. For example, if it is known that x(t) = 2e2t and y(t) = 3et then all that is required is plot([2*exp(2*t),3*exp(t),t=0..1],labels=[x,y]);

4A.2 Three-variable case Plotting trajectories in 3-dimensional phase space is fundamentally the same, with just a few changes to the commands used. Equation (4.36) with Mathematica The input instructions are Clear[x,y,z] sol=DSolve[{x’[t]==x[t],y’[t]==x[t]+3y[t]-z[t], z[t]==2y[t]+3x[t],x[0]==1,y[0]==1,z[0]==2}, {x[t],y[t],z[t]},t] solx=sol[[1,1,2]] soly=sol[[1,2,2]] solz=sol[[1,3,2]] x[t-]:=solx y[t-]:=soly z[t-]:=solz traj=ParametricPlot3D[{x[t],y[t],z[t]},{t,0,5}]

If the equations for x(t), y(t) and z(t) are already known, then only the last instruction need be given. For example, if it is known that x(t) = et , y(t) = 2et − e2t + 2tet and z(t) = 4tet − e2t + 3et then all that is required is traj=ParametricPlot3D[{et ,2et -e2t +2tet ,4tet -e2t +3et }, {t,0,5}]

Equation (4.36) with Maple The input instructions are restart; with(plots): sys:={diff(x(t),t)=x(t), diff(y(t),t)=x(t)+3*y(t)-z(t), diff(z(t),t)=2*y(t)+3*x(t),x(0)=1,y(0)=1,z(0)=2}; vars:={x(t),y(t),z(t)}: sol:=dsolve(sys,vars,numeric); odeplot(sol,[x(t),y(t),z(t)],0..5,labels=[x,y,z]);

Systems of ﬁrst-order differential equations If the equations for x(t), y(t) and z(t) are already known, then we use the spacecurve command, as illustrated in the following instructions: traj=spacecurve([exp(t),2*exp(t)-exp(2*t)+2*t*exp(t), 4*t*exp(t)-exp(2*t)+3*exp(t)], t=0..5,labels=[x,y,z]);

Exercises 1.

2.

(i) Show that y y (x) = 2x is a separable function, and solve assuming x(0) = 2 and y(0) = 3. (ii) Verify your result using either Mathematica or Maple. For the system x˙ = x − 3y y˙ = −2x + y use a software package to derive the trajectories of the system for the following initial values: (a) (b) (c) (d)

3.

(x0 , y0 ) = (4, 2) (x0 , y0 ) = (4, 5) (x0 , y0 ) = (−4, −2) (x0 , y0 ) = (−4, 5)

For the system x˙ = −3x + y y˙ = x − 3y (i) Show that points (x0 , y0 ) = (4, 8) and (x0 , y0 ) = (4, 2) remain in quadrant I, as in ﬁgure 4.9. (ii) Show that points (x0 , y0 ) = (−4, −8) and (x0 , y0 ) = (−4, −2) remain in quadrant III, as in ﬁgure 4.9. (iii) Show that points (x0 , y0 ) = (2, 10) and (x0 , y0 ) = (−2, −10) pass from one quadrant into another before converging on equilibrium. (iv) Does the initial point (x0 , y0 ) = (2, −5) have a trajectory which converges on the ﬁxed point without passing into another quadrant?

4.

For the system x˙ = −2x − y + 9 y˙ = −y + x + 3

5.

establish the trajectories for each of the following initial points (i) (x0 , y0 ) = (1, 3), (ii) (x0 , y0 ) = (2, 8), and (iii) (x0 , y0 ) = (3, 1), showing that all trajectories follow a counter-clockwise spiral towards the ﬁxed point. Given the dynamic system x˙ = 2x + 3y y˙ = 3x + 2y

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Economic Dynamics (i) Show that the characteristic roots of the system are r = 5 and s = −1. (ii) Derive the eigenvectors associated with the eigenvalues obtained in (i). (iii) Show that the solution values are: x(t) = c1 e5t + c2 e−t y(t) = c1 e5t − c2 e−t and verify that 1 1 c1 e5t , c2 e−t 1 −1 are linearly independent. (iv) Given x(0) = 1 and y(0) = 0, show that x(t) = 12 e5t + 12 e−t y(t) = 12 e5t − 12 e−t 6.

For the dynamic system x˙ = x + 3y y˙ = 5x + 3y Show: (i) that the two eigenvalues are r = 6 and s = −2 (ii) that the two eigenvectors are 1 1 and vs = vr = 5/3 −1 (iii) and that the general solution satisfying x(0) = 1 and y(0) = 3 is x(t) = 32 e6t − 12 e−2t

7.

y(t) = 52 e6t + 12 e−2t # " Let V = v1 v2 denote a matrix formed from the eigenvectors. Thus, if 1 1 1 2 and v = v = −2 2 then V=

1 −2

1 2

The determinant of this matrix is called the Wronksian, i.e., W(v1 , v2 ) = det(V). Then v1 and v2 are linearly independent if and only if W(v1 , v2 ) is nonzero. Show that for the system x˙ = x + y y˙ = −2x + 4y the Wronksian is nonzero.

Systems of ﬁrst-order differential equations 8.

Given x˙ = x y˙ = 2x + 3y + z z˙ = 2y + 4z (i) Find the eigenvalues and eigenvectors. (ii) Provide the general solution. (iii) Show that the Wronksian is nonzero.

9.

For each of the following systems (a) ﬁnd the eigenvalues and eigenvectors; (b) solve the system by ﬁnding the general solution; (c) obtain the trajectories for the speciﬁed initial points; and (d) classify the ﬁxed points. (i)

x˙ = −3x + y y˙ = x − 3y

initial points = (1, 1), (−1, 1), (−1, −1), (1, −1), (2, 0), (3, 1), (1, 3) (ii)

x˙ = 2x − 4y y˙ = x − 3y

initial points = (1, 1), (−1, 1), (4, 1), (−4, −1), (0, 1), (0, −1), (3, 2), (−3, −2) (iii)

x˙ = y y˙ = −4x

initial points = (0, 1), (0, 2), (0, 3) (iv)

10.

x˙ = −x + y y˙ = −x − y

initial points = (1, 0), (2, 0), (3, 0), (−1, 0), (−2, 0), (−3, 0). For the following Holling–Tanner predatory–prey model 6xy x − x˙ = x 1 − 6 (8 + 8x) 0.4y y˙ = 0.2y 1 − x (i) Find the ﬁxed points. (ii) Do any of the ﬁxed points exhibit a stable limit cycle?

11.

Consider the R¨ossler attractor x˙ = −y − z y˙ = x + 0.2y z˙ = 0.2 + z(x − 2.5) (i) Show that this system has a period-one limit cycle. (ii) Plot x(t) against t = 200 to 300, and hence show that the system settles down with x having two distinct amplitudes.

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Economic Dynamics 12.

Consider the following Walrasian price and quantity adjustment model φ (Y) = 0.5 + 0.25Y D(p) = −0.025p3 + 0.75p2 − 6p + 40 p˙ = 0.75[D(p, φ(Y)) − Y] Y˙ = 2[p − φ (Y)] (i) What is the economically meaningful ﬁxed point of this system? (ii) Does this system have a stable limit cycle?

13.

Reconsider the system in question 12, but let the quantity adjustment equation be given by Y˙ = β[p − φ (Y)]

14.

Let β = 2, 2.5, 3 and 3.2. What do you conclude about the long-run behaviour of this system? Consider the following system φ (Y) = 0.5 + 0.25Y D(p) = −0.025p3 + 0.75p2 − 6p + 40 p˙ = α[D(p, φ(Y)) − Y] Y˙ = 2[p − φ (Y)]

15.

Let α = 0.5, 0.75 and 1. What do you conclude about the long-run behaviour of this system? Set up the R¨ossler attractor x˙ = −y − z y˙ = x + ay z˙ = b + z(x − c) on a spreadsheet with step size t = 0.01 and a = 0.4, b = 2 and c = 4. Plot the system for initial point (x, y, z) = (0.1, 0.1, 0.1) in (i) (x,y)-plane (ii) (x,z)-plane (iii) (y,z)-plane Additional reading

Additional material on the contents of this chapter can be obtained from Arrowsmith and Place (1992), Beavis and Dobbs (1990), Borrelli et al. (1992), Boyce and DiPrima (1997), Braun (1983), Chiang (1984), Flaschel et al. (1997), Giordano and Weir (1991), Jeffrey (1990), Lynch (2001), Mas-Colell (1986), Percival and Richards (1982), Schwalbe and Wagon (1996), Shone (2001) and Tu (1994).

CHAPTER 5

Discrete systems of equations

5.1 Introduction In chapter 3 we considered linear difference equations for a single variable, such as xt = 2xt−1 ,

xt = 4xt−1 + 4xt−2 ,

xt = axt−1 + b

Each of these equations is linear and autonomous. But suppose we are interested in such systems as the following: (i) (ii) (iii)

xt = axt−1 + byt−1 yt = cxt−1 + dyt−1 xt = 4xt−1 + 2 yt = −2yt−1 − 3xt−1 + 3 xt = 2xt−1 + 3yt−1 + 4zt−1 yt = 0.5xt−1 zt = 0.7yt−1

All these are examples of systems of linear autonomous equations of the ﬁrst order. As in previous chapters, we shall here consider only autonomous equations (i.e. independent of the variable t), but we shall also largely restrict ourselves to linear systems. If all the equations in the system are linear and homogeneous, then we have a linear homogeneous system. If the system is a set of linear equations and at least one equation is nonhomogeneous, then we have a linear nonhomogeneous system. If the equations are homogeneous but at least one equation in the system is nonlinear, then we have a nonlinear homogeneous system. If at least one equation is nonlinear and at least one equation in the system is nonhomogeneous, then we have a nonlinear nonhomogeneous system. In this chapter we shall concentrate on linear homogeneous equation systems. In terms of the classiﬁcation just given, systems (i) and (iii) are linear homogeneous systems, while (ii) is a linear nonhomogeneous system. A more convenient way to express linear systems is in matrix form. Hence the three systems can equally be written in the form: a b xt−1 xt = (or ut = Aut−1 ) (i) yt yt−1 c d

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Economic Dynamics

(ii)

(iii)

4 0 xt−1 0 = + (or ut = Aut−1 + b) yt−1 −3 −2 3 2 3 4 xt−1 xt yt = 0.5 0 0 yt−1 (or ut = Aut−1 ) zt zt−1 0 0.7 0 xt yt

In general, therefore, we can write ﬁrst-order linear homogeneous systems as ut = Aut−1

(5.1)

and a ﬁrst-order linear nonhomogeneous system as ut = Aut−1 + b

(5.2)

where u is a n × 1 vector, A a n × n square matrix and b a n × 1 vector. Consider the system xt a b xt−1 = yt yt−1 c d Then in equilibrium xt = xt−1 = x∗ for all t and yt = yt−1 = y∗ for all t. Hence ∗ ∗ a b x x = y∗ y∗ c d or u∗ = Au∗ An equilibrium solution exists, therefore, if u∗ − Au∗ = 0 i.e. (I − A)u∗ = 0 or u∗ = (I − A)−1 0 = 0 An equilibrium for a ﬁrst-order linear homogeneous system is, therefore, u∗ = 0. This is a general result. For a ﬁrst-order linear nonhomogeneous system ut = Aut−1 + b an equilibrium requires ut = ut−1 = u∗ for all t, so that u∗ = Au∗ + b (I − A)u∗ = b u∗ = (I − A)−1 b and so an equilibrium exists so long as (I − A)−1 exists. The solution u∗ = (I − A)−1 b is the general equilibrium solution for a ﬁrst-order linear nonhomogeneous system.

Discrete systems of equations Example 5.1 xt = 2xt−1 + 3yt−1 yt = −2xt−1 + yt−1 or

xt yt

=

2 −2

3 1

xt−1 yt−1

i.e. ut = Aut−1 where

−3 0 −1 and (I − A)u∗ = 2

−1 I−A= 2

−3 0

x∗ y∗

=

0 0

the only values for x and y satisfying this system are x∗ = 0 and y∗ = 0. Example 5.2 xt = 4xt−1 + 2 yt = −2yt−1 − 3xt−1 + 3 i.e.

xt yt

4 = −3

0 −2

xt−1 2 + yt−1 3

Then u∗ = (I − A)−1 b −1 −3 0 2 −2/3 = = 3 3 3 5/3 i.e. x∗ = −2/3 and y∗ = 5/3. Having established that an equilibrium exists, however, our main interest is establishing the stability of such systems of equations. In establishing this we need to solve the system. This is fairly straightforward. For the ﬁrst-order linear homogeneous equation system we have ut = Aut−1 = A(Aut−2 ) = A2 ut−2 = A2 (Aut−3 ) = A3 ut−3 .. .

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Economic Dynamics with solution

(5.3)

ut = At u0 where u0 is the initial values of the vector u. Given u0 and the matrix A, then we could compute u100 = A100 u0 , or any such time period. Similarly, with the ﬁrst-order nonhomogeneous linear equation system we have ut = Aut−1 + b = A(Aut−2 + b) + b = A2 ut−2 + Ab + b = A2 (Aut−3 + b) + Ab + b = A3 ut−3 + A2 b + Ab + b .. . with solution

(5.4)

ut = At u0 + (I + A + A2 + . . . + At−1 )b Although solution (5.3) and (5.4) are possible to solve with powerful computers, it is not a useful way to proceed. We require to approach the solution from a different perspective. It will be recalled from our analysis of differential equation systems in chapter 4 that a linear nonhomogeneous system can be reduced to a linear homogeneous system by considering deviations from equilibrium. Thus for ut = Aut−1 + b, with equilibrium vector u∗ we have u∗ = Au∗ + b. Subtracting we obtain ut − u∗ = A(ut−1 − u∗ ) or zt = Azt−1 which is a linear ﬁrst-order homogeneous system. In what follows, therefore, we shall concentrate more on linear homogeneous systems with no major loss.

5.2 Basic matrices with Mathematica and Maple Both Mathematica and Maple have extensive facilities for dealing with matrices and matrix algebra. The intention in this section is to supply just the briefest introduction so that the reader can use the packages for the matrix manipulations required in this book. It is assumed that the reader is familiar with matrix algebra. Both programmes treat matrices as a list of lists – a vector is just a single list. While most of the basic matrix manipulations are built into Mathematica, it is necessary to load one or even two packages in Maple. The two packages are (1) linalg and (2) LinearAlgebra, and are loaded with the instructions: with(linalg): with(LinearAlgebra):

The lists in Mathematica use curly braces, while those in Maple use straight (table 5.1). Both programmes have palettes that speed up the entry of vectors and matrices, although Mathematica’s is far more extensive than that of Maple.

Discrete systems of equations Table 5.1 Representations of matrices in Mathematica and Maple Mathematica

Maple

Vector

{a, b, c}

[a, b, c]

Matrix

{{a, b}, {c, d}}

[ [a, b], [c, d] ]

5.2.1

Matrices in Mathematica

To illustrate Mathematica’s package, let mA =

Conventional representation a [a, b, c] or b c a b c d

3 2 1 −2

4 , −3

0 −1 1 mB = , 2 3 0

2 1 mC = −1 0 2 3

then in Mathematica use: mA={{3,2,4},{1,-2,-3}} mB={{0,-1,1},{2,3,0}} mC={{2,1},{-1,0},{2,3}} mA+mB (to add) mA-mB (to subtract) mA.mC (to multiply)

Notice that mA cannot be multiplied by mB. Any such attempt leads to an error message indicating that the matrices have incompatible shapes. Square matrices have special properties. For illustrative purposes, let 2 1 −1 0 2 mA = 3 −1 2 1 Typical properties are shown in table 5.2. A special square matrix is the identity matrix. To specify a 3 × 3 identity matrix in Mathematica one uses Identity Matrix[3]. To construct the characteristic polynomial for the above square matrix, then we use1 mA-λIdentityMatrix[3]

and the characteristic equation is obtained using Det[mA-λIdentityMatrix[3]]==0

which in turn can be solved using Solve[Det[mA-λIdentityMatrix[3]]==0] 1

The characteristic polynomial can be obtained directly with the command CharacteristicPolynomial[mA].

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Economic Dynamics Table 5.2 Properties of matrices and Mathematica input Property

Mathematica input

Trace Transpose Inverse Determinant Eigenvalues Eigenvectors Characteristic polynomial Matrix Power (power n)

Tr[mA] Transpose[mA] Inverse[mA] Det[mA] Eigenvalues[mA] or Eigenvalues[N[mA]] Eigenvectors[mA] or Eigenvectors[N[mA]] CharacteristicPolynomial[mA,λ] MatrixPower[mA,n]

or Solve[N[Det[mA-λIdentityMatrix[3]]]==0]

As one gets familiar with the package, long strings of instructions can be entered as a single instruction, as in the ﬁnal solve. To verify the results of example 4.12 in chapter 4, input the following, where we have added the instruction ‘// MatrixForm’ to display the matrix in more familiar form mA={{-2,1},{1,-2}} Det[mA] mA-λIdentityMatrix[2] //MatrixForm Eigenvalues[mA] Eigenvectors[mA]

All results are indeed veriﬁed.

5.2.2

Matrices in Maple

To illustrate Maple’s package, let mA =

3 1

2 −2

4 , −3

mB =

0 2

−1 3

1 , 0

2 1 mC = −1 0 2 3

then in Maple use: mA:=matrix([[3,2,4],[1,-2,-3]]); mB:=matrix([[0,-1,1],[2,3,0]]); mC:=matrix([[2,1],[-1,0],[2,3]]); evalm(mA+mB) (to add) evalm(mA-mB) (to subtract) evalm(mA&*mC) (to multiply)

Notice that mA cannot be multiplied by mB. Any such attempt leads to an error message indicating that the matrices have non-matching dimensions.

Discrete systems of equations Table 5.3 Properties of matrices and Maple input Property

Maple input

Trace Transpose Inverse Determinant Eigenvalues Eigenvectors Characteristic polynomial Matrix Power (power n)

trace(mA); transpose(mA); inverse(mA); det(mA); eigenvals(mA); or evalf(eigenvals(mA)); eigenvects(mA); or evalf(eigenvects(mA)); charpoly(mA,’lambda’); evalm(mA^n)

Square matrices have special properties. For illustrative purposes, let 2 1 −1 mA = 3 0 2 −1 2 1 Typical properties are shown in table 5.3. The characteristic polynomial in Maple simply requires the input charpoly(mA,’lambda’);

which in turn can be solved using solve(charpoly(mA,’lambda’)=0);

or fsolve(charpoly(mA,’lambda’)=0,lambda,complex);

As one gets familiar with the package, long strings of instructions can be entered as a single instruction, as in the ﬁnal fsolve. Notice too that the ﬁnal fsolve required the option ‘complex’ to list all solutions. Using fsolve(charpoly(mA,’lambda’)=0);

gives only the real solution. To verify the results of example 4.12 in chapter 4, input the following: with(linalg): with(LinearAlgebra): mA:=matrix([[-2,1],[1,-2]]); det(mA); evalm(mA-lambda*IdentityMatrix(2)); eigenvals(mA); eigenvects(mA);

All results are indeed veriﬁed, when it is realised that vr = [ 1 −1 ] is fundamentally the same as vr = [ −1 1 ].

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5.3 Eigenvalues and eigenvectors Let us concentrate on the ﬁrst-order linear homogeneous equation system

xt yt

a = c

b d

xt−1 yt−1

or ut = Aut−1 with solution ut = At u0 . We invoke the following theorem. THEOREM 5.1 If the eigenvalues of the matrix A are r and s obtained from |A − λI| = 0 such that r = s, then there exists a matrix V = [ vr vs ] composed of the eigenvectors associated with r and s, respectively, such that D=

0 = V−1 AV s

r 0

We shall illustrate this theorem by means of an example.

Example 5.3 Let A=

2 1

1 2

The characteristic equation is |A − λI| = 0, i.e. 2 − λ 1

1 = (2 − λ)2 − 1 = λ2 − 4λ + 3 = 0 2 − λ

Hence, λ = r = 1 and λ = s = 3. For λ = r = 1 we have the equation (A − rI)vr = 0 or

i.e.

2 1

r 0 1 1 0 v1 = − vr2 0 2 0 1 r 1 1 v1 0 = r 1 1 v2 0

Hence vr1 + vr2 = 0. Let vr1 = 1 then vr2 = −vr1 = −1. Thus,

1 v = −1 r

Discrete systems of equations

209

For λ = s = 3 we have s 2 1 3 0 v1 0 − = 1 2 0 3 vs2 0 s −1 1 v1 0 i.e. = 1 −1 vs2 0 Hence, −vs1 + vs2 = 0. Let vs1 = 1, then vs2 = vs1 = 1. Thus, the second eigenvector is 1 vs = 1 Our matrix, V, is therefore "

V= v

r

s

v

#

1 = −1

1 1

From the theorem we have D = V−1 AV, i.e. −1 1 1 2 1 1 1 1 −1 = V AV = −1 1 1 2 −1 1 0

0 3

which is indeed the matrix D formed from the characteristic roots of A. Since D = V−1 AV then VDV−1 = V(V−1 AV)V−1 = A Furthermore A2 = (VDV−1 )(VDV−1 ) = VD2 V−1 A3 = (VDV−1 )(VD2 V−1 ) = VD3 V−1 .. . At = (VDV−1 )(VDt−1 V−1 ) = VDt V−1 Hence ut = At u0 = VDt V−1 u0 or

t r ut = V 0

0 −1 V u0 st

We can summarise the procedure as follows: (1)

(2)

(3)

Given a ﬁrst-order linear homogeneous equation system ut = Aut−1 , where u is a 2 × 1 vector and A is a 2 × 2 matrix, with solution ut = At u0 , obtain the eigenvectors r and s (assumed to be distinct). Derive the eigenvector vr associated with the eigenvalue r and the eigenvector vs associated with the eigenvalue s, and form the matrix V = [vr , vs ]. From (2) we have the general solution ut = art vr + bst vs

(5.5)

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Economic Dynamics and we can ﬁnd a and b for t = 0 from u0 = avr + bvs where u0 is known. This can either be done by direct substitution, or using the fact that # a " r r s s = u0 av + bv = v v b a i.e. V = u0 b a or = V−1 u0 b (4)

Write the solution ut = art vr + bst vs

But we can do this whole process in one step. First we note # rt 0 " r a t r t s s ut = ar v + bs v = v v b 0 st a = VDt b But

a = V−1 u0 b

Hence ut = VDt V−1 u0 which is the result we proved above. The gain, if there is one, in doing the four steps is the need to solve for a and b. Since this can often be done by direct substitution, then the four steps involve no inverse matrix computation. Example 5.4 Let xt+1 = −8 − xt + yt yt+1 = 4 − 0.3xt + 0.9yt setting xt+1 = xt = x∗ and yt+1 = yt = y∗ for all t, the ﬁxed point is readily shown to be (x∗ , y∗ ) = (6.4, 20.8). Now consider the system in terms of deviations from equilibrium, then xt+1 − x∗ = −(xt − x∗ ) + ( yt − y∗ ) yt+1 − y∗ = −0.3(xt − x∗ ) + 0.9( yt − y∗ ) or ut = Aut−1

Discrete systems of equations where

−1 A= −0.3

1 0.9

Solving for the eigenvalues from −1 − λ 1 A − λI = −0.3 0.9 − λ we have |A − λI| = −(1 + λ)(0.9 − λ) + 0.3 = λ2 + 0.1λ − 0.6 = 0 giving r = 0.7262 and s = −0.8262. Given r = 0.7262 then (A − 0.7262I)vr = 0 so r −1.7262 1 v1 0 = −0.3 0.1738 vr2 0 i.e. −1.7262vr1 + vr2 = 0 −0.3vr1 + 0.1738vr2 = 0 Let vr2 = 1 then vr1 = 0.5793. For s = −0.8262 s −0.1738 1 v1 0 = vs2 −0.3 1.7262 0 i.e. −0.1738vs1 + vs2 = 0 −0.3vs1 + 1.7262vs2 = 0 Let vs2 = 1 then vs1 = 5.7537. Hence 0.5793 5.7537 V = [ vr vs ] = 1 1 and

(0.7262)t ut = V 0

0 V−1 u0 (−0.8262)t

Suppose x0 = 2 and y0 = 8, i.e. −4.4 u0 = −12.8 Then

xt+1 − x∗ yt+1 − y∗

0.5793 = 1

0.5793 1

5.7537 1 5.7537 1

(0.7262)t 0

−1

−4.4 −12.8

0 (−0.8262)t

211

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Economic Dynamics

Figure 5.1.

i.e. xt+1 − x∗ = −7.7526(0.7262)t + 3.3526(−0.8262)t yt+1 − y∗ = −13.3827(0.7262)t + 5.827(−0.8262)t This procedure does give insight into the dynamics and it is possible to plot the solutions. However, if interest is purely in the dynamics of the trajectory, this can be obtained immediately using a spreadsheet. Once the equations for xt+1 and yt+1 have been entered in the ﬁrst cells they are then copied down for as many periods as is necessary and the {xt ,yt } coordinates plotted on the x-y line plot, as shown in ﬁgure 5.1. This simple procedure also allows plots of x(t) and y(t) against time.2 The solution generalises to more than two equations. If A is a 3 × 3 matrix with distinct roots q, r and s, then the solution is t q 0 0 ut = V 0 rt 0 V−1 u0 0 0 st # " here V = vq vr vs . Example 5.5 3 In this example we shall also illustrate how Mathematica or Maple can be employed as an aid. Let xt−1 1 2 1 xt yt = −1 1 0 yt−1 3 −6 −1 zt zt−1

2 3

See Shone (2001) and section 5.5 below. Adapted from Sandefur (1990, chapter 6).

Discrete systems of equations Then

1−λ 2 A − λI = −1 1 − λ 3 −6

1 0 −1 − λ

Within Mathematica carry out the following instructions, where we have replaced λ by a m = {{1-a,2,1}, {-1,1-a,0}, {3,-6,-1-a}} sols = Solve[ Det[m]==0, a]

or in Maple m:=matrix( [ [1-a,2,1], [-1,1-a,0], [3,-6,-1-a] ] ); sols:=solve(det(m)=0,a);

which gives the three eigenvalues4 q = 0, r = −1 and s = 2. The next task is to obtain the associated eigenvectors. For q = 0, then q 0 1 2 1 v1 q r (A − 0I)v = −1 1 0 v2 = 0 q v3 0 3 −6 −1 which leads to the equations q

q

q

q

q

v1 + 2v2 + v3 = 0 −v1 + v2 = 0 q 3v1

−

q 6v2

q

− v3 = 0

We can solve this system within Mathematica with the instruction Solve[ {x+2y+z==0, -x+y==0, 3x-6y-z==0}, {x,y,z}]

or in Maple with the instruction solve( {x+2*y+z=0, -x+y=0, 3*x-6*y-z=0}, {x,y,z});

which provides solutions x = 1, y = 1 and z = −3, where x is set arbitrarily at unity. Carrying out exactly the procedure for r = −1 and s = 2 we obtain the results r = −1

implies

x = 2, y = 1 and z = −6

s=2

implies

x = 1, y = −1 and z = 3

Hence our three eigenvectors and the matrix V are: 1 2 1 vq = 1 , vr = 1 , vs = −1 , −3 −6 3

4

1 V= 1 −3

2 1 −6

1 −1 3

This could be obtained directly using the command Eigenvalues[m] in Mathematica or eigenvals(m) in Maple.

213

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Economic Dynamics Hence our solution is t (0) ut = V 0 0

0 (−1)t 0

0 0 V−1 u0 (2)t

Suppose

3 u0 = −4 3 Then

t −1 xt 1 2 1 0 1 2 1 3 0 0 yt = 1 1 −1 0 (−1)t 0 1 1 −1 −4 zt −3 −6 3 0 0 2t −3 −6 3 3 i.e. xt = 6(−1)t + 2(2t ) yt = 3(−1)t − 2(2t ) zt = −18(−1)t + 6(2t ) or 2 1 xt yt = 3(−1)t 1 + 2t+1 −1 −6 3 zt

5.4 Mathematica and Maple for solving discrete systems Mathematica and Maple can be used in a variety of ways in helping to solve systems of discrete equations. Here we consider two: (i) (ii)

Solving directly using the RSolve/rsolve command. Solving using the Jordan form.

5.4.1

Solving directly

Mathematica’s RSolve command and Maple’s rsolve command can each handle systems of linear difference equations besides a single difference equation. In each case the procedure is similar to that outlined in chapter 3, section 3.13. Suppose we wish to solve example 5.4 with initial condition (x0 , y0 ) = (2, 8), i.e., the system xt+1 = −8 − xt + yt yt+1 = 4 − 0.3xt + 0.9yt x0 = 2,

y0 = 8

Discrete systems of equations then the instructions in each case are: Mathematica equ={x[t+1]==-8-x[t]+y[t], y[t+1]==4-0.3x[t]+0.9y[t], x[0]==2, y[0]==8} var={x[t],y[t]} RSolve[equ,var,t]

Maple equ:=x(t+1)=-8-x(t)+y(t), y(t+1)=4-0.3*x(t)+0.9*y(t); init:=x(0)=2, y(0)=8; var:={x(t),y(t)}; rsolve({equ,init},var);

The output from each programme looks, on the face of it, quite different – even after using the evalf command in Maple to convert the answer to ﬂoating point arithmetic. Maple gives a single solution to both x(t) and y(t). Mathematica, however, gives a whole series of possible solutions depending on the value of t being greater than or equal to 1, 2 and 3, respectively, and further additional conditional statements. In economics, with t representing time, the value of t must be the same for all variables. This means we can ignore the additional conditional statements. What it does mean, however, is that only for t ≥ 3 will the solution for x(t) provided by Mathematica and Maple coincide; while y(t) will coincide for t ≥ 2. This should act as a warning to be careful in interpreting the output provided by these packages. Turning to the three-equation system (example 5.5) with initial condition (x0 , y0 , z0 ) = (3, −4, 3) xt = xt−1 + 2yt−1 + zt−1 yt = −xt−1 + yt−1 zt = 3xt−1 − 6yt−1 − zt−1 x0 = 3,

y0 = −4,

z0 = 3

then we would enter the following commands in each programme: Mathematica equ={x[t]==x[t-1]+2y[t-1]+z[t-1], y[t]==-x[t-1]+y[t-1], z[t]==3x[t-1]-6y[t-1]-z[t-1] x[0]==3, y[0]==-4,z[0]==3} var={x[t],y[t],z[t]} RSolve[equ,var,t]

Maple equ:=x(t)=x(t-1)+2*y(t-1)+z(t-1), y(t)=-x(t-1)+y(t-1), z(t)=3*x(t-1)-6*y(t-1)-z(t-1); init:=x(0)=3, y(0)=-4, z(0)=3; var:={x(t),y(t),z(t)}; rsolve({equ,init},var);

215

216

Economic Dynamics In this instance the output in both programmes is almost identical. Mathematica, however, qualiﬁes the solution for z[t] by adding 15 If[t==0,1,0]. If t is time, then this will not occur, and so this conditional statement can be ignored, in which case the two programmes give the same solution – which is also the one provided on p. 214. 5.4.2

Solving using the Jordan form

In section 5.3 we found the eigenvalues of the matrix A and used these to ﬁnd the matrix V formed from the set of linearly independent eigenvectors of A. The diagonal matrix (5.6)

J = diag(λ1 , . . . , λn ) is the Jordan form of A and V is the transition matrix, such that

(5.7)

V−1 AV = J From this result we have At = VJt V−1

(5.8)

and since the solution to the system ut = Aut−1 is ut = At u0 , then t λ1 0 · · · 0 0 λt2 · · · 0 t −1 t ut = VJ V u0 where J = . .. .. .. . . 0 0 · · · λtn So our only problem is to ﬁnd the matrices J and V. Both Mathematica and Maple have commands to supply these matrices directly. In Mathematica one uses the command JordanDecomposition[mA]; while in Maple it is necessary to ﬁrst load the linalg package, and then to use the command jordan(mA, 'V'), where mA denotes the matrix under investigation and V is the transition matrix. To illustrate how to use these commands consider example 5.3, where 2 1 mA = 1 2 Mathematica mA={{2,1},{1,2}} {V,J}=JordanDecomposition[mA] MatrixForm /@ {V,J} MatrixForm[[Inverse[V].mA.V]]

Maple with(linalg): mA:=matrix( [ [2,1],[1,2] ]); J:=jordan(mA,’V’); print(V); evalm(V^(-1)&*mA&*V);

Discrete systems of equations In each of these instructions the last line is a check that undertaking the matrix multiplication does indeed lead to the Jordan form of the matrix. In each package we get the Jordan form 1 0 J= 0 3 However, the transition matrix in each package on the face of it looks different. More speciﬁcally, −1 1 Mathematica V = 1 1

1 1 Maple

V=

2

− 12

2 1 2

But these are fundamentally the same. We noted this when deriving the eigenvectors in the previous section. We arbitrarily chose values for vr1 or vr2 (along with the values associated with the eigenvalue s). In Maple, consider the ﬁrst column, which is the ﬁrst eigenvector. Setting vr2 = 1, means multiplying the ﬁrst term by −2, which gives a value for vr1 = −1. Similarly, setting vs1 = 1 in Maple, converts vs2 also to the value of unity. Hence, the two matrices are identical. In each case the last instruction veriﬁes that V−1 AV = J. Using Maple veriﬁes all the results in section 5.3. However, Mathematica seems to give inconsistent results for a number of the problems. In particular, it appears the transition matrices provided by Mathematica for examples 5.4, 5.6 and 5.7 are not correct. This shows up with the last instruction, since for these examples MatrixForm[Inverse[V].ma.V] does not give the matrix J! It should be noted that all the examples in section 5.3 involve real and distinct roots. Even in the case of complex roots, these are distinct. A more general theorem than Theorem 5.1 is the following: THEOREM 5.2 If A is a n × n square matrix with distinct eigenvalues λ1 , . . . , λn , then the matrix A is diagonalisable, such that V−1 AV = J and J = diag(λ1 , . . . , λn ). Since λ1 , . . . , λn are distinct eigenvalues of the matrix A, then it is possible to ﬁnd n linearly independent eigenvectors v1 , . . . , vn to form the transition matrix V. Systems that have repeated roots involve linear dependence. Such systems involve properties of Jordan blocks, which is beyond the scope of this book. However, a complete study of the stability of discrete systems would require an understanding of Jordan blocks, see Elaydi (1996) and Simon and Blume (1994). When the matrix A has repeated roots, then it is not diagonalisable. It is, however, possible to ﬁnd an ‘almost diagonalisable’ matrix which helps in solving systems with repeated roots. As indicated in the previous paragraph, for a general system

217

218

Economic Dynamics of n equations, this requires knowledge of Jordan blocks. Here we shall simply state a result for a 2 × 2 system. THEOREM 5.3 If A is a 2 × 2 matrix, then there is a transition matrix V such that

(a) (b) (c)

r 0 for real distinct roots r and s V AV = J1 = 0 s λ 1 V−1 AV = J2 = for repeated root λ 0 λ α + βi 0 V−1 AV = J3 = 0 α − βi −1

for complex conjugate roots λ = α ± βi In each case, Ji is the Jordan form of the particular matrix A. We shall use theorem 5.3 when discussing the stability of discrete systems in section 5.6. Section 5.2 dealt with case (a) in detail. Here we shall consider just one example of cases (b) and (c), using both Mathematica and Maple.

Example 5.6 Consider the matrix in example 4.10, which is 1 −1 A2 = 1 3 then the instructions in each programme are: Mathematica A2={{1,-1},{1,3}} Eigenvalues[A2] {V2,J2}=JordanDecomposition[A2] MatrixForm /@ {V2,J2} MatrixForm[Inverse[V2].A2.V2]

Maple with(linalg): A2:=matrix([[1,-1],[1,3]]); eigenvals(A2); J2:=jordan(A2,’V2’); print(V2); evalm(V2^(-1)&*A2&*V2);

With each programme we get the Jordan form as 2 1 J2 = 0 2

Discrete systems of equations

219

Example 5.7 Next consider the matrix in example 4.11, which is −3 4 A3 = −2 1 then the instructions in each programme are: Mathematica A3={{-3,4},{-2,1}} Eigenvalues[A3] {V3,J3}=JordanDecomposition[A3] MatrixForm /@ {V3,J3} MatrixForm[Inverse[V3].A3.V3]

Maple with(linalg): A3:=matrix([[-3,4],[-2,1]]); eigenvals(A3); J3:=jordan(A3,’V3’); print(V3); evalm(V3^(-1)&*A3&*V3);

With each programme we get the Jordan form as −1 + 2i 0 J3 = 0 −1 − 2i Verifying the results in theorem 5.3. When considering the stability of the system ut = Aut−1

(5.9)

we can approach this from a slightly different perspective, which can provide some valuable insight into the phase portrait of discrete systems. What we intend to do is to transform the system using the matrix V. Thus, deﬁne zt = V−1 ut

(5.10)

This implies ut = Vzt . We can therefore write system (5.9) in the form Vzt = AVzt−1 premultiplying by the matrix V−1 , we have zt = V−1 AVzt−1 = Jzt−1

where J =

r 0

0 s

System zt = Jzt−1 is referred to as the canonical form of the system ut = Aut−1 . The important point is that the stability properties of (5.11) are the same as those of (5.9). The solution to the canonical form is simply t r 0 t where z0 = V−1 u0 z zt = J z0 = 0 st 0

(5.11)

220

Economic Dynamics When considering the phase space of this canonical form it is useful to consider the following: t r 0 z10 z1t = z2t z20 0 st Now take the ratio of z2 /z1 , then s t z z2t st z20 20 = t = z1t r z10 r z10 and so the path of the system is dominated by the value/sign of s/r.

5.5 Graphing trajectories of discrete systems The mathematics of solving simultaneous equation systems is not very straightforward and it is necessary to obtain the eigenvalues and the eigenvectors. However, it is possible to combine the qualitative nature of the phase plane discussed in the previous section and obtain trajectories using a spreadsheet or the recursive features of Mathematica and Maple. 5.5.1

Trajectories with Excel

Example 5.8 Consider the following system of equations xt = −5 + 0.25xt−1 + 0.4yt−1 yt = 10 − xt−1 + yt−1 x0 = 10, y0 = 5 In cells B8 and C8 we place the initial values for x and y, namely x0 = 10 and y0 = 5. In cells B9 and C9 we place the formulas. These are B9 = -5 + 0.25* B8 + 0.4* C8 C8 = 10 - B8 + C8

These cell entries contain only relative addresses. Cells B8 and C8 are then copied to the clipboard and pasted down in cells B10:C28. Once the computations for (xt , yt ) have been obtained, then it is a simple matter of using the x-y plot to plot the trajectory. Given the discrete nature of the system the trajectories are not the regular shapes indicated by the phase plane diagram. They constitute discrete points that are joined up. Even so, the nature of the system can readily be investigated. Figure 5.2 shows the initial values of x0 = 10 and y0 = 5. Always a good check that the equations have been entered correctly is to place the equilibrium values as the initial values. The equilibrium point is (x∗ , y∗ ) = (10, 31.25). Placing these values in cells B8 and C8 leads to them being repeated in all periods. One of the advantages of this approach, besides its simplicity, is the ready investigation of the system for various initial conditions. The graphics plot can sometimes change quite dramatically! This procedure allows quite complex discrete dynamic systems of two equations to be investigated with the minimum mathematical knowledge. Of course, to fully

Discrete systems of equations

221 Figure 5.2.

appreciate what is happening requires an understanding of the material in many of the chapters of this book. Consider the following nonlinear system, which is used to produce the H´enon map and which we shall investigate more fully in chapter 7. Example 5.9 The system is 2 + yt−1 xt = 1 − axt−1

yt = bxt−1 Our purpose here is not to investigate the properties of this system, but rather to see how we can display trajectories belonging to it. We begin with the spreadsheet, as shown in ﬁgure 5.3. We place the values of a and b in cells E3 and E4, where a = 1.4 and b = 0.3. In cells B8 and C8 we place the initial values for x and y, which are x0 = 0.01 and y0 = 0. The formulas for the two equations are placed in cells B9 and C9, respectively. These take the form B9

1-$E$3*B8^2+C8

C9

$E$4*B8

The cells with dollar signs indicate absolute addresses, while those without dollar signs indicate relative addresses. Cells B9 and C9 are then copied to the clipboard and pasted down. After blocking cells B8:C28 the graph wizard is then invoked and the resulting trajectory is shown in the inserted graph. The most conspicuous feature of this trajectory is that it does not have a ‘pattern’. In fact, given the parameter values there are two equilibrium points: (x1∗ , y∗1 ) = (−1.1314, −0.3394) and (x2∗ , y∗2 ) = (0.6314, 0.1894), neither of which is approached within the ﬁrst twenty periods. Why this is so we shall investigate in chapter 7.

222

Economic Dynamics

Figure 5.3.

5.5.2

Trajectories with Mathematica and Maple

The spreadsheet is ideal for displaying recursive systems and the resulting trajectories. But occasionally it is useful to display these trajectories within Mathematica or Maple. In doing this care must be exercised in writing the simultaneous equations for computation so that the programmes remember earlier results and do not recompute all previous values on each round. This leads to more cumbersome input instructions – which is why the spreadsheet is so much easier for many problems. We shall consider once again examples 5.8 and 5.9. Example 5.8 (cont.) The input instructions for each programme are Mathematica Clear[x,y,t] x[0]:=10; y[0]:=5; x[t-]:=x[t]=-5+0.25x[t-1]+0.4y[t-1] y[t-]:=y[t]=10-x[t-1]+y[t-1] data:=Table[{x[t],y[t]},{t,0,20}]; ListPlot[data,PlotJoined->True,PlotRange->All]

Maple t:=’t’: x:=’x’: y:=’y’: x:=proc(t) option remember;-5+0.25*x(t-1)+0.4*y(t-1)end: y:=proc(t) option remember; 10-x(t-1)+y(t-1) end: x(0):=10: y(0):=5: data:=[seq([x(t),y(t)],t=0..20)]; plot(data);

The Maple instructions join the points by default. If just a plot of points is required

Discrete systems of equations with Maple, then the last line becomes plot(data, plotstyle=point);

The resulting trajectories are similar to that shown in the chart in ﬁgure 5.2 (p. 221). As one might expect, both Mathematica and Maple allow more control over the display of the trajectories than is available within Excel. Furthermore, both these programmes allow more than one trajectory to be displayed on the same diagram. This is not possible within spreadsheets. Spreadsheets can display only one (x, y)trajectory at a time. Example 5.9 (cont.) The input instructions for each programme for producing discrete plot trajectories are Mathematica Clear[x,y,t,a,b] x[0]:=0.01; y[0]:=0; a:=1.4; b:=0.3; x[t-]:=x[t]=1-a x[t-1]^2+y[t-1] y[t-]:=y[t]=b x[t-1] data=Table[{x[t],y[t]},{t,0,20}]; ListPlot[data,PlotJoined->True,PlotRange->All]

Maple t:=’t’: x:=’x’: y:=’y’: a:=’a’: b:=’b’: x:=proc(t) option remember; 1-a*x(t-1)^2+y(t-1) end: y:=proc(t) option remember; b*x(t-1) end: x(0):=0.01: y(0):=0: a:=1.4: b:=0.3: data:=[seq([x(t),y(t)],t=0..20)]; plot(data);

The resulting trajectories are similar to that shown in the chart in ﬁgure 5.3. Spreadsheets do not allow three-dimensional plots, but it is very easy to adapt the instructions just presented for Mathematica and Maple to do this. The only essential difference is the ﬁnal line in each programme. Assuming ‘data’ records the list of points {x(t), y(t), z(t)}, then a three-dimensional plot requires the instruction Mathematica ListPlot3D[data,PlotJoined->True]

Maple plot3d(data);

5.6 The stability of discrete systems 5.6.1

Real distinct roots

For systems with real distinct roots, r and s, which therefore have linearly independent eigenvectors, we can establish the stability properties of such systems by

223

224

Economic Dynamics considering the general solution

ut = ar v + bs v t r

t s

where

x − x∗ ut = t yt − y∗

If |r| < 1 and |s| < 1 then art vr → 0 and bst vs → 0 as t → ∞ and so ut → 0 and consequently the system tends to the ﬁxed point, the equilibrium point. Return to example 5.4 where r = 0.7262 and s = −0.8262. The absolute value of both roots is less than unity, and so the system is stable. We showed this in terms of ﬁgure 5.1, where the system converges on the equilibrium, the ﬁxed point. We pointed out above that the system can be represented in its canonical form, and the same stability properties should be apparent. To show this our ﬁrst task is to compute the vector z0 . Since z0 = V−1 u0 , then

z10 z20

0.5793 5.7537 = 1 1

−1

−4.4 −13.3827 = −12.8 0.5827

and z1t = (0.7262)t (−13.3827) z2t = (−0.8262)t (0.5827) Setting this up on a spreadsheet, we derive ﬁgure 5.4. The canonical form has transformed the system into the (z1 , z2 )-plane, but once again it converges on the ﬁxed point, which is now the origin. Now consider the situation where |r| > 1 and |s| > 1 then ut → ±∞ depending on the sign of the characteristic roots. But this result occurs so long as at least one root is greater than unity in absolute value. This must be so. Let |r| > 1 and let |s| < 1. Then over time bst vs → 0 as t → ∞, while art vr → ±∞ as t → ∞, which means ut → ±∞ as t → ∞. The system is unstable. Return to example 5.3 where r = 1 and s = 3. Both roots are distinct, positive and at least one is greater than unity. It follows from our earlier argument that this system must be unstable. We have already demonstrated that the diagonal matrix Figure 5.4.

Discrete systems of equations (the Jordan form) and the transition matrix are: 1 0 1 1 J= , V= 0 3 −1 1 Suppose for this system (u10 , u20 ) = (5, 2), then t −1 1 1 1 1 5 u1t 1 0 = 0 3t u2t −1 1 −1 1 2 i.e. 3 7 t + 3 2 2 3 7 u2t = − + 3t 2 2 u1t =

Therefore, as t increases u1t → +∞ and u2t → +∞. Turning to the canonical form, z0 = V−1 u0 , hence z10 3/2 = z20 7/2 and

3 3 = 2 2 t 7 z2t = 3 2 z1t = 1t

Furthermore, z2t = z1t

t 3 7/2 7 = 3t 1 3/2 3

and so for each point in the canonical phase space, the angle from the origin is increasing, and so the direction of the system is vertically upwards, as illustrated in ﬁgure 5.5(b). Once again, the same instability is shown in the original space, ﬁgure 5.5(a), and in its canonical form, ﬁgure 5.5(b). The fact that the trajectory in ﬁgure 5.5(b) is vertical arises from the fact that root r = 1. Suppose one root is greater than unity in absolute value and the other less than unity in absolute value. Let these be |r| > 1 and |s| < 1. The system remains unstable because it will be governed by the root |r| > 1, and the system will be dominated by the term ar t v r . This means that for most initial points u0 , the system diverges from the equilibrium, from the steady state (x∗, y∗ ). But suppose a = 0, then ut = bst vs and since |s| < 1, then the system will converge on (x∗, y∗ ). There exist, therefore some stable trajectories in the phase plane. Example 5.10 xt+1 = −0.85078xt − yt yt+1 = xt + 2.35078yt

225

226

Economic Dynamics

Figure 5.5.

The ﬁxed point is at the origin, i.e., x∗ = 0 and y∗ = 0. This system takes the matrix form xt+1 −0.85078 −1 = yt+1 1 2.35078 The eigenvalues of the matrix A of this system are readily found to be r = 2 and s = −0.5. We therefore satisfy the condition |r| > 1 and |s| < 1. Furthermore, we can obtain the eigenvectors of this system as follows (A − 2I)vr = 0 −2.85078 i.e. 1

−1 0.35078

vr1 vr2

0 = 0

Discrete systems of equations or −2.85078vr1 − vr2 = 0 vr1 + 0.35078vr2 = 0 Let vr1 = 1 then vr2 = −2.8508. The second eigenvector is found from (A − 0.5I)vs = 0 −0.35078 i.e. 1

−1 2.85078

vs1 vs2

0 = 0

or −0.35078vs1 − vs2 = 0 vs1 + 2.85078vs2 = 0 Let vs2 = 1 then vs1 = −2.8508. Hence 1 −2.8508 , vs = vr = −2.8508 1 One eigenvector represents the stable arm while the other represents the unstable arm. But which, then, represents the stable arm? To establish this, convert the system to its canonical form, with zt+1 = V−1 ut+1 . Now take a point on the ﬁrst eigenvector, i.e., point (1, −2.8508), then −1 1 1 1 −2.8508 −1 = z0 = V u0 = −2.8508 0 −2.8508 1 Hence, z1t = 2t (1) z2t = (−0.5)t (0) = 0 Therefore z1t → +∞ as t → ∞. So the eigenvector vr must represent the unstable arm. Now take a point on the eigenvector vs , i.e., the point (−2.8508, 1), then −1 −2.8508 0 1 −2.8508 −1 = z0 = V u0 = 1 1 −2.8508 1 Hence, z1t = 2t (0) = 0 z2t = (−0.5)t (1) Therefore z2t → 0 as t → ∞. So the eigenvector vs must represent the stable arm of the saddle point. Figure 5.6 illustrates the trajectory of the original system starting from point (−2.8508, 1), and shows that it indeed converges on the point (x∗ , y∗ ) = (0, 0).5 Before leaving this example, notice that when we considered point (1, −2.8508) on the eigenvector associated with r = 2, the point became (1, 0) in terms of its 5

The system is, however, sensitive. A plot beyond t = 10 has the system moving away from the ﬁxed point.

227

228

Economic Dynamics

Figure 5.6.

canonical representation. Similarly, point (−2.8508, 1) on the eigenvector associated with s = −0.5 became point (0, 1) in its canonical representation. In other words, the arms of the saddle point equilibrium became transformed into the two rectangular axes in z-space. This is a standard result for systems involving saddle point solutions. So long as we have distinct characteristic roots these results hold. Here, however, we shall conﬁne ourselves to the two-variable case. To summarise, if r and s are the characteristic roots of the matrix A for the system ut = Aut−1 and derived from solving |A − λI| = 0, then (i) (ii) (iii)

if |r| < 1 and |s| < 1 the system is dynamically stable if |r| > 1 and |s| > 1 the system is dynamically unstable if, say, |r| > 1 and |s| < 1 the system is dynamically unstable.

In the case of (iii) the system will generally be dominated by the largest root and will tend to plus or minus inﬁnity depending on its sign. But given the ﬁxed point is a saddle path solution, there are some initial points that will converge on the ﬁxed point, and these are values that lie on the stable arm of the saddle point. As we shall see in part II, such possible solution paths are important in rational expectations theory. Under such assumed expectations behaviour, the system ‘jumps’ from its initial point to the stable arm and then traverses a path down the stable arm to equilibrium. Of course, if this initial ‘jump’ did not occur, then the trajectory would tend to plus or minus inﬁnity and be driven away from equilibrium. 5.6.2

Repeating roots

When there is a repeating root, λ, the system’s dynamics is dominated by the sign/value of this root. If |λ| < 1, then the system will converge on the equilibrium value: it is asymptotically stable. If |λ| > 1 then the system is asymptotically unstable. We can verify this by considering the canonical form. We have already

Discrete systems of equations showed that the canonical form of ut = Aut−1 is zt = Jt z0 In the case of a repeated root this is t λ tλt−1 zt = z0 0 λt Hence z1t = λt z10 + tλt−1 z20 z2t = λt z20

Therefore if |λ| < 1, then λt → 0 as t → ∞, consequently z1t → 0 and z2t → 0 as tt → ∞. The system is asymptotically stable. If, on the other hand, |λ| > 1, then λ → ∞ as t → ∞, and z1t → ±∞ and z2t → ±∞ as t → ∞. The system is asymptotically unstable. We can conclude for repeated roots, therefore, that if |λ| < 1 the system is asymptotically stable if |λ| > 1 the system is asymptotically unstable.

(a) (b)

Example 5.11 Consider the following system xt+1 = 4 + xt − yt yt+1 = −20 + xt + 3yt ∗

Then x = 12 and y∗ = 4. Representing the system as deviations from equilibrium, we have xt+1 − x∗ = (xt − x∗ ) − (yt − y∗ ) yt+1 − y∗ = (xt − x∗ ) + 3(yt − y∗ ) This system can be represented in the form ut = Aut−1 and the matrix of the system is 1 −1 A= 1 3 We have already considered this in example 5.6 with results 2 1 −1 1 J= V= 0 2 1 0 The solution is then ut = VJt V−1 u0

where

Jt =

2t 0

t2t−1 2t

Since the stability properties of ut = Aut−1 are the same as those of its canonical form, let us therefore consider t 2 t2t−1 zt = z0 0 2t

229

230

Economic Dynamics

Figure 5.7.

i.e. z1t = 2t z10 + t2t−1 z20 z2t = 2t z20 Clearly, z1t → ∞ and z2t → ∞ as t → ∞ regardless of the initial point. This is illustrated for the original system using a spreadsheet, as shown in ﬁgure 5.7 for the initial point (0.1, 0.1). Example 5.12 Consider xt+1 = 8 + 1.5xt − yt yt+1 = −15 + xt − 0.5yt Then x∗ = 108 and y∗ = 62. Taking deviations from equilibrium xt+1 − x∗ = 1.5(xt − x∗ ) − (yt − y∗ ) yt+1 − y∗ = (xt − x∗ ) − 0.5(yt − y∗ ) and so the matrix of this system is 1.5 −1 A= 1 −0.5 Using either Mathematica or Maple we can establish that the eigenvalues are λ = 0.5 (repeated twice) and the Jordan form and transition matrices are 0.5 1 1 1 J= V= 0 0.5 1 0

Discrete systems of equations

231 Figure 5.8.

Hence,

0.5t J = 0 t

t0.5t−1 0.5t

and the canonical form of the system is t 0.5 t0.5t−1 z0 zt = 0 0.5t Hence z1t = 0.5t z10 + t(0.5t−1 )z20 z2t = 0.5t z20 Therefore, z1t → 0 and z2t → 0 as t → ∞ regardless of the initial point. This is illustrated for the original system using a spreadsheet, as shown in ﬁgure 5.8 for the initial point (5, 5).

5.6.3

Complex conjugate roots

Although complex conjugate roots involve distinct roots and independent eigenvectors, it is still necessary to establish the conditions for stability where r = α + βi and s = α − βi. For the system ut = Aut−1 where A is a 2 × 2 matrix with conjugate roots α ± βi, then the Jordan form is α + βi 0 0 (α + βi)t t J= and J = 0 α − βi 0 (α − βi)t

232

Economic Dynamics Considering the canonical form zt = Vzt−1 , then 0 (α + βi)t zt = Jt z0 = z 0 (α − βi)t 0 To investigate the stability properties of systems with complex conjugate roots, we employ two results (see Simon and Blume 1994, appendix A3): (i) (ii)

α ± iβ = R(cos θ ± i sin θ) (α ± iβ)n = Rn [cos(nθ) ± i sin(nθ)]

(De Moivre’s formula)

where R=

α2 + β 2

and

tan θ =

β α

From the canonical form, and using these two results, we have z1t = (α + βi)t z10 = Rt [cos(tθ) + i sin(tθ)] z2t = (α − βi)t z20 = Rt [cos(tθ) − i sin(tθ)] It follows that such a system must oscillate because as t increases, sin(tθ) and cos(tθ) range between +1 and −1. Furthermore, the limit of z1t and z2t as t → ∞ is governed by the term Rt = |R|t . If |R| < 1, then the system is an asymptotically stable focus; if |R| > 1, then the system is an unstable focus; while if |R| = 1, then we have a centre.6 We now illustrate each of these cases. Example 5.13 |R| < 1 Consider the system xt = 0.5xt−1 + 0.3yt−1 yt = −xt−1 + yt−1 Then the matrix of the system is 0.5 0.3 A= −1 1 with characteristic roots r = 0.75 + 0.48734i and s = 0.75 − 0.48734i . Hence R = (0.75)2 + (0.48734)2 = 0.8944 Such a system should therefore have an asymptotically stable focus. We illustrate this in ﬁgure 5.9, where the original system is set up on a spreadsheet and the initial point is given by (x0 , y0 ) = (10, 5). As can be seen from the inserted graph, the system tends in the limit to the origin.

6

See section 3.8 for the properties of R in the complex plane, especially ﬁgure 3.15.

Discrete systems of equations

233 Figure 5.9.

Example 5.14 |R| > 1 Consider the system xt = xt−1 + 2yt−1 yt = −xt−1 + yt−1 Then the matrix of the system is 1 2 A= −1 1

√ √ with characteristic roots r = 1 + i 2 and s = 1 − i 2. Hence √ √ R = (1)2 + ( 2)2 = 3 = 1.73205 Such a system should therefore have an unstable focus. We illustrate this in ﬁgure 5.10, where the original system is set up on a spreadsheet and the initial point is given by (x0 , y0 ) = (0.5, 0.5). As can be seen from the inserted graph, the system is an unstable focus, spiralling away from the origin. Example 5.15 |R| = 1 Consider the system xt = 0.5xt−1 + 0.5yt−1 yt = −xt−1 + yt−1 Then the matrix of the system is 0.5 0.5 A= −1 1

234

Economic Dynamics

Figure 5.10.

Figure 5.11.

with characteristic roots r = 0.75 + 0.661438i and s = 0.75 − 0.661438i. Hence R=

(0.75)2 + (0.661438)2 = 1

Such a system should oscillate around a centre, where the centre is the origin. We illustrate this in ﬁgure 5.11, where the original system is set up on a spreadsheet and the initial point is given by (x0 , y0 ) = (5, 5). As can be seen from the inserted graph the system does indeed oscillate around the origin.

Discrete systems of equations

5.7 The phase plane analysis of discrete systems Consider the following system outlining discrete changes in x and y. xt+1 = a0 + a1 xt + a2 yt yt+1 = b0 + b1 xt + b2 yt where xt+1 = xt+1 − xt and yt+1 = yt+1 − yt . In equilibrium, assuming it exists, at x = x∗ and y = y∗ , we have 0 = a0 + a1 x∗ + a2 y∗ 0 = b0 + b1 x∗ + b2 y∗ Hence, the system in terms of deviations from equilibrium can be expressed xt+1 = a1 (xt − x∗ ) + a2 (yt − y∗ ) yt+1 = b1 (xt − x∗ ) + b2 (yt − y∗ ) We can approach the problem in terms of the phase plane. As with the continuous model we ﬁrst obtain the equilibrium lines xt+1 = 0 and yt+1 = 0. Consider the following system with assumed signs on some of the parameters xt+1 = a0 + a1 xt + a2 yt

a 1 , a2 > 0

yt+1 = b0 + b1 xt + b2 yt

b1 > 0, b2 < 0

Then

a1 −a0 − xt a2 a2 −b0 b1 = 0 then yt = − xt b2 b2

If xt+1 = 0 then yt = If yt+1

Consider now points either side of xt+1 = 0. a1 −a0 − xt If xt+1 > 0 then yt > a2 a2 a1 −a0 If xt+1 < 0 then yt < − xt a2 a2

a2 > 0

Similarly for points either side of yt+1 = 0 −b0 b1 If yt+1 > 0 then yt < − xt b2 b2 b1 −b0 If yt+1 < 0 then yt > − xt b2 b2

b2 < 0

a2 > 0

b2 < 0

The vector forces are illustrated in ﬁgures 5.12(a) and (b). Combining the information we have the phase plane diagram for a discrete system, illustrated in ﬁgure 5.13. The combined vector forces suggest that the equilibrium is a saddle point.

235

236

Economic Dynamics

Figure 5.12.

Example 5.16 Given xt+1 = −12 + 0.3xt + 3yt yt+1 = 4 + 0.25xt − 1.5yt Then xt+1 = 0 implies yt = 4 − 0.1xt yt+1 = 0 implies yt = 2.6667 + 0.1667xt and we obtain the equilibrium values x∗ = 5

y∗ = 3.5

Discrete systems of equations

237 Figure 5.13.

In terms of deviations from the equilibrium we have for x xt+1 = −12 + 1.3xt + 3yt x∗ = −12 + 1.3x∗ + 3y∗ (xt+1 − x∗ ) = 1.3(xt − x∗ ) + 3(yt − y∗ ) and for y we have yt+1 = 4 + 0.25xt − 0.5yt y∗ = 4 + 0.25x∗ − 0.5y∗ (yt+1 − y∗ ) = 0.25(xt − x∗ ) − 0.5( yt − y∗ ) Therefore the system expressed as deviations from equilibrium is xt+1 − x∗ = 1.3(xt − x∗ ) + 3(yt − y∗ ) yt+1 − y∗ = 0.25(xt − x∗ ) − 0.5( yt − y∗ ) The dynamics of the system in the neighbourhood of (x∗ , y∗ ) is therefore determined by the properties of 1.3 3 A= 0.25 −0.5 First we require to obtain the eigenvalues of the matrix A: 1.3 − λ 3 = λ2 − 0.8λ − 1.4 = 0 |A − λI| = 0.25 −(0.5 + λ) with distinct eigenvalues r = 1.649 and s = −0.849.

238

Economic Dynamics Next we need to obtain the associated eigenvectors, vr and vs, respectively. Consider r = 1.649, then −0.349 3 A − rI = 0.25 −2.149 Then (A − rI)vr = 0 and r 0 −0.349 3 v1 = vr2 0 0.25 −2.149 then −0.349vr1 + 3vr2 = 0 0.25vr1 − 2.149vr2 = 0 Let vr2 = 1, then vr1 = 8.596, which arises from either equation. Similarly, for s = −0.849 then 2.149 3 A − sI = 0.25 0.349 and

(A − sI)vs =

2.149 0.25

3 0.349

vs1 vs2

giving 2.149vs1 + 3vs2 = 0 0.25vs1 + 0.349vs2 = 0 Let vs2 = 1, then vs1 = −1.396. Hence the two eigenvectors are 8.596 −1.396 r s , v = v = 1 1 with associated matrix 8.596 −1.396 V= 1 1 Hence, our solution is (1.649)t ut = V 0

0 V−1 u0 (−0.849)t

Given some initial values (x0 , y0 ) we could solve explicitly for ut. But we can gain insight into the dynamics of this system by looking closely at the phase plane. We have already established that for xt+1 = 0 then yt = 4 − 0.1xt yt+1 = 0 then yt = 2.6667 + 0.1667xt These are drawn in ﬁgure 5.14.

Discrete systems of equations

239 Figure 5.14.

We can also readily establish that if xt+1 > 0 then yt > 4 − 0.1xt and x is rising if xt+1 < 0 then yt < 4 − 0.1xt and x is falling and if yt+1 > 0 then yt > 2.6667 + 0.1667xt and y is rising if yt+1 < 0 then yt < 2.6667 + 0.1667xt and y is falling These vector forces are also illustrated in ﬁgure 5.14. The ﬁgure also illustrates that a saddle point equilibrium is present with a stable arm S1 S1 and an unstable arm S2 S2 . In general, the system will move away from the equilibrium point, except for initial values lying on the stable arm S1 S1 . Although the vector force diagram of discrete systems can provide much information on the dynamics of the system, the stability properties usually can be obtained only from its mathematical properties.

5.8 Internal and external balance We can illustrate the use of the phase plane by considering an important policy issue that has been discussed in the literature, namely internal and external balance.7 Having set up a macroeconomic model, a ﬁxed target policy is then imposed on it. Two ﬁxed targets are chosen: the level of real income and the balance on the balance of payments. Real income is considered set at the full employment level, 7

See Shone (1989, chapter 11) and the seminal article by Mundell (1962).

240

Economic Dynamics which denotes the condition of internal balance.8 External balance represents a zero balance on the combined current and capital account of the balance of payments. Following Tinbergen’s analysis (1956), there are two policy instruments necessary for achieving the two policy objectives. These are government spending, which is used to achieve internal balance, and the interest rate, which is used to achieve external balance (by inﬂuencing explicitly net capital ﬂows). Suppose we set up an adjustment on the part of the two instruments that assumes that the change in the policy variable is proportional to the discrepancy between its present level and the level to achieve its target. More formally we have gt+1 = gt+1 − gt = k1 (gt − g∗t ) k1 < 0 rt+1 = rt+1 − rt = k2 (rt − rt∗ )

k2 < 0

g∗t

where is the target level of government spending in period t, and rt∗ is the target interest rate in period t. Example 5.17 Following Shone (1989) we have the following two equations relating gt and rt derived from a macroeconomic model, IB

rt = −3.925 + 0.5gt

XB

rt = 7.958 + 0.186gt

where IB denotes internal balance and XB denotes external balance. In the case of internal balance we require g∗t which is equal to g∗t = 7.85 + 2rt , while for external balance we require rt∗ which is equal to rt∗ = 7.958 + 0.186gt . Then gt+1 = k1 (gt − 7.85 − 2rt )

k1 < 0

rt+1 = k2 (rt − 7.958 − 0.186gt ) k2 < 0 The isoclines are where gt+1 = 0 and rt+1 = 0 and the equations of which no more than represent the internal balance and external balance lines, respectively. The stationary values of g and r are where the two isoclines intersect, giving g∗ = 37.84 and r∗ = 15. The situation is illustrated in ﬁgure 5.15. The vectors of force are readily established (noting k1 and k2 are negative): if gt+1 > 0 then rt > −3.925 + 0.5gt and gt is rising if gt+1 < 0 then rt < −3.925 + 0.5gt and gt is falling Also if rt+1 > 0 then rt < 7.958 + 0.186gt and rt is rising if rt+1 < 0 then rt > 7.958 + 0.186gt and rt is falling which are also illustrated in ﬁgure 5.15. The use of the spreadsheet is convenient in considering this problem. To do this we need to express gt+1 and rt+1 in terms of gt and rt . We assume that k1 = −0.5 8

Internal balance can also be considered as a suitable income–inﬂation combination. See Shone (1979).

Discrete systems of equations

241 Figure 5.15.

Figure 5.16.

and k2 = −0.75. We then have gt+1 = 3.925 + 0.5gt + rt rt+1 = 5.9685 + 0.1395gt + 0.25rt The quadrant that typiﬁed the UK economy for periods in the 1960s is quadrant IV (ﬁgure 5.15), which has the economy with a balance of payments deﬁcit (below the XB curve) and unemployment (above the IB curve). Suppose, then, that the economy begins at point (g0 , r0 ) = (20, 9). The trajectory the economy follows is shown in ﬁgure 5.16. It is very easy to use this spreadsheet to investigate the path of the economy in any of the four quadrants, and we leave this as an exercise. What can readily be established is that, regardless of the initial point, the economy moves towards the equilibrium point where the two isoclines intersect.

242

Economic Dynamics Example 5.18 But we have presupposed that government spending is used to achieve internal balance and the rate of interest is used to achieve external balance. Suppose we assume the opposite assignment: set interest rates to achieve internal balance and government spending to achieve external balance. Then rt∗ = −3.925 + 0.5gt and g∗t = −42.785 + 5.376rt Then rt+1 = k3 (rt − rt∗ ) = k3 (rt + 3.925 − 0.5gt ) gt+1 = k4 (gt −

g∗t )

k3 < 0

= k3 (gt + 42.785 − 5.376rt ) k4 < 0

Setting rt+1 = 0 and gt+1 = 0 gives rise to the internal balance isocline and the external balance isocline, respectively, leading to the same equilibrium point (g∗ , r∗ ) = (37.85, 15), as shown in ﬁgure 5.17. However, the vector forces are now different: if rt+1 > 0 then rt < −3.925 + 0.5gt and rt is rising if rt+1 < 0 then rt > −3.925 + 0.5gt and rt is falling Similarly if gt+1 > 0 then rt > 7.958 + 0.186gt and gt is rising if gt+1 < 0 then rt < 7.958 + 0.186gt and gt is falling From ﬁgure 5.17 it is apparent that the equilibrium point E is a saddle point. Under this assignment economies ﬁnding themselves in sectors I and III will converge on the equilibrium only so long as they remain in these sectors, although this is Figure 5.17.

Discrete systems of equations

243 Figure 5.18.

not guaranteed (see exercise 9). Any point in sectors II and IV, other than the equilibrium point, however, will move away from equilibrium (ﬁgure 5.17). Given this alternative assignment, and assuming k3 = −0.75 and k4 = −0.5 we obtain the two equations gt+1 = −21.3925 + 0.5gt + 2.688rt rt+1 = −2.94375 + 0.375gt + 0.25rt Taking a point once again in quadrant IV, but now close to the equilibrium, point (37, 14), the spreadsheet calculations quite readily show the economy diverging from the equilibrium, as illustrated in ﬁgure 5.18, where we plot only up to period 9.9 Again it is very easy to use this spreadsheet to investigate the path of the economy for any initial point in any of the four quadrants. This we leave as an exercise. Comparing ﬁgures 5.15 and 5.17 leads to an important policy conclusion. It is not the slopes of the internal and external balance lines per se which governs the dynamics, but rather the policy assignment. This was Mundell’s conclusion (1962). He put it differently and claimed that stability requires pairing the instrument with the target over which it has the greatest relative impact: the principle of effective market classiﬁcation. Government spending has the greatest relative impact on income and hence on achieving internal balance. This immediately implies that the interest rate has the greatest relative impact on the balance of payments, and hence on achieving external balance. Consequently, the assignment represented in ﬁgure 5.15 is stable while that in ﬁgure 5.17 is unstable. This approach to dynamics is readily generalised. Consider again internal and external balance, but now using the two instruments government spending, g, and the exchange rate, S. The situation is shown in ﬁgure 5.19 (Shone 1989, chapter 11). We can capture this situation with the two linear equations

9

IB

St = a0 − a1 gt

a1 > 0

XB

St = b0 + b1 gt

b1 > 0

Note how sensitive discrete systems can be to initial conditions. Placing the ‘equilibrium’ values (37.847,14.9985) as the initial conditions, still has the system diverging!

244

Economic Dynamics

Figure 5.19.

where a0 > b0 . Equilibrium is readily found to be g∗ =

a0 − b0 a1 + b1

S∗ =

a0 b1 + a1 b0 a1 + b1

However, in order to consider the dynamics of this point we need to specify some assignment. Suppose we assign government spending to internal balance and the exchange rate to external balance, satisfying gt+1 = gt+1 − gt = k1 (gt − g∗t ) k1 < 0 St+1 = St+1 − St = k2 (St − St∗ ) k2 < 0 The equations for g∗t and St∗ are g∗t = (a0 /a1 ) − (1/a1 )St St∗ = b0 + b1 gt Hence IB

gt+1 = k1 [gt − (a0 /a1 ) + (1/a1 )St ]

XB

St+1 = k2 [St − b0 − b1 gt ]

It is readily established (noting k1 and k2 are negative) that: if gt+1 > 0 then St < a0 + a1 gt and gt is rising if gt+1 < 0 then St > a0 + a1 gt and gt is falling if St+1 > 0 then St < b0 + b1 gt and St is rising if St+1 < 0 then St > b0 + b1 gt and St is falling

Discrete systems of equations

245 Figure 5.20.

with resulting forces illustrated in ﬁgure 5.19. This quite clearly illustrates an anticlockwise motion. What we cannot establish is whether the system converges on the equilibrium or moves away from it. Example 5.19 To illustrate this suppose we have for internal and external balance the equations IB

St = 20 − 2gt

XB

St = −4 + 4gt

with equilibrium g∗ = 2.5 and S∗ = 10. Further, suppose adjustment is of the form gt+1 = −0.75(gt − g∗t ) St+1 = −0.5(St − St∗ ) where g∗t = −2.5 + 0.5St St∗ = 20 − 4gt then gt+1 = −1.875 + 0.25gt + 0.375St St+1 = 10 − 2gt + 0.5St Given an initial point (g0 , S0 ) = (4, 12), ﬁgure 5.20 shows the typical anticlockwise spiral trajectory that is tending towards the equilibrium.

5.9 Nonlinear discrete systems Just as we can encounter nonlinear equations of the form xt = f (xt−1 ), so we can have nonlinear systems of equations of the form xt = f (xt−1 , yt−1 ) yt = g(xt−1 , yt−1 )

246

Economic Dynamics where we assume just a one-period lag. A steady state (x∗ , y∗ ) exists for this system if it satisﬁes x∗ = f (x∗ , y∗ ) y∗ = g(x∗ , y∗ ) It is possible to investigate the stability properties of this nonlinear system in the neighbourhood of the steady state so long as f and g are continuous and differentiable. Under such conditions we can expand the system in a Taylor expansion about (x∗ , y∗ ), i.e. xt − x ∗ =

∂ f (x∗ , y∗ ) ∂ f (x∗ , y∗ ) (xt−1 − x∗ ) + ( yt−1 − y∗ ) ∂xt−1 ∂yt−1

yt − y∗ =

∂g(x∗ , y∗ ) ∂g(x∗ , y∗ ) (xt−1 − x∗ ) + ( yt−1 − y∗ ) ∂xt−1 ∂yt−1

Let a11 =

∂ f (x∗ , y∗ ) , ∂xt−1

a12 =

∂ f (x∗ , y∗ ) ∂yt−1

a21 =

∂g(x∗ , y∗ ) , ∂xt−1

a22 =

∂g(x∗ , y∗ ) ∂yt−1

then xt − x∗ = a11 (xt−1 − x∗ ) + a12 (yt−1 − y∗ ) yt − y∗ = a21 (xt−1 − x∗ ) + a22 ( yt−1 − y∗ ) or

xt − x∗ yt − y ∗

a = 11 a21

a12 a22

xt−1 − x∗ yt−1 − y∗

i.e. ut = Aut−1 which is no more than a ﬁrst-order linear system with solution ut = VDt V−1 u0 and where D is the diagonal matrix with distinct eigenvalues r and s on the main diagonal, V = [ vr vs ] is the matrix of eigenvectors associated with r and s, and u0 is a vector of initial values. It should be noted that the matrix A is simply the Jacobian matrix, J, of the nonlinear system evaluated at a ﬁxed point (x∗ , y∗ ). Under certain restrictions on A (or J), the linear system ‘behaves like’ the nonlinear system. The situation can be illustrated by means of ﬁgure 5.21. The nonlinear system, N, can be mapped into an equivalent linear system, L, by the mapping F, such that the qualitative properties of the linear system in the neighbourhood of 0 are the same as that of the nonlinear system in the neighbourhood of 0.10 In other words, the two systems are topologically equivalent. 10

F is then said to be a diffeomorphism.

Discrete systems of equations

247 Figure 5.21.

F

What are these restrictions? We require tr(J) = 0 and det(J) = 0.11 Since the two systems are topologically equivalent, then the dynamics of N in the neighbourhood of 0 can be investigated by means of the linear system in the neighbourhood of 0. We do this by establishing the eigenvalues and eigenvectors of J. More speciﬁcally, we require conditions to be imposed on the eigenvalues. If r and s are the two eigenvalues then the systems are topologically equivalent if: (1) (2)

r and s are distinct real roots and |r| < 1 and |s| < 1. r and s are distinct but complex and lie strictly inside the unit circle.12

If r and s in the neighbourhood of (x∗ , y∗ ) satisfy this second condition, then (x∗ , y∗ ) is said to be a hyperbolic ﬁxed point.

Exercises 1.

2.

3.

4.

11 12

Convert example 5.4 into a system of difference equations and establish the qualitative properties of the system in terms of the isoclines and vector forces. Does this conﬁrm the stability established in section 5.2? Convert example 5.6 into a system of difference equations and establish the qualitative properties of the system in terms of the isoclines and vector forces. Does this illustrate that the equilibrium point is a saddle path solution? For example 5.4 use a spreadsheet to verify that the system converges on the ﬁxed point (x∗ , y∗ ) = (6.4, 20.8) for each of the following initial points: (3,10), (3,30), (10,10), (10,30). Set up example 5.5 on a spreadsheet and investigate its characteristics. In particular, consider: (a) different initial values (b) plot x(t), y(t), and z(t) against t (c) plot y(t) against x(t) (d) plot z(t) against x(t) (e) plot x(t) against z(t).

An alternative way to state the condition is that J must be invertible. See sub-section 3.8.1, p. 110.

248

Economic Dynamics 5. 6.

7.

Use either Mathematica or Maple and do a 3D plot of the solution to example 5.5 for the initial point u0 = (3, −4.3). Use either Mathematica or Maple and derive a 3D directional ﬁeld for example 5.5. Does this give you any more insight into the dynamics of the system over what you gained from question 4? For each of the following systems: (a) ﬁnd the eigenvalues of the system (b) ﬁnd the eigenvectors of the system (c) establish the diagonal matrix of the system. (i) xt+1 = yt yt+1 = −xt (ii) xt+1 = −2xt + yt yt+1 = xt + 2yt (iii) xt+1 = 3xt − 4yt yt+1 = xt − 2yt

8.

For the model IB:

rt = −3.925 + 0.5gt

XB:

rt = 7.958 + 0.186gt

where gt+1 = −0.5(gt − g∗t ) rt+1 = −0.75(rt − rt∗ )

9.

use a spreadsheet to plot the trajectories for the following initial values: (a) (g0 ,r0 ) = (20,12) (b) (g0 ,r0 ) = (20,20) (c) (g0 ,r0 ) = (50,10) (d) (g0 ,r0 ) = (50,20) For the assignment of interest rates to achieve internal balance and government spending to achieve external balance, we have the set of equations gt+1 = −21.3925 + 0.5gt + 2.688rt rt+1 = −2.94375 + 0.375gt + 0.25rt

10.

(a) Take four initial points, one in each of the sectors represented in ﬁgure 5.17, and use a spreadsheet to investigate the economy’s trajectory. (b) Take a variety of points in sectors I and III and establish whether the trajectories remain in these sectors. (c) Take a point on the stable arm and, using a spreadsheet, establish that the trajectory tends to the equilibrium point. For the system IB:

St = 5 + 2gt

XB:

St = 20 − 4gt

Discrete systems of equations where gt+1 = −0.75(gt − g∗t ) St+1 = −0.5(St − St∗ ) and g∗t = −2.5 + 0.5St St∗ = 20 − 4gt

11.

12.

for the target equations, using a spreadsheet establish the trajectories for the following initial conditions: (a) (g0 ,S0 ) = (2.5,12) (b) (g0 ,S0 ) = (3,10) (c) (g0 ,S0 ) = (1,5) (d) (g0 ,S0 ) = (1,12) Let 2 3 4 −2 3 2 1 , mB = , mC = mA = 1 −2 1 −1 −1 0 3 Using any software package, perform the following operations (i) Trace and determinant of mA × mB (ii) Transpose of mA × mC (iii) Inverse of mB (iv) Eigenvalues and eigenvectors of mA and mB (v) Characteristic polynomial of mA. Solve the following system using a software package xt+1 = −5 + xt − 2yt yt+1 = 4 + xt − yt x0 = 1,

13.

14.

15.

y0 = 2

What is the Jordan form, J, and the transition matrix, V, of the following matrix? 1 −2 mA = 1 −1 Hence show that V−1 .mA.V = J. For the system in question 12, set this up on a spreadsheet. (i) What is the ﬁxed point of the system? (ii) Plot the trajectory from the initial point. Does this trajectory converge on the ﬁxed point? For the following system, establish the Jordan form and the transition matrix. Represent the original system and its canonical form on a spreadsheet, and hence show whether the system is asymptotically stable. xt = 5.6 − 0.4xt−1 yt = 3.5 + 0.4xt−1 − 0.5yt−1 x0 = 2,

y0 = 1

249

250

Economic Dynamics Additional reading Discrete systems of equations are discussed less frequently than continuous systems of equations, but additional material on the mathematical contents of this chapter can be found in Azariades (1993), Chiang (1984), Elaydi (1996), Goldberg (1961), Grifﬁths and Oldknow (1993), Holmgren (1994), Kelley and Peterson (2001), Lynch (2001), Sandefur (1990), Shone (2001), Simon and Blume (1994) and Tu (1994). On internal and external balance, references will be found in the main body of the chapter in section 5.8.

CHAPTER 6

Optimal control theory

6.1 The optimal control problem Consider a ﬁsh stock which has some natural rate of growth and which is harvested. Too much harvesting could endanger the survival of the ﬁsh, too little and proﬁts are forgone. Of course, harvesting takes place over time. The obvious question is: ‘what is the best harvesting rate, i.e., what is the optimal harvesting?’ The answer to this question requires an optimal path or trajectory to be identiﬁed. ‘Best’ itself requires us to specify a criterion by which to choose between alternative paths. Some policy implies there is a means to inﬂuence (control) the situation. If we take it that x(t) represents the state of the situation at time t and u(t) represents the control at time t, then the optimal control problem is to ﬁnd a trajectory {x(t)} by choosing a set {u(t)} of controls so as to maximise or minimise some objective that has been set. There are a number of ways to solve such a control problem, of which the literature considers three: (1) (2) (3)

Calculus of variations Dynamic programming Maximum principle.

In this chapter we shall deal only with the third, which now is the dominant approach, especially in economics. This approach is based on the work of Pontryagin et al. (1962), and is therefore sometimes called the Pontryagin maximum principle. Since minimising some objective function is the same as maximising its negative value, then we shall refer in this chapter only to maximising some objective function. Second, our control problem can either be in continuous time or in discrete time. To see the difference and to present a formal statement of the optimal control problem from the maximum principle point of view, consider table 6.1. In each case, the objective is to maximise J and so ﬁnd a trajectory {x(t)} by choosing a suitable value {u(t)}. What table 6.1 presents is the most general situation possible for both the continuous and discrete formulations of the optimal control problem under the maximisation principle. There are some special cases, the most important being the distinction between ﬁnite and inﬁnite horizon models. In the latter case the terminal time period is at inﬁnity. All the problems we shall discuss in this chapter involve autonomous systems, and so t does not enter explicitly into V, f or F. An important aspect of control problems is that of time preference. The

252

Economic Dynamics Table 6.1 The control problem Continuous max J = {u(t)}

t1

Discrete V(x, u, t)dt + F(x1 , t)

t0

x˙ = f (x, u, t) x(t0 ) = x0 x(t1 ) = x1 {u(t)} ∈ U

max J = {ut }

T−1

V(xt , ut , t) + F(xT , t)

t=0

xt+1 − xt = f (xt , ut , t) xt = x0 when t = 0 xt = xT when t = T ut ∈ U

t0 (or t = 0) is initial time t1 (or T) is terminal time x(t) = {x1 (t), . . . , xn (t)} or xt = {x1t , . . . , xnt } n-state variables x(t0 ) = x0 or xt = x0 for t = 0 is the initial state x(t1 ) = x1 or xt = xT for t = T is the ﬁnal state (or terminal state) u(t) = {u1 (t), . . . , um (t)} or ut = {u1t , . . . , umt } m-control variables {u(t)} is a continuous control trajectory t0 ≤ t ≤ t1 {ut } is a discrete control trajectory 0 ≤ t ≤ T U is the set of all admissible control trajectories x(t) ˙ = f (x, u, t) or xt+1 − xt = f (xt , ut , t) denote the equations of motion J is the objective function V(x(t), u(t), t) or V(xt , ut , t) is the intermediate function F(x1 , t) or F(xT , t) is the ﬁnal function

simplest models involve no discounting. It is sometimes easier to consider a model with no discounting, and then to consider the more realistic case of the same model with discounting. In many models the terminal value F(xT ) is zero, but this need not always be so. A typical continuous optimal control problem incorporating the assumptions of (1) a ﬁnite time horizon, T, (2) only autonomous equations, (3) a zero function in the terminal state and, (4) only one state variable and one control variable is T V(x, u)dt max J = {u(t)}

(6.1)

x˙ = f (x, u) x(0) = x0 x(T) = xT where the state variable, x and the control variable, u, are both functions of time t. The situation is illustrated in ﬁgure 6.1. The paths u∗ and u∗∗ both constitute solutions to the differential equation x˙ = f (x, u). The problem, however, is to choose one path that maximises the relation J and that satisﬁes the terminal condition x(t∗ ) = xT and x(t∗∗ ) = xT .

6.2 The Pontryagin maximum principle: continuous model As just pointed out, the objective is to ﬁnd a control trajectory {u(t)} that maximises J and takes the system from its present state x0 to its terminal state xT. What is required, therefore, is a ‘set of weights’ that allows a comparison of the different trajectories of alternative controls. Also note that the emphasis of this formulation

Optimal control theory

253 Figure 6.1.

xT

of the control problem is to ﬁnd the optimal control trajectory {u(t)}. Once this is known the optimal state trajectory {x(t)} can be computed. The ‘weights’ are achieved by deﬁning a Hamiltonian for the control problem (6.1). As with Lagrangian multipliers, let λ(t) denote the Lagrangian multiplier for the constraint x˙ = f (x, u). This is referred to as the costate variable or adjoint variable. Then T T ˙ V(x, u)dt + λ[ f (x, u) − x]dt L= 0

=

0 T

˙ [V(x, u) + λ f (x, u) − λx]dt

The Hamiltonian function is deﬁned as H(x, u) = V(x, u) + λ f (x, u) Hence

T

L=

˙ [H(x, u) − λx]dt

(6.2)

(6.3)

Equation (6.3) can be further transformed by noting that (see exercise 2) T T ˙ − [λ(T )x(T ) − λ(0)x(0)] λdt = xλdt − 0

(6.4)

which allows us to express L as T ˙ [H(x, u) + λx]dt − [λ(T)x(T) − λ(0)x(0)] L= 0

Consider what happens to the state variable when the control variable changes, i.e., let {u(t)} change to {u(t) + u(t)} with the result on the state trajectory from {x(t)} to {x(t) + x(t)}. Then the change in the Lagrangian, L, is T ∂H ∂H ˙ L = dx + du + λdx dt − λ(T)dxT ∂x ∂u 0 T ∂H ˙ ∂H du + + λ dx dt − λ(T)dxT = ∂u ∂x 0

(6.5)

254

Economic Dynamics For a maximum L = 0. This implies the necessary conditions: (i) (ii) (iii)

∂H =0 0≤t≤T ∂u ∂H λ˙ = − 0≤t≤T ∂x λ(T) = 0 (or x(T) = xT if xT is known)

Condition (i) states that the Hamiltonian function is maximised by the choice of the control variable at each point along the optimum trajectory – where we are assuming an interior solution and no constraint on the control variable. Condition (ii) is concerned with the rate of change of the costate variable, λ. It states that the rate of change of the costate variable is equal to the negative of the Hamiltonian function with respect to the corresponding state variable.1 Condition (iii) refers to the costate variable in the terminal state, and indicates that it is zero; or if the terminal value x(T) = xT is given then dxT = 0. From the deﬁnition of the Hamiltonian function, the differential equation for the state variable can be expressed in terms of it as follows x˙ = f (x, u) =

∂H ∂λ

We therefore arrive at the following procedure. Add a costate variable λ to the problem and deﬁne a Hamiltonian function H(x, u) = V(x, u) + λf (x, u) and solve for trajectories {u(t)}, {λ(t)}, and {x(t)} satisfying:

(iv)

∂H =0 0≤t≤T ∂u ∂H 0≤t≤T λ˙ = − ∂x ∂H = f (x, u) x˙ = ∂λ x(0) = x0

(v)

λ(T) = 0

(i) (ii) (6.6)

(iii)

(or x(T) = xT )

These results can be generalised for x1 (t), . . . , xn (t) state variables, λ1 (t), . . . , λn (t) costate variables and u1 (t), . . . , um (t) control variables: (i) (6.7)

(ii) (iii)

1

∂H =0 i = 1, . . . , m 0≤t≤T ∂ui ∂H λ˙ = − i = 1, . . . , n 0≤t≤T ∂xi ∂H x˙ = = f (x, u) i = 1, . . . , n ∂λi

If there were, for example, two state variables x1 and x2 and two corresponding costate variables λ1 and λ2 , then λ˙ 1 = −∂H/∂x1 0≤t≤T 0≤t≤T λ˙ 2 = −∂H/∂x2

Optimal control theory (iv)

xi (0) = xi0

i = 1, . . . , n

(v)

λi (T) = 0

i = 1, . . . , n

(or xi (T) = xiT

i = 1, . . . n)

We shall now illustrate the continuous control problem by considering three examples. In each case we have the initial value and the terminal value for the state variable, i.e., x(0) = x0 and x(T) = xT are given, as of course is T. Example 6.1 In this ﬁrst example we consider a boundary solution. The control problem is: 1 max 5x dx {u}

x˙ = x + u x(0) = 2, u(t) ∈ [0, 3]

x(1) free

The Hamiltonian for this problem is H(x, u) = V(x, u) + λ f (x, u) = 5x + λ(x + u) = (5 + λ)x + λu With ﬁrst-order conditions:

(iii)

∂H =λ ∂u ∂H = −(5 + λ) λ˙ = − ∂x x˙ = x + u

(iv)

x(0) = 2

(v)

λ(1) = 0

(i) (ii)

Condition (i) is no help in determining u∗ . If λ > 0 then H is a maximum at u = 3 the boundary, hence u∗ (t) = 3, as shown in Figure 6.2(a). From (ii) we have λ˙ = −λ − 5 λ∗ (t) = ke−t − 5 λ∗ (1) = ke−1 − 5 = 0 k = 5e1 ...

λ∗ (t) = 5e1−t − 5

Since u∗ (t) = 3 x˙∗ = x∗ + 3 x∗ (t) = −3 + ket x(0) = −3 + ke0 = 2 ... k = 5

255

256

Economic Dynamics

Figure 6.2.

Hence x∗ (t) = −3 + 5et Although the control variable remains constant throughout, the state variable increases from x(0) = 2, as shown in ﬁgure 6.2(b). Example 6.2 The control problem is 1 max u2 dt {u}

x˙ = −u x(0) = 1 x(1) = 0 The Hamiltonian for this problem is H(x, u) = V(x, u) + λ f (x, u) = u2 + λ(−u)

Optimal control theory with ﬁrst-order conditions: (i) (ii) (iii) (iv) (v)

∂H = 2u − λ = 0 ∂u ∂H =0 λ˙ = − ∂x x˙ = −u x(0) = 1 x(1) = 0

From (i) 2u = λ u = 12 λ Thus x˙ = −

λ 2

λ˙ = 0 Solving these with a software package we obtain x(t) = c1 −

λt 2

λ(t) = c2 But x(0) = 1 so 1 = c1 −

0 2

c1 = 1

or

Similarly x(1) = 0 λ =0 2 ... λ = 2 or x(1) = 1 −

c2 = 2

2t =1−t x∗ = 1 − 2 u∗ = 12 λ = 1 These optimal paths are illustrated in ﬁgure 6.3. Example 6.3 The control problem is 1 1 2 (x + u2 )dt max − 4 {u}

x˙ = x + u x(0) = 2,

x(1) = 0

257

258

Economic Dynamics

Figure 6.3.

The Hamiltonian for this problem is H(x, u) = V(x, u) + λ f (x, u) −(x2 + u2 ) + λ(x + u) 4 With ﬁrst-order conditions =

(i) (ii) (iii)

∂H u =− +λ=0 implying u = 2λ ∂u 2 −x ∂H λ˙ = − =− + λ = 12 x − λ ∂x 2 x˙ = x + u

implying

x˙ = x + 2λ

Substituting (i) into (iii) and eliminating u, we arrive at two differential equations in terms of x and λ x˙ = x + 2λ λ˙ = 12 x − λ Although a simple set of differential equations, the solution values are rather involved, especially when solving for the constants of integration. The general solution is2 √

√

x(t) = c1 e 2 t + c2 e− 2 t √ √ c1 √ c2 √ λ(t) = ( 2 − 1)e 2 t − ( 2 + 1)e− 2 t 2 2 However we can solve for c1 and c2 by using the conditions x(0) = 2 and x(1) = 0 as follows x(0) = c1 + c2 = 2 x(1) = c1 e 2

√

2t

√

+ c2 e−

2t

=0

The software packages give, on the face of it, quite different solutions. They are, however, identical. The results provided here are a re-arrangement of those provided by Maple.

Optimal control theory

259 Figure 6.4.

Solving we get c1 = −0.1256 and c2 = 2.1256. All this can be done with the help of computer software programs, with the resulting trajectories for x∗ and u∗ shown in ﬁgure 6.4(a) and (b). What these examples show is a pattern emerging for solving the control problem. The steps are: (1) (2) (3) (4) (5) (6)

Specify the Hamiltonian and obtain the maximisation conditions Use the equation ∂H/∂u to solve for u in terms of the costate variable λ Obtain two differential equations: one for the state variable, x, and one for the costate variable, λ Solve the differential equations deriving general solutions Use the conditions on x(0) and x(T) to obtain values for the coefﬁcients of integration Substitute the optimal path for λ∗ into the equation for u to obtain the optimal path u∗ for the control variable.

6.3 The Pontryagin maximum principle: discrete model The discrete time control model based on the maximum principle of Pontryagin takes a similar approach to the continuous time formulation so we can be brief,

260

Economic Dynamics although some care must be exercised in the use of time periods. Again we let x denote the only state variable, u the only control variable and λ the costate variable. Our problem amounts to: max J = {ut }

(6.8)

T−1

V(xt , ut )

t=0

xt+1 − xt = f (xt , ut ) x0 = a The Lagrangian is then L=

(6.9)

T−1

{V(xt , ut ) + λt+1 [ f (xt , ut ) − (xt+1 − xt )]}

t=0

Deﬁne the discrete form Hamiltonian function H(xt , ut ) = V(xt , ut ) + λt+1 f (xt , ut )

(6.10)

then L=

T−1

[H(xt , ut ) − λt+1 (xt+1 − xt )]

t=0

which can be maximised by satisfying the conditions ∂L ∂H = =0 t = 0, . . . , T − 1 ∂ut ∂ut ∂L ∂H = + λt+1 − λt = 0 t = 1, . . . , T − 1 ∂xt ∂xt ∂H ∂L = − (xt+1 − xt ) t = 0, . . . , T − 1 ∂λt+1 ∂λt+1 ∂L = −λT = 0 ∂xT More succinctly: (i) (ii) (6.11)

(iii)

∂H =0 ∂ut

t = 0, . . . , T − 1

∂H t = 1, . . . , T − 1 ∂xt ∂H xt−1 − xt = = f (xt , ut ) t = 0, . . . , T − 1 ∂λt+1 λt+1 − λt = −

(iv)

λT = 0

(v)

x0 = a

It is useful to verify these conditions for, say, T = 3, most especially noting the range for t for condition (ii). But how do we go about solving such a model? Unlike the continuous time model it is not simply solving two differential equations. It is true that in each time period we have two difference equations for the state and costate variables that require solving simultaneously. One solution method is to program the problem, as

Optimal control theory in Conrad and Clark (1987). A simpler method in the case of numerical examples is to use a spreadsheet. To illustrate the solution method by means of a spreadsheet, consider the following example. Example 6.4 3 Iron ore sells on the market at a constant price p per period but costs ct = by2t/xt , where xt denotes the remaining reserves at the beginning of period t and yt is the production in period t. The mine is to be shut down in period 10. What is the optimal production schedule {y∗t } for t = 0, . . . , 9 given p = 3, b = 2 and the initial reserves x0 = R = 600 tons? (Assume no discounting over the period.) Let us ﬁrst set up the model in general terms, replacing ut by yt . The objective function V(xt , yt ) is no more than the (undiscounted) proﬁt, namely by2t byt yt V(xt , yt ) = pyt − = p− xt xt Next we note that if xt denotes the remaining reserves at the beginning of period t, then xt+1 = xt − yt or xt+1 − xt = −yt . Thus, our Hamiltonian function is byt yt − λt+1 yt H(xt , yt ) = p − xt Our optimality conditions are therefore:

(iii)

2byt ∂H =p− − λt+1 = 0 t = 0, . . . , 9 ∂yt xt 2 byt ∂H t = 1, . . . , 9 =− λt+1 − λt = − ∂xt xt2 xt−1 − xt = −yt t = 0, . . . , 9

(iv)

x0 = R

(v)

λT = 0

(i) (ii)

To solve this problem for a particular numerical example, let p = 3, b = 2 and R = 600. The computations are set out in detail in ﬁgure 6.5. In doing these computations we begin in period 10 and work backwards (see exercise 1 on backward solving). Since λ10 = 0 then from (i) we know y9 3−4 =0 x9 which allows us to compute y9 /x9 . Having solved for y9 /x9 we can then use condition (ii) to solve for λ9 . We do this repeatedly back to period 0. This gives us columns 2 and 3 of the spreadsheet. Since x0 = R = 600, we have the ﬁrst entry in the x(t) column. Then y0 is equal to x0 ( y0 /x0 ) and ﬁnally x1 = x0 − y0 . This allows us to complete the ﬁnal two columns. The optimal production path {y∗t } is therefore given by the ﬁnal column in ﬁgure 6.5 and its path, along with that of the reserves, is shown in ﬁgure 6.6(a). 3

This is adapted from Conrad and Clark (1987, p. 20).

261

262

Economic Dynamics

Figure 6.5.

Given the computations the trajectory for (λ∗t , xt∗ ) can also be plotted, which is shown in ﬁgure 6.6(b), which are direct plottings from a spreadsheet. In this example we solved the discrete optimisation problem by taking account of the ﬁrst-order conditions and the constraints. We employed the spreadsheet merely as a means of carrying out some of the computations. However, spreadsheets come with nonlinear programming algorithms built in. To see this in operation, let us re-do the present example using Excel’s nonlinear programming algorithm, which is contained in the Solver add-on package.4 The initial layout of the spreadsheet is illustrated in ﬁgure 6.7. It is important to note that when setting out this initial spreadsheet we place in cells B7 to B16 some ‘reasonable’ numbers for extraction. Here we simply assume a constant rate of extraction of 60 throughout the 10 periods t = 0 to t = 9. Doing this allows us to compute columns D and E. Column D sets λ10 = 0 and then copies backwards the formula 2 byt λt = λt+1 + xt2 for cells D16 to D8 (no value is placed in cell D7). The values in column E are the values for the objective function V(xt , yt ). The value for L, which is the sum of the values in column E for periods t = 0 to t = 9, is placed in cell E19. At the moment this stands at the value 1448.524. 4

On using Excel’s Solver see Whigham (1998), Conrad (1999) and Judge (2000).

Optimal control theory

263 Figure 6.6.

Figure 6.7.

264

Economic Dynamics

Figure 6.8.

Of course it would be most unlikely if L were at a maximum with such arbitrary numbers for extraction. The maximum control problem is to maximise the value in cell E19, i.e., maximise L, subject to any constraints and production ﬂows. The constraints are already set in the spreadsheet, although we do require others on the sign of variables. First move the cursor to cell E19 and then invoke the solver. By default this is set to maximise a cell value, namely cell E19. We next need to inform the programme which is the control variable and hence which values can be changed, i.e., what cells it can change in searching for a maximum. These are cells B7 to B16. In specifying the above problem we implicitly assumed xt and yt were both positive. In particular, we assumed the level of production, the control variable, was positive. We need to include this additional constraint in the Solver so that any negative values are excluded from the search process. The Solver window is shown in ﬁgure 6.8. Once all this information has been included the Solver can do its work. The result is shown in the spreadsheet in ﬁgure 6.9. As can be observed this gives more or less the same results as ﬁgure 6.5, as it should. The value of the objective function has also increased from 1448.52 to 1471.31. It should be noted in ﬁgure 6.9 that in period 10 we have λ(10) = 0 and at this value x(10) = 9.9811. We have to assume that the reserves in period 10 are therefore 9.9811 and that these are simply left in the ground. In other words, x(T) is free. The shadow price of a free product is zero, hence λ(10) = 0, and this implies it is not optimal to mine the remaining reserves. Hence F(xT ) = 0 or xT is free. We have spent some time on this problem because it illustrates the use of spreadsheets without having to handle algebraically the ﬁrst-order conditions. It also has the advantage that it can handle corner solutions.5 Most important of all, it provides a way of solving real-life problems. 5

Corner solutions would require setting out the Kuhn–Tucker conditions for optimisation. See Chiang (1984), Simon and Blume (1994) and Huang and Crooke (1997).

Optimal control theory

265 Figure 6.9.

6.4 Optimal control with discounting We have noted that a major feature of the control problem is to maximise the objective function V(x, u). However, for many economic problems V(x, u) would represent such things as proﬁts or net beneﬁts. The economist would not simply maximise such an income stream without ﬁrst discounting it to the present. Thus, if δ were the rate of discount then the aim of the control would be to T e−δt V(x, u)dt max J = {u(t)}

(6.12)

subject to various conditions which are unaffected by the discounting. Thus, the typical continuous time maximisation principle problem with discounting is the control problem T e−δt V(x, u)dt max J = {u(t)}

x˙ = f (x, u) x(0) = x

(6.13)

x(T) = xT while the discrete form is max J = {ut }

T−1

ρ t V(xt , ut )

t=0

xt+1 − xt = f (xt , ut ) x0 = a where ρ = 1/(1 + δ) and ρ is the discount factor while δ is the discount rate.

(6.14)

266

Economic Dynamics Let us ﬁrst consider the discrete form. The Lagrangian is L=

(6.15)

T−1

ρ t {V(xt , ut ) + ρλt+1 [ f (xt , ut ) − (xt+1 − xt )]}

t=0

Notice in this expression that λt+1 is discounted to period t by multiplying it by the discount factor ρ. But then the whole expression {.} is discounted to the present by multiplying by the term ρ t. We now introduce a new concept: the current value Hamiltonian function, denoted Hc (x, u). This is deﬁned, for the discrete case, as Hc (xt , ut ) = V(xt , ut ) + ρλt+1 f (xt , ut )

(6.16)

and in all other respects the optimisation conditions are similar, i.e. (i) (ii) (6.17)

(iii)

∂Hc =0 ∂ut

t = 0, . . . , T − 1

∂Hc t = 1, . . . , T − 1 ∂xt ∂Hc xt−1 − xt = = f (xt , ut ) t = 0, . . . , T − 1 ∂ρλt+1 ρλt+1 − λt = −

(iv)

λT = 0

(v)

x0 = a

We can illustrate this with the mine example (example 6.4), but now assume a discount rate of 10%. With a discount rate of 10% the discount factor ρ = 1/(1 + 0.1) = 0.909091. Example 6.5 Given p = 3, R = 600 and ρ = 0.909091 byt ρt p − yt max J = {yt } xt xt+1 − xt = −yt x0 = R The current value Hamiltonian is byt Hc (xt , yt ) = p − yt − ρλt+1 yt xt with optimality conditions: (i) (ii) (iii)

∂Hc 2byt =p− − ρλt+1 = 0 t = 0, . . . , 9 ∂yt xt 2 by ∂Hc ρλt+1 − λt = − = − 2t t = 1, . . . , 9 ∂xt xt ∂Hc = −yt t = 0, . . . , 9 xt−1 − xt = ∂ρλt+1

Optimal control theory

267 Figure 6.10.

(iv)

λT = 0

(v)

x0 = R = 600

It should be noted that the only difference between this and the undiscounted conditions is in terms of condition (i), where λt+1 is multiplied by the discount factor. Once again we use Excel’s Solver to handle the computations of this problem, with the results shown in ﬁgure 6.10, which should be compared with ﬁgure 6.9. Notice once again that in period 10 we have λ(10) = 0 and that at this value x(10) = 2.9362. This level of reserves in period 10 is simply left in the ground, x(T) is free. The shadow price of a free good is zero, hence λ(10) = 0, and this implies it is not optimal to mine the remaining reserves. Put another way, it is cheaper to leave the remaining reserves unmined than incur the costs of mining them. What these computations show is a similar trajectory for optimal production but starting from a much higher level of production. This is understandable. The future in a discounting model is weighted less signiﬁcantly than the present. The comparison is shown in ﬁgure 6.11. Consider now discounting under a continuous time model. Consider the control problem T e−δt V(x, u)dt max J = {u(t)}

x˙ = f (x, u) x(0) = 0 x(T) = xT

The Lagrangian is T ˙ {e−δt V(x, u) + λ[ f (x, u) − x]}dt L= 0

(6.18)

268

Economic Dynamics

Figure 6.11.

and the Hamiltonian is H(x, u) = e−δt V(x, u) + λf (x, u) Deﬁne the current value Hamiltonian function, Hc , by Hc (x, u) = V(x, u) + µ f (x, u) then Hc = Heδt µ = λeδt

or or

H = Hc e−δt λ = µe−δt

Now reconsider our ﬁve optimality conditions. Since eδt is a constant for a change in the control variable, then condition (i) is simply ∂Hc /∂u = 0. The second condition is less straightforward. We have ∂Hc −δt ∂H =− e λ˙ = − ∂x ∂x From λ = µe−δt λ˙ = µe ˙ −δt − δµe−δt Equating these we have ∂Hc −δt e = µe ˙ −δt − δµe−δt ∂x ∂Hc + δµ or µ ˙ =− ∂x Condition (iii) is −

x˙ =

∂Hc −δt ∂H ∂Hc = e = = f (x, u) ∂λ ∂λ ∂µ

while condition (iv) becomes λ(T) = µ(T)e−δt = 0 and condition (v) remains unchanged.

Optimal control theory

269

To summarise, deﬁne the current value Hamiltonian and current value Lagrangian multiplier, i.e. Hc (x, u) = H(x, u)eδt = V(x, u) + µ f (x, u) where λ = µe−δt . Then the optimality conditions are: (i) (ii) (iii)

∂Hc =0 0≤t≤T ∂u ∂Hc µ ˙ =− + δµ 0≤t≤T ∂x ∂Hc x˙ = = f (x, u) ∂µ

(iv)

x(0) = x0

(v)

µ(T)e−δt = 0

(or x(T) = xT )

These optimality conditions allow us to eliminate the control variable u using condition (i) and to obtain two differential equations: one for the state variable, x, and the other for the current value costate variable, µ. Example 6.6

10

max J = − {u}

u2 e−0.1t dt

x˙ = u x(0) = 0 x(10) = 1000 and ﬁnd the optimal path x∗ (t). The current value Hamiltonian is Hc = −u2 + µu with optimality conditions

(ii)

∂Hc = −2u + µ = 0 ∂u µ ˙ = 0 + 0.1µ

(iii)

x˙ = u

(i)

From (i) we have u = 0.5µ, which when substituted into (iii) gives x˙ = 0.5µ. Thus we have two differential equations x˙ = 0.5µ µ ˙ = 0.1µ Solving we obtain x(t) = c1 + c2 e0.1t µ(t) = 0.2c2 e0.1t

(6.19)

270

Economic Dynamics Given x(0) = 0 and x(10) = 1000 then we can solve for c1 and c2 by solving c1 + c2 = 0 0.2c2 e = 1000 which gives c1 = −581.9767 and c2 = 581.9767. Hence x∗ (t) = −581.9767 + 581.9767e0.1t = 581.9767(e0.1t − 1)

6.5 The phase diagram approach to continuous time control models First let us reconsider examples 6.1–6.3. Example 6.1 (cont.) In example 6.1 we derived the two differential equations x˙ = x + u λ˙ = −(5 + λ) In this instance, ∂H/∂u = λ which is of no help in eliminating u. We did, however, establish that H is a maximum when u = 3 and that this variable remains constant throughout. Therefore, x˙ = x + 3 λ˙ = −5 − λ and so we have two isoclines. The x-isocline at x = −3 and the λ-isocline at λ = −5. Furthermore, x˙ > 0 λ˙ < 0

implies

x > −3

implies

λ > −5

so we know that the optimal trajectory starting from x(0) = 2 will lead to a rise in the state variable x and a fall in the costate variable λ. This is veriﬁed in ﬁgure 6.12. The system begins from point (x(0), λ(0)) = (2, 8.5914), satisfying the initial condition on the state variable x; and has a terminal point (x(1), λ(1)) = (10.5914, 0), which satisﬁes the terminal condition on the costate variable, λ. Of all possible trajectories in the phase plane, this is the optimal trajectory. Example 6.2 (cont.) In example 6.2 we derived the following two differential equations x˙ = − 12 λ λ˙ = 0 There is only one isocline for this problem. When x˙ = 0 then λ = 0 and so the x-isocline coincides with the x-axis. Our initial point is (x(0), λ(0)) = (1, 2) and

Optimal control theory

271 Figure 6.12.

Figure 6.13.

for λ > 0 we have x˙ < 0 and so the trajectory is moving to the left. Earlier we demonstrated that λ remains at the value of 2 throughout the trajectory. When t = 1 then x(1) = 0, which satisﬁes the condition on the terminal point, which in the phase plane is the point (x(1), λ(1)) = (0, 2). As can be seen in terms of ﬁgure 6.13, the optimal trajectory in the phase plane is the horizontal line pointing to the left.

Example 6.3 (cont.) The two differential equations we derived for example 6.3 were x˙ = x + 2λ λ˙ = 1 x − λ 2

272

Economic Dynamics When x˙ = 0 then λ = − 12 x and when λ˙ = 0 then λ = 12 x. We have therefore two distinct isoclines in this example. Furthermore, if x˙ > 0 then x + 2λ > 0 implying λ > − 12 x Hence, above the x-isocline, x is rising while below it is falling. Similarly, if λ˙ > 0 then 12 x − λ > 0 implying λ < 12 x Hence, below the λ-isocline, λ is rising while above it is falling. This suggests that we have a saddle-point solution. This is also readily veriﬁed by considering the eigenvalues of the system. The matrix of the system is 1 2 A= 1 −1 2 √ √ with eigenvalues r = 2 and s = − 2. Since these are real and of opposite sign, then we have a saddle point solution. When t = 0 we already have x(0) = 2 but we need to solve for λ(0). But λ(0) =

c2 √ c1 √ ( 2 − 1) − ( 2 + 1) 2 2

and we know that c1 = −0.1256 and c2 = 2.1256. Substituting these values we get λ(0) = −2.6. The initial point (x(0), λ(0)) = (2, −2.6) therefore begins below the x-isocline, and so the vector forces are directing the system up and to the left. The optimal trajectory is shown in ﬁgure 6.14. It is apparent from example 6.3 and 6.6 that the maximisation approach of Pontryagin gives us ﬁrst-order conditions in terms of the Hamiltonian which, in the present simple models, leads to two differential equations in terms of the state variable x and the costate variable λ (or µ). Control problems, however, pose two difﬁculties: (1) (2)

the differential equations are often nonlinear in economics functional forms are often unspeciﬁed.6

Even the most simple control problem can lead to nonlinear differential equations, and although we have developed techniques elsewhere for dealing with these,7 until the advent of the computer they were largely left to the mathematician. When the functional forms are not even speciﬁed then there are no explicit differential equations to solve. However, the qualitative properties of the ﬁxed points can still be investigated by considering the system’s qualitative properties in the phase plane. First consider a simple example for which we have an explicit solution. 6 7

What we often know are certain properties. Thus we may have a production function y = f (k) where f (k) is unspeciﬁed other than being continuous, differentiable and where f (k) > 0 and f (k) < 0. See sections 2.7 and 3.9.

Optimal control theory

273 Figure 6.14.

Example 6.7 8 Our problem is

max J = {u}

∞

(20 ln x − 0.1u2 )dt

x˙ = u − 0.1x x(0) = 80 The Hamiltonian for this problem is H = 20 ln x − 0.1u2 + λ(u − 0.1x) with ﬁrst-order conditions ∂H = −0.2u + λ = 0 ∂u 20 ∂H =− − 0.1λ λ˙ = − ∂x x x˙ = u − 0.1x which can be reduced to two differential equations in terms of x and λ x˙ = −0.1x + 5λ −20 λ˙ = + 0.1λ x The ﬁxed point of this system is readily found by setting x˙ = 0 and λ˙ = 0, giving x∗ = 100 and λ∗ = 2. Furthermore, the two isoclines are readily found to be λ = 0.02x

(x˙ = 0)

200 λ= (λ˙ = 0) x and illustrated in ﬁgure 6.15. 8

Adapted from Conrad and Clark (1987, pp. 46–8).

274

Economic Dynamics Figure 6.15 also shows the vector of forces in the four quadrants, which readily indicate a saddle point solution. This can be veriﬁed by considering a linearisation about the ﬁxed point (x∗ , λ∗ ) = (100, 2). This gives the linear equations x˙ = −0.1(x − x∗ ) + 5(λ − λ∗ ) λ˙ = 0.002(x − x∗ ) + 0.1(λ − λ∗ ) The resulting matrix of the linear system is −0.1 5 A= 0.002 0.1 with eigenvalues r = 0.14142 and s = −0.14142, conﬁrming a saddle point solution. To establish the equations of the arms of the saddle point solution, take ﬁrst the eigenvalue r = 0.14142. Then (A − rI)vr = 0 i.e.

or

−0.1 0.002

5 1 − 0.14142 0.1 0

−0.24142 0.002

5 −0.04142

Using the ﬁrst equation, −0.24142vr1 + 5vr2 = 0 Figure 6.15.

vr1 vr2

0 1

0 = 0

vr1 vr2

0 = 0

Optimal control theory Let vr1 = 1, then 5vr2 = 0.24142 vr2 = 0.048284 Therefore, (λ − λ∗ ) = 0.048284(x − x∗ ) (λ − 2) = 0.048284(x − 100) i.e. λ = −2.8284 + 0.48284x and since this is positively sloped it represents the equation of the unstable arm. Now consider the second eigenvalue, s = −0.14142 s −0.1 5 1 0 v1 0 s + 0.14142 = (A − rI)v = 0.002 0.1 0 1 vs2 0 i.e.

0.04142 0.002

5 0.24142

vs1 vs2

0 = 0

Using again the ﬁrst equation, then 0.04142vs1 + 5vs2 = 0 Let vs1 = 1, then 5vs2 = −0.04142 vs2 = −0.008284 Therefore, (λ − λ∗ ) = −0.008284(x − x∗ ) (λ − 2) = −0.008284(x − 100) i.e. λ = 2.8284 − 0.00828284x and since this is negatively sloped it represents the equation of the stable arm. If x(0) = 80 then the value of λ on the stable arm is λ(0) = 2.16568. The trajectory, along with isoclines and the stable arm, are shown in ﬁgure 6.16. Although the point begins on the stable arm, it gets pulled away before it reaches the equilibrium! What this diagram reveals is that this system is very sensitive to initial conditions. But the direction ﬁeld does show a clear saddle point equilibrium. Example 6.8 (Ramsey growth model) In this example we shall consider the Ramsey growth model,9 which is the basis of much of the optimal growth theory literature. We shall consider the model in terms 9

Ramsey (1928). See also Burmeister and Dobell (1970), Barro and Sala-i-Martin (1995) and Romer (2001).

275

276

Economic Dynamics

Figure 6.16.

of continuous time. We begin with simple deﬁnitions of income and investment, namely Y(t) = C(t) + I(t) ˙ + δK(t) I(t) = K(t)

(6.20)

Hence ˙ C(t) K(t) δK(t) Y(t) = + + L(t) L(t) L(t) L(t) ˙ K(t) i.e. y(t) = c(t) + + δk(t) L(t) But

˙ ˙ ˙ ˙ ˙k = d K = LK − K L = K − K L dt L L2 L L L L˙ K˙ = −k L L

˙ = n hence We assume population grows at a constant rate n, so that L/L K˙ = k˙ + kn L and ˙ + (n + δ)k(t) y(t) = c(t) + k(t) If we have a homogeneous of degree one production function then we can express output, y, as a function of k. Thus, y = f (k). Dropping the time variable for convenience, we therefore have the condition (6.21)

k˙ = f (k) − (n + δ)k − c In order to consider the optimal growth path we require to specify an objective. Suppose U(c) denotes utility as a function of consumption per head. The aim is

Optimal control theory to maximise the discounted value of utility subject to the equation we have just derived, i.e. ∞ e−βt U(c)dt max J = {c}

k˙ = f (k) − (n + δ)k − c k(0) = k0 0 ≤ c ≤ f (k) The current value Hamiltonian function is Hc = U(c) + µ[ f (k) − (n + δ)k − c] with ﬁrst-order conditions: ∂Hc = U (c) − µ = 0 (i) ∂c (ii) µ ˙ = −µ f (k) + µ(n + δ) + βµ (iii)

k˙ = f (k) − (n + δ)k − c

or (i)

U (c) = µ

(ii)

µ ˙ = −µf (k) + (n + δ + β)µ

(iii)

k˙ = f (k) − (n + δ)k − c

As they stand these equations are not easy to interpret or solve. We can, however, with some rearrangement, derive two differential equations in terms of the state variable k and the control variable c. From (i) differentiate with respect to time. Then d[U (c)] =µ ˙ dt dc U (c) = µ ˙ = −µf (k) + (n + δ + β)µ dt i.e. U (c)c˙ = −µ[ f (k) − (n + δ + β)] or −

U (c) c˙ = f (k) − (n + δ + β) U (c)

(since µ = U (c))

Now deﬁne Pratt’s measure of relative risk aversion10 cU (c) σ (c) = − U (c) then σ (c) c˙ = f (k) − (n + δ + β) c 10

Pratt (1964), see also Shone (1981, application 2, section A2.4).

277

278

Economic Dynamics or c˙ =

1 [ f (k) − (n + δ + β)]c σ (c)

We therefore have two differential equations c˙ =

1 [ f (k) − (n + δ + β)]c σ (c)

k˙ = f (k) − (n + δ)k − c If c˙ = 0 then f (k∗ ) = n + δ + β. On the other hand, if k˙ = 0 then c∗ = f (k∗ ) − (n + δ)k∗ . Furthermore, if c˙ > 0 then f (k∗ ) > (n + δ + β) which implies k < k∗ as seen in terms of the upper diagram of ﬁgure 6.17. Hence, to the left of the c˙ = 0 isocline, c is rising; to the right of c˙ = 0, then c is falling. Similarly, if k˙ > 0 then f (k∗ ) − (n + δ)k∗ > c. Thus below the k˙ = 0 isocline k is rising, while above the k˙ = 0 isocline k is falling. The vector forces clearly indicate that (k∗ , c∗ ) is a saddle point solution. The only optimal trajectory is that on the stable arm. For any k0 the only viable level of consumption is that represented by the associated point on the stable arm. Given the initial point on the stable arm, the system is directed towards the equilibrium. Notice that in equilibrium k is constant and so capital is growing at the same rate as the labour force. Furthermore, since k is constant in equilibrium then so is y, and hence Y is also growing at the same rate as the labour force. We have, therefore, a balanced-growth equilibrium. Example 6.9 (Ramsey growth model: a numerical example) Consider the optimal growth problem ∞ e−βt U(c)dt max J = {c}

k˙ = f (k) − (n + δ)k − c k(0) = k0 0 ≤ c ≤ f (k) where β = 0.02, f (k) = k0.25 , n = 0.01, δ = 0.05, k(0) = 2 and U(c) =

c1−θ 1−θ

√ If θ = 12 , then U(c) = 2 c.11 Then our maximisation problem is ∞ √ e−0.02t 2 c dt max J = {c}

k˙ = k0.25 − 0.06k − c k(0) = 2 11

Notice that this utility function has a relative measure of risk aversion equal to θ.

Optimal control theory

279 Figure 6.17.

The current value Hamiltonian is √ Hc = 2 c + µ(k0.25 − 0.06k − c) with ﬁrst-order conditions

∂Hc (i) = 2 12 c−1/2 − µ = 0 ∂c (ii) µ ˙ = −µ(0.25)k−0.75 + 0.08µ (iii)

k˙ = k0.25 − 0.06k − c

280

Economic Dynamics From the ﬁrst condition we have c−1/2 = µ. Differentiating this with respect to time, then − 12 c−3/2 c˙ = µ ˙ Using condition (ii) we have − 12 c−3/2 c˙ = −µ(0.25)k−0.75 + 0.08c−1/2 But µ = c−1/2 hence − 12 c−3/2 c˙ = −c−1/2 (0.25)k−0.75 + 0.08c−1/2 Dividing throughout by c−1/2 we obtain − 12 c−1 c˙ = −(0.25)k−0.75 + 0.08 i.e. c˙ = 2c(0.25)k−0.75 − 2(0.08)c = (0.5k−0.75 − 0.16)c We now have two differential equations for the state variables c and k, which are c˙ = (0.5k−0.75 − 0.16)c k˙ = k0.25 − 0.06k − c The ﬁrst thing to note about these equations is that they are nonlinear and therefore not easy to solve without some software.12 Using either Mathematica or Maple (or Excel as indicated in n. 12), the following equilibrium values are obtained k∗ = 4.5688,

c∗ = 1.1879

Second we note that the consumption-isocline is given by the formula c = k0.75 − 0.06k. Differentiating this with respect to k and setting this equal to zero allows us to solve for the value of k at which consumption is at a maximum c = k0.25 − 0.06k dc = 0.25k−0.75 − 0.06 = 0 dk kmax = 6.7048 At this value of k then consumption takes the value cmax = 1.2069.13 12

If you do not have a software package like Mathematica or Maple, you can use Excel’s Solver to solve for the equilibrium values. Place an arbitrary value of k in one cell; say, our starting value of 2. Suppose this is cell C3. Now place the formula = (0.5∗ $C$3ˆ(−0.75) − 0.16)∗ ($C$3ˆ0.25 − 0.06∗ $C$3)

13

in the target cell. In the Solver window declare the cell where the formula is located as the target cell and set this to have a value of zero; allow cell $C$3 to be the cell whose values are changed. In order to avoid the problem of a zero solution, place a constraint on $C$3 that it should be greater than or equal to unity. Having calculated the equilibrium value of k in this manner, it is a simple matter then to solve for the equilibrium value of consumption, c. Note that the c-isocline cuts the k-axis at 0 and the value 42.5727.

Optimal control theory To establish the properties of the equilibrium, we can linearise the system around the point (k∗ , c∗ ) = (4.5688, 1.1879). Let c˙ = f (c, k) = (0.5k−0.75 − 0.16)c k˙ = g(c, k) = k0.25 − 0.06k − c The system can then be written in the linearised form c˙ = fc (c∗ , k∗ )(c − c∗ ) + fk (c∗ , k∗ )(k − k∗ ) k˙ = gc (c∗ , k∗ )(c − c∗ ) + gk (c∗ , k∗ )(k − k∗ ) with fc (c∗ , k∗ ) = 0,

fk (c∗ , k∗ ) = −0.0312

gc (c∗ , k∗ ) = −1,

gk (c∗ , k∗ ) = 0.02

and so the matrix of the system is 0 −0.0312 A= −1 0.02 with eigenvalues r = 0.1869 and s = −0.1669. Since these are opposite in sign, then the equilibrium is a saddle point solution. Given that we are dealing with a numerical example then we can approximate the saddle path equations utilising the linear approximation to the system. First take the eigenvalue r = 0.1869 (A − rI)vr = 0 i.e.

−0.1869 −0.0312 −1 −0.1669

vr1 vr2

0 = 0

then −vr1 − 0.1669vr2 = 0 vr1 = −0.1669vr2 Let vr2 = 1 then vr1 = −0.1669. This saddle path is therefore negatively sloped and denotes the unstable arm. Turn next to the eigenvalue s = −0.1669 then s 0 0.1669 −0.0312 v1 = vs2 0 −1 0.1869 and −vs1 + 0.1869vs2 = 0 vs1 = 0.1869vs2 Let vs2 = 1 then vs1 = 0.1869. This saddle path is positively sloped and represents the stable arm. The equation of the stable arm can be found from c − c∗ = 0.1869(k − k∗ ) c − 1.1879 = 0.1869(k − 4.5688) c = −0.33399 + 0.1869k

281

282

Economic Dynamics

Figure 6.18.

All these results are illustrated in ﬁgure 6.18 in the neighbourhood of the ﬁxed point. As can be seen from ﬁgure 6.18, however, although the trajectory does begin on the saddle path, the system is extremely sensitive to initial conditions, and given the initial point (k(0), c(0)) = (2, 0.70779), the trajectory begins to move away from the saddle path before it reaches the equilibrium! The direction ﬁeld shown in ﬁgure 6.18 does, however, illustrate the existence of the stable arm with trajectories tending to the balanced-growth path equilibrium. What these examples show is that we can eliminate the control variable using the ﬁrst-order conditions and derive two differential equations, one for the state variable and another for the costate variable. These can more generally be expressed x˙ = R(x, λ) λ˙ = S(x, λ) We can then deﬁne two isoclines, one for x˙ = 0 (or R = 0), and another for λ˙ = 0 (or S = 0). As these examples illustrate, however, such isoclines do not always exist. When they both exist, the state space is separated into four quadrants. Each quadrant exerts different dynamic forces on any trajectory beginning in it. In most of these examples we know the initial point and terminal point. The derived dynamic equations maximise the objective function, satisfy the equation of motion and satisfy initial and terminal states. So we know the optimal trajectory. When we have two state variables, as in example 6.8 (and its numerical version, example 6.9), then we sketch the state-space only and the optimal trajectory {x(t)}. If we have a discrete system then xt+1 − xt = R(xt , λt ) λt+1 − λt = S(xt , λt ) and the isoclines remain R = 0 (xt+1 = 0) and S = 0 (λt+1 = 0).

Optimal control theory For many problems we do not have speciﬁc functional forms either for the objective function or for the equations of motion. This was the situation in the general Ramsey growth model (example 6.8). It is in such circumstances that deriving isoclines and establishing properties of the state space provides qualitative insight into the optimal path. For instance, example 6.8 illustrates that the trajectory of the economy is along the stable arm eventually resulting in a balanced-growth equilibrium. Such a path will maximise discounted consumption over the inﬁnite time horizon.

Exercises 1.

2.

Given the following stages of production labelled I, II, III and IV, a variety of possible processes can be followed, as shown in the accompanying ﬁgure, where the cost of transforming from one stage into another is indicated in the circle while the two states are labelled A, B, C, etc.

(i) Compute all possible solution paths and show which is the minimum. (ii) ‘Back solve’ by starting at the terminal state J and minimise at each node arrived at. Is this the same solution path you derived in (i) when then looked at forward? (iii) Why is it sensible to ‘back solve’ but not to ‘forward solve’? Prove that − t0

3.

t1

˙ = λxdt

t1

˙ − [λ(t1 )x(t1 ) − λ(t0 )x(t0 )] xλdt

t0

(i) For the model in section 6.2 derive the optimal path for production under both no discounting and discounting at 10% under the following alternative assumptions: (a) p = 5 (b) b = 3 (c) R = 800 (ii) What conclusions do you draw?

283

284

Economic Dynamics 4.

5.

6.

7.

Consider the model in example 6.5. Suppose the manager is unsure of the discount rate. He decides to choose three rates: 5%, 10% and 15%. Compare the optimal production schedules under each assumption. Consider examples 6.4 and 6.5 under the assumption that price is expected to rise at 5% per period (i.e. inﬂation is 5%), but costs are not subject to any rise; derive the optimal production schedule in each case assuming p0 = 3. Consider example 6.4 under the assumption that price is expected to rise at 10% per period and costs are expected to rise at 15% per period. Derive the optimal production schedule assuming p0 = 3 and b0 = 2. Set up the following problem as a control problem. A government has an objective function that indicates it wants to maximise votes, v, by pursing policies towards unemployment, u, and inﬂation π . The party is constrained in its behaviour by the existence of an augmented Phillips curve of the form π = −α(u − un ) + π e

α>0

and expectations take the form of adaptive expectations, i.e. π˙ e = β(π − π e )

8.

β>0

The government has just won the election at t = 0 and the next election is in 5 years’ time. It assumes that voters have poor memories, and weight more heavily the economic situation the closer it is to the election. It accordingly assumes a weighting factor of e0.05t . Solve the following control problem 1 2 u2 x − dt max − {u} 4 9 0 x˙ = −x + u x(0) = 5,

9.

x(1) = 10

Plot the trajectory in (x, λ)-space. Solve the following control problem 1 max (3x2 − u2 )dt {u}

x˙ = 2x + u x(0) = 10, 10.

x(1) = 15

Plot the trajectory in (x, λ)-space Solve the equilibrium for the following Ramsey model. Linearise the system about the equilibrium and establish its stability properties ∞ max J = e−0.03 U(c)dt {c}

Optimal control theory k˙ = k0.3 − (n + δ)k − c k(0) = 1 where U(c) = 4c1/4 , n = 0.02, δ = 0.03 Additional reading Beavis and Dobbs (1990), Blackburn (1987), Bryson, Jr. and Ho (1975), Burmeister and Dobell (1970), Chiang (1992), Conrad (1999), Conrad and Clark (1987), Fryer and Greenman (1987), Intriligator (1971), Kirk (1970), L´eonard and Long (1992), Pontryagin et al. (1962), Ramsey (1928), Romer (2001) and Takayama (1994).

285

CHAPTER 7

Chaos theory

7.1 Introduction The interest and emphasis in deterministic systems was a product of nineteenthcentury classical determinism, most particularly expressed in the laws of Isaac Newton and the work of Laplace. As we pointed out in chapter 1, if a set of equations with speciﬁed initial conditions prescribes the evolution of a system uniquely with no external disturbances, then its behaviour is deterministic and it can describe a system for the indeﬁnite future. In other words, it is fully predictable. This view has dominated economic thinking, with its full embodiment in neoclassical economics. Furthermore, such systems were believed to be ahistoretic. In other words, such systems were quite reversible and would return to their initial state if the variables were returned to their initial values. In such systems, history is irrelevant. More importantly from the point of view of economics, it means that the equilibrium of an economic system is not time-dependent. Although the physical sciences could in large part undertake controlled experiments and so eliminate any random disturbances, this was far from true in economics. This led to the view that economic systems were subject to random shocks, which led to indeterminism. Economic systems were much less predictable. The random nature of time-series data led to the subject of econometrics. The subject matter of econometrics still adheres to the view that economic systems can be captured by deterministic components, which are then augmented by either additive or multiplicative error components. These error components pick up the stochastic nature of the data series, most especially time-series data. For instance, the classical linear model takes the form (7.1)

y = Xβ + ε ε ∼ N(0, σ 2 I) Not only is Xβ assumed linear but, more signiﬁcantly from the point of view of our present discussion, it is assumed to be deterministic. All randomness is attributed to the error term. Even where the error term has distributions that are not normally distributed, the econometric approach effectively partitions the problem into a deterministic component and a random component. Randomness cannot and does not arise from the deterministic component in this approach. Such a view of the world is a ‘shotgun wedding of deterministic theory with “random shocks” ’ (Mirowski 1986, p. 298). Mandelbrot (1987), in particular, was an ardent critic of the way econometrics simply borrowed classical determinism and added a random

Chaos theory

287

component. The emphasis of econometrics on the Central Limit Theorem was, in Mandelbrot’s view, ﬂawed. It is not our intention here, however, to expand on these views. Sufﬁce it to say that the dichotomy between deterministic economic elements of a system and additive random components is still the mainstay of the econometric approach. Three important considerations came out of these early discussions that are relevant for this chapter. 1.

2.

3.

Linearity, far from being the norm, is the exception. Linear economic models lead to unique equilibrium points, which are either globally stable or globally unstable. On the other hand, nonlinear systems can lead to multiple equilibria and hence, local stability and instability. However, nonlinear systems also tend to lead to complexity. Until the development of chaos theory, there was little formal way of handling complex systems. On a more practical note, the study of complex systems could not have occurred without the development of computers and computer software. Many economic time series are generated by discrete processes. The second consideration, once again highlighted by Mandelbrot, is that many economic time series are generated by discrete processes, and therefore should not be modelled as continuous processes. The emphasis of continuous processes comes, once again, from the physical sciences. We already noted in earlier chapters that a discrete equivalent of a continuous system could exhibit instability while its continuous version is stable! Although this is not always the case, it does highlight the importance of modelling systems with discrete models if discrete processes generate the time series. The occurrence of bifurcations. A third strand was consideration of a system’s equilibrium to changes in the value of important parameters. It became clear that for some systems, their behaviour could suddenly change dramatically at certain parameter values. This led to the study of bifurcation.

This chapter is concerned with how deterministic systems can exhibit chaos, and to all intents and purposes have the characteristics of randomness: a randomness, however, which does not occur from random shocks to the system. We ﬁnd that such behaviour occurs when parameters of the system take certain values. The parameter value at which the system’s behaviour changes is called a bifurcation value. We therefore begin our discussion with bifurcation theory.

7.2 Bifurcations: single-variable case In this and the next section we shall conﬁne ourselves to studying some of the properties of ﬁrst-order systems that depend on just one parameter. We shall represent this with xt+1 = f (xt , λ)

(7.2)

288

Economic Dynamics in which f is nonlinear. Two examples are1 : (i) (ii)

xt+1 = 1.5xt (1 − xt ) − λ xt+1 = λxt (1 − xt )

and we shall make great use of these two equations to illustrate bifurcation theory and chaos. But what do we mean by the terms ‘bifurcation’ and ‘chaos’? Bifurcation theory is the study of points in a system at which the qualitative behaviour of the system changes. In terms of our general representation, f (x∗ , λ) denotes an equilibrium point, a stationary point, whose value depends on the precise value of the parameter λ. Furthermore, the stability properties of the equilibrium point will also depend on the value of λ. At certain values of λ the characteristics of the system change, sometimes quite dramatically. In other words, the qualitative behaviour of the system either side of such values is quite different. These points are called bifurcation points. The types of bifurcations encountered in dynamic systems are often named according to the type of graph they exhibit, e.g., cusp bifurcation and pitchfork bifurcation, to name just two. But such classiﬁcations will become clearer once we have described how to construct a bifurcation diagram. As we shall note in a moment when we consider the two examples in detail, for certain values of the parameter λ a system may settle down to a periodic cycle: cycles of 2, 4, 8, etc. or cycles of odd-numbered periods, like 3. However, there comes a point, a value of λ beyond which there is no regular cycle of any period. When this happens the system becomes irregular or chaotic. As we shall see, the bifurcation diagram is most useful in showing the occurrence of chaotic behaviour of dynamic systems. Example 7.1 We shall now consider the ﬁrst example in detail to highlight some points about bifurcation theory, and consider the logistic equation in detail in the next section. First, we need to establish the ﬁxed points of the system. These are found by solving x∗ = 1.5x∗ (1 − x∗ ) − λ or solving 15x∗2 − 5x∗ + 10λ = 0 i.e.

√

1 − 24λ 6 If 1 − 24λ < 0, i.e., λ > 1/24, then no equilibrium exists. If 1 − 24λ > 0, i.e., λ < 1/24, then two equilibria exist √ √ 1 − 1 − 24λ 1 + 1 − 24λ ∗ ∗ x1 = and x2 = 6 6 ∗

x =

1

1±

The ﬁrst example is adapted from Sandefur (1990), while the second example has been widely investigated by mathematicians. A good starting point, however, is May (1976).

Chaos theory

289

In order to investigate the stability of these equilibria, we need to consider f (x∗ ). This is f (x∗ ) = 1.5 − 3x∗ . Substituting the lower value we have √ f (x1∗ ) = 1 + 0.5 1 − 24λ > 1 for all λ < 1/24 Hence, x1∗ is unstable or repelling. Next consider the stability of x2∗ √ f (x2∗ ) = 1 − 0.5 1 − 24λ < 1

for all λ < 1/24

The system is stable or attracting if −1 < f (x2∗ ) < 1, i.e., if −0.625 < λ < 1/24 or −0.625 < λ < 0.041667. The third and ﬁnal situation is where λ = 1/24. In this case the two ﬁxed points are the same with value 1/6. Furthermore, f (1/6) = 1, and so the stability of this ﬁxed point is inconclusive or semistable. The value λ = 1/24 is a bifurcation value. We can combine all this information about the equilibrium points and their attraction or repelling on a diagram which has the parameter λ on the horizontal axis, and the equilibrium point x∗ on the vertical axis. Such a diagram is called a bifurcation diagram, and such a diagram is shown in ﬁgure 7.1 for the present problem. It is to be noted that the heavy (continuous and dotted) line denoting the equilibrium values for various values of the parameter λ is a parabola, which satisﬁes the equation (6x∗ − 1)2 = 1 − 24λ Figure 7.1.

290

(7.3)

Economic Dynamics with vertex (λ, x∗ ) = (1/24, 1/6) occurring at the bifurcation value λ = 1/24 with corresponding equilibrium value x∗ = 1/6. The vertical arrows show the stability properties of the equilibria. Notice that inside the parabola the arrows point up while outside of the parabola the arrows point down. In particular, for λ > 1/24 the system will tend to minus inﬁnity; if λ = 1/24 then there is only one equilibrium value (x∗ = 1/6) which is semistable from above; and if −0.625 < λ < 0.041667 then there are two equilibrium values, the greater one of which is stable and the lower one unstable. We can be more precise. Let Nλ denote the number of equilibrium values of the system when the parameter is equal to λ, then for any interval (λ0 − ε, λ0 + ε) if Nλ is not constant, λ0 is called a bifurcation value, and the system is said to undergo a bifurcation as λ passes through λ0 . For the example we have been discussing 2, for λ < 1/24 Nλ = 1, for λ = 1/24 0, for λ > 1/24 and so λ = 1/24 is a bifurcation. Furthermore, this is the only value of λ for which Nλ is not a constant, and so the system has just this one bifurcation value. Figure 7.1 also illustrates what is called a saddle node bifurcation. It is called this because at the value λ0 the ﬁxed points of the system form a U-shaped curve that is opened (in this instance opened to the left). Example 7.2 (saddle-node bifurcation) In this example we shall take a similar case, but consider a continuous model. Let

(7.4)

x (t) = λ − x(t)2 For equilibrium we have 0 = λ − x∗2 √ x∗ = λ

(7.5)

If λ 0 there are two ﬁxed points one for + λ and another for − λ. In order to consider the stability conditions for continuous systems we need to consider f (x∗ ) in the neighbourhood of the ﬁxed ∗ point. If f (x∗ ) < 0, then x∗ is locally stable; and if√f (x∗ ) > 0, √ then x is locally ∗ ∗ ∗ unstable. Since√f (x ) = −2x , then f (x1 ) = f (+ λ) = −2 √λ < 0 for√λ > 0, On the other hand, f (x2∗ ) = f (− λ) = +2 λ > 0 and so x1∗ = + λ is stable. √ for λ > 0, and so x2∗ = − λ is unstable. At λ = 0 the two ﬁxed points coincide and the ﬁxed point is stable from above. The situation is shown in ﬁgure 7.2. Summarising in the neighbourhood of the point λ = 0, 2, for λ > 0 Nλ = 1, for λ = 0 0, for λ < 0 and once again we have a saddle node bifurcation occurring this time at λ = 0.

Chaos theory

291 Figure 7.2.

Example 7.3 (transcritical bifurcation) Consider another example of a continuous nonlinear dynamical system x (t) = λx − x2 = x(λ − x)

(7.6)

with ﬁxed points x1∗ = 0

and

x2∗ = λ

Obviously, the two ﬁxed points become identical if λ = 0. Summarising in the neighbourhood of λ = 0, we have 2, for λ < 0 Nλ = 1, for λ = 0 2, for λ > 0 and so λ = 0 is a bifurcation value. Turning to the stability properties, we have f (x∗ ) = λ − 2x∗ and

f (0) = λ

$

> 0 for λ > 0 hence unstable < 0 for λ < 0 hence stable

For the second ﬁxed point we have f (λ) = −λ

$

< 0 for λ > 0 hence stable > 0 for λ < 0 hence unstable

Another way to view this is to consider x1∗ = 0 being represented by the horizontal axis in ﬁgure 7.3, and x2∗ = λ being represented by the 45◦ -line. The two branches intersect at the origin and there takes place an exchange of stability. This is called a transcritical bifurcation. The characteristic feature of this bifurcation point is that the ﬁxed points of the system lie on two intersecting curves, neither of which bends back on themselves (unlike the saddle-node bifurcation).

(7.7)

292

Economic Dynamics

Figure 7.3.

Example 7.4 (pitchfork bifurcation) Consider the following continuous nonlinear dynamical system x (t) = λx(t) − x(t)3 = x(t)(λ − x(t)2 )

(7.8)

This system has three critical points: √ √ x2∗ = + λ x3∗ = − λ x1∗ = 0,

(7.9)

where the second and third ﬁxed points are deﬁned only for positive λ. Summarising in the neighbourhood of λ = 0 we have $ 1 for λ ≤ 0 Nλ = 3 for λ > 0 and so λ = 0 is a bifurcation value. Since f (x∗ ) = λ − 3x∗2 , then at each ﬁxed point we have $ < 0 for λ < 0 hence stable f (0) = λ > 0 for λ > 0 hence unstable √ f (+ λ) = −2λ < 0 for λ > 0 hence stable √ f (− λ) = −2λ < 0 for λ > 0 hence stable The characteristic feature of this bifurcation is that at the origin we have a U-shaped curve, which here is open to the right, and another along the horizontal axis that crosses the vertex of the U. It forms the shape of a pitchfork, as shown in ﬁgure 7.4. It is therefore called a pitchfork bifurcation.2 It is important to recall in all this discussion of bifurcation points that the properties, especially the stability/instability properties, are deﬁned only for neighbourhoods of the bifurcation point. There may be other bifurcation points belonging to the system with different properties. 2

Example 7.4 illustrates what is sometimes referred to as a ‘supercritical pitchfork’. If we have the same continuous nonlinear system but with −x3 replaced with +x3 , then we have a ‘subcritical pitchfork’.

Chaos theory

293 Figure 7.4.

7.3 The logistic equation, periodic-doubling bifurcations and chaos Chaos theory is a new and growing branch of mathematics that has some important implications for economic systems. As an introduction to this subject return to the generic form of the logistic difference equation xt+1 = f (xt , λ) = λxt (1 − xt )

0≤λ≤4

For this generic form of the logistic equation x must lie in the closed interval [0,1]. But this places an upper limit on the value of λ. To see this, ﬁrst establish the value of x at which f is a maximum. This is f (x) = λ − 2λx = 0 1 x= 2 At this value of x, f ( 12 ) = λ/4 and since x cannot exceed 1, then λ cannot exceed 4. Therefore the generic logistic equation is deﬁned for 0 ≤ λ ≤ 4. It would appear that this simple nonlinear equation would show an orbit, a time path {xt }, which would be quite simple. As ﬁgure 7.5 shows, however, this is far from true. In this ﬁgure we have drawn the orbit for 100 periods for three different values of λ, (a) λ = 3.2, (b) λ = 3.85 and (c) λ = 4. In each case the initial value is x0 = 0.1. In ﬁgure 7.5(a) the time path settles down to a two-cycle very quickly. When λ = 3.85 there is some initial chaotic behaviour, but then the series settles down to a three-cycle. This is an example of transient chaos (Hommes 1991). In ﬁgure 7.5(c) there is no periodic behaviour shown at all. The series is aperiodic or chaotic. It is apparent, therefore, that this simple equation can exhibit very different time paths depending on the value of λ. The question is can we identify when the system changes from one type of orbit into another? Put another way, can we identify any bifurcation points for the logistic equation? In order to do this, ﬁrst establish the equilibrium points where xt+1 = xt = x∗ for all t. This is where x∗ = λx∗ (1 − x∗ ) λx∗2 + (1 − λ)x∗ = 0 x∗ [λx∗ + (1 − λ)] = 0

(7.10)

294 Figure 7.5.

Economic Dynamics

Chaos theory

295

i.e. λ−1 λ To investigate the stability of these solutions we shall employ the linear approximation discussed in chapters 2 and 3, namely x∗ = 0

or

x∗ =

(7.11)

xt+1 = f (x∗ , λ) + f (x∗ , λ)(xt − x∗ ) But f (x∗ , λ) = 0 or f (x∗ , λ) = (λ − 1)/λ, and f (x∗ , λ) = λ − 2λx∗ . Thus, λ for x∗ = 0 ∗ f (x , λ) = λ−1 2 − λ for x∗ = λ Consider x∗ = 0. Then we have xt+1 = λxt with solution xt = λ t x 0 which is stable if 0 < λ < 1. Next consider x∗ = (λ − 1)/λ, then xt+1 = x∗ + (2 − λ)(xt − x∗ ) or ut+1 = (2 − λ)ut where ut+1 = xt+1 − x∗ and ut = xt − x∗ . This has the solution ut = (2 − λ)t u0 which is stable so long as |2 − λ| < 1 or −1 < 2 − λ < 1, giving a range for λ of 1 < λ < 3. What we observe is that for 0 ≤ λ < 1 the only solution is x∗ = 0 and this is locally stable. For 1 < λ < 3 we have an equilibrium solution x∗ = (λ − 1)/λ, which varies with λ. The situation is shown in ﬁgure 7.6. At λ = 1, where the two solution curves intersect, there is an exchange of stability from one equilibrium solution to the other. Figure 7.6.

296

Economic Dynamics

Figure 7.7.

Of course, λ is not necessarily restricted to a range below 3. The question is: What happens to the solution values as λ is allowed to increase? We began to answer this question in chapter 3. In section 3.9 we investigated the range for λ that led to a two-cycle result. This range we established as 3 < λ < 3.449. In other words, at λ = 3 the solution becomes unstable but gives rise to a stable two-cycle result, as shown in ﬁgure 7.6. Consequently, the value λ = 3 is a dividing value between a unique solution (other than zero) and a two-cycle solution. The value λ = 3 is a pitchfork bifurcation.3 Again, the actual values of the two limit points depend on the precise value of λ. For example, in section 3.9 we established the two limit values of 0.799455 and 0.513045 for λ = 3.2 (see also appendices 3.1 and 3.2). The reason for this apparent instability of one solution and the occurrence of a stable two-cycle is illustrated in ﬁgure 7.7(a)–(d), where f (x) = λx(1 − x) and f 2 (x) = f ( f (x)). In ﬁgure 7.7(a) λ = 0.8 and so the only solution is x∗ = 0. Any (positive) value x0 ‘close to’ zero will be attracted to x∗ = 0. Also note that the twocycle curve f 2 (x) lies wholly below the 45◦ -line and so no two-cycles occur. For this solution, f (x∗ = 0, λ = 0.8) = 0.8 < 1, and so x∗ = 0 is stable. In ﬁgure 7.7(b) λ = 2.5 and we have solution x∗ = (λ − 1)/λ = 0.6 and f (x∗ = 0.6, λ = 2.5) = 2 − 2.5 = −0.5, with | f (x∗ = 0.6, λ = 2.5)| < 1 and so x∗ = 0.6 is stable. Also note that f 2 (x) cuts the 45◦ -line just the once, at E0 and so no two-cycles occur. The situation begins to change in ﬁgure 7.7(c) where λ = 3. Here x∗ = (λ − 1)/λ = 2/3 and f (x∗ = 2/3, λ = 3) = 2 − 3 = −1, and so x∗ = 2/3 is semistable. The fact that f (x∗ = 2/3, λ = 3) = −1 is semistable is shown in ﬁgure 7.7(c) by the feature that the two-cycle f 2 (x) is tangent to the 45◦ -line at point E0 . Because 3

It is sometimes called a period-doubling bifurcation, since it denotes the value of λ at which a stable two-period cycle occurs.

Chaos theory

297 Figure 7.8.

x∗ = 2/3 lies on f (x) where this intersects the 45◦ -line, then it is stable for λ < 3. However if λ > 3 then f 2 (x) begins to intersect the 45◦ -line in three places. The instability of the single solution and the stability of the two-cycle is more clearly shown in ﬁgure 7.7(d), where λ = 3.4. A number of features should be noted about this ﬁgure. First, f (x) and f 2 (x) intersect at the common point E0 = 0.70588. This is unstable. We can establish this by computing f 2 (x0∗ ), which has a value of 1.96, and since f 2 (x0∗ ) > 1, x0∗ = 0.70588 is unstable. Second, there are two stable equilibrium points E1 and E2 with the following characteristics E1 : E2 :

x1∗ = 0.451963 f (x∗ , λ) = −0.76 x2∗ = 0.842154 f (x∗ , λ) = −0.76

since f 2 (x1∗ ) = f 2 (x2∗ ) = −0.76, then both E1 and E2 are stable. However, at λ = 3.449 the two-cycle becomes unstable. But what occurs at and beyond the point where λ = 3.449? What occurs is two period-doubling bifurcation points, i.e., each of the two solutions themselves become unstable but divide into two stable solutions, leading to a total of four solutions. In other words, we have a four-cycle solution. The range for a fourcycle result can be found in a similar manner to the range for a two-cycle result, i.e., by ﬁnding the values of a which satisfy the equation a = f ( f ( f ( f (a))))

or

a = f 4 (a)

(7.12)

and whose stability is established by solving4 −1 < f (a1 ) f (a2 ) f (a3 ) f (a4 ) < 1 This is even more tedious than the two-cycle result, and can only sensibly be solved with the help of a computer. However, we illustrate the result over the range 3.44 < λ < 3.55 in ﬁgure 7.8. Figure 7.8 shows that the four-cycle itself becomes unstable and bifurcates into an eight-cycle. What is more surprising, however, is that the value of λ at which 4

See theorem 3.2, p. 93 and appendices 3.1 and 3.2.

(7.13)

298

Economic Dynamics

Figure 7.9.

further bifurcations occur approaches a limit of approximately λ = 3.57. After this value, the system exhibits chaotic behaviour, i.e., behaviour that, although generated by a deterministic system, has all the characteristics of random behaviour. There are no regular cycles. A typical situation is illustrated in ﬁgure 7.9 for λ = 3.65. In this ﬁgure we plot the series (7.14)

xt+1 = 3.65xt (1 − xt ) for two different starting values: x0 = 0.1 and x0 = 0.105. These starting values are very close to one another, however, the two systems diverge from one another after about ten periods. This is because of the chaotic nature of the system. With chaotic systems we have sensitive dependence to initial conditions. But another characteristic arises in the case of a series entering the chaotic region for its parameter value. Consider the following example provided by Baumol and Benhabib (1989):

(7.15)

xt+1 = 3.94xt (1 − xt )

x0 = 0.99

whose graph is derived using a spreadsheet, and shown in ﬁgure 7.10. Although the series is chaotic, it is not purely random, and in particular exhibits sudden changes. In ﬁgure 7.10 the series suddenly changes from showing an oscillatory behaviour to one that is almost horizontal (which it does for about ten periods) and then just as suddenly, and for no obvious reason, begins to oscillate once again. For the logistic curve the characteristics listed in table 7.1 have been established. It should be noted that a three-period cycle is also present, and ﬁrst shows itself at the point λ = 3.8284. In fact, we derived a three-period cycle in chapter 3 (p. 97). Just as there are repeating even-numbered cycles, so there are repeated oddnumbered cycles. Two- and three-period cycles (and even a four-period cycle) can be shown quite clearly using the cobweb diagram we will develop in chapter 8. On the vertical axis we have xt+1 and on the horizontal axis xt . The 45◦ -line represents the equilibrium condition with xt+1 = xt . The curve is f (xt ) = λxt (1 − xt ). If λ = 3.2 and x0 = 0.513045 a two-cycle results, as shown in ﬁgure 7.11(a). On the other hand, if λ = 3.839 and x0 = 0.48912 then a stable three-cycle results, as shown in

Chaos theory

299 Figure 7.10.

Table 7.1 Properties of the logistic curve Description

Value of λ

Comments

Exchange of stability Fixed point becomes unstable (2-cycles appear)

1 3

2-cycle becomes unstable (4-cycles appear) 4-cycle becomes unstable (8-cycles appear) Upper limit value on 2-cycles (chaos begins) First odd-cycle appears Cycles with period 3 appear Chaotic regions ends

3.44949 3.54409 3.57 3.6786 3.8284 4

First bifurcation point Second bifurcation point e.g. λ = 3.2, section 3.9, p. 118 Established in section 3.9

e.g. λ = 3.84, section 3.9, p. 118

ﬁgure 7.11(b). When λ = 3.59 and x0 = 0.4 many periods occur (ﬁgure 7.11(c)), but the values that x takes are bounded. When λ = 4 and x0 = 0.2 then the series is chaotic. This is revealed in ﬁgure 7.11(d) by the fact that all values of x in the closed interval [0,1] occur, i.e., the web covers the whole possible graph. What we observe, then, is that from a very simple deterministic equation a whole spectrum of patterns emerge, and in particular an apparent random series arises for a parameter value of λ > 3.57. With such a diversity of equilibrium solutions depending on the value of λ, it would be interesting to know what the bifurcation diagram for the logistic equation would look like over the range 0 ≤ λ ≤ 4. We need to plot the relationship between x∗ and λ between the values of 0 and 4. Before the advent of computers, this would be virtually impossible, but now it is relatively easy. The result is shown in ﬁgure 7.12,5 while a closer look at the range 3 ≤ λ ≤ 4 is shown in ﬁgure 7.13. Notice in ﬁgure 7.13 the ‘windows’ occurring. Three are marked on the diagram. The ﬁrst window marked is for the occurrence of a six-period cycle; the second is the occurrence of a ﬁve-period cycle, while the third is for a three-period cycle. 5

The bifurcation diagram plots only stable equilibrium points, so in the range 3 < λ < 3.449 only two curves are plotted. The bifurcation point λ = 3 is not a saddle node bifurcation but rather a pitchfork bifurcation, as shown in ﬁgure 7.6. It is therefore more useful to think of this point as a period-doubling bifurcation.

300

Economic Dynamics

Figure 7.11.

Figure 7.12.

Such windows represent stable periodic orbits that are surrounded by chaotic behaviour (the dark regions). These two diagrams show an amazing diversity of equilibria for such a simple deterministic equation. Such results direct attention to three observations: (1) (2) (3)

the presence of nonlinearity can give rise to deterministic chaos, in the presence of chaos there exists the sensitive dependence to initial conditions, and in the presence of chaos prediction, even for a simple deterministic system, is virtually impossible.

Bifurcation diagrams require a considerable amount of routine computations and there are now a growing supply of such routines written by mathematicians

Chaos theory

301 Figure 7.13.

and computer programmers, both in Mathematica and Maple. This is also true of strange attractors, such as the H´enon map and the Lorenz strange attractor, which we discuss in section 7.7. The interested reader should consult the various sources at the end of this chapter. In general for the present text, the bifurcation diagrams in Mathematica utilise the routine provided by Gray and Glynn (1991), while a number of the chaos diagrams in Maple utilise the routines provided by Lynch (2001).6

7.4 Feigenbaum’s universal constant In discussing the logistic equation we noted that the ﬁrst bifurcation occurred at √ value 3, the second at value 1 + 6 = 3.44949, while a third occurs at value 3.54409. We can think of these values as representing the point at which a 2k -cycle ﬁrst appears. Thus for k = 0 a 20 -cycle occurs at λ0 = 3 for k = 1 a 21 -cycle occurs at λ1 = 1 +

√ 6 = 3.44949

for k = 2 a 22 -cycle occurs at λ2 = 3.54409 and so on. If λk denotes such occurrences of a 2k -cycle, then three results have been shown to hold for large k: √ (i) λk+1 ≈ 1 + 3 + λk k = 2, 3, . . . (ii) (iii)

lim λk = 3.57

k→∞

If dk =

λk − λk−1 λk+1 − λk

k = 2, 3, 4, . . .

then lim dk = δ = 4.669202 k→∞

6

The printing of such diagrams can be problematic and will certainly depend on the internal RAM of the printer.

302

Economic Dynamics Table 7.2 Approximations for the occurrence of the 2k-cycle k

1

2

3

4

5

6

λk dk

3

3.44949 4.75148

3.54409 4.655512

3.56441 4.671264

3.56876 4.677419

3.56969 4.65

3.56989

δ is called the Feigenbaum constant after its discoverer, and is important because it is a universal constant. The ﬁrst result is only approximate, and there are a number of ways of approximating the value of λk+1 (see exercise 2). Although these approximations are supposed to hold only for large k, they are reasonable even for small k. Using a procedure provided by Gray and Glynn (1991, p. 125) we derive a more accurate estimate of the two-cycle bifurcation points. In particular, table 7.2 provides the ﬁrst seven points and computes λk and dk . As can be seen, λk is rapidly approaching the limit of 3.57, while dk approaches the limit of 4.6692, although not so rapidly and not uniformly.

7.5 Sarkovskii theorem In this section we have a very limited objective. Our intention is to present the background concepts necessary to understand the signiﬁcance of the Sarkovskii theorem, which is central to periodic orbits of nonlinear systems. The ideas and concepts are explained by means of the logistic equation, most of the properties of which we have already outlined. In table 7.3 we list the ﬁrst 30 positive integers, where we have expressed some of the numbers in a way useful for interpreting a Sarkovskii ordering. Using the numbers in table 7.3 as a guide, we can identify the following series, where a b means a precedes b in the order and ‘odd number’ means the odd numbers except unity (table 7.4). Every possibly positive integer is accounted for only once by all series taken together. A Sarkovskii ordering is then S0 S1 S2 . . . Sk . . . 24 23 22 2 1 We are now in a position to state the theorem. THEOREM 7.1 (Sarkovskii) Let f be a continuous function deﬁned over a closed interval [a,b] which has a periodic point with prime period n. If n m in a Sarkovskii ordering, then f also has a periodic point with prime period m. Another theorem occurring just over a decade later is the following.7

7

Sarkovskii’s paper of 1964 was not known to Western mathematicians until the publication of Li and Yorke’s paper in 1975.

Chaos theory Table 7.3 Expressing the ﬁrst 30 integers for a Sarkovskii ordering 1 2 3 4 22 5 6 2.3 7 8 23 9 10 2.5

11 12 13 14 15 16 17 18 19 20

22 .3 2.7 24 2.9 22 .5

21 22 23 24 25 26 27 28 29 30

2.11 23 .3 2.13 22 .7 2.15

Table 7.4 The series of a Sarkovskii ordering Series

Numbers in the series

Description of the series

S0 S1 S2 .. . Sk –

3 5 7 ... 2.3 2.5 2.7 . . . 22 .3 22 .5 22 .7 . . . .. . 2k .3 2k .5 2k .7 . . . 24 23 22 2 1

odd numbers 2.(odd numbers) 22 .(odd numbers) .. . 2k .(odd numbers) Powers of 2 in descending order∗

Note: *Recall 21 = 2 and 20 = 1.

THEOREM 7.2 (Li–Yorke) If a one-dimensional system can generate a three-cycle then it can generate cycles of every length along with chaotic behaviour. The Li–Yorke theorem is a corollary of the Sarkovskii theorem. If m = 3 in the Sarkovskii theorem, then n = 5, say (n m) also has a periodic point. Therefore for all n m f will have a periodic point. Hence, if a one-dimensional system can generate a three-cycle, it must be capable of generating a cycle of any length. The windows in ﬁgure 7.13 represent period-6, period-5 and period-3 cycles, respectively. The period-5 lies to the left of period-3, with period-3 being the highest ordering. But why is period-6 to the left of period-5? Period-6 is equivalent to period-2.3 in the Sarkovskii ordering and so belongs to the series S1 . All periods in S1 are to the left of all periods in S0 . Hence, period-6 is to the left of period-5, which in turn is to the left of period-3. Suppose a continuous function f over the closed interval [a,b] has a period-5 cycle, then according to the Sarkovskii theorem it has cycles of all periods with the possible exception of period-3. Notice that the possibility of a period-3 is not ruled out. Similarly, if f has no point of period-2, then there do not exist higher-order periodicities, including chaos. The Sarkovskii theorem, and to some extent the Li–Yorke theorem, demonstrates that even systems that exhibit chaotic behaviour still have a structure. The word

303

304

Economic Dynamics ‘chaos’ conjures up purely unsystematic patterns and unpredictability. Although the movement of an individual series may be aperiodic, chaotic systems themselves have structural characteristics that can be identiﬁed.

7.6 Van der Pol equation and Hopf bifurcations We met the Van der Pol equation in chapter 4 when we considered limit points. In this section our interest is in the Van der Pol equation and its bifurcation features. The nonlinear equation is a second-order differential equation of the form (7.16)

x¨ = µ(1 − x2 )˙x − x Second-order differential equations can be expressed in the form of a system of ﬁrst-order differential equations with suitable transformations. Let x˙ = y then y˙ = x¨ , hence we have the system of ﬁrst-order equations:

(7.17)

x˙ = y y˙ = µ(1 − x2 )y − x and the only unknown parameter is µ. The ﬁxed points of the system are established by setting x˙ = 0 and y˙ = 0, which is the singular point P = (0, 0). Furthermore, the linearisation of the system can be expressed as x x˙ 1 0 = 2 y˙ −(1 + 2µxy) µ(1 − x ) y Expanding the system around the ﬁxed-point, P = (0, 0), we have x˙ 1 0 x = y˙ −1 µ y Hence the matrix of the linearised system is 1 0 A= −1 µ

(7.18)

whose eigenvalues are µ − µ2 − 4 λ1 = 2

and

λ2 =

µ+

µ2 − 4 2

Using these eigenvalues we can identify ﬁve cases, as shown in table 7.5. Figure 7.14 illustrates each of these cases. If we concentrate on the equilibrium values for x and y, say x∗ and y∗ , then for µ < 0, x∗ = 0 and y∗ = 0 and the system moves along the µ-axis. At µ = 0 the system changes dramatically taking on the shape of a circle at this value. Then, as µ continues in the positive direction the system takes on a limit cycle in the x-y plane for any particular positive value of µ, the shape of which is no longer a circle. All of these are schematically illustrated in ﬁgure 7.15, which also shows the movement of the system by means of arrows. Clearly the system exhibits a bifurcation at the value µ = 0. This is an example of a Hopf bifurcation.

Chaos theory

305

Table 7.5 Properties of the Van der Pol equation Cases

Parameter values

Properties

I II III IV

µ ≤ −2 −2 < µ < 0 µ=0 0 λs , the system also has a stable large-amplitude limit cycle. The bifurcation diagram for this system is illustrated in ﬁgure 7.16.9 As can be seen in ﬁgure 7.16, over the range λs < λ < 0, there exist two steadystate solutions. A number of important implications follow from this feature. First, which steady state is approached depends on the initial conditions. If the equations represented some economic system, then in all probability the welfare attached to one of the stable equilibrium would be quite different from that of the other. Policy-makers may, therefore, attempt to push the system in the direction of one particular stable equilibrium by changing the initial conditions. Second, the system exhibits hysteresis. Suppose the value of λ started at λ0 < λs , ﬁgure 7.16. As λ is increased then x remains at x∗ = 0 until λ = 0. At λ = 0, however, there is a sudden jump to the large-amplitude limit cycle. This is so because λ = 0 is a subcritical Hopf bifurcation. As λ continues to increase, the value of x∗ follows the upper path, along RS. The path traversed is then PQRS. Now suppose the value of λ is decreased to its former level λ0 . The system moves down along the upper path SRT. Once λ = λs , the system jumps down to x∗ = 0 (point U) and then remains there as λ is decreased further. The ‘return trip’ is therefore SRTUP, which is quite different from its outward journey. The system is hysteretic.

7.7 Strange attractors We noted in the last section how with a two-dimensional system a Hopf bifurcation could arise. In the Van der Pol equation, once µ ≥ 0 then the system gets attracted to a limit cycle. But other two- or higher-dimensional systems can have ‘strange’ attractors. We shall discuss the concept of strange attractors by way of examples. The examples we consider are the H´enon map and the Lorenz attractor.10 9

10

Part of the bifurcation diagram can be constructed using the implicit plot routines in either Mathematica or Maple. Just do the implicit plot of λ + x2 − x4 = 0 for −1 < λ < 2 and 0 < x < 2, and the result is as portrayed in ﬁgure 7.16. See also the R¨ossler attractor in exercises 9 and 10.

308

Economic Dynamics 7.7.1

The H´enon map

The H´enon map arises from a set of two equations involving two parameters, which are real numbers xt+1 = 1 − axt2 + yt a > 0 |b| < 1 yt+1 = bxt

(7.20)

(7.21)

We can think of this as the function 1 − axt2 + yt Ha,b (xt , yt ) = bxt or simply H. To establish some of the properties of the H´enon map, let f (x, y) = 1 − ax2 + y g(x, y) = bx Then the Jacobian, J, is −2ax fx fy = J= gx gy b

1 0

whose determinant is −b and eigenvalues λ = −ax ± a2 x2 + b

(7.22)

which is readily obtained using either Mathematica or Maple – in fact all the mathematical properties we are about to discuss are obtained using √either of these software programmes. Hence, the eigenvalues are real only if a2 x2 + b ≥ 0. Furthermore, the ﬁxed points of the H´enon map are found to be 1 b − 1 + (1 − b)2 + 4a x = 2a P1 = , b y = b − 1 + (1 − b)2 + 4a 2a 1 b − 1 − (1 − b)2 + 4a x = 2a P2 = b y = b − 1 − (1 − b)2 + 4a 2a which exist if a ≥ − 14 (1 − b)2 . Turning now to the stability properties of the ﬁxed points, we recall that the ﬁxed point is attracting if the eigenvalue is less than unity in absolute value. It can be established (Gulick 1992, pp. 171–2) that (1) (2)

If a < − 14 (1 − b)2 , then H has no ﬁxed points If − 14 (1 − b)2 < a < 34 (1 − b)2 and a = 0, then H has two ﬁxed points, P1 and P2 , of which P1 is attracting and P2 is a saddle point

If the parameter b is set ﬁxed over the interval [0,1] and a is allowed to vary, then there will be two bifurcation values for a: one at −(1 − b)2/4 and another at 3(1 − b)2 /4, with the system changing from an attracting ﬁxed point to one of a saddle point. Figure 7.17 shows the H´enon map with parameters a = 1.4 and b = 0.3 and with initial point (x0 , y0 ) = (0.1, 0). The two ﬁxed points are P1 = (0.6314, 0.1894)

Chaos theory

309 Figure 7.17.

Figure 7.18.

and P2 = (−1.1314, −0.3394), and the ﬁgure illustrates the existence of a strange attractor. Why then is it called a strange attractor? For b = 0.3 the Feigenbaum constant of a for the H´enon map is 1.0580459 (Gulick 1992, p. 173). One would assume that for a > 1.06 with chaos present that the iterates would virtually ﬁll the whole map. But this is not the case. For example, given the parameter values a = 1.4 and b = 0.3, then no matter what the starting values for x and y, the sequence of points {x(t), y(t)} is attracted to the orbit shown in ﬁgure 7.17, and such an orbit seems rather a ‘strange’ shape. At the same time, however, the trajectory {x(t), y(t)} is very sensitive to initial conditions. This is illustrated in ﬁgure 7.18 in the case of variable x, for the same parameter values and (x0 , y0 ) = (0.001, 0), where we plot the ﬁrst 100 observations.11 So although the two sequences converge on the attractor illustrated in ﬁgure 7.17, they approach it quite differently. Because 11

This plot is derived using Excel rather than Mathematica or Maple, since spreadsheets are good for plotting time-series or discrete trajectories quickly and easily. See section 5.5.

310

Economic Dynamics this attractor is sensitive to initial conditions, then it is called a chaotic attractor. The word ‘strange’ refers to the geometrical shape of the attractor, while the word ‘chaotic’ indicates sensitivity to initial conditions and therefore refers to the dynamics of the attractor (Hommes 1991). We have already noted that when a series is chaotic then it is very sensitive to initial conditions. We have also noted in terms of the H´enon map, that regardless of the initial value, the orbit will settle down to that indicated in ﬁgure 7.17 if a = 1.4 and b = 0.3. Now if two series are chaotic then they will diverge from one another, and such a divergence will increase exponentially, as illustrated in ﬁgure 7.18. If it is possible to measure the divergence between two series then we can obtain some measure of chaos. Furthermore, looking at the H´enon map, if we divide the rectangular box into very tiny rectangles then, by means of a computer, it is possible to establish how many times points in the attractor are visited by trajectories of various points. The Lyapunov dimension or Lyapunov number does just this. For instance, the Lyapunov number for the H´enon map is 1.26. The following theorem has been demonstrated (Gulick 1992). THEOREM 7.3 (1) If an attractor has a non-integer Lyapunov number then it is a strange attractor. (2) If an attractor has sensitive dependence on initial conditions then it has a Lyapunov number greater than unity. The H´enon map satisﬁes (1), so it is a strange attractor; but it also satisﬁes (2), so it is also called a chaotic attractor. It is possible to have a strange attractor that is not chaotic and a chaotic attractor that is not strange. However, most strange attractors are also chaotic. The Lyapunov dimension is important in the area of empirical chaos, and is used in the economics literature to detect chaos.12 7.7.2

The Lorenz attractor

The Lorenz attractor was probably the ﬁrst strange attractor to be discussed in the literature, and was certainly the origin of the term ‘butterﬂy effect’.13 The Lorenz system is composed of three differential equations x˙ = −σ (x − y) y˙ = rx − y − xz

(7.23)

z˙ = xy − bz Solving for the steady state, we obtain three ﬁxed points P1 = (0, 0, 0) √ √ P2 = ( b(r − 1), b(1 − r), r − 1) √ √ P3 = (− b(r − 1), − b(1 − r), r − 1)

(7.24)

12 13

There are other measures of chaos, such as Brock’s residual test (Brock 1986). The title of Lorenz’s address to the American Association for the Advancement of Sciences in 1979 was: ‘Predictability: Does the ﬂap of the butterﬂy’s wings in Brazil set off a tornado in Texas.’

Chaos theory

311 Figure 7.19.

The critical point P0 holds for all values of r, while critical points P1 and P2 only hold for r ≥ 1, where P0 = P1 = P2 for r = 1. There is therefore a bifurcation at r = 1. As r passes through this value the critical point P0 bifurcates and the critical points P1 and P2 come into existence. A typical plot of the Lorenz system is shown in ﬁgure 7.19 for σ = 3, r = 26.5 and b = 1. Starting at the value (x0 , y0 , z0 ) = (0, 1, 0) the system ﬁrst gets drawn to point P2 then after moving around and away from this point it gets drawn to point P1 . This process keeps repeating itself. In order to consider the stability properties of the Lorenz system we need to consider the linear approximation. The coefﬁcient matrix of the linearised system is −σ σ 0 A = r − z −1 −x y x −b This leads to a set of cubic characteristic equations, one for each of the three critical points. There has been much investigation into the properties of these for various values of the parameters, which is beyond the scope of this book. Asymptotic stability is assured, however, if the real parts of the eigenvalues are negative. It can be shown that if 1 < r < rH where σ (σ + b + 3) σ −b−1 then the real parts of the eigenvalues are negative and so the three critical points are asymptotically stable. Furthermore, if r = rH then two of the roots are imaginary and there occurs a Hopf bifurcation. When r > rH there is one negative real eigenvalue and two complex eigenvalues with positive real parts and the three critical points are unstable. Figure 7.20 shows the paths of two Lorenz systems. Both are drawn for σ = 10 and b = 8/3. With these values then rH = 24.7368. In ﬁgure 7.20(a) we have r = 22.4 with initial point (7,7,20), which is ‘close to’ the strange attractor; and in ﬁgure 7.20(b) we have r = 28. The ﬁrst shows what an asymptotically stable rH =

312

Economic Dynamics

Figure 7.20.

system looks like, with the system being attracted to one of the ﬁxed points. What is not so clear is what an unstable Lorenz system would look like. Figure 7.20(b) shows one such possibility, with the system starting from point (5,5,5) constantly being ﬁrst attracted to one ﬁxed point and then the other repeatedly. The Lyapunov dimension for the Lorenz system is 2.07. Then by theorem 7.3 it is a strange attractor (the Lyapunov number is noninteger) and it is also chaotic (the Lyapunov number exceeds unity).

7.8 Rational choice and erratic behaviour In this ﬁrst economic application of chaos theory we consider the situation where preferences depend on experience, and is based on the paper by Benhabib and

Chaos theory

313

Day (1981). In such a situation choices can show cyclical patterns or even erratic (chaotic) patterns. The type of situations envisaged in which preferences depend on experience is where an individual reads a novel, then sees the movie rather than read the book a second time, but as a consequence of seeing the movie is stimulated to reread the novel. This consumption pattern is intertemporal. Furthermore, although habit is a strong pattern of human behaviour, so is novelty. So from time to time we do something quite different. For example, the person who holidays each year in Majorca, but then suddenly decides a holiday in the Alps is what is required. Or the individual who alternates between a beach holiday and one in the mountains or the country. The feature here is a shift in consumption pattern that then shifts back. Such choice behaviour is ruled out in neoclassical consumer theory. Neoclassical consumer theory cannot handle novelty in choice behaviour. The analysis begins with the typical Cobb–Douglas utility function that is maximised subject to the budget constraint. Here x and y denote the consumption of the two goods, p and q their prices, respectively, and m is the individual’s level of income max U = xa y1−a

(7.25)

s.t. px + qy = m Setting up the Lagrangian L = xa y1−a − λ(m − px − qy) leads to the ﬁrst-order conditions ∂L/∂x = aU/x − λp = 0 ∂L/∂y = (1 − a)U/y − λq = 0 ∂L/∂λ = m − px − qy = 0 From these conditions we readily establish the demand curves x=

m a, p

y=

m (1 − a) q

(7.26)

Now assume that the parameter a in the utility function, which represents a property of preferences, depends endogenously on past choices. More speciﬁcally, assume at+1 = bxt yt

(7.27)

The parameter b in equation (7.27) denotes an ‘experience-dependent’ parameter. The greater the value of b the greater the value of the parameter a in the next period and so the more preferences swing in favour of good x. Substituting (7.27) into the demand equations (7.26), then xt+1 =

m bxt yt , p

yt+1 =

m (1 − bxt yt ) q

Concentrating on xt+1 , then from the budget equation we have yt =

m − pxt q

(7.28)

314

Economic Dynamics so xt+1 =

mbxt (1 − pxt ) q

Taking the short run and normalising prices at unity, so that p = q = 1, then xt+1 = bmxt (m − xt )

(7.29)

The ﬁxed point of equation (7.29) is found by solving x∗ = bmx∗ (m − x∗ ) which is x∗ =

bm2 − 1 bm

So such consumption will only be positive if bm2 > 1. Furthermore, since xt+1 = f (xt ) = bmxt (m − xt ) represents consumption of good x, then this is at a maximum when f (x) = 0, i.e. f (x) = bm2 − 2bmx = 0 m x= 2 and maximum consumption is f (m/2) = bm3/4. Since maximum consumption cannot exceed total income (recall p = 1), then f (m/2) ≤ m, or bm3 ≤m 4

implying

bm2 ≤ 4

To summarise, we have established two sets of constraints bm2 > 1 and bm2 ≤ 4. Putting these together, then 1 < bm2 ≤ 4

(7.30)

Benhabib and Day (1981) in considering equation (7.29) and the constraints (7.30), establish the following results: (1) (2)

A three-period cycle exists and so by the Li–Yorke theorem, period cycles of every order exist, including chaos. Chaos begins at a critical value c = 2.57 over the interval x ∈ [0, m] where c < bm2 ≤ 4.

Benhabib and Day also make the following observation. The smaller the experience parameter b, the greater the income endowment m must be in order to generate chaotic behaviour. What this suggests is that for individuals with low income, longrun patterns of consumption tend to be stable. However, as income grows, so does the possibility of instability, and erratic (chaotic) behaviour becomes more likely at very high levels of income.14 14

Does this explain the erratic consumption patterns of individuals like the DJ Chris Evans, Sir Elton John or that of Victoria and David Beckham?

Chaos theory

315 Figure 7.21.

7.9 Inventory dynamics under rational expectations This section discusses a disequilibrium inventory model by Hommes (1991), where a full discussion can be found (1991, chapter 28). In this model the short side of the market determines market values. We discuss this concept in chapter 8, but the situation is illustrated in ﬁgure 7.21 for both the labour market and the product market. Labour supply, Lts , is assumed constant at the value of c. Labour demand, Ltd , is less straightforward and we shall approach a discussion of this by considering ﬁrst aggregate supply. For the moment, we assume it is a downward sloping curve, as shown in ﬁgure 7.21.15 Actual labour employed, Lt , is then given by the short side of the market, so Lt = min{Ltd , Lts }, and is shown by the heavy line in ﬁgure 7.21(a). Figure 7.21(b) shows aggregate demand, ydt , and aggregate supply, yst . It denotes the level of inventories, and is positive when there is excess demand, otherwise it is zero, i.e., It = max{0, yst − ydt }. We denote expected aggregate demand by E(ydt ) and the desired level of inventories by Itd . It is assumed that desired inventories are proportional to expected aggregate demand, Itd = βE(ydt ). Production is assumed proportional to labour employed, δLt , and so δ denotes labour productivity. Aggregate supply, therefore, is inventories over from the last period plus current production, i.e., yst = It−1 + δLt . On the other hand, aggregate supply is based on expected aggregate demand plus desired inventories. Thus yst = It−1 + δLt

ydt = E ydt + Itd = E ydt + βE ydt

= (1 + β)E ydt Setting these equal to each other gives

(1 + β)E ydt − It−1 Lt = δ 15

Figure 7.21 presents the terms in a more familiar setting, but wages and prices do not enter this modelling framework.

(7.31)

316

(7.32)

Economic Dynamics Hence labour demand is given by & '

d − I (1 + β)E y t−1 t Ltd = max 0, δ Aggregate demand in the economy is assumed to be a linear function of labour employed, ydt = a + bLt , where b can be thought of as the marginal propensity to consume, and we assume labour productivity is greater than the marginal propensity to consume, δ > b. The ﬁnal element of the model is our rational expectations assumption. We assume E(ydt ) = ydt , i.e., perfect foresight. The model can therefore be summarised by six equations & '

(1 + β)E ydt − It−1 d (1) Lt = max 0, δ

(7.33)

(2)

Lts = c where c is a constant

(3)

ydt = a + bLt

(4)

yst = It−1 + δLt ) ( It = max 0, yst − ydt

E ydt = ydt

(5) (6)

(7.34)

Consider now the labour market in terms of ﬁgure 7.21(a). At a wage rate, w0 , say, actual labour employed, Lt must be positive and equal to labour demand. Therefore

(1 + β)E ydt − It−1 (1 + β)(a + bLt ) − It−1 = Lt = δ δ Solving (7.34) for Lt gives Lt =

(7.35)

(1 + β)a − It−1 δ − b(1 + β)

This value must lie between 0 and c. Assume δ − b(1 + β) > 0, then (1 + β)a − It−1 γ1 . Similarly, if Lt > 0 then It−1 < a(1 + β). Let γ2 = a(1 + β), then putting these together, we have that if Lt lies between 0 and c, then γ1 < It−1 < γ2 . We need, however, to consider three situations: (i) It−1 ≤ γ1 , (ii) γ1 < It−1 < γ2 , and (iii) It−1 ≥ γ2 . (i)

It−1 ≤ γ1

If It−1 ≤ γ1 then Lt = c since this is the short side of the market in these circumstances. Then It = yst − ydt = It−1 + δc − a − bc

Chaos theory

317

i.e. It = It−1 + (δ − b)c − a (ii)

(7.36)

γ1 < It−1 < γ2

Then It = yst − ydt = It−1 + δLt − a − bLt (1 + β)a − It−1 −a = It−1 + δ − b δ − b(1 + β) On expanding and simplifying It = (iii)

aδβ −bβIt−1 + δ − b(1 + β) δ − b(1 + β)

(7.37)

It−1 ≥ γ2

Under these circumstances Lt given by equation (7.35) is negative and so Lt = Ltd = 0. Hence It = yst − ydt = It−1 − a

(7.38)

Combining all three cases, then It = f (It−1 ) is a piecewise function of the form It−1 ≤ γ1 It−1 + (δ − b)c − a −bβI aδβ t−1 It = + γ1 < It−1 < γ2 δ − b(1 + β) δ − b(1 + β) It−1 − a It−1 ≥ γ2 and describes the inventory dynamics of the present model. Equilibrium investment is deﬁned only for the range γ1 < It−1 < γ2 . In this instance I∗ = i.e.

aδβ −bβI ∗ + δ − b(1 + β) δ − b(1 + β) aδβ I∗ = δ−b

and since we have assumed δ > b, then this is positive. We shall now pursue this model in terms of a numerical example. Example 7.6 Let a = 0.2,

b = 0.75,

c = 1,

δ=1

and for the moment we shall leave β unspeciﬁed. If β = 0.2, then γ1 = 0.14 and γ2 = 0.24. Equilibrium investment is given by I ∗ = 0.16, which lies between the two parameter values. The piecewise difference equation is then given

(7.39)

318

Economic Dynamics

Figure 7.22.

Figure 7.23.

by

It−1 + 0.05 It = −1.5It−1 + 0.4 It−1 − 0.2

It−1 ≤ 0.14 0.14 < It−1 < 0.24 It−1 ≥ 0.24

The situation is shown in ﬁgure 7.22. The equation It = f (It−1 ) is clearly nonlinear, even though it is made up of linear segments. Although the equilibrium level of investment, I ∗ = 0.16, exists and is identiﬁed in terms of ﬁgure 7.22 where the diagonal cuts the function f (It−1 ), it is not at all obvious that this ﬁxed point will be approached in the present case. In fact, as ﬁgure 7.23 illustrates, the system is chaotic, as shown by the cobweb not settling down at the ﬁxed point from a starting value of I0 = 0.14.16 In constructing ﬁgure 7.23 we assumed that β = 0.2. If, as we just indicated, this results in chaotic behaviour, then it would be useful to see the bifurcation diagram for I ∗ against β. In doing this we allow β to range over the interval 0 to 1/3, this 16

The procedure for constructing cobwebs for piecewise functions is covered in chapter 8. However, Mathematica can be used to produce this function with the Which command as follows f[x-]:=Which[x0.14 && x=0.24,x-0.2] Maple’s deﬁnition for the piecewise function is f:=x->piecewise(x0.14 and x=0.24,x-0.2);

Chaos theory

319 Figure 7.24.

latter value comes from setting δ − b(1 + β) = 0 for the values given above. In constructing the bifurcation diagram it is important to realise that the intervals for the piecewise function vary with the change in β. Given the values above, we have the piecewise difference equation It−1 +0.05 It−1 ≤ 0.2(1+β)−[1−0.75(1+β)] −0.75βIt−1 0.2β It = + 0.2(1+β)−[1−0.75(1+β)]0 α>0

Demand and supply models

327

where quantities demanded and supplied, qd and qs , and price, p, are assumed to be continuous functions of time. The ﬁxed point, the equilibrium point, of this system is readily found by setting dp/dt = 0, which gives p∗ =

a−c b+d

(8.2)

and equilibrium quantity q∗ =

ad + bc b+d

(8.3)

For a solution (a ﬁxed point, an equilibrium point) to exist in the positive quadrant, then a > c and ad + bc > 0. We can solve for the price path by substituting the demand and supply equations into the price adjustment equation giving the following ﬁrst-order linear nonhomogeneous differential equation dp + α(b + d)p = α(a − c) dt with solution a−c a−c + p0 − e−α(b+d)t p(t) = b+d b+d which satisﬁes the initial condition p(0) = p0 . It may be thought that the dynamics of this model are quite explicit, but this is not in fact the case. To see this consider the movement of the quantity over time. The ﬁrst thing we must note is that there are two different quantities qd (t) and qs (t). So long as the price is not the equilibrium price, then these quantities will differ, and it is this difference that forces the price to alter. Thus a−c ad + bc + b p0 − qd (t) = e−α(b+d)t b+d b+d and

a−c ad + bc + d p0 − q (t) = e−α(b+d)t b+d b+d s

The situation is illustrated in ﬁgure 8.2 for an initial price below the equilibrium price. What this model lacks is a statement about the quantity transacted or traded on the market. To make a statement about the quantity traded we must make an additional assumption. Suppose we make the following assumption: ASSUMPTION 1 In disequilibrium, the short side of the market is transacted. Let q(t) denote the quantity traded, then this assumption amounts to q(t) = qs (t)

for

p 0 ≤ p∗

q(t) = q (t)

for

p0 ≥ p∗

d

(8.4)

(8.5)

(8.6)

(8.7)

328

Economic Dynamics

Figure 8.2.

or, more succinctly q(t) = min(qd (t), qs (t)) The logic behind assumption 1 is that the model contains no stocks, and when the price is below the equilibrium price, the current production is all that can be supplied onto the market. Some demand will go unsatisﬁed. If, on the other hand, the price is above the equilibrium price, and current production is in excess of demand, suppliers cannot force people to purchase the goods, and so will sell only what is currently demanded. Although when price is below the equilibrium price and the quantity traded is what is currently produced, the excess demand gives a signal to suppliers to increase future production and to raise the price. The model is less satisfactory when interpreting what is happening when the price is above the equilibrium price. In this instance there will be unsold goods from current production. The model has nothing to say about what happens to these goods. All we can say is that in the future suppliers will decrease their current production and lower the price. If stocks were included in the model and the stocks, which accumulated when the price was above the equilibrium price, could be sold off when there is excess

Demand and supply models

329

demand, then we move away from assumption 1 to a different assumption. By way of example suppose we make the following assumption: ASSUMPTION 2 Stocks are sufﬁciently plentiful to allow all demands to be met at any price, and price adjusts in proportion to the change in stock levels. Thus, if i(t) denotes the inventory holding of stocks at time t, then di = q s − qd dt t

i = i0 +

(qs − qd )dt

and price adjusts according to di dp = −α = −α(qs − qd ) = α(qd − qs ) α > 0 dt dt On the face of it this appears to be the same model, resulting as it does in the same ﬁrst-order linear nonhomogeneous differential equation. It is true that the solution for the price gives exactly the same solution path. The difference in the model arises in terms of the quantity traded. The time path of the quantity demanded, and the time path of the quantity supplied, are as before. For a price above the equilibrium price it is still the case that the quantity traded is equal to the quantity demanded (the short side of the market), and the unsold production goes into stock holdings. It is this rise in stocks that gives the impetus for suppliers to drop the price. On the other hand, when the price is below the equilibrium price, the quantity traded is equal to the quantity demanded (the long side of the market!). The excess demand over current production is met out of stocks. The fall in stock holdings is the signal to producers to raise the price and raise the level of production. Under this second assumption, therefore, we have q(t) = qd (t) for all p What is invariably missing from elementary discussions of demand and supply is the dynamics of the quantity traded. At any point in time there can be only one quantity traded, and whether this is equal to the quantity demanded, the quantity supplied or some other quantity depends on what is assumed about the market process when in a disequilibrium state. To illustrate the signiﬁcance of the arguments just presented let us consider the labour market. Here we are not concerned with the derivation of the demand for labour and the supply of labour, we shall simply assume that labour demand, LD, is negatively related to the real wage rate, w, and labour supply, LS, is positively related to the real wage rate. Since we are concerned here only with the dynamic process of market adjustment, we shall assume labour demand and labour supply are linear functions of the real wage rate, and we set up the model in continuous time. Thus LD(t) = a − bw(t)

b>0

LS(t) = c + dw(t)

d>0

LD(t) = LS(t)

(8.8)

330

Economic Dynamics

Figure 8.3.

The ﬂexible wage theory asserts that wages are highly ﬂexible and will always alter to achieve equilibrium in the labour market, i.e., the wage rate will adjust to establish LD(t) = LS(t). Let us be a little more speciﬁc. A typical textbook version (Parkin and King 1995, chapter 29) is that if the real wage is below the equilibrium wage (w < w∗ ), then the demand for labour is above the supply of labour. Employers will raise the wage rate in order to attract labour, and that this will continue until a wage rate of w∗ is established. On the other hand, if the wage rate is above the equilibrium wage rate, then households will not be able to ﬁnd jobs and employers will have many applicants for their vacancies. There will be an incentive for employers to lower the wage rate, and for labour to accept the lower wage in order to get employed. This will continue until the wage rate of w∗ is established. The dynamic adjustment just mentioned is assumed to take place very quickly, almost instantaneously. For this reason the model is often referred to as a market clearing model.1 At any point in time the wage rate is equal to the equilibrium wage rate, and the quantity of employment is equal to labour demand that is equal to labour supply. Let employment at time t be denoted E(t), then in the ﬂexible wage theory, the wage will adjust until LD(t) = LS(t) = E(t). A shift in either the demand curve for labour or the supply curve of labour will alter the equilibrium wage rate and the level of employment. Thus a rise in the capital stock will increase the marginal product of labour and shift the labour demand curve to the right. This will lead to a rise in the real wage rate and a rise in the level of employment, as illustrated in ﬁgure 8.3. A different wage theory, however, is also prevalent in the literature, called the sticky wage theory. This asserts that money wage rates are ﬁxed by wage contracts and adjust only slowly. If this assumption is correct, then it does not follow that the real wage will adjust to maintain equality between labour demand and labour 1

This is the basic assumption underlying the approach by Barro and Grilli (1994) in their elementary textbook, European Macroeconomics.

Demand and supply models

331 Figure 8.4.

supply. Suppose at the ruling money wage and price level the real wage is above the market clearing wage, as shown in ﬁgure 8.4 at the real wage rate w1 . The labour market is in disequilibrium and there is no presumption in this theory that the real wage will fall, at least in the short and (possibly) medium term. At this real wage rate there is an excess supply of labour. But employment will be determined by the demand for labour, and the excess supply will simply increase the level of unemployment. Employment will be at the level E1 and unemployment will be increased by U1 . It is just like our earlier model of stock accumulation. However, in this model it is not so easy for employers to reduce the wage on offer. At a real wage rate of w2 , which is below the market clearing wage rate of w∗ , there is an excess demand for labour. Employment, however, is no longer determined by the demand curve of labour. If individuals are not willing to put themselves on to the labour market at that real wage, then employers will simply be faced with vacancies. Employment will be at the level E2 and vacancies will rise by V2 . Of course, if such a situation prevails there will be pressure on the part of ﬁrms to increase the nominal wage, and hence increase the real wage, in order to attract individuals into the labour market. There will be dynamic forces present which will push up the nominal, and hence the real, wage rate. What we observe, then, is that the short side of the market determines the level of employment. But the labour market illustrates another implicit dynamic assumption. When the real wage is below the market clearing wage then there will be pressure on ﬁrms to raise the nominal wage in order to ﬁll their vacancies. When the real wage is above the market clearing wage employers may wish to reduce their nominal, and hence their real, wage but may be prevented from doing so because of contractual arrangements. In other words, there is an asymmetric market adjustment: real wages may rise quicker when there is excess demand for labour than they will fall when there is an equivalent excess supply of labour. Such asymmetric market adjustment takes us yet further into realms of additional assumptions speciﬁc to the labour

332

Economic Dynamics market. What appeared to be a straightforward demand and supply model takes on a rather complex pattern depending on the assumptions of dynamic adjustment.

8.3 The cobweb model In the previous section we referred to the possibility that price would be altered in the next period in the light of what happened in this period. Whenever decisions in one period are based on variables in another period we inherently have a dynamic model. The simplest of such models is the cobweb model of demand and supply. This model is also set out most usually in a discrete form. This is understandable. The model was originally outlined for agriculture (Ezekiel 1938), and concentrated on the decision-making of the farmer. First we shall consider a simple linear version of the model. Demand at time t is related to the price ruling at time t. Letting qdt denote the quantity demanded at time t and pt the actual price at time t, then we have qdt = a − bpt

b>0

However, the farmer when making a decision of what to grow or what to produce will need to make a decision much earlier, and he will make a decision of what quantity to supply, qst , based on what price he expects to receive at time t, i.e., pet . Accordingly, the supply equation takes the form qst = c + dpet

d>0

At any point in time, the market is considered to be in equilibrium and so the quantity traded, qt , is equal to the quantity demanded, which is equal to the quantity supplied. The model is, then qdt = a − bpt (8.9)

qst = c + dpet qdt = qst = qt As the model stands it cannot be solved because of the unobservable variable pet . We therefore have to make a further assumption about how the supplier forms his or her expectation or makes a decision about the expected price. The simplest assumption of all is that he or she expects the price at time t to be what it was in the previous period. This assumption, of course, amounts to assuming pet = pt−1 . The model now becomes qdt = a − bpt

(8.10)

qst = c + dpt−1 qdt = qst = qt Substituting the demand and supply equations into the equilibrium condition we obtain

(8.11)

a − bpt = c + dpt−1 a−c d pt = − pt−1 b b

Demand and supply models

333

which is a ﬁrst-order nonhomogeneous dynamic system. Notice that it is also an autonomous dynamic system because it does not depend explicitly on the variable t. The system is in equilibrium when the price remains constant for all time periods, i.e., pt = pt−1 = . . . = p∗ . Thus p∗ =

a−c b+d

where p∗ ≥ 0 if a ≥ c

(8.12)

With linear demand and supply curves, therefore, there is only one ﬁxed point, one equilibrium point. However, such a ﬁxed point makes economic sense (i.e. for price to be nonnegative) only if the additional condition a ≥ c is also satisﬁed. To solve this model, as we indicated in part I, we can reduce the nonhomogeneous difference equation to a homogeneous difference equation by taking deviations from the equilibrium. Thus d a−c − pt−1 pt = b b a−c d ∗ − p p∗ = b b d ∗ pt − p = − ( pt−1 − p∗ ) b with solution d t pt − p = − ( p0 − p∗ ) b ∗

which satisﬁes the initial condition pt = p0 when t = 0. More fully pt =

a−c b+d

a−c d t p0 − + − b b+d

With the usual shaped demand and supply curves, i.e., b > 0 and d > 0, then d/b > 0, hence (−d/b)t will alternate in sign, being positive for even numbers of t and negative for odd numbers of t. Furthermore, if 0 < |−d/b| < 1 then the series will become damped, and in the limit will tend towards the equilibrium price. On the other hand, if |−d/b| > 1 then the system will diverge from the equilibrium price. Finally, if |b| = |d| (or |−d/b| = 1), then the system will neither converge nor diverge and will exhibit a two-period cycle. These results were veriﬁed by means of a simple numerical example and solved by means of a spreadsheet in chapter 3, ﬁgure 3.11. Example 8.1 By way of variation, here we shall solve a numerical version of the system using Mathematica and Maple (see Eckalbar 1993). The input instructions for each

(8.13)

334

Economic Dynamics package are Mathematica {a,b,c,d}={20,4,2,2.5} A=-b/d pt1[pt-]:=A*pt price[t-,p0-]:=ListPlot[NestList[pt1,p0,t], PlotJoined->True, PlotRange->All, AxesOrigin->{0,0}] price[20,1]

Maple A:=20: b:=4: c:=2: d:=2.5: A:=-d/b; sol:=rsolve(p(t)=A*p(t-1),p): equ:=subs(p(0)=1,sol): f:=t->equ: points:=[seq([t,f(t)],t=0..20): plot(points);

In Mathematica, we ﬁrst set values for the parameters a, b, c and d, namely {a, b, c, d} = {20, 4, 2, 2.5}. We then deﬁne the ratio (−d/b) = A, resulting in A = −0.625, and deﬁne the recursive function pt1[pt ] := A*pt. It now remains to generate the series of values for the price using the NestList command, and to plot the resulting series using the ListPlot command. The result is ﬁgure 8.5. Notice that this version of the cobweb allows different periods to be speciﬁed and a different initial price. Thus, price[50,2] would indicate a plot of 50 periods with an initial price of 2. Figure 8.5 can also be generated using the Maple commands given above. Again, we input the values for the four parameters and deﬁne A. In Maple we approach the next part slightly differently. We solve the difference equation, using the rsolve command. Then deﬁne the equation for the initial price being unity, and use this to deﬁne the mapping f . Next we create a series of points using the sequence command and plot these to create ﬁgure 8.5. It is readily observed from ﬁgure 8.5 that this system is dynamically stable, with the price converging on the equilibrium price – since deviations from equilibrium converge on zero. Using a software package such as Mathematica or Maple different values for the parameters can readily be investigated. For instance, parameter

Figure 8.5.

Demand and supply models

335 Figure 8.6.

values {20, 4, 2, 6} and price [20,1] will readily be shown to exhibit an unstable system with price diverging from its equilibrium. This stability and instability is purely dependent on the value of A. In ﬁgure 8.5 the system is stable and A = −0.625 while the alternative parameter values gives A = −1.5 leading to an unstable system. All this we investigated in chapter 3. Nothing in the analysis so far would indicate why it is referred to as the ‘cobweb’ model. This is because we have concentrated solely on the time path of price in the system. There are two ways to exhibit the cobweb. One is to show the sequence of points on a demand and supply diagram (ﬁgure 8.6(a))2 and the other is to plot the differential equation for price in relation to the 45◦ -line (ﬁgure 8.6(b)). In many ways ﬁgure 8.6(a) is more revealing because it shows the behaviour of both price and quantity. Here we have a convergent cobweb. The same convergence pattern is shown in ﬁgure 8.6(b), but here concentration is on the price sequence. The second approach, however, is more useful for mathematical investigation (especially of nonlinear systems). So far we have concentrated on the very simple linear model. A number of avenues can be explored. Within the conﬁnes of the linear model, it is possible to specify a different behaviour for the expected price. It may be argued that expecting the price in the current period to be what it was in the previous period is very na¨ıve and takes no account of the trend in prices. It is possible, therefore, to specify an 2

Notice that quantity is on the vertical axis and price on the horizontal axis.

336

Economic Dynamics adaptive expectation in which the expected price is an adjustment of the forecast error in the previous guess. More speciﬁcally, we can write pet = pet−1 − λ(pet−1 − pt−1 )

(8.14)

If λ = 1 then this amounts to our previous model. In other words, the previous model can be considered as a special case of the present model in which λ = 1. The model in full is qdt = a − bpt qst pet qdt

(8.15)

b>0

= c + dpt−1 d > 0

= pet−1 − λ pet−1 − pt−1

0≤λ≤1

= qst = qt

We can solve this model by noting pet = (1 − λ)pet−1 + λpt−1 qd − c a − bpt − c qst − c = t = d d d b a − c − pt−1 ... pet−1 = d d

pet =

Hence

b pt−1 + dλpt−1 d λd a−c + 1−λ− pt−1 pt = λ d b

a − bpt = c + d(1 − λ) (8.16)

i.e.

(8.17)

a−c d

−

Setting pt = pt−1 = . . . = p∗ for all t readily gives the same equilibrium price, namely p∗ = (a − c)/(b + d). As in our earlier and simpler model, taking deviations from equilibrium readily gives the solution t λd p t − p∗ = 1 − λ − ( p0 − p∗ ) b Let the term in square brackets be denoted B, then for this model to exhibit a convergent oscillatory solution it is necessary for −1 < B < 0, in other words it is necessary to satisfy the condition

(8.18)

d 2 1 −1< < −1 λ b λ Of course, there can be many other speciﬁcations for price expectation, each giving rise to a different model. For instance, we can postulate the following (Goodwin 1947)

(8.19)

pet = pt−1 + η( pt−1 − pt−2 ) where η is the coefﬁcient of expectations. If η = 0 then pet = pt−1 , which is our original formulation. If η > 0 then price is expected to move in the same direction as in the past; while if η < 0, then price is expected to reverse itself. The extent of

Demand and supply models

337

these price movements is very dependent on the magnitude of η. The full model is qdt = a − bpt qst pet qdt

b>0

= c + dpt−1

d>0

= pt−1 + η( pt−1 − pt−2 )

(8.20)

= qst = qt

The model can be solved algebraically, but it can also be easily investigated by means of a spreadsheet. To do this we ﬁrst do some algebraic manipulation. Substituting the expectations equation into the supply equation, and then equating this with demand, we ﬁnd a − bpt = c + d[pt−1 + η( pt−1 − pt−2 )] a−c dη d i.e. pt = − (1 + η)pt−1 + pt−2 b b b

(8.21)

which is a second-order difference equation.

Example 8.2 Suppose we set up the initial spreadsheet with the following parameter values, and the necessary two initial prices a = 100 b=2 p0 = 20

c = −20 d = 1.25 p1 = 24

η = −1.2

The model is illustrated in ﬁgure 8.7. Having set up the model, it is quite easy to change the value of the parameters, but most especially η, and see the result on the price series. In most spreadsheets the price plot will change interactively as the parameter values are changed. It is of course possible to solve the model algebraically and investigate the restrictions on the parameter values (see Gandolfo 1971, pp. 91–6), but any student of economics can investigate the properties of this model by means of a spreadsheet. Figure 8.7.

338

Economic Dynamics

8.4 Cobwebs with Mathematica and Maple The intention of this section is to provide routines for creating cobwebs using Mathematica and Maple. There are many ways to do this and here we shall provide only one for each package. Mathematica’s routine is based on Gray and Glynn (1991, chapter 7), while Maple’s routine is an adaptation of this. The Maple routine is written as a procedure, a mini-programme. The reason for this is so many cobwebs can be then created with the minimum of input instructions. A different Maple routine can be found in Lynch (2001).3 Both the routines below can handle linear and nonlinear equations and can also be adapted to deal with stepwise functions. Since the Mathematica routine is the basis for the two instructions, a detailed explanation of this is provided in appendix 8.1. This appendix also provides the Maple routine just as a set of input instructions rather than as a procedure. We consider only the recursive form of the cobweb, as illustrated in ﬁgure 8.6(b) and begin with the linear recursive equation xt = f (xt−1 ) = a + bxt−1

(8.22)

The instructions for each programme are as follows, where we use the recursive equation in ﬁgure 8.6(b) as an illustration: Mathematica4 f[x-]:=a+b*x {a=4.5,b=-0.625} x0=1 points=Rest[Partition[Flatten[Transpose[ {NestList[f,x0,20],NestList[f,x0,20]}] ],2,1]]; web=ListPlot[points,PlotJoined->True] lines=Plot[{f[x],x},{x,0,5}] cobweb=Show[web,lines]

Maple restart; with(linalg): with(plots): cobweb:=proc(f,x0,n,xmin,xmax) local fk,list1,list2,list3,list4,web,lines; fk:=(x,k)->simplify(([emailprotected]@k)(x)); list1:=transpose(array([[seq(fk(x0,k),k=0..n)], [seq(fk(x0,k),k=0..n)]])): list2:=convert(convert(list1,vector),list): list3:=convert(transpose(array( [list2[1..nops(list2)-1], list2[2..nops(list2)]])),listlist): list4:=[list3[2..nops(list3)]]: 3 4

See Lynch (2001, pp. 249–50 ). Here Lynch provides a routine for the tent function, which is a stepwise function, but this is readily adapted for any linear or nonlinear function. Intermediate displays in Mathematica can be suppressed by including the instruction: DisplayFunction->Identity, then in the ﬁnal display using the Show command, include DisplayFunction->$DisplayFunction. See ﬁgure 8.8(a) and appendix 8.1.

Demand and supply models web:=plot(list4): lines:=plot({f(x),x},x=xmin..xmax,colour=blue): display({web,lines}); end: f:=x->4.5-0.625*x; cobweb(f,1,20,0,5);

The cobweb procedure for Maple requires you to ﬁrst deﬁne the function f and then to supply the values for x0, n, xmin and xmax. The penultimate line therefore deﬁnes the function and then the input instruction cobweb( f ,1,20,0,5) indicates to use the procedure for the deﬁned function, give x0 a value of unity, n a value of 20, and xmin and xmax values of 0 and 5, respectively. These instructions produce a similar plot to the Mathematica instructions given above. Nonlinear equations too are readily handled. If we are considering, for example, the nonlinear logistic equation xt = f (xt−1 ) = rxt−1 (1 − xt−1 )

x0 = 0.1

with r = 3.85, then in Mathematica we replace the ﬁrst three lines with f[x-]:=r*x*(1-x) {r=3.85} x0=0.1

and we also need to change the range for x in the ‘lines’ input to {x,0,1}. In Maple, on the other hand, we simply replace the last two lines with f:=x->3.85*x*(1-x); cobweb(f,0.1,20,0,1);

Figure 8.8 shows screen shots of the ﬁnal output for each programme. As can be seen from the ﬁgure, the results are virtually the same. It is, of course, possible to include instructions for labelling the axes and provide headings, but we have not done that here. It is also fairly straightforward to deal with piecewise functions, and we shall illustrate exactly how in the next section.

8.5 Cobwebs in the phase plane Return to our linear cobweb model in which supply is based on last period’s price. The resulting difference equation is d a−c pt = − pt−1 b b or pt = A − Bpt−1 d a−c and B = where A = b b Hence, the function pt = f ( pt−1 ) = A − Bpt−1 is linear in the phase space in which pt−1 is on the horizontal axis and pt is on the vertical axis. Figure 8.9 illustrates

339

340

Economic Dynamics

Figure 8.8.

two possibilities in which demand and supply have conventional slopes. The linear mapping is shown by the line denoted L. The 45◦ -line, denoted E, satisﬁes the condition pt = pt−1 ∗

for all t

and p denotes a ﬁxed point, an equilibrium point.

Demand and supply models

341 Figure 8.9.

Whether such a ﬁxed point is stable, unstable or periodic can be established from the slope of L. If 0 < |−B| < 1, as in ﬁgure 8.9(a), then p∗ is an attractor, and the price sequence {pt }, starting at p0 , converges on p∗ . Starting at p0 , then, p1 = A − Bp0 , which is read off the line L. This is the same as p1 on the E-line. At p1 , then p2 = A − Bp1 , which again is read off the line L. The sequence will continue until p∗ is reached. On the other hand, the sequence {pt } starting at p0 in ﬁgure 8.9(b), diverges from p∗ . This is because |−B| > 1, and so p∗ is a repellor. What happens when |−B| = 1, where the slope of the demand curve in absolute value is equal to the slope of the supply curve? We then have pt = A − pt−1 The situation is illustrated in ﬁgure 8.10. In chapter 3, section 3.4, we deﬁned a solution yn as periodic if yn+m = yn , and the smallest integer for m is the period

342

Economic Dynamics

Figure 8.10.

of the solution. In the present example, beginning at p0 , we have p1 = A − p0 p2 = A − p1 = A − (A − p0 ) = p0 p 3 = A − p 2 = A − p0 p4 = A − p3 = A − (A − p0 ) = p0 If follows, therefore, that p0 = p2 = p4 = . . .

and

p1 = p3 = p5 = . . .

We have a two-period cycle solution. In fact, with linear demand and supply, with equal slopes in absolute value, there can only be a two-period cycle and no higher one is possible. Discrete systems have come under increasing investigation in recent years because of the possibility of chaotic behaviour to which they can give rise (see chapter 7). By way of introduction to this analysis we shall continue with our simple linear example and consider how Mathematica can be used to investigate cobweb models. It will be found that the phase plane plays an important part in this approach. Although a little repetitive, we shall provide both Mathematica and Maple instructions for deriving the cobwebs. However, since the Maple procedure never changes, we shall simply write ‘cobweb procedure’. Example 8.3 We shall illustrate the technique by means of the following simple cobweb model qdt = 24 − 5pt qst = −4 + 2pt−1 qdt = qst = qt

Demand and supply models

343

which has equilibrium values p∗ = 4 and q∗ = 4. The resulting difference equation is pt = 5.6 − 0.4pt−1 The objective is to plot a sequence of points in the phase plane. Joining up these points forms the web of the cobweb. Superimposed on this web is the line f (p) = 5.6 − 0.4p and the 45◦ -line. A full explanation of the instructions for Mathematica and Maple are given in appendix 8.1. Mathematica f[p-]:=5.6-0.4p p0=1 points=Rest[Partition[Flatten[Transpose[ {NestList[f,p0,20],NestList[f,p0,20]} ] ], 2, 1 ] ]; web=ListPlot[points,PlotJoined->True] lines=Plot[ {f[p],p}, {p,0,10} ] cobweb=Show[web,lines, AxesLabel->{"pt-1","pt"} ]

Maple `cobweb procedure’ f:p->5.6-0.4*p cobweb(f,1,20,0,10)

Notice in the Maple instructions that although the procedure deﬁnes the function in terms of the variable x, we deﬁne f in terms of p. We can do this, and we do it repeatedly throughout this chapter, because the variables within the procedure are deﬁned locally. The result is shown in ﬁgure 8.11. To illustrate the generality of this approach, and to highlight a nonlinear cobweb, consider the following model. Example 8.4 qdt = 4 − 3pt qst = p2t−1 qdt = qst = qt Figure 8.11.

344

Economic Dynamics

Figure 8.12.

The equilibrium, assuming a positive price, is given by p∗ = 1 and q∗ = 1, and the difference equation is given by pt = (4/3) − (1/3)p2t−1 The question now arises as to whether the solution p∗ = 1 is stable or not. Using Mathematica we can input the following instructions (where we have included solving for equilibrium price): Mathematica f[p-]:=(4/3)-(1/3)p^2 EquPrice=Solve[p==f[p],p] p0=1.5 points=Rest[Partition[Flatten[Transpose[ {NestList[f,p0,20],NestList[f,p0,20]} ] ], 2, 1 ] ]; web=ListPlot[points,PlotJoined->True] lines=Plot[ {f[p],p}, {p,0,2} ] cobweb=Show[web,lines, AxesLabel->{"pt-1","pt"} ]

Maple’s instructions are Maple `cobweb procedure’ f:=p->(4/3)-(1/3)*p^2 EquPrice:=solve(p=f(p),p); cobweb(f,1.5,20,0,2);

The resulting cobweb is illustrated in ﬁgure 8.12. The model, therefore, has a locally stable solution for a positive price equilibrium. It should be noted5 that at the equilibrium point we have f ( p∗ ) = 2(−1/3)p∗ = −2/3 i.e. f (p∗ ) < 1

5

See chapter 3, section 3.4.

Demand and supply models which veriﬁes the local stability of p∗ = 1. Although this particular nonlinearity leads to a stable solution, some authors (e.g. Waugh 1964) have argued that other types will lead to a two-period cycle and may well be the norm. One especially important nonlinearity can arise where there is a price ceiling set (Waugh, 1964). Consider the following linear demand and supply model. Example 8.5 qdt = 42 − 4pt qst = 2 + 6pt−1 qdt = qst = qt The difference equation from this model is pt = 10 − 1.5pt−1 with equilibrium values p∗ = 4 and q∗ = 26. Since the slope of the function f (p) = 10 − 1.5p is greater than unity in absolute terms, then the cobweb is unstable. This is readily veriﬁed starting with a price of p0 = 3.5. Suppose a price ceiling pc = 6 is set, then the price in any period cannot exceed this ceiling. What we have, then, is a function which is kinked at p = 8/3, the value where pc = f ( p). This can be expressed as $ 6 p < 8/3 f (p) = 10 − 1.5p p ≥ 8/3 Within Mathematica there is only a slight difference in deﬁning the function. We deﬁne it using the If command, i.e. f[p-]:=If[ ppiecewise(p{"pt-1","pt"} ]

Maple `cobweb procedure’ f:p->piecewise(p 0

sct = c1 dtc = sct

d1 > 0

+

d1 pct−1

Hog market dth = a2 − b2 pht sht = c2 + dth = sht

(8.24)

d2 pht−1

b2 > 0 +

epct−1

d1 > 0,

eTrue] ListPlot[dataRARB,PlotStyle->PointSize[0.02]];

Maple RA:=’RA’: RB:=’RB’: t:=’t’: a:=’a’: b:=’b’: mA:=’mA’: mB:=’mB’: nA:=’nA’: nB:=’nB’: a:=0.1: b:=-3.8: mA:=100: mB:=100: nA:=40: nB:=40:

412

Economic Dynamics RA:=proc(t) option remember; max(0,evalf(RA(t-1)+(2*a*mA/Pi)*(arctan (RA(t-1)))(2*a*mB/Pi)*(arctan(RB(t-1)))-a*(nA-nB))) end; RB:=proc(t) option remember; max(0,evalf(RB(t-1)+(2*b*mB/Pi)*(arctan (RB(t-1)))(2*b*mA/Pi)*(arctan(RA(t-1)))-b*(nB-nA))) end; RA(0):=10: RB(0):=15: dataRA:=[seq([t,RA(t)],t=200..250)]: dataRARB:=[seq([RA(t),RB(t)],t=200..250)]: plot(dataRA); plot(dataRARB,style=point);

The only item in these instructions that needs to be changed in producing ﬁgure 9.18 is the value of the parameter b, which takes on the four values b = −3.8, −4.0, −4.18 and −4.5. 9.5.2

Learning

The model is further extended by the authors to take account of learning. The possibility of adaptation is taken into account by allowing the policy parameters a and b to change. In establishing when to change the policy a simple rule is chosen. Let f denote the frequency of choosing when to change policy (assumed constant), e.g., every year or every quarter. Then deﬁne M, the sum of the difference in product quality standard between the last decision point and the present one, assumed to be at time t, i.e. M=

t

StA − StB

t−f +1

If M > 0 then the current policy is assumed satisfactory and no change in policy is made. If, however, M < 0, then a change is considered necessary. Since a positive value of M for ﬁrm A implies a negative value of M for ﬁrm B and vice versa, then at any decision point one ﬁrm will always be altering its policy. The authors consider two policy adaptations. Proportional policy adaptation A change amounting to a proportion of the existing policy, i.e. as = αas−1 bs = βbs−1

00

Hence show (i) equilibrium q∗1 and q∗2 are A − 2a1 + a2 3B A + a1 − 2a2 ∗ q2 = 3B q∗1 =

4.

(ii) that no matter what the initial value for (q10 , q20 ), the system always converges on the equilibrium. For the duopoly model p=9−Q Q = q1 + q2 TC1 = a1 q1 TC2 = a2 q2

5.

(i) Establish the equilibrium for q1 and q2 in terms of a1 and a2 . Show that if a1 < a2 then q∗1 > q∗2 . (ii) Let a1 = 3 and a2 = 5 and consider initial points (a) ﬁrm 1 the monopolist (b) ﬁrm 2 the monopolist From which initial point does the system reach equilibrium sooner? Consider the model set out in equation (9.5). Let the costs, however, be TCi = 5qi , i = 1, 2, 3. (i) Is the equilibrium point (q∗1 , q∗2 , q∗3 ) closer to the origin? (ii) Establish the reaction curves for this model. (iii) Does this system also oscillate with constant amplitude?

Dynamic theory of oligopoly 6.

Consider the model p=9−Q Q = q 1 + q 2 + q3 TC1 = 3q1 TC2 = 2q2 TC3 = q3

7.

(i) Establish the Cournot equilibrium. (ii) Plot trajectories from initial points where each ﬁrm is a monopolist. (iii) Under the assumption that each ﬁrm maximises its proﬁts under the conjectural variation that the other ﬁrms are holding their output levels constant, solve the system’s difference equations. Consider the model p = 15 − 2Q Q = q1 + q2 + q3 TC1 = 5q1 TC2 = 3q2 TC3 = 2q3

8.

(i) Establish the Cournot solution. (ii) Plot trajectories from initial points where each ﬁrm is a monopolist. (iii) Under the assumption that each ﬁrm maximises its proﬁts under the conjectural variation that the other ﬁrms are holding their output levels constant, solve the system’s difference equations. (iv) Is the system dynamically stable? Consider the following four models (a)

p = 20 − 3Q Q = q1 + q2 TC1 = 4q1 TC2 = 4q2

(b)

p = 20 − 3Q Q = q 1 + q2 + q 3 TC1 = 4q1 TC2 = 4q2 TC3 = 4q3

(c)

p = 20 − 3Q Q = q1 + q2

(d)

p = 20 − 3Q Q = q 1 + q2 + q 3

TC1 = 4q21

TC1 = 4q21

TC2 = 4q22

TC2 = 4q22 TC3 = 4q23

(i) Establish the Cournot solution for each. (ii) What, if anything do you observe about the dynamic behaviour in comparing n = 2 as against n = 3 for constant MC? (Take initial points from the monopoly position in each case.) (iii) What, if anything, do you observe about the dynamic behaviour in comparing model (a) with model (b) and model (c) with model (d)? (Take initial points from the monopoly position in each case.)

421

422

Economic Dynamics 9.

Consider the continuous model p(t) = 20 − 3Q(t) Q(t) = q1 (t) + q2 (t) TC1 (t) = 4q1 (t) TC2 (t) = 4q2 (t) q˙ 1 (t) = 0.2(x1 (t) − q1 (t)) q˙ 2 (t) = 0.2(x2 (t) − q2 (t))

10.

where xi (t) i = 1, 2 is the desired output level that maximises proﬁts under the assumption that the other ﬁrm does not alter its output level. (i) Find the Cournot solution. (ii) Is the system dynamically stable? (iii) Construct a phase diagram which includes the direction ﬁeld and trajectories for initial conditions: (a) ﬁrm 1 a monopolist (b) ﬁrm 2 a monopolist (c) (0,0) Construct a phase diagram with direction ﬁeld for the continuous model p(t) = 20 − 5Q(t) Q(t) = q1 (t) + q2 (t) TC1 (t) = 4q21 (t) TC2 (t) = 4q22 (t) q˙ 1 (t) = 0.2(x1 (t) − q1 (t)) q˙ 2 (t) = 0.2(x2 (t) − q2 (t))

11. 12.

13.

14.

15.

along the lines of sub-section 9.4.2. Is it true for this model that all paths converge on the equilibrium regardless of the initial value? Re-do ﬁgure 9.18 but plotting the trajectories {ktA , ktB } and show that the same pattern emerges. Show the path RAt for t = 200 . . . 250 for the same values given in ﬁgure 9.18(d). Show that the series is sensitive to initial conditions by setting RA0 = 10.1 and show this series for t = 200 . . . 250 on the same graph. Re-do ﬁgure 9.18 under the following alternative assumptions. Treat each one separately mB = 110 (a) mA = 100 A nB = 50 (b) n = 40 What do you conclude? Consider the example of the proportional adaptation model in sub-section 9.5.2 but with f = 20. What do you conclude about increasing the length of the decision span? Show that the two adaptation policies in sub-section 9.5.2 are special cases of the following more general adaptation policy as = α0 + α1 as−1 bs = β0 + β1 bs−1

Dynamic theory of oligopoly Derive the paths for RAt and RBt given the following adaptive policies on the part of ﬁrms A and B as = −0.05 + 0.8as−1 bs = +0.2 + 0.9bs−1

given (a0 , b0 ) = (0.13, −5.0), RA0 , RB0 = (10, 5) and f = 10. Additional reading Friedman (1983), Gandolfo (1997), Gehrig (1981), Henderson and Quandt (1971), McMannus (1962), Okuguchi (1970, 1976), Okuguchi and Szidarovsky (1988, 1990), Parker, Whitby and Tobias (2000) and Theocharis (1960).

423

CHAPTER 10

Closed economy dynamics

The IS-LM model is still one of the main models with which to introduce macroeconomics.1 In its static form it comprises an IS curve, which denotes real income and interest rate combinations which lead to equilibrium in the goods market, and an LM curve, which denotes real income and interest rate combinations which lead to equilibrium in the money market. Overall equilibrium is established where the IS curve cuts the LM curve. It is then common to consider comparative statics, which involves changing one or more exogenous variables or changing some parameter of the model. Very rarely do we observe any detailed analysis of what happens out of equilibrium, and yet this is what we are more likely to be observing around us. In this chapter we will reconsider this model from a dynamic point of view, beginning with a simple linear version and extending the analysis to more complex formulations and nonlinear speciﬁcations. In the ﬁrst two sections we consider the goods market and then the goods market along with the money market using simple discrete dynamic models of the macroeconomy. In these formulations we introduce dynamics through the goods market equilibrium condition. Rather than assume aggregate income equals aggregate expenditure in the same period, we make the assumption that income in period t is equal to total expenditure in the previous period. On the other hand, we assume that the money market clears in the same time period, i.e., the demand and supply of real money balances in any time period t are equal. Next we consider continuous versions of the IS-LM model introducing differential speeds of adjustment in the goods market and the money market. Similar to the discrete models, we assume that the goods market is slower to adjust to equilibrium than the money market. In the case of the money market we consider instantaneous adjustment and noninstantaneous adjustment. In section 10.3 we speciﬁcally assume real investment is negatively related to the rate of interest only. However, in section 10.4 we allow investment to be positively related to the level of real income. This is found to be signiﬁcant for the issue of stability, and it is this topic that we pay attention to in section 10.4. All the models in sections 10.1–10.4 are linear. In section 10.5 we turn to a nonlinear IS-LM model. Even here, however, we consider the dynamics of the model only after using a linear approximation. Finally, in section 10.6, we outline the Tobin–Blanchard model. The IS-LM model determines income and interest rates and has nothing to say about the impact the behaviour of the stock market has 1

We consider a dynamic IS-LM-BP model in chapter 12.

Closed economy dynamics

425

on these. Using the q-theory of investment, the Tobin–Blanchard model provides some alternative dynamic behaviour.

10.1 Goods market dynamics By way of introduction to lags in the IS-LM model, consider the simplest of goods market models in discrete form, with investment constant, i.e., It = I for all t Ct = a + bYt Et = Ct + It

(10.1)

Yt = Et where C = consumption Y = income E = total expenditure I = investment This is a static model with equilibrium Y∗ =

a+I 1−b

(10.2)

Two generalisations are possible which give this model some dynamic character. One is to assume consumption is related to lagged income. The model is, then Ct = a + bYt−1 Et = Ct + I

(10.3)

Yt = Et which immediately gives the difference equation Yt = (a + I) + bYt−1

(10.4)

An alternative formulation is to assume a lag between production, Yt , and expenditure Et . Suppose that production in time t is related to overall expenditure in time t − 1. Then we have the model Ct = a + bYt Et = Ct + I

(10.5)

Yt = Et−1 This also gives rise to the same difference equation, namely Yt = (a + I) + bYt−1

(10.6)

The equilibrium level of income remains what it was in the static model. Since Yt = Yt−1 = Y ∗ for all t in equilibrium, then Y ∗ = (a + I) + bY ∗ or Y∗ =

a+I 1−b

(10.7)

426

Economic Dynamics However, given some initial level of income we can plot the path of income. More explicitly, we can solve this model as follows Yt = (a + I) + bYt−1 Y ∗ = (a + I) + bY ∗ Yt − Y ∗ = b(Yt−1 − Y ∗ ) Deﬁning yt = Yt − Y ∗ , then yt = byt−1 with solution yt = bt y0 or

(10.8)

(10.9)

(10.10)

Yt =

a+I a+I + b t Y0 − 1−b 1−b

So long as 0 < b < 1, then income will converge on the equilibrium value. If we begin in equilibrium and shock demand, say with a rise in investment to I1 = I0 + I, then a + I1 a + I1 + b t Y0 − Yt = 1−b 1−b and income converges on the new equilibrium level, Y1∗ = (a + I1 )/(1 − b). Given this path of income we immediately have the path for consumption, namely a + bI1 a + I1 t+1 +b Ct = Y0 − 1−b 1−b Example 10.1 The situation is illustrated in table 10.1. The table is based on the relationships Ct = 110 + 0.75Yt I = 300 Et = Ct + I Yt = Et−1

(10.11)

Equilibrium income is initially £1640 million. Investment rises by £20 million, resulting in a new equilibrium level of income of £1720 million. However, income takes time to converge on this level of income, as shown in table 10.1. The model is illustrative of the inﬂuence of lags. It also shows the dynamic multiplier in operation. The income multiplier with respect to a change in investment is Y k= I which in the present example is 4. This is the multiplier from one equilibrium to the next. But with income changing we deﬁne Yt = Yt − Y0∗ , the deviation

Closed economy dynamics

427

Table 10.1 Dynamic multiplier t

Yt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .. . ∞

1640 1660 1675 1686.25 1694.69 1701.02 1705.76 1709.32 1711.99 1713.99 1715.50 1716.62 1717.47 1718.10 1718.58 1718.93 1719.20 1719.40 1719.55 1719.66 1719.75 .. . 1720

kt 1 1.75 2.31 2.73 3.05 3.29 3.47 3.60 3.70 3.77 3.83 3.87 3.90 3.93 3.95 3.96 3.97 3.98 3.98 3.99 .. . 4

of income from the initial equilibrium level, and so we have a period multiplier deﬁned by: kt =

Yt I

What is clearly revealed from table 10.1 is that kt → k as t → ∞. This is understandable since Yt → Y1∗ as t → ∞. In this na¨ıve model income converges steadily on the new equilibrium level of income. So long as 0 < b < 1 this must be so. Furthermore, income cannot oscillate. In section 3.10, however, we considered the multiplier–accelerator model. In its discrete form this is Ct = a + bYt−1 It = I0 + v(Yt−1 − Yt−2 ) Et = Ct + It

(10.12)

Yt = Et Then Yt = a + bYt−1 + I0 + vYt−1 − vYt−2 i.e. Yt − (b + v)Yt−1 + vYt−2 = a + I0

(10.13)

428

Economic Dynamics with equilibrium income of a + I0 1−b while the solution to (10.13) is Y∗ =

Y t = Y ∗ + c 1 r t + c2 s t where r=

(b + v) +

(b + v)2 − 4v , 2

s=

(b + v) −

(b + v)2 − 4v 2

Example 10.2 Ct = 110 + 0.75Yt−1 It = 300 + 1.5(Yt−1 − Yt−2 ) Et = Ct + It Yt = Et then (10.14)

Yt − 2.25Yt−1 + 1.5Yt−2 = 410 and Y ∗ = 1640. The roots to x2 − 2.25x + 1.5 = 0 are complex conjugate, with r = 1.125 + 0.484123i and s = 1.125 − 0.484123i, and so income will oscillate. Furthermore, R = α 2 + β 2 = 1.22474 > 1 and so income will diverge from the new equilibrium. This is shown in ﬁgure 10.1. The initial level of income is the original equilibrium level of £1,640. In period 1 autonomous investment is raised by £20, which is maintained for all periods thereafter, so income in period 1 is £1,660. Income in period 3 and beyond is then speciﬁed according to the recursive equation (10.14). However, as ﬁgure 10.1 reveals, income never reaches the new equilibrium of £1,720. Of course, this is not the only possibility and the resulting path of income depends very much on whether (b + v)2 − 4v is real or complex.

Figure 10.1.

Closed economy dynamics

429

10.2 Goods and money market dynamics2 The previous section considered only the goods market, and even then only in simple terms. The essence of the IS-LM model is the interaction between the goods market and the money market. This interaction is even more signiﬁcant when there are lags in the system. Again we illustrate this by introducing a lag into the goods market of the form Yt = Et−1 . On the other hand, we assume the money market adjusts in the same time period t, so that the demand for real money balances in time t is equal to the supply of real money balances in time t. This is a reasonable assumption. Algebraically, our model is Goods market 0 0, u > 0

=m

e = real total expenditure md = the demand for real money balances

= mst

m = the supply of real money s

balances On substitution, we arrive at the difference equation m0 − m kh + b(1 − t1 ) − yt−1 yt = (a − bt0 + i0 + g) − b u u

(10.16)

Or more simply yt = A + Byt−1 where

A = a − bt0 + i0 + g − h kh B = b(1 − t1 ) − u

m0 − m u

setting yt = yt−1 = y∗ the equilibrium level of income is found to be m0 − m (a − bt0 + i0 + g) − h u y∗ = kh 1 − b(1 − t1 ) + u 2

The model in this section is based on Teigen (1978, introduction to chapter 1).

(10.17)

430

Economic Dynamics Deﬁne xt = yt − y∗ , then xt = Bxt−1 with solution xt = Bt x0 or yt = y∗ + Bt (y0 − y∗ ) The stability of the equilibrium now depends on whether B < 1 or B > 1. If B < 1 then this amounts to 1 − b(1 − t1 ) k < − h u But why express the condition in this way? The equation for the IS curve is the solution for goods market equilibrium. This takes the form yt = a − bt0 + i0 + g + b(1 − t1 )yt − hrt or a − bt0 + i0 + g [1 − b(1 − t1 )]yt − h h The LM curve is the solution for the money market. This takes the form k m0 − m + yt rt = u u rt =

Hence, the stability condition k [1 − b(1 − t1 )] < h u amounts to the slope of the IS curve being less steep than the slope of the LM curve. This is deﬁnitely satisﬁed for the usual case where the IS curve is negatively sloped and the LM curve is positively sloped. With 0 < b < 1 and 0 < t1 < 1, then 0 < 1 − b(1 − t1 ) < 1. With h > 0 then −[1 − b(1 − t1 )]/ h < 0 while k/u > 0. A shift, say, in the IS curve to the right will lead to a rise in income over time, converging on the new equilibrium level. B 0,

0 < b < 1,

0 < t1 < 1,

where e = real expenditure a = autonomous expenditure b = marginal propensity to consume t1 = marginal rate of tax

h>0

(10.18)

432

Economic Dynamics y = real income h = coefﬁcient of investment in response to r r = nominal interest rate The demand for real money balances is assumed to be positively related to real income and negatively related to the nominal interest rate

(10.19)

md (t) = ky(t) − ur(t)

k, u > 0

The nominal money supply is assumed exogenous at Ms = M0 and the price level is assumed constant. Hence, real money balances are exogenous at m0 = M0 /P. It is now necessary to be more precise on the adjustment assumptions in each of the markets. We assume that in the goods market, income adjusts according to the excess demand in that market and that interest rates adjust according to the excess demand in the money market, i.e. (10.20)

y˙ = y (t) = α(e(t) − y(t))

r˙ = r (t) = β(m (t) − m0 ) d

α>0 β>0

These differential equations can be expressed explicitly in terms of y and r, where we now assume these variables are continuous functions of time, and that we drop the time variable for convenience (10.21)

y˙ = α[b(1 − t1 ) − 1]y − αhr + αa r˙ = βky − βur − βm0 The equilibrium lines in the (y,r)-phase plane are established simply by setting y˙ = 0 and r˙ = 0 respectively. For y˙ = 0 we derive the equilibrium line

(10.22)

−α[1 − b(1 − t1 )] y − αhr + αa = 0 a − [1 − b(1 − t1 )]y i.e. r = h which is no more than the IS curve. This equilibrium line has a positive intercept (a/h) and a negative slope (−(1 − b(1 − t1 ))/h). Similarly, for r˙ = 0 we derive the equilibrium line which is no more than the LM curve. This equilibrium line has a negative intercept −m0 /u and a positive slope k/u. The model has just one ﬁxed point for which y˙ = 0 and r˙ = 0. This is the point a + (h/u)m0 −(m0 /u)(1 − b(1 − t1 )) + (k/u)a , (y∗ , r∗ ) = 1 − b(1 − t1 ) + (kh/u) 1 − b(1 − t1 ) + (kh/u) and is shown by point E0 in ﬁgure 10.1. More importantly, we need to consider the dynamic forces in operation when each of the markets are not in equilibrium. First consider the goods market. For points to the right of the IS curve, as drawn in ﬁgure 10.2, we have a − [1 − b(1 − t1 )]y h 0 > a + b(1 − t1 )y − hr − y r>

implying y˙ < 0. Hence, to the right of the IS curve income is falling. By the same reasoning it is readily established that for points to the left of the IS curve income

Closed economy dynamics

433 Figure 10.2.

is rising. Considering next the money market, for points to the right of the LM curve ky − m0 u 0 > ky − ur − m0 r>

implying r˙ > 0, and so interest rates are rising. Similarly, to the left of the LM curve it is readily established that interest rates are falling. The implied vectors of force in the four quadrants are illustrated in ﬁgure 10.2, which clearly indicate a counter-clockwise movement. Suppose the economy is in all-round equilibrium, shown by point E0 in ﬁgure 10.3. Now consider the result of a fall in the nominal money supply. This will shift the money market equilibrium line to the left. The new equilibrium will be at point E1 . But what trajectory will the economy take in getting from E0 to E1 ? Four possible paths are drawn, labelled T1 , T2 , T3 and T4 , respectively. Trajectory T1 makes a very extreme assumption on the part of adjustment in the money market and the goods market. It assumes that the money market adjusts instantaneously, with interest rates adjusting immediately to preserve equilibrium in the money market. With such immediate adjustment, then in the ﬁrst instance the economy must move from E0 vertically up to point A. This is because income has not yet had a chance to change, and is still at the level y0 . With the sharp rise in interest rates, investment will fall, and through the multiplier impact on income, income will fall. As income falls, the demand for money declines, and so too does the rate of interest. The interest rate will fall always in such a manner that equilibrium is preserved in the money market. This means that the adjustment must take place along the new LM curve, as shown by trajectory T1 . Under this assumption of instantaneous adjustment in the money market, the interest rate

434

Economic Dynamics

Figure 10.3. A monetary contraction Note: Vector forces are with respect to E1 and not E0

overshoots its new equilibrium value and then settles down at the new equilibrium rate. Real income, on the other hand, falls continually until the new equilibrium level is reached. Trajectory T2 , on the other hand, indicates that both markets adjust imperfectly in such a manner that the economy gradually moves from E0 to E1 , with interest rates rising gradually until they reach the new level of r1 , and income falling gradually until it reaches its new level of y1 . If the economy conforms to this trajectory, then no overshooting occurs. But our analysis in part I indicates that there is no reason to assume that this is the only possible trajectory – given the vector of forces present. For instance, trajectory T3 shows a sharper rise in interest rates than in trajectory T2 , and overshooting of interest rates and income, with a resulting counter-clockwise spiral towards the new equilibrium E1 . If we assume that the money market, although not adjusting instantaneously, is very quick to adjust, and that the goods market is also adjusting quickly, then trajectory T3 is more likely. This is an important observation. A spiralling trajectory to the new equilibrium (trajectory T3 ) is more likely if both markets have quick adjustment speeds, and consequently the more likely overshooting will be observed in both endogenous variables y and r. Even so, a counter-clockwise spiral is not the most likely outcome; it is more likely to be trajectory T4 . This is because, in general, the money market is relatively much quicker to adjust than the goods market and the adjustment path will be contained within the triangle E0 AE1 , being drawn towards trajectory T1 . A similar analysis holds for a monetary expansion, shown in ﬁgure 10.4, where the economy is initially at equilibrium point E0 . Under instantaneous adjustment in the money market, the trajectory is T1 . Interest rates fall to point A on the new LM curve. The sharp fall in interest rates stimulates investment, which, through the multiplier, stimulates the level of income. As income rises the demand for money rises and so too do interest rates, but in such a manner that the money market clears

Closed economy dynamics

435 Figure 10.4. A monetary expansion Note: Vector forces are with respect to E1 and not E0

continually. Hence the economy moves along the new LM curve until equilibrium E1 is reached. Once again, interest rates overshoot their new equilibrium level, but the level of income adjusts gradually until its new equilibrium level is achieved. If both markets show a fair degree of adjustment, then path T2 will be followed. However, this would require the goods market to adjust quite quickly. In this instance, interest rates fall gradually until the new level of r1 is reached, and income rises gradually until the new level of y1 is reached. There is no overshooting either of the interest rate or of income. If both the money market and the goods market are quick to adjust, then the economy is more likely to follow the trajectory illustrated by T3 in ﬁgure 10.4. In other words, a spiral path to the new equilibrium, moving in a counter-clockwise direction, and such that both the rate of interest and the level of income overshoot their equilibrium values. However, with the dominance of adjustment in the money market, a counter-clockwise movement will be observed but it is not likely to be a spiral path. The most likely trajectory is T4 . It is apparent from this discussion that the speed of adjustment is very much to do with the values of the reaction coefﬁcients α and β in the dynamic system. The higher the value of the coefﬁcient, the quicker the market responds to a disequilibrium. To some extent, it is the relative values of these coefﬁcients that will determine which trajectory the economy will take. To clarify this point, let us consider a numerical example. Example 10.4 Since throughout the price level is constant, we shall assume that this has a value of unity. The assumed parameter values and the initial level of the money stock are a = 50

k = 0.25

b = 0.75

m0 = 8

436

Economic Dynamics

Figure 10.5.

t1 = 0.25

u = 0.5

h = 1.525 The economy’s equilibrium is (y0 , r0 ) = (62, 15), shown by point E0 in ﬁgure 10.5(a). A fall in the real money stock to m1 = 5 leads to the new equilibrium point3 (y1 , r1 ) = (54, 17) and shown by point E1 . The resulting differential

3

More exactly (y1 , r1 ) = (54.375, 17.1875).

Closed economy dynamics equation system, with unspeciﬁed values for α and β, is y˙ = −0.4375αy − 1.525αr + 50α r˙ = 0.25βy − 0.5βr − 5β The trajectory the economy takes to the new equilibrium will depend very much on the values of α and β. Consider three possible combinations, leading to three possible trajectories T1:

α = 0.05

T2 :

β = 0.8

α = 0.1

T3 : α = 0.5

β = 0.8

β = 0.8

If the money market is quicker to adjust than the goods market, as is the most likely situation, then typical trajectories are T1 and T2 in ﬁgure 10.5(a). In these cases the economy will exhibit overshooting of the interest rate, ﬁrst rising above the equilibrium level and then falling, with the new equilibrium interest rate higher than initially, as shown in ﬁgure 10.5(a). On the other hand, there will be a gradual decrease in the level of income to the new lower equilibrium level. A counterclockwise spiral pattern, as shown by trajectory T3 , will occur only if both the money market and the goods market are quick to adjust, as illustrated in ﬁgure 10.5(c). Although a counter-clockwise spiral is possible, therefore, it is not the most likely outcome of this dynamic system because the goods market is not likely to be quick to adjust.

10.4 Trajectories with Mathematica, Maple and Excel Figure 10.5(a) set out three trajectories employed in example 10.4. In this and later chapters we shall be producing a number of trajectories for both continuous and discrete systems of equations. We shall therefore take a digression and outline exactly how to do this with three different software packages: Mathematica, Maple and (for discrete systems) Excel.4 Figure 10.5 will be used throughout as an example. 10.4.1

Mathematica

To produce trajectories and other plots with Mathematica, two commands of importance are used, namely the NDSolve command and the ParametricPlot command. The ﬁrst command is used to obtain a numerical solution to the differential equation system, which it does by producing an InterpolatingFunction. The second command is then used to plot the values of the InterpolatingFunction. The input instructions are as follows: sol1=NDSolve[ y’[t]==2.5-0.07625r[t]-0.021875y[t], r’[t]==-4.0-0.4r[t]+0.2y[t], y[0]==62,r[0]==15}, {y,r},{t,0,50}] tr1=ParametricPlot[ {y[t],r[t]} /. sol1, {t,0,50}, PlotPoints->200]; 4

See Shone (2001) for a demonstration of how to produce trajectories on a spreadsheet for continuous systems of two equations employing Euler’s approximation.

437

438

Economic Dynamics sol2=NDSolve[ y’[t]==5-0.1525r[t]-0.04375y[t], r’[t]==-4.0-0.4r[t]+0.2y[t], y[0]==62,r[0]==15}, {y,r},{t,0,50}] tr2=ParametricPlot[ {y[t],r[t]} /. sol2, {t,0,50}, PlotPoints->200]; sol3=NDSolve[ y’[t]==25-0.7625r[t]-0.21875y[t], r’[t]==-4.0-0.4r[t]+0.2y[t], y[0]==62,r[0]==15}, {y,r},{t,0,50}] tr3=ParametricPlot[ {y[t],r[t]} /. sol3, {t,0,50}, PlotPoints->200]; trajectories=Show[tr1,tr2,tr3]; pathy1=Plot[ y[t] /.sol1, {t,0,50}, PlotPoints->200]; pathy2=Plot[ y[t] /.sol2, {t,0,50}, PlotPoints->200]; pathy3=Plot[ y[t] /.sol3, {t,0,50}, PlotPoints->200]; pathy=Show[pathy1,pathy2,pathy3]; pathr1=Plot[ r[t] /.sol1, {t,0,50}, PlotPoints->200]; pathr2=Plot[ r[t] /.sol2, {t,0,50}, PlotPoints->200]; pathy3=Plot[ r[t] /.sol3, {t,0,50}, PlotPoints->200]; pathr=Show[pathr1,pathr2,pathr3];

Note: 1. 2.

3. 4. 5.

6.

We use the NDSolve rather than DSolve because we are deriving a numerical solution. The simultaneous equations include the two initial values for y and r, which in the present example denotes the initial equilibrium before a disturbance. The parameter values include the fall in the money supply to m0 = 5 and we are deriving trajectory T1 , so α = 0.05 and β = 0.8. ParametricPlot is a built in command in Mathematica v2.0 and higher, and so can be employed without recourse to other subroutines. There is no comma after {y[t],r[t]} because these coordinates are speciﬁed for the solution values derived earlier. Thus, the qualiﬁer ‘/. sol1’ instructs the programme to plot the coordinates using each value derived from the output of sol1. Interim displays can be suppressed by including the option Display Function->Identity in each and then in the Show command include the option DisplayFunction-> $DisplayFunction. For instance trajectory 1 can be written tr1=ParametricPlot[ {y[t],r[t]} /. sol1, {t,0,50}, PlotPoints->200, DisplayFunction->Identity]

and trajectories can be written trajectories=Show[tr1,tr2,tr3, DisplayFunction->$DisplayFunction];

Closed economy dynamics 10.4.2

Maple

In some respects it is easier to produce trajectories in Maple, but more involved to produce the values for plotting y(t) and r(t). The reason for this is because we can use Maple’s phaseportrait command to produce the trajectories. This implicitly uses the numerical solution for the differential equations. Thus, the three trajectories and their combined display for ﬁgure 10.5 is as follows: with(DEtools): with(plots): tr1:=phaseportrait( [D(y)(t)=2.5-0.07625*r(t)-0.021875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=black, arrows=none, thickness=2): tr2:=phaseportrait( [D(y)(t)=5-0.1525*r(t)-0.04375*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=red, arrows=none, thickness=2): tr3:=phaseportrait( [D(y)(t)=25-0.7625*r(t)-0.21875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=blue, arrows=none, thickness=2): display(tr1,tr2,tr3);

Notes: 1. 2. 3. 4.

It is necessary to load the DEtools and plots subroutines ﬁrst. Using phaseportrait implicitly uses a numerical solution to the differential equations. A small stepsize, here 0.05, produces a smoother plot. Having arrows set at none means the direction ﬁeld is not included.

439

440

Economic Dynamics Figures 10.5(b) and 10.5(c) can be produced with a similar set of instructions, except now we use DEplot with the option ‘scene’. The instructions are: pathy1=DEplot( [D(y)(t)=2.5-0.07625*r(t)-0.021875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[t,y]): pathy2=DEplot( [D(y)(t)=5-0.1525*r(t)-0.04375*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[y,t]): pathy3=DEplot( [D(y)(t)=25-0.7625*r(t)-0.21875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=blue, arrows=none, thickness=2, scene=[t,y]): display(pathy1,pathy2,pathy3); pathr1=DEplot( [D(y)(t)=2.5-0.07625*r(t)-0.021875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[t,r]): pathr2=DEplot( [D(y)(t)=5-0.1525*r(t)-0.04375*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50,

Closed economy dynamics [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=black, arrows=none, thickness=2, scene=[r,t]): pathr3=DEplot( [D(y)(t)=25-0.7625*r(t)-0.21875*y(t), D(r)(t)=-4-0.4*r(t)+0.2*y(t)], [y(t),r(t)], t=0..50, [[y(0)=62,r(0)=15]], stepsize=.05, linecolour=blue, arrows=none, thickness=2, scene=[t,r]): display(pathr1,pathr2,pathr3);

10.4.3

Excel

Discrete trajectories can also be derived using Excel, although there are some limitations. Consider a discrete variant of example 10.4. yt+1 − yt = −0.4375αyt − 1.525αrt + 50α rt+1 − rt = 0.25βyt − 0.5βrt − 5β or the recursive form yt+1 = (1 − 0.4375α)yt − 1.525αrt + 50α rt+1 = 0.25βyt + (1 − 0.5β)rt − 5β This numerical example is set out in the spreadsheet shown in ﬁgure 10.6. The spreadsheet shows the data computations which can be used to produce a given trajectory or a multiple time plot of y(t) or r(t). The initial values are the equilibrium values y∗ = 62 and r∗ = 15. Cells B13 and C13 write out the formulas using both absolute addresses for the parameters α and β and relative addresses for y(0) and r(0). These cells are then copied to the clipboard and pasted down up to t = 50. A similar procedure is done for columns F and G along with columns J and K. Unfortunately spreadsheets cannot plot more than one trajectory on the same graph. Selecting cells B12 : C62 and invoking the chart wizard and selecting the x-y plot option produces a plot of trajectory T1 . Similarly, selecting cells F12 : G62 produces trajectory T2 and selecting J12 : K62 produces trajectory T3 . To produce the discrete equivalent of ﬁgure 10.5(b) ﬁrst select cells A12 : B62 and while holding down the Ctrl-key, select cells F12 : F62 and, while continuing to hold down the Ctrl-key, select cells J12 : J62. Invoking the chart wizard and selecting the x-y plot option produces ﬁgure 10.5(b). In the same manner ﬁgure 10.5(c) can be produced for a multiple plot of r(t) against t.

441

442

Economic Dynamics

Figure 10.6.

The instructions provided in this section allow the reproduction of all twodimensional trajectories provided in this book. They can be used to produce trajectories for any similar set of differential or difference equations.

10.5 Some important propositions Similar results can be derived for an increase in the money supply (see exercise 7). The most likely trajectory to the new equilibrium point is for the economy to exhibit an overshoot with regard to its interest rate response (falling sharply and then rising somewhat), while income will gradually rise to its new higher equilibrium level. Does the economy exhibit the same type of dynamic behaviour for a shock to the goods market, i.e., a shift in the IS curve? The situation is shown in ﬁgures 10.7 and 10.8. Consider ﬁrst a ﬁscal expansion (a rising from 50 to 55) which shifts the IS curve from IS0 to IS1 , as illustrated in ﬁgure 10.7. The economy moves from equilibrium point E0 to equilibrium point E1 . But what dynamic path does it take to the new equilibrium? If we again assume that the money market adjusts instantaneously, then there will be a gradual rise in income as the multiplier impact of the expansion moves through the economy. The increase in income will raise the demand for money and hence raise the rate of interest. This rise in interest rate will be such as to maintain equilibrium in the money market. Hence, the economy will move along the LM curve until the new equilibrium is reached. There is no overshooting either of income or of interest rates. With less than instantaneous adjustment in the money market (β = 0.8), and a sluggish adjustment in the goods market (α = 0.1), then the economy will follow trajectory T2 , with interest rates rising gradually until the new level r1 is reached, and income adjusting gradually until the new level of y1 is reached. Again the economy exhibits no overshooting. Only in the unlikely event that the goods market adjusts very rapidly (e.g. α = 0.5) along with the money market will the economy exhibit a spiral path following a counter-clockwise movement to the new equilibrium, trajectory T3 , and with the economy exhibiting overshooting behaviour (see exercise 9). In the case of a ﬁscal contraction (a falling from 50 to 45), illustrated in ﬁgure 10.8, the economy will follow trajectory T1 with instantaneous adjustment

Closed economy dynamics

443 Figure 10.7.

Figure 10.8.

in the money market, with interest rates and income declining steadily until the new equilibrium is reached. Similarly, if the money market is quick to adjust (but not instantaneous, e.g., β = 0.8) and the goods market is sluggish in its adjustment (α = 0.1), then path T2 will be followed. Only in the unlikely event that the goods market is very quick to adjust (e.g. α = 0.5) as well as the money market (e.g. β = 0.8) will a spiral path like T3 be followed (see exercise 10). We can make a number of important propositions about the dynamic behaviour of (closed) economies concerning money market shocks and goods market shocks. PROPOSITION 1 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a monetary shock will most likely lead to a counterclockwise movement with the interest rate overshooting its equilibrium value and income gradually changing to its new equilibrium level. COROLLARY 1 A counter-clockwise spiral to a new equilibrium arising from a monetary shock is only likely to occur in the event that both the money market and goods market are quick to adjust to disequilibrium states.

444

Economic Dynamics PROPOSITION 2 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a goods market shock will most likely lead to a gradual movement of the economy to its new equilibrium, with the economy exhibiting no overshooting of either interest rates or income. COROLLARY 2 A counter-clockwise spiral to a new equilibrium arising from a ﬁscal shock is only likely to occur in the event that both the money market and goods market are quick to adjust to disequilibrium states. Can we make any observations about the dynamic behaviour of this economy when there is a combined ﬁscal and monetary shock? In carrying out this particular analysis we shall simply assume that the money market is quick to adjust, but not instantaneous, and that the goods market is sluggish in its adjustment. In ﬁgure 10.9 we illustrate a ﬁscal and monetary expansion, a rising from 50 to 55 and m rising from 8 to 12. In ﬁgure 10.10 we illustrate a ﬁscal and monetary contraction, a falling from 50 to 45 and m falling from 8 to 5. Under the assumption made about relative adjustment, it is very likely that the trajectory of the economy in each case is a counter-clockwise movement to the new equilibrium, with major overshooting of interest rates and a gradual change in income to the new equilibrium level. Overshooting of income will, once again, occur only if the goods market adjusts

Figure 10.9.

Figure 10.10.

Closed economy dynamics

445

quickly to a disequilibrium along with the money market, trajectory T3 . Similarly, a combined ﬁscal and monetary contraction, which is illustrated in ﬁgure 10.10, leads to a sharp rise in interest rates in the short period, and as income begins to fall, interest rates too are brought down. There is unlikely to be any overshooting of income. Only in the unlikely event that the goods market adjusts quickly to a disequilibrium along with the money market will this occur, trajectory T3 . These results should not be surprising. The initial impact on interest rates comes about because of the shift in the LM curve. Only when income begins to adjust will this effect be reversed. In the case of ﬁscal and monetary shocks opposing each other, and under the same assumption about relative adjustment behaviour, the dynamic path to the new equilibrium can have various possibilities depending on which shock is the greater. Figure 10.11 illustrates a ﬁscal expansion and a monetary contraction, with equilibrium points E0 and E1 , respectively. If the ﬁscal expansion is the more

Figure 10.11. Monetary contraction and fiscal expansion

446

Economic Dynamics

Figure 10.12. Monetary expansion and fiscal contraction

dominant of the two shocks (ﬁgure 10.11(a)), then the economy will traverse a smooth path from E0 to E1 with a rise in interest rates and a rise in income. On the other hand, if the monetary contraction dominates (ﬁgure 10.11(b)), then the economy will move counter-clockwise, with interest rates overshooting their new equilibrium level and income gradually falling. Similarly, in ﬁgure 10.12 we show a ﬁscal contraction and a monetary expansion. If the ﬁscal contraction dominates (ﬁgure 10.12(a)), then the economy will decline gradually from equilibrium point E0 to E1 . On the other hand, if the monetary expansion dominates, the decline in the interest rate may very well overshoot its equilibrium level, although income will gradually rise. We arrive, then, at two further propositions: PROPOSITION 3 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a ﬁscal expansion (contraction) combined with

Closed economy dynamics

447

a monetary expansion (contraction) will more likely lead to a counterclockwise movement, with interest rates rising (falling) initially and then falling (rising) into the medium and long term; while income will gradually rise (fall) until its new equilibrium position is reached. PROPOSITION 4 If the money market is quick to adjust and the goods market is sluggish in its adjustment, then a ﬁscal expansion (contraction) combined with a monetary contraction (expansion) will give rise to a gradual change in interest rates and income if the ﬁscal shock dominates, but will exhibit interest rate overshooting if the monetary shock dominates. There is one important observation we can draw from this analysis about the dynamic behaviour of the economy. Given the assumption about relative speeds of adjustment, then interest rate volatility is far more likely to be observed than income volatility.

10.6 IS-LM continuous model: version 2 In this section we shall extend the investment function to include real income. In other words, business will alter the level of investment according to the level of income; the higher the level of income the more business undertakes new investment. We shall continue with a simple linear model, but this simple extension will lead to the possibility that the IS curve is positively sloped. In considering the implications of this we shall consider some explicit numerical examples in order to see the variety of solution trajectories. In one case we shall derive an explicit saddle path solution. Since the formal derivation is similar to the previous section we can be brief. The model is5 e = a + b(1 − t)y − hr + jy md = ky − ur y˙ = α(e − y)

(10.23)

r˙ = β(md − m0 ) where a > 0, 0 < b < 1, 0 < t < 1, h > 0, j > 0, k > 0, u > 0, α > 0, β > 0 which gives the two differential equations y˙ = α[b(1 − t) + j − 1]y − αhr + αa r˙ = βky − βur − βm0 with the IS curve obtained from setting y˙ = 0 as r= 5

a − [1 − b(1 − t) − j]y h

Although we use t for the marginal rate of tax, there should be no confusion with the same letter standing for time.

(10.24)

448

Economic Dynamics and an LM curve obtained from setting r˙ = 0 as r=

ky − m0 u

The major difference between this version and the one in the previous section is that now the IS curve can have either a negative slope (if b(1 − t) + j < 1) or a positive slope (if b(1 − t) + j > 1), and that the positive slope is more likely the larger the value of the coefﬁcient j. Since we dealt with a negatively sloped IS curve in the previous section, let us consider here the implications of a positively sloped IS curve, i.e., we assume b(1 − t) + j > 1. For a positively sloped IS curve, points to the left of this line represent a − [1 − b(1 − t) − j]y h 0 > a + [b(1 − t) + j]y − hr − y r>

implying y˙ < 0. Hence, to the left of the IS curve income is falling. Similarly, by the same reasoning, for points to the right of the IS curve income is rising. There is no change for the LM curve, and we have already established that for points to the right of the LM curve interest rates are rising while to the left of the LM curve interest rates are falling. With a positively sloped IS curve, there are two possibilities: (i) (ii)

the IS curve is less steep than the LM curve the IS curve is steeper than the LM curve.

The two possibilities, along with the vector of forces outlined above, are illustrated in ﬁgure 10.13(a) and (b). Figure 10.13(a) reveals a counter-clockwise trajectory while ﬁgure 10.13(b) reveals an unstable situation, although it does indicate that a trajectory might approach the equilibrium point. Neither situation is straightforward. Although ﬁgure 10.13(a) indicates a counter-clockwise trajectory, is the trajectory tending towards the equilibrium or away from it? There is nothing within the model as laid down so far to indicate which is the case. In order to see what the difﬁculty is, consider the following two numerical examples. Example 10.5 a=2 b = 0.75 t = 0.25 h = 1.525 j = 0.8

Example 10.6 k = 0.25 u = 0.5 m0 = 8

α = 0.05 β = 0.8

a=2 b = 0.8 t = 0.2 h = 1.525 j = 0.95

Solution

Solution

y∗ = 66 r∗ = 17

y∗ = 55.3 r∗ = 22.3

k = 0.25 u = 0.25 m0 = 8

α = 0.2 β = 0.3

Closed economy dynamics

449 Figure 10.13.

Intercepts and slopes

Intercepts and slopes

IS intercept = 1.3 IS slope = 0.238 LM intercept = −16 LM slope = 0.5

IS intercept = 1.3 IS slope = 0.387 LM intercept = −32 LM slope = 1

Both examples typify the situation in ﬁgure 10.13(a), with a positive IS curve, and the IS curve less steep than the LM curve. However, the dynamics of both these examples is different. Both lead to a counter-clockwise path. However, example 10.5 leads to a stable path which appears to traverse a straight line path after a certain time period, while example 10.6 leads to an unstable spiral, as illustrated in ﬁgure 10.14(a) and 10.14(b) (see exercise 11). But, then, what is it that is different between these two examples? To answer this question we need to consider the differential equation system in terms of deviations from equilibrium, and then to consider the trace and determinant of the dynamic system.6 Return to the general speciﬁcation of the differential equations y˙ = α[b(1 − t) + j − 1]y − αhr + αa r˙ = βky − βur − βm0 6

See chapter 4.

450

Economic Dynamics

Figure 10.14.

and consider the equilibrium values of the variables, i.e. 0 = α[b(1 − t) + j − 1]y∗ − αhr∗ + αa 0 = βky∗ − βur∗ − βm0 Subtracting the second set from the ﬁrst we have (10.25)

(10.26)

y˙ = α[b(1 − t) + j − 1](y − y∗ ) − αh(r − r∗ ) r˙ = βk(y − y∗ ) − βu(r − r∗ ) and the matrix of this system is α[b(1 − t) + j − 1] A= βk

−αh −βu

whose trace and determinant are (10.27)

tr(A) = α[b(1 − t) + j − 1] − βu det(A) = −αβu[b(1 − t) + j − 1] + αβkh Using these results we can summarise the properties of the systems in examples 10.5 and 10.6 in terms of the values for their trace and determinant. These are

Closed economy dynamics Example 10.5 tr(A) = −0.382 det(A) = 0.008

where

tr(A)2 > 4 det(A)

where

tr(A)2 < 4 det(A)

Example 10.6 tr(A) = 0.043 det(A) = 0.014

In terms of table 4.1 in part I (p. 180), it is clear that example 10.5 satisﬁes the conditions for an asymptotically stable node while example 10.6 satisﬁes the condition of an unstable spiral, verifying what is shown in ﬁgure 10.13(a). Notice that in both examples the determinant of the system is positive. Although this is necessary for a spiral path, it is not sufﬁcient to determine whether the path is stable or unstable. This requires information on the sign of the trace. A stable spiral requires the trace to be negative; while an unstable spiral arises if the trace is positive – and in both cases the condition that tr(A)2 < 4 det(A) needs to be satisﬁed. Unfortunately, there is no geometric representation of the trace requirement. It can be ascertained only from the system itself. Even so, a comparison of the two examples indicates quite clearly that for an unstable spiral to be more likely, it is necessary for the coefﬁcient of induced spending (b(1 − t) + j) to be high and for there to be quick adjustment in both markets (large values for the reaction coefﬁcients α and β). If the IS curve is positively sloped, and is less steep than the LM curve, the most likely result is a counter-clockwise stable spiral. Let us now consider a third example for which the IS curve is positively sloped but is steeper than the LM curve. Example 10.7 Parameter values a = −25 k = 0.22 b = 0.75 u = 0.75 t1 = 0.25 m0 = 8 h=1 j = 0.95 tr(A) = −0.574375 det(A) = −0.006575

α = 0.05 β = 0.8

Solution y∗ = 65.4 r∗ = 10.5

Intercepts and slopes IS intercept = −25 IS slope = 0.5125 LM intercept = −10.7 LM slope = 0.293

where tr(A)2 > 4 det(A)

This example typiﬁes the situation in ﬁgure 10.13(b). But in order to see what is happening, we need to derive the characteristic equations of the system. In terms of deviations from the equilibrium we have the general results indicated already in terms of the equation system given above. Substituting the numerical values given in example 10.7, we obtain the following differential equation system y˙ = 0.025625(y − y∗ ) − 0.05(r − r∗ ) r˙ = 0.176(y − y∗ ) − 0.6(r − r∗ )

451

452

Economic Dynamics whose characteristic roots can be obtained from |A − λI| = 0 i.e.

0.025625 − λ −0.05 =0 0.176 −0.6 − λ

Which leads to the quadratic equation λ2 + 0.574375λ − 0.006575 = 0 with solutions r = 0.0112277

s = −0.585603

Using the ﬁrst solution, we have y − y∗ 0.025625 −0.05 y − y∗ = 0.0112277 r − r∗ r − r∗ 0.176 −0.6 which leads to the relationship r − r∗ = 0.287945( y − y∗ ) On the other hand, using the second characteristic root, and following through the same procedure, we ﬁnd r − r∗ = 12.224555( y − y∗ ) These two results indicate two saddle paths; one of which is stable and the other is unstable. To verify this, we use Mathematica to plot ten trajectories of example 10.7, which are illustrated in ﬁgure 10.15. Given the vectors of force already established for ﬁgure 10.13(b), which typiﬁes example 10.7, it is clear that the ﬁrst characteristic root leads to an unstable saddle path, while the second characteristic root leads to a stable saddle path. The dynamics of this system, then, is schematically illustrated in ﬁgure 10.16, showing the saddle paths (denoted S1 S 1 and S2 S 2 associated with r and s, respectively) in relation to the IS and LM curves. The equilibrium of this system, then, is unstable except for the unlikely event that the initial point lies on the stable saddle path denoted S2 S 2 in ﬁgure 10.16. Also notice Figure 10.15.

Closed economy dynamics

453 Figure 10.16.

from ﬁgure 10.16 that one of the saddle paths is almost identical to the LM curve. This is a result of the assumption of rapid adjustment in the money market relative to the goods market. With perfect adjustment in the money market, then one saddle path would be identical with the LM curve, and this would be the unstable saddle path in the present context.

10.7 Nonlinear IS-LM model In this section we shall consider a nonlinear version of the IS-LM model. We can be brief because much of the analysis has already been carried out. In this version consumption spending in real terms is related to real income (where we assume disposable income has been eliminated); investment is inversely related to the nominal rate of interest (we assume expected inﬂation is zero) and positively to the level of real income; government spending is assumed exogenous. Our expenditure function, in real terms, is then e = c(y) + i(r, y) + g

0 < cy < 1, ir < 0, iy > 0

(10.28)

The demand for real money balances, md , is assumed to be positively related to real income (the transactions demand for money) and inversely related to the rate of interest (the speculative demand for money). Thus md = l( y, r) ly > 0, lr < 0

(10.29)

The dynamics are in terms of excess demand in the goods market and excess demand for real money balances, i.e. y˙ = α(e − y)

α>0

r˙ = β(l( y, r) − m0 )

β>0

where m0 is the supply of real money balances, and m0 is assumed exogenous. Equilibrium in the goods market requires y˙ = 0 or e = y, while equilibrium in the money market requires r˙ = 0 or l(y, r) = m0 . Suppose such a ﬁxed point exists and is denoted (y∗ , r∗ ). The question arises is whether such an equilibrium is dynamically stable. Given the nonlinear nature of the system, and the fact that no explicit functional forms are speciﬁed, then it is not possible to establish this in any absolute sense. We can, however, use the linearisation technique discussed in

(10.30)

454

Economic Dynamics part I to establish the stability in the neighbourhood of the equilibrium point.7 As we mentioned in part I, when employing such a linearisation, only local stability can be established. But even this is better than having nothing to say on the matter. Expanding the above system around the ﬁxed point (y∗ , r∗ ) gives ∂(e − y) ∂(e − y) y˙ = α ( y − y∗ ) + (r − r∗ ) ∂y ∂r ∂(l − m0 ) ∂(l − m0 ) ∗ ∗ (y − y ) + (r − r ) r˙ = β ∂y ∂r But ∂(e − y) = cy + iy − 1, ∂y

∂(e − y) = ir ∂r

∂(l − m0 ) = ly , ∂y

∂(l − m0 ) = lr ∂r

Hence y˙ = α(cy + iy − 1)( y − y∗ ) + αir (r − r∗ )

(10.31)

(10.32)

(10.33)

r˙ = βly ( y − y∗ ) + βlr (r − r∗ ) which can be written as a matrix dynamic system in the form y˙ α(cy + iy − 1) αir y − y∗ = βly βlr r − r∗ r˙ and where the matrix of the system is α(cy + iy − 1) αir A= βly βlr The dynamics of the system can now be determined from the properties of A. These are tr(A) = α(cy + iy − 1) + βlr det(A) = αβ(cy + iy − 1)lr − αβir ly

(10.34)

= −αβ[lr (1 − cy − iy ) + ir ly ] Can we interpret any economic meaning to the tr(A) and the det(A)? To see if we can, let us consider the slopes of the IS and LM curves. For the IS curve we have y = e, hence y = c( y) + i(y, r) + g Totally differentiating this expression with respect to y and r we obtain dy = cy dy + iy dy + ir dr and so the slope of the IS curve, denoted dr/dy, is given by (1 − cy − iy )dy = ir dr 1 − cy − i y dr or = dy ir 7

See section 2.7.

Closed economy dynamics

455

The slope of the LM curve is established in the same manner (and noting that m0 is exogenous) 0 = ly dy + lr dr −ly dr = dy lr If the IS curve is less steep than the LM curve, then −ly 1 − cy − iy < ir lr i.e. lr (1 − cy − iy ) + ir ly < 0 −αβ[lr (1 − cy − iy ) + ir ly ] > 0 Hence, det(A) > 0. This is certainly satisﬁed in the usual case of a negatively sloped IS curve and a positively sloped LM curve. But we have already established in the previous section that a stable solution will occur if both the IS and the LM curves are positively sloped but that the IS curve is less steep than the LM curve and that the trace of the system is negative in sign.

10.8 Tobin–Blanchard model 10.8.1

The model in outline8

There has been some interest by economists as to whether stock market behaviour can inﬂuence income and interest rates – at least in the short run. The IS-LM model so far outlined does not allow for any such link. It is plausible to think that investment will, in some way, be inﬂuenced by stock market behaviour. Such a link was considered by Blanchard (1981) following on the approach to investment suggested by Tobin (1969), and what is referred to as the q-theory of investment. The variable q represents the market value of equities as a ratio of the replacement cost. It can be understood as follows.9 If all future returns are equal, and denoted R, and are discounted at the interest rate r, then the present value of equities, V say, is equal to R/r. On the other hand, ﬁrms will invest until the replacement cost of any outstanding capital stock, RC, is equal to the return on investment, R/ρ, where ρ is the marginal efﬁciency of capital. Then q=

R/r ρ V = = RC R/ρ r

Consequently, net investment is a positive function of q, which still means that it is inversely related to r. In the long run r = ρ and so q = 1, and there is no net investment. The upshot of this approach is that investment, rather than being inversely related to r is positively related to q. This in turn means aggregate expenditure (and hence aggregate demand) is positively related to q. 8 9

A different treatment than the one presented here, also utilising phase diagrams, is provided in Romer (2001, chapter 8). See also Obstfeld and Rogoff (1999, section 2.5.2). See Stevenson, Muscatelli and Gregory (1988, pp. 156–9) for a fuller discussion.

(10.35)

456

Economic Dynamics We can accordingly express aggregate expenditure, e, as a(t) = a1 y(t) + a2 q(t) + g

0 < a1 < 1, a2 > 0

where g is real government spending. The goods market is assumed to adjust with a lag, with reaction coefﬁcient σ > 0, thus ˙ = σ (e(t) − y(t)) y(t)

σ >0

The money market, on the other hand, is assumed to adjust instantaneously, and so the demand for real money balances is equal to the supply of real money balances, i.e. ky(t) − ur(t) = m0

k > 0, u > 0

The next equation relates the rate of interest (on bonds) to the yield on equities, which are equal because it is assumed that bonds and equities are perfect substitutes, i.e. r(t) =

b1 y(t) + q˙e (t) q(t)

where b1 y constitutes the ﬁrms’ proﬁts, which are assumed proportional to output, and q˙e constitutes expected capital gains. Finally, we assume rational expectations, which in the present model is equivalent to perfect foresight, and so ˙ Suppressing the time variable, then the model can be stated in terms of ﬁve q˙e = q. equations e = a1 y + a2 q + g m0 = ky − ur (10.36)

y˙ = σ (e − y) b1 y + q˙e r= q q˙e = q˙ which can be reduced to two nonlinear nonhomogeneous differential equations, namely

(10.37)

y˙ = σ (a1 − 1)y + σ a2 q + σg kq qm0 − b1 y − q˙ = u u First we need to establish the existence of a ﬁxed point, an equilibrium point. We do this by setting y˙ = 0 and q˙ = 0, and solving for y and q. This is no more than where the two isoclines intersect. So let us ﬁrst look at these separately. First consider the y˙ = 0 isocline, which we shall refer to as the IS curve since it implies goods market equilibrium. We have y˙ = σ (a1 y + a2 q + g − y) = 0 −(1 − a1 )y + a2 q + g = 0 (1 − a1 )y − g i.e. q = a2

Closed economy dynamics

457

which is linear with intercept on the q-axis of −g/a2 and slope of (1 − a1 )/a2 . Since we have assumed that a1 lies between zero and unity, then the slope of this line is positive. Next consider the q˙ = 0 isocline, which we shall refer to as the LM curve since it implies money market equilibrium. We have qm0 kq − b1 y − =0 q˙ = u u ky m0 − q = b1 y u u ub1 y i.e. q = (ky − m0 ) which is nonlinear, and has an asymptote at y = m0 /k, which means that q is positive only if y > m0 /k. This we shall assume to be the case. Also, as y → ∞, then q → ub1 /k. Although it is possible to solve for y and q, the solution involves a quadratic and does not reveal anything new. What we have here, however, is a nonlinear nonhomogeneous differential equation system. To establish the nature of the equilibrium we need to consider the vectors of forces in the four quadrants. We have already established that the y˙ = 0 isocline is positively sloped. Furthermore, if y˙ > 0 then q>

(1 − a1 )y − g a2

Hence, above the y˙ = 0 isocline, y is rising while below it y is falling, as illustrated in ﬁgure 10.17. In establishing the nature of the forces either side of the q˙ = 0 isocline we ﬁrst need to establish its slope. We ﬁnd this with a little manipulation as follows q=

ub1 y (ky − m0 )

Figure 10.17.

458

Economic Dynamics k u2 b 1 y (ky − m0 )ub1 − ub1 yk dq b1 u = − = dy (ky − m0 )2 (ky − m0 )/u (ky − m0 )2 k b1 − q u = r The slope of the LM curve in (q,y)-space is therefore ambiguous. The slope is positive if b1 > qk/u and negative if b1 < qk/u. To interpret these two situations, consider a rise in income, shown by the movement from point A to point B in ﬁgures 10.18(a) and (b). From the money market equation this will raise the rate of interest, r; from the yield on equities equation, this will raise proﬁts and hence the equity yield. If the rise in income raises the yield on equities by less than it raises r then q must fall in order to re-establish equilibrium between r and the ˙ as shown in ﬁgure 10.18(a) by the movement from yield on equities ((b1 y + q)/q), point B to point C. This Blanchard called the ‘bad news’ case because the rise in

Figure 10.18.

Closed economy dynamics

459

income led to a fall in stock market prices. On the other hand, if the increase in ˙ then q must income increases r by less than the yield on equities ((b1 y + q)/q), rise, as shown in ﬁgure 10.18(b) by the movement from point B to point C. This Blanchard called the ‘good news’ case, since the rise in income leads to a rise in stock market prices. Whether the q˙ = 0 isocline is negatively sloped (‘bad news’) or positively sloped (‘good news’), if q˙ > 0 then q>

ub1 y (ky − m0 )

and so above the q˙ = 0 isocline q is rising while below it q is falling, as shown by the arrows in ﬁgure 10.18. The combined vector forces in both the ‘bad news’ case and the ‘good news’ case are illustrated in ﬁgures 10.18(a) and (b). In each case, the vector forces indicate

Figure 10.19.

460

Economic Dynamics

Figure 10.20.

a saddle path solution. Given the assumption of rational expectations, and given that for any value of y there is a unique point, (a unique value of q) on the saddle path, then the economy will be at this value of q and will, over time, converge on the equilibrium.10 Example 10.8 Let us illustrate the model with a numerical example. In this example we consider only the ‘bad news’ case. The model is e = 0.8y + 0.2q + 7 8 = 0.25y − 0.2r y˙ = 2(e − y) 0.1y + q˙ r= q leading to the two nonlinear nonhomogeneous differential equations y˙ = 14 − 0.4y + 0.4q q˙ = 1.25qy − 0.1y − 40q with equilibrium values y∗ = 35.76 and q∗ = 0.76 (and r∗ = 4.7). The solution with vector forces is shown in ﬁgure 10.20. Let us take this numerical example further and consider the linear approximation. Taking a Taylor expansion around the equilibrium, we have y˙ = −0.4(y − y∗ ) + 0.4(q − q∗ ) q˙ = 1.25q∗ ( y − y∗ ) − 0.1( y − y∗ ) − 40(q − q∗ ) + 1.25y∗ (q − q∗ ) i.e. y˙ = −0.4(y − y∗ ) + 0.4(q − q∗ ) q˙ = 0.85(y − y∗ ) + 4.7(q − q∗ ) 10

There is a problem if the LM curve is everywhere steeper than the IS curve (see Scarth 1996).

Closed economy dynamics

461 Figure 10.21.

The matrix of the system is −0.4 0.4 A= 0.85 4.7 with characteristic equation λ2 − 4.43λ − 2.22 = 0 and characteristic roots r = 4.7658 and s = −0.4658. The fact that the characteristic roots have opposite signs veriﬁes the saddle point equilibrium (as does the fact that det(A) is negative, i.e., det(A) = −2.22). The general solution is y(t) = y∗ + c1 e4.7658t + c2 e−0.4658t q(t) = q∗ + c3 e4.7658t + c4 e−0.4658t The saddle paths are readily found by solving (A − rI)vr = 0 and (A − sI)vs = 0 giving the two respective eigenvectors 1 1 r s , v = v = 12.9145 −0.1645 where vs is the stable arm of the saddle point. These results, using the above linearisation, are shown in ﬁgure 10.21, which includes the direction ﬁeld for the linearisation.11

11

In this example, the stable arm is almost identical with the linear approximation to q˙ = 0 at (y∗ , q∗ ), see exercise 13.

462

Economic Dynamics 10.8.2

Unanticipated ﬁscal and monetary expansion

We are now in a position to consider the effects of ﬁscal and monetary policy. In this sub-section we shall concentrate on unanticipated changes in policy, leaving anticipated changes to sub-section 10.8.3.

Fiscal expansion Consider ﬁrst a ﬁscal expansion, a rise in g. This has no impact on the q˙ = 0 isocline but decreases the intercept of the y˙ = 0 isocline, i.e., it shifts this isocline right (down). The situation for both the ‘bad news’ case and the ‘good news’ case is illustrated in ﬁgure 10.22 (where we assume that the q˙ = 0 isocline is less steep than the y˙ = 0 isocline). In each case the initial equilibrium is at point E1 where y˙1 = 0 intersects q˙ = 0. The associated stable arm of the saddle point is S1 S1 . A rise in g shifts the IS curve down to y˙2 = 0. Initially income does not alter, and the system ‘jumps’ to the new saddle path at point A, and then over time Figure 10.22.

Closed economy dynamics

463

moves along S2 S2 to the new equilibrium point E2 . Although income rises in both situations, in the ‘bad news’ case asset prices decline, while in the ‘good news’ case they rise.

Monetary expansion Consider next monetary expansion, a rise in m0 . This has no impact on the IS curve, but shifts the q˙ = 0 isocline up, since b1 y dq = (ky − m0 )2 > 0 dm0 q=0 u ˙ The system ‘jumps’ from E1 to point A on the new saddle path S2 S2 and then moves along this until the new equilibrium point E2 is reached, as shown in ﬁgure 10.23. In each case income rises and asset prices rise from one equilibrium point

Figure 10.23.

A

464

Economic Dynamics to the next but the path of asset prices is different between the ‘bad news’ case and the ‘good news’ case.12 Even though the ﬁscal and monetary changes were unanticipated, it is assumed that the moment they are implemented the economy ‘jumps’ from the initial equilibrium to a point on the saddle path, and then adjusts over time along the stable arm of the saddle point. But what happens if the changes are announced in advance?

10.8.3

Anticipated ﬁscal and monetary policy

Suppose some policy change is announced at time t0 and to be implemented in some future time t1 . In this instance the policy change is anticipated, and some response can occur now in anticipation of what is known to occur once the policy is actually implemented. However, what occurs now is governed by the dynamics of the original equilibrium, since the new equilibrium has yet to come about.

Fiscal expansion Consider ﬁrst a ﬁscal expansion. We have already established that this will not shift the q˙ = 0 isocline but will shift the IS curve down. In anticipation of what will happen to stock market prices, the system will move from point E1 to point A (where A falls short of point A on the saddle path), as shown in ﬁgure 10.24. In the ‘bad news’ case, stock market prices fall while in the ‘good news’ case they rise. This movement, of course, is simply anticipating the ﬁnal implication of the policy change. But from the time the policy is announced until the time the policy is implemented, the economy is driven by the dynamic forces associated with the initial equilibrium point E1 . Hence, the system moves from point A to point B (on the saddle path S2 S2 ). The policy is now carried out, and the system moves along S2 S2 from point B to point E2 . The impact on income in the two cases is now different. In the ‘bad news’ case income falls and then rises, while in the ‘good news’ case it continually rises over time. On the other hand, asset prices gradually fall (if rather irregularly) in the ‘bad news’ case, and gradually rise (if rather irregularly) in the ‘good news’ case.

Monetary expansion Finally consider the case of monetary expansion, which is announced in advance, and shown in ﬁgure 10.25. As in the previous situation, in the ﬁrst instance the asset price will move part way towards its new equilibrium value, shown by point A . It will then be governed by forces associated with the initial equilibrium point E1 , and so will move along the trajectory with points A B . Point B is associated with the time the policy is implemented. Thereafter, the system will move along the stable arm of the saddle point, i.e., along S2 S2 , until point E2 is reached. 12

For a fuller discussion of what is taking place over the adjustment path, see Blanchard (1981).

Closed economy dynamics

465 Figure 10.24.

Comparing ﬁgure 10.24 with 10.22 and ﬁgure 10.25 with 10.23 shows quite a different behaviour and that it makes quite a difference to the dynamic path of the economy whether policies are announced (anticipated) or not. This is important. There has been a growing tendency on the part of policy-makers to announce in advance their policy intentions – and, at least in the UK, this applies to both monetary and ﬁscal policy.

10.9 Conclusion Although the IS-LM model is considered in some detail in intermediate macroeconomics, little attention has been paid to its dynamic characteristics. In this chapter we have concentrated on discussing the dynamics of the IS-LM model – both in discrete terms and by means of continuous time variables. Such a treatment has allowed us to consider possible trajectories for income and the rate of interest. Although we have not dealt with other endogenous variables, it is quite clear

466

Economic Dynamics

Figure 10.25.

that we can obtain their paths from a knowledge of y(t) and r(t). For instance, given y(t) we can compute tax(t) = t0 + t1 y(t), which in turn allows us to compute disposable income, yd (t) = y(t) − tax(t). This in turn allows us to compute consumption, c(t) = a + byd (t), and so on. However, this is possible only when we have explicit functional forms for all relationships in the model. What we observe from this chapter is the importance of different adjustment speeds in the goods market relative to the money market, where the latter adjusts more quickly than the former. Although a number of trajectories exhibit a counter-clockwise movement towards equilibrium, a counter-clockwise spiral, although possible, is not so likely given a quick adjustment in the money market. Overshooting, however, especially of the rate of interest, is likely to be a common occurrence, as is interest rate volatility. With investment related to both the rate of interest and the level of income, it is possible to have a positively sloped IS curve. If the IS curve is steeper than the LM curve then the most likely outcome is an unstable saddle path. This result not only depends on investment being signiﬁcantly and positively related to income, but also on the (realistic) assumption that the money market is quicker to adjust

Closed economy dynamics than the goods market. Although we have considered this possibility in the conﬁnes of a simple (linear) model, it does beg the question of whether it will occur in a more complex linear model or even in a nonlinear model. We do not, however, investigate these questions in this text. Finally, we extended the IS-LM analysis to allow for stock market behaviour employing the Tobin–Blanchard model. Once again the differential speeds of adjustment in the goods market relative to the asset market was shown to be important for dynamic trajectories. This model also highlighted the importance of unanticipated against anticipated policy changes.

Exercises 1.

Consider the model Ct = 110 + 0.75Yt It = 300 Et = Ct + It Yt = Et−1

2.

Plot the solution path for income and consumption for three different adjustment lags, j=1,2,3 for a permanent increase in investment of £10 million beginning in period 1. Consider the numerical model in example 10.2, but assume that it = 320 − 4rt−1

3.

4.

Show that this leads to a second-order difference equation for income. Either solve this second-order equation for y0 = 2000, y1 = 2010 and y2 = 2010. Hence plot the path of y(t) and r(t); or else set the problem up on a spreadsheet and plot y(t) and r(t). Reconsider the model in exercise 1 and establish the dynamic multiplier for each of the three time lags in response to a rise in investment of £20 million. What can you conclude from these results? Consider the model Ct = 110 + 0.75Yt It = 4(Yt − Yt−1 ) Et = Ct + It Yt = Et−1 (i) Show that this results in a second-order difference equation for income. Solve this equation. (ii) Suppose It = 4(Ct − Ct−1 )

5.

Does this lead to a different time path for income? For the numerical model in example 10.2 set up a spreadsheet and derive the solution path for all endogenous variables resulting from a rise in

467

468

Economic Dynamics

6.

real money balances of £20 million. Compare your results with those provided in table 10.2. Set up a spreadsheet to derive trajectories for the discrete model yt+1 − yt = α[b(1 − t) − 1]yt − αhrt + αa rt+1 − rt = βkyt − βurt − βm0

7.

8.

9.

10.

11.

12. 13.

Derive the equilibrium income and interest rate, by setting yt+1 = yt = . . . and rt+1 = rt = . . . and place cells on the spreadsheet to compute such equilibria. Use the parameter values in the text to derive the three trajectories T1 (α = 0.05, β = 0.8), T2 (α = 0.1, β = 0.8) and T3 (α = 0.5, β = 0.8). How would you use this speciﬁcation to: (i) show a goods market shock? (ii) show a money market shock? Show that for the same system outlined for ﬁgure 10.4 that a monetary expansion from m0 = 8 to m1 = 12 leads to a new equilibrium point (y, r) = (72.2, 12.1). Using either Mathematica or Maple, establish three trajectories for the same combinations of α and β as in exercise 6. Or, using the discrete form of the model outlined in exercise 6, set up the model on a spreadsheet and obtain the three trajectories. Use the spreadsheet model of exercise 6 to investigate the implications for the three trajectories T1 , T2 and T3 of (i) a higher marginal propensity to consume (ii) a lower marginal rate of tax (iii) investment being more interest-sensitive (higher h) (iv) a higher income velocity of circulation of money (lower k) (v) a more interest-sensitive demand for money (higher u). Use your model in exercise 6 and verify that if the parameter a rises from 50 to 55 the adjustment path exhibits overshooting of both y and r if α = 0.5 and β = 0.8. Use your model in exercise 6 and verify that if the parameter a falls from 50 to 45 the adjustment path exhibits overshooting of both y and r if α = 0.5 and β = 0.8. Use Mathematica or Maple to derive the trajectory {y(t),r(t)} for examples 10.5 and 10.6 in section 10.6. Verify the statements in the text, namely (i) Example 10.5 leads to a stable path which appears to traverse a straight line path after a certain time period. (ii) Example 10.6 leads to an unstable spiral. Reconsider example 10.6 in section 10.6. Does the same saddle path result if α = 0.1 and β = 0.8? (i) Show that the linear approximation to the q˙ = 0 isocline in the Tobin–Blanchard model is b1 m0 u ∗ (y − y∗ ) q=q − (ky∗ − m0 )2

Closed economy dynamics (ii)

In example 10.8 show that the equation for the linear approximation to q˙ = 0 at (y∗ , q∗ ) is q = 7.23352 − 0.181008y while the equation for the saddle path is q = 6.64988 − 0.164683y

14.

Consider the following IS-LM model e = a + b(1 − t)y − hr + jy md = ky − ur y˙ = α(e − y) r˙ = β(md − m0 )

15.

a=5 b = 0.75

k = 0.5 u = 0.3

t = 0.25 h = 0.3 j = 0.4

α = 0.25 β = 0.4 m 0 = 10

(i) Find y∗ and r∗ . (ii) What are the equations for the IS curve and the LM curve? (iii) Obtain the trace and determinant of the system, and hence establish whether a stable or unstable spiral is present. Given the Tobin–Blanchard model e = 0.8y + 0.2q + 7 16 = 0.5y − 0.25r y˙ = 2(e − y) 0.15y + q˙ r= q (i) Find y∗ and q∗ . (ii) Show that the ﬁxed point (y∗ , q∗ ) is a saddle point equilibrium. (iii) Derive the equation of the stable arm of the saddle point. Additional reading

Additional material on the contents of this chapter can be obtained from Blanchard (1981), McCafferty (1990), Obstfeld and Rogoff (1999), Romer (2001), Scarth (1996), Shone (1989, 2001), Stevenson, Muscatelli and Gregory (1988), Teigen (1978) and Tobin (1969).

469

CHAPTER 11

The dynamics of inﬂation and unemployment

11.1 The Phillips curve At the heart of most discussions of inﬂation is the Phillips curve which, in its modern formulation, stipulates a relationship between price inﬂation, π, and unemployment, u, augmented for inﬂationary expectations, π e . Thus π = f (u) + ξ π e

(11.1)

0 0

(11.2)

This is no more than a continuous version of adaptive expectations. When the actual rate of inﬂation exceeds the expected rate, expectations are revised upwards and when the actual rate is below the expected rate, then expectations are revised downwards. Suppose the government attempts to maintain unemployment at some constant level, u∗ .1 We further suppose that they are successful and so f (u∗ ) is a constant and known. To establish the implications of such a policy, differentiate the Phillips curve relationship (11.1) with respect to time under the assumption that u = u∗ and substitute in equation (11.2). Then π˙ = ξ π˙ e = ξβ(π − π e ) = β(ξ π − ξ π e ) But ξ π e = π − f (u∗ ), and so π˙ = β[ξ π − π + f (u∗ )] i.e. π˙ = βf (u∗ ) − β(1 − ξ )π

(11.3)

which is linear with intercept βf (u∗ ) and slope −β(1 − ξ ). Relationship (11.3) is illustrated in ﬁgure 11.1. 1

u∗ is often assumed to be the level of unemployment associated with full employment. This was the type of policy pursued in the UK between 1945 and 1979.

The dynamics of inﬂation and unemployment

471 Figure 11.1.

First we need to establish whether a ﬁxed point exists. For such a point π˙ = 0, i.e. βf (u∗ ) − β(1 − ξ )π ∗ = 0 or π∗ =

f (u∗ ) 1−ξ

(11.4)

Only if 0 < ξ < 1, however, will π ∗ exist. In particular, if ξ = 1 then π ∗ is undeﬁned. Second, if 0 < ξ < 1, then π ∗ is asymptotically globally stable since the relationship between π˙ and π is negatively sloped. Third, if ξ = 1 inﬂation is always correctly anticipated and π = π e and π˙ e = 0. In this instance the rate of unemployment is constant regardless of the rate of inﬂation. This unemployment rate, following the work of Friedman and Phelps, is referred to as the natural rate of unemployment (or the non-accelerating inﬂation rate of unemployment, NAIRU), and denoted un . This rate satisﬁes f (un ) = 0. The situation is illustrated in the more conventional diagram, ﬁgure 11.2. However, we can go further. Since π˙ = βf (u) and f (u) < 0, if u = u∗ < un then it follows that π˙ > 0; while if u = u∗ > un , then π˙ < 0. This implies that if the government maintains the level of unemployment below the natural level (here we ignore u∗ > un ) then permanent inﬂation will be the result. This is because expected inﬂation always lags behind actual inﬂation and the economy is forever trying to catch up with what it observes. It is common in a number of studies to assume a relationship between inﬂation, π, and real income, y, and expected inﬂation, π e . In particular it is common to express this relationship in the form π = α(y − yn ) + π e α > 0 where y and yn are in natural logarithms and yn is the natural level of income (the income level associated with un ), and where the relationship is referred to as ‘the

(11.5)

472

Economic Dynamics

Figure 11.2.

Phillips curve’. This is not the relationship between inﬂation and unemployment and in fact embodies two reaction functions.2 It is worth spelling these out in detail because of the common occurrence of this equation. Following the original formulation of the Phillips curve, we postulate a relationship of the form π = −γ1 (u − un ) + π e

γ1 > 0

This is the ﬁrst reaction function indicating the response of price inﬂation to the unemployment gap. It implies a speciﬁc functional form for f (u). The second reaction function is a formulation of Okun’s law3 and is given by u − un = −γ2 (y − yn ) γ2 > 0 Substituting this into the previous equation we obtain π = γ1 γ2 ( y − yn ) + π e or π = α( y − yn ) + π e

(11.6)

α>0

Although both π, in terms of the unemployment gap, and π, in terms of the output gap, are both referred to as ‘the Phillips curve’, the second is more suspect because it involves an additional behavioural relationship, namely Okun’s law – which is far from being a law. We shall, however, conform to common usage and refer to both as the Phillips curve.

11.2 Two simple models of inﬂation Macroeconomic modelling has generally incorporated the Phillips curve within an IS-LM framework. In this section we shall consider the simplest of these models to 2 3

See Shone (1989, chapter 3) for a more detailed discussion on this. See Shone (1989, appendix 3.2).

The dynamics of inﬂation and unemployment

473

highlight the dynamics. Basically, the goods market and money market combine to give the aggregate demand curve (see Shone (1989, chapter 2). To see this, consider the following simple linear model, where variables (other than inﬂation and rates of interest) are in logarithms. Goods market c = a + b(1 − t)y i = i0 − h(r − π e ) y=c+i+g

(11.7)

Money market md = ky − ur ms = m − p md = ms

(11.8)

where c = real consumption y = real GDP i = real investment r = nominal rate of interest π e = expected inﬂation g = real government spending md = real money demand ms = real money supply m = nominal money stock p = price level Solving for y and r we obtain (a + i0 + g) + (h/u)(m − p) + hπ e 1 − b(1 − t) + (hk/u) ∗ − (m − p) ky r∗ = u The main focus of attention is on y∗ , the equilibrium level of real income. It should be noted that this is a linear equation in terms of m − p and π e , i.e. y∗ =

y = a0 + a1 (m − p) + a2 π e

a1 > 0, a2 > 0

and this represents the aggregate demand curve, the AD curve. Why? Because it denotes equilibrium in both the goods market and the money market. In other words, all points along the AD curve denote equilibrium in both the goods market and the money market. We can express the aggregate demand curve in the usual way as a relationship between p and equilibrium y, with p on the vertical axis and y on the horizontal axis. Then 1 a2 a 0 + a1 m − y+ πe p= a1 a1 a1 i.e.

p = c0 − c1 y + c2 π e

(11.9)

(11.10)

474

Economic Dynamics

Figure 11.3.

where c0 =

a 0 + a1 m , a1

c1 =

1 , a1

c2 =

a2 a1

which clearly indicates an inverse relationship between the price level, p, and the level of real income, y. It is at this point we introduce inﬂation, π. We assume that the rate of inﬂation is proportional to the output gap and adjusted for expected inﬂation, as outlined in the previous section, i.e. π = α( y − yn ) + π e

α>0

yn is the output level for which π = π e = 0. It represents the long-run situation where prices are completely ﬂexible. Under this condition the equilibrium price level is p∗ and y = yn regardless of p and so the long-run aggregate supply curve is vertical at yn . The situation is illustrated in ﬁgure 11.3. Although ﬁgure 11.3 expresses p as a function of y, the more interesting and revealing relationship is that between y and real money balances, m − p, i.e., y = a0 + a1 (m − p) + a2 π e . Only when there is a change in real money balances (a change in m − p) will there be a shift in AD. This is important. In elementary courses in economics it is quite usual to say something like ‘a decrease in the money supply shifts LM left, raising r and reducing y’. But money supply has hardly ever decreased! What has decreased is the growth in the money supply. This leads to a fall in p. So long as m falls more than p, then real money balances will fall, i.e., m − p < 0. It is this which shifts the LM curve to the left.4 In other words, only when m − p = 0 will the aggregate demand curve shift. 4

See chapter 10 on the IS-LM model.

The dynamics of inﬂation and unemployment

475 Figure 11.4.

Example 11.1 To illustrate this model, let π e = 0 and let y = 9 + 0.4(m − p) m=5 yn = 6 π = 0.2( y − yn ) then p = 27.5 − 2.5y In equilibrium y = yn hence π = 0, i.e., y∗ = 6 and p∗ = 12.5. The situation is illustrated in ﬁgure 11.4. At a price level below (or above) p∗ = 12.5, forces will come into play to move the economy towards equilibrium. To illustrate these dynamic forces, consider the following discrete version of the model. We have (noting we have the natural logarithm of prices) yt−1 = 9 + 0.4(mt−1 − pt−1 ) πt = pt − pt−1 = 0.2( yt−1 − yn ) i.e. πt = 0.2[9 + 0.4(mt−1 − pt−1 ) − 6] = 0.6 + 0.08(mt−1 − pt−1 )

476

Economic Dynamics

Figure 11.5.

But mt−1 = mt = 5 for all t and so πt = 1 − 0.08pt−1 Since prices are in natural logarithms, then πt = pt − pt−1 , hence pt − pt−1 = 1 − 0.08pt−1 i.e. pt = 1 + 0.92pt−1 We can either use the original formulation of the model in a spreadsheet, or this linear relationship5 pt = f (pt−1 ) = 1 + 0.92 pt−1 , as shown in ﬁgure 11.5. Either way, we can ﬁrst solve for the ﬁxed point pt = p∗ for all t so that p∗ = 1 + 0.92p∗ p∗ = 12.5 The spreadsheet representation is illustrated in ﬁgure 11.6, which shows the system converging on equilibrium for an initial price of p0 = 5. Convergence to equilibrium in the neighbourhood of p∗ = 12.5 is assured because | f (p∗ )| < 1 (see n. 5). The model just discussed has a major weakness and that is that in the long run the only acceptable level of inﬂation is zero, since only this is consistent with the (assumed) zero expectations value of inﬂation. But can a situation arise in which π = π e at some positive value and the economy is in long-run equilibrium with income at the natural level? To answer this question, ﬁrst return to our aggregate demand relation y = a0 + a1 (m − p) + a2 π e If we take the time derivative of this relationship6 we obtain the demand pressure curve, with formula y˙ = a1 (m˙ − π ) + a2 π˙ e

5 6

Given pt = f (pt−1 ) = 1 + 0.92 pt−1 then f (p) = 0.92 < 1, which is the requirement for stability as indicated in part I. We assume that the variables are in logarithms and so dp/dt = d lnP/dt = π .

The dynamics of inﬂation and unemployment

477 Figure 11.6.

Figure 11.7.

˙ is exogenously given. We can now combine this with where monetary growth, m, the Phillips curve and a dynamic adjustment for inﬂationary expectations, giving the model y˙ = a1 (m˙ − π ) + a2 π˙ e a1 > 0, a2 > 0 α >0 π = α( y − yn ) + π e π˙ e = β(π − π e ) β >0 The model is captured in terms of ﬁgure 11.7 in its more traditional form. The demand pressure curve intersects the short-run Phillips curve on the long-run Phillips curve. Since in this situation π = π e , then it follows y = yn and π˙ e = 0, which implies m˙ = π.

(11.11)

478

Economic Dynamics To consider the dynamics of the model, it can be reduced to two differential equations.7 From the Phillips curve and the dynamic adjustment equations we immediately obtain π˙ e = αβ( y − yn ) For the demand pressure curve we substitute the short-run Phillips curve for π and the result just obtained for π˙ e i.e. y˙ = a1 (m˙ − π) + a2 π˙ e = a1 m˙ − a1 [α( y − yn ) + π e ] + a2 αβ(y − yn ) = a1 m˙ − α(a1 − a2 β)( y − yn ) − a1 π e Thus, we have the two differential equations π˙ e = αβ(y − yn )

(11.12)

y˙ = a1 m˙ − α(a1 − a2 β)( y − yn ) − a1 π e ∗

which can be solved for y∗ and π e . Notice that the model solves for the time path of expected inﬂation, but the time path of actual inﬂation is readily obtained from the short-run Phillips relationship, i.e. π(t) = α(y(t) − yn ) + π e (t) To solve for equilibrium, a steady state, we set π˙ e = 0 and y˙ = 0. From the ﬁrst condition it immediately follows that y = yn . Combining this result with π˙ e = 0 ∗ ˙ (In what follows we shall suppress and y˙ = 0 immediately gives the result π˙ e = m. the asterisk.) First consider the π˙ e = 0 isocline. In this instance it readily follows that y = yn and so the isocline is vertical at the natural level of income. If y > yn then π˙ e > 0 and hence π e is rising, and so to the right of the vertical isocline we have vector forces pushing up expected inﬂation. Similarly, when y < yn then π˙ e < 0 and there are forces pushing down the rate of expected inﬂation. These forces are illustrated in ﬁgure 11.8(a). Consider next the y˙ = 0 isocline. In this case a1 m˙ − α(a1 − a2 β)( y − yn ) = a1 π e a2 β ... π e = m˙ − α 1 − ( y − yn ) a1 which is negatively sloped if 1 − (a2 β/ a1 ) > 0, which we assume to be the case. If y˙ > 0 then a2 β π e < m˙ − α 1 − ( y − yn ) a1

7

This is a simpler version of a similar model discussed in McCafferty (1990, chapter 7).

The dynamics of inﬂation and unemployment

479 Figure 11.8.

and so to the left (below) the y˙ = 0 isocline there are forces present increasing y. Similarly, to the right (above) this isocline there are forces decreasing y. These forces are illustrated in ﬁgure 11.8(b). Combining the two isoclines leads to four quadrants with vector forces as shown in ﬁgure 11.9. What this shows is a counter-clockwise movement of the system. Hence, starting at any point such as point A, the system will move in an anticlockwise direction either converging directly on the equilibrium point, as shown by trajectory T1 , or converging on the equilibrium point with a counter-clockwise spiral, as shown by trajectory T2 . Which of these two trajectories materialises depends on the values of the exogenous variables and parameters of the dynamic system. Of course, there is nothing in the qualitative dynamics preventing the counter-clockwise spiral diverging from the equilibrium. All we know from ﬁgure 11.9 is that the equilibrium is a spiral node. We can illustrate the model with a numerical example. We shall present this model ﬁrst in continuous time and then in discrete time. The discrete time version has the merit that the system’s dynamics can readily be investigated on a spreadsheet.

480

Economic Dynamics

Figure 11.9.

Example 11.2 Consider the numerical model y˙ = 10(15 − π) + 0.5π˙ e π = 0.2(y − 15) + π e π˙ e = 1.5(π − π e ) Equilibrium income and expected inﬂation is readily found to be y∗ = 15 and ∗ π e = 15, which is equal to the actual rate of inﬂation and to the growth of the money supply. We have already established that the π˙ e = 0 isocline is vertical at the natural level of income, namely y∗ = yn = 15. On the other hand, the demand pressure curve y˙ = 0 is given by a2 β e π = m˙ − α 1 − (y − yn ) a1 i.e.

π e = 17.775 − 0.185y

and it is readily veriﬁed that π e∗ = 15 when y∗ = 15. Furthermore, the two differential equations take the form y˙ = 177.75 − 1.85y − 10π e π˙ e = −4.5 + 0.3y which in terms of deviations from equilibrium are ∗

y˙ = −1.85( y − y∗ ) − 10(π e − π e ) π˙ e = 0.3( y − y∗ ) Hence, the matrix of this system is −1.85 −10 A= 0.3 0

The dynamics of inﬂation and unemployment

481 Figure 11.10.

Figure 11.11.

with tr(A) = −1.85 and det(A) = 3. From chapter 4, table 4.1 (p. 180), since tr(A) < 0, det(A) > 0 and tr(A)2 < 4 det(A) then we have a spiral node. Furthermore, the characteristic roots of A are r, s = −0.925 ± 1.4644i and since α in the characteristic roots r, s = α ± βi is negative, then the system is asymptotically stable. We verify this by using a software package to derive the direction ﬁeld of this system along with a trajectory beginning at point ( y0 , π0e ) = (12, 12), as shown in ﬁgure 11.10. Consider the system in equilibrium at π ∗ = π e = 15 and y = yn = 15. Now let monetary growth decline from m˙ 0 = 15 to m˙ 1 = 12. The result is shown in ﬁgure 11.11. In line with our previous analysis, we have an anticlockwise spiral that converges on the new equilibrium point E1 . We noted above that although the model solves for π e (t) we can derive π (t) from the short-run Phillips curve. What is the difference between the path of π e (t) and the path of π(t)? These paths for a reduction in monetary growth just analysed

482

Economic Dynamics

Figure 11.12.

are shown in ﬁgure 11.12, which also shows the path of y(t) over part of the adjustment period. What the lower diagram illustrates is not only the cycle nature of actual and expected inﬂation, but that actual inﬂation is initially below expected inﬂation. This is because actual income initially falls short of the natural level and so dampens inﬂation. When, however, income is above the natural level then actual inﬂation is above expected inﬂation and so pushes up actual inﬂation. Example 11.3 Next consider a discrete version of the model with the same parameter values. The model is

e yt − yt−1 = 10(m˙ t−1 − πt−1 ) + 0.5 πte − πt−1 πt = 0.2( yt − yn ) + πte

e e πte − πt−1 = 1.5 πt−1 − πt−1 which leads to the two difference equations e yt = 177.75 − 0.85yt−1 − 10πt−1 e πte = −4.5 + 0.3yt−1 + πt−1

The dynamics of inﬂation and unemployment

483 Figure 11.13.

These are readily set out on a spreadsheet as shown in ﬁgure 11.13, which includes the dynamic path of the system from a starting value of ( y0 , π0e ) = (12, 12). The system, however, now diverges from the ﬁxed point in a counter-clockwise direction! Why is this? The matrix of the system is −0.85 −10 A= 0.3 1 with characteristic equation λ2 − 0.15λ + 2.15 = 0 and complex roots, r, s = α ± βi, i.e. r = 0.075 + 1.4644i s = 0.075 − 1.4644i For discrete systems, stability requires8 2 α + β 2 < 1 However, in this example α 2 + β 2 = 1.4663 and so the system, illustrated in ﬁgure 11.13, is explosive. This example should act as a warning. It is not possible to attribute the same properties to discrete systems as occur in continuous systems. The more complex the system the more likely the discrete system will exhibit different properties from its continuous counterpart.

8

See section 3.8 and Azariadis (1993, pp. 36–8).

484

Economic Dynamics

11.3 Deﬂationary ‘death spirals’9 At the time of writing (mid-2001), Japan was in a recession and the USA began to experience a serious downturn – enough for some economists to wonder whether a major deﬂation worldwide was likely. In explaining such a possibility, interest has returned to the concept of the liquidity trap. Not in the sense of the early literature that considered a low positive nominal interest rate so that the demand for real money balances became inﬁnitely elastic at this value, but because the nominal interest rate cannot be negative. These two types of liquidity trap are conceptually different. The ﬂoor of zero on the nominal interest rate leads to what Groth (1993) has called a dynamic liquidity trap. Here we shall present a simpliﬁed version of the model outlined in Groth (1993) and similar to the one utilised by Krugman (1999). The model is in natural logarithms, except for all inﬂation rates and the nominal interest rate. (1) (2) (3) (4) (5) (6) (7) (8)

c = a + b(1 − t)y i = i0 − h(r − π e ) y=c+i+g md = ky − ur ms = m − p md = ms π = α(y − yn ) + π e π˙ e = β(π − π e )

c = consumption y = income i = investment r = nominal interest rate π e = expected inﬂation md = demand for real money balances ms = supply of real money balances m = nominal money supply p = price level yn = natural level of income π = inﬂation π˙ e = dπ e /dt

g, yn and m are assumed constant, as are all autonomous expenditures (a and i0 ) and all parameters (b, t, h, k, u, α and β). The ﬁrst six equations are the familiar IS-LM model, equation (7) is the expectations augmented Phillips curve and equation (8) speciﬁes adaptive expectations. The dynamics of the model is analysed in terms of (ms , π e )-phase space, i.e., we need to derive two equations of the form m˙ s = f (ms , π e ) π˙ e = g(ms , π e ) Although the algebra is a little tedious, it does allow us to investigate various numerical versions of the model. From equation (5), and noting m is constant, we have m˙ s = −π and substituting equation (7) into this we have m˙ s = −[α(y − yn ) + π e ]

(11.13)

9

I am grateful to Christian Groth, University of Copenhagen, for drawing my attention to the literature on which this section is based.

The dynamics of inﬂation and unemployment

485

From equation (7) we immediately have π − π e = α( y − yn ), which on substitution into equation (8), gives π˙ e = αβ( y − yn ) In order to eliminate income, y, in each dynamic equation, we require to solve the IS-LM component of the model embedded in equations (1)–(6). Combining (1), (2) and (3) we derive the IS-curve in exactly the same way we did in chapter 10. This is a + i0 + g [1 − b(1 − t)]y + πe − r= h h

(11.14)

(11.15)

From equations (4), (5) and (6) we obtain the LM-curve r=

ky −ms + u u

(11.16)

Substituting equation (11.16) into equation (11.15) we derive an expression for equilibrium income y∗ =

(a + i0 + g) + hπ e + (h/u)ms 1 − b(1 − t) + (kh/u)

(11.17)

Substituting equation (11.17) into equation (11.13) we obtain * α(h/u)ms −α(a + i0 + g) + αyn − m˙ s = 1 − b(1 − t) + (kh/u) 1 − b(1 − t) + (kh/u) * $ αh + 1 πe − 1 − b(1 − t) + (kh/u) $

(11.18)

which is a linear function of ms and π e . Substituting equation (11.17) into equation (11.14) we obtain $ π˙ e =

* αβ(a + i0 + g) αβ(h/u)ms − αβyn + 1 − b(1 − t) + (kh/u) 1 − b(1 − t) + (kh/u)

+

αβhπ e 1 − b(1 − t) + (kh/u)

which is also linear in ms and π e . We shall simplify these linear equations by writing them in the form m˙ s = A + Bms + Cπ e π˙ e = D + Ems + Fπ e Using these equations we can deﬁne the (ms , π e )-phase plane with isoclines m˙ s = 0 and π˙ e = 0. We shall now pursue this model by means of a numerical example.

(11.19)

486

Economic Dynamics Example 11.4 Consider the model c = 60 + 0.75(1 − 0.2)y i = 430 − 4(r − π e ) y=c+i+g md = 0.25y − 10r ms = 450 − p md = ms π = 0.1(y − 2000) + π e π˙ e = 0.08(π − π e )

a = 60 b = 0.75 i0 = 430 g = 330 k = 0.25 m = 450 yn = 2000 α = 0.1 β = 0.08

t = 0.2 h=4 u = 10 p=0

Then m˙ s = 36 − 0.08ms − 1.8π e π˙ e = −2.88 + 0.0064ms + 0.064π e Setting m˙ s = 0 and π˙ e = 0 we derive the two isoclines (11.20)

m˙ s = 0

π e = 20 − 0.0444ms

π˙ e = 0

π e = 45 − 0.1ms

with ﬁxed point (m∗s , π e∗ ) = (450, 0). The two isoclines identify four quadrants, as shown in ﬁgure 11.14. To derive the vector forces in each quadrant, we note that m˙ s > 0 implies π e < 20 − 0.0444ms therefore below m˙ s = 0 and ms is rising while above it it is falling. Similarly, π˙ e > 0 implies π e > 45 − 0.1ms therefore above π˙ e = 0 and π e is rising while below it it is falling. The vector forces, therefore, indicate a counter-clockwise movement around the ﬁxed point. To consider (local) stability, consider the linear system in terms of deviations from equilibrium, then m˙ s = −0.08(ms − m∗s ) − 1.8(π e − π e∗ ) π˙ e = 0.0064(ms − m∗s ) + 0.064(π e − π e∗ ) Figure 11.14.

The dynamics of inﬂation and unemployment

487 Figure 11.15.

whose matrix is −0.08 A= 0.0064

−1.8 0.064

with tr(A) = −0.016 det(A) = 0.0064 which indicates local stability.10 In fact, as Groth (1993) indicates, stability is guaranteed if βu/m∗s < 1, and in the present example βu/m∗s = 0.00178 and so stability is assured. Before continuing with this example, it is useful to display the results in the more familiar IS-LM model. The equations for the IS-curve and the LM-curve are IS : LM :

r = 205 − 0.1y r = −45 + 0.025y

which intersect at the point ( y∗ , r∗ ) = (2000, 5) with π = π e = 0, as shown in ﬁgure 11.15. So far, however, we have not taken account of the nominal interest rate ﬂoor of zero. If the equilibrium interest rate is r = 0, then md = kyn = 500, which is equal to the money supply, ms . But if r = 0, then π e must be equal to minus the real rate of interest, where rreal = r − π e . Therefore in our numerical example it follows that π e = −5. This is illustrated by the dotted line in ﬁgure 11.15, which passes through point y = yn = 2000 for r = 0. 10

The eigenvectors are −0.008 ± 0.0796i and since the real part is negative, the system is asymptotically stable. See chapter 4.

488

Economic Dynamics The resulting kink in the money demand curve at r = 0 results in a kink in both isoclines m˙ s = 0 and π˙ e = 0. To establish exactly where these kinks occur we note that the equilibrium interest rate for the general model is $ * (h/u)(k/u) 1 (k/u)(a + i0 + g) + − ms r∗ = 1 − b(1 − t) + (kh/u) 1 − b(1 − t) + (kh/u) u

(11.21)

+

(kh/u)π e 1 − b(1 − t) + (kh/u)

i.e. r∗ = G + Hms + Jπ e For our numerical example, this expression is r∗ = 41 − 0.08ms + 0.2π e and so the relationship between π e and ms when r∗ = 0 is given by (11.22)

π e = −205 + 0.4ms Equating equation (11.22) with each equation in (11.20) gives the kinks at the following values m˙ s = 0

(ms , π e ) = (506.3, −2.48)

π˙ e = 0

(ms , π e ) = (500, −5)

At these values the isoclines become horizontal, as illustrated in ﬁgure 11.16. Note in particular, that π˙ e is equal to the real rate of interest that we established above, namely −5. Now return to equation (11.14) where π˙ e = αβ(y − yn ). It immediately follows that π˙ e = 0

implies

y = yn

π˙ > 0

implies

y > yn

π˙ < 0

implies

y < yn

e e

Figure 11.16.

The dynamics of inﬂation and unemployment

489 Figure 11.17.

But we established earlier that for π˙ e < 0 the economy is below the π˙ e = 0 isocline. So the recessionary region is shown by the area below this isocline, as illustrated in ﬁgure 11.17, and identiﬁed by the shaded area. Consider a situation where the economy is in recession, and at point A in ﬁgure 11.17, where y < yn and there is excess capacity. Suppose the line marked T1 shows the trajectory of the economy. But at point B, the economy hits the nominal interest rate ﬂoor, and thereafter moves in the southeast direction and always away from the ﬁxed point. It cannot get out of the dynamic liquidity trap. The output gap feeds expectations of deﬂation, and since the nominal interest rate cannot fall any further below zero, this implies a rise in the real interest rate. This in turn worsens the output gap. The economy falls into a deﬂationary spiral that it cannot escape. More signiﬁcantly, raising the money supply to expand the economy will not alleviate the situation. Now consider an independent Central Bank’s solution to the economy’s problem at point A. Given the economy is in recession, it could expand the money supply. At point A the nominal rate of interest is positive. If it expands the money supply immediately, we may suppose the economy moves along trajectory T2 . It passes into the corridor (what Krugman calls the ‘window of opportunity’) and can manoeuvre the economy to equilibrium. On the other hand, if it misses the corridor and follows trajectory T1 , then deﬂation passes the point of no return. What Krugman argues is that if the Central Bank increases monetary growth rapidly then trajectory T1 is more likely. It is interesting in this regard to comment on the behaviour of the European Central Bank (ECB) in April–May 2001. The USA was concerned about a recession and the Fed (Federal Reserve) lowered interest rates in a set of steps. The independent Bank of England also lowered UK interest rates. However, the ECB kept interest rates constant, i.e., refused to expand the money supply. Europe at the time was in a situation of below full capacity ( y < yn ), with high levels of unemployment, especially in Germany. If, then, the economy (Europe in this instance)

490

Economic Dynamics follows path T1 by the time the ECB decides to act, it may be too late. As Krugman (1999) says, conservative monetary policy may seem prudent and responsible to the European Central Bank today, just as it did to the Bank of Japan not long ago, but in retrospect that supposed prudence may look like disastrous folly.

11.4 A Lucas model with rational expectations In line with earlier sections, our aim in this one is to introduce a simple macroeconomic model to illustrate how rational expectations are employed in macroeconomic modelling. From the outset we need to be absolutely clear about variables for which expectations are formed. In particular, we need to specify the date the expectation was formed, and second the future time period about which the expectation is being formed. To be more precise, suppose we have a variable X about which expectations are being formed. If the expectation is made at time t, then we write Et to denote an expectation being formed at time t. But it is possible to formulate an expectation about X one period ahead, i.e., Et Xt+1 , or two periods ahead, Et Xt+2 , etc. In fact, we can formulate an expectation for any future time period. By the same reasoning, an expectation about Xt + 1 may have been made two periods ago, i.e., Et−1 Xt+1 is an expectation made at time t − 1 about variable X at time t + 1. So far we are simply specifying a notation to express expectations. No statement has been made about how such expectations are formed. Thus, if π denotes inﬂation, then Et πt+1 denotes expected inﬂation11 next period having been made in period t. Since prices pt are usually expressed in natural logarithms, as we shall be doing in this section, then Et πt+1 = Et pt+1 − pt The model we shall investigate is a0 > 0, a1 > 0 ydt = a0 + a1 (mt − pt ) + εt s yt = yn + b1 (pt − Et−1 pt ) + νt b1 > 0 ydt = yst = yt

ν ∼ N 0, σν2 ε ∼ N 0, σε2 ,

(11.23)

This model has a variety of new features that are worth commenting on. First, aggregate demand is the same as our earlier section but has a random component added to it. Second, the Phillips curve is in the Lucas form, i.e., the natural level of income is adjusted by deviations of prices from expected prices. Third, the aggregate supply also involves random shocks. Fourth, the random components are normally distributed with zero mean and constant variance. For those readers less familiar with stochastic equations, the random terms simply act like shocks to the AD and AS curves. On average, since their means are zero, the likely expected curve is the respective deterministic component. 11

Although it is common to write π et+1 , this does not make it explicit when the expectation was formed. It is implicitly assumed to be at time t.

The dynamics of inﬂation and unemployment

491

First we solve the model under the assumption that expectations are given. Thus, we can express the equations in matrix form as follows yt a0 + a1 m + εt 1 a1 = 1 −b1 pt yn − b1 Et−1 pt + νt Using Cramer’s rule we can solve for yt and pt a0 b1 + a2 yn a1 b1 mt a1 b1 Et−1 pt b1 εt + a1 νt + − + a1 + b1 a 1 + b1 a 1 + b1 a1 + b1 a0 − yn a1 mt b1 Et−1 pt εt − νt pt = + + + a1 + b1 a 1 + b1 a 1 + b1 a1 + b1 yt =

(11.24)

These are the reduced form equations under the assumption that expected prices are exogenous. The next step in the rational expectations procedure is to take the expected value at time t−1 for the variable pt . In other words, the expectation of the variable p is derived in the same manner that determines the variable p itself. Thus Et−1 pt =

a0 − yn a1 Et−1 mt b1 Et−1 pt Et−1 εt − Et−1 νt + + + a1 + b1 a1 + b1 a1 + b1 a1 + b1

But Et−1 εt = Et−1 νt = 0, hence Et−1 pt =

a 0 − yn a1 Et−1 mt b1 Et−1 pt + + a 1 + b1 a1 + b1 a1 + b1

a0 − yn ... Et−1 pt = + Et−1 mt a1 which is the rational expectations solution for Et−1 pt . Now having solved for Et−1 pt we can substitute this into the reduced form equations. Doing so, and simplifying, we obtain the solutions for yt and pt as follows a1 b1 (mt − Et−1 mt ) b1 εt + a1 νt + a1 + b1 a 1 + b1 a0 − yn a1 mt − b1 Et−1 mt εt − νt pt = + + a1 a1 + b1 a1 + b1 yt = yn +

To see that this model is consistent with our earlier results, consider the following two cases: (i) (ii)

constant money supply and correct expectations constant monetary growth and correct expectations.

To analyse these two cases we ﬁrst need to obtain the rate of inﬂation πt = pt − pt−1 pt =

a0 − yn a1 mt − b1 Et−1 mt εt − νt + + a1 a1 + b1 a1 + b1

pt−1 =

a0 − yn a1 mt−1 − b1 Et−2 mt−1 εt−1 − νt−1 + + a1 a1 + b1 a1 + b1

(11.25)

492

Economic Dynamics But πt = pt − pt−1 , hence a1 (mt − mt−1 ) b1 (Et−1 mt − Et−2 mt−1 ) + a 1 + b1 a1 + b1 (εt − εt−1 ) − (νt − νt−1 ) + a 1 + b1 Under the condition that mt = mt−1 and Et−1 mt = Et−2 mt−1 , then πt =

(11.26)

(εt − εt−1 ) − (νt − νt−1 ) a1 + b1 with no random shocks (εt = εt−1 = 0 and νt = νt−1 = 0) then πt = 0, which was the ﬁrst model we analysed in section 11.2. Under the condition of constant monetary growth, λ, which is expected, then πt =

mt − mt−1 = λ Et−1 mt − Et−2 mt−1 = λ so that πt = (11.27)

a1 λ b1 λ (εt − εt−1 ) − (νt − νt−1 ) + + a1 + b1 a1 + b1 a 1 + b1

(εt − εt−1 ) − (νt − νt−1 ) a1 + b1 with no random shocks inﬂation is equal to monetary growth, λ, the result next analysed in section 11.2. It is worth summarising a number of features of this model. i.e. πt = λ +

(1)

Since yt = yn +

(2)

(3) (4)

a1 b1 (mt − Et−1 mt ) b1 εt + a1 νt + a 1 + b1 a1 + b1

then the correct forecast on the part of market participants means that income can still deviate from the natural level in the short run, but only due to random factors either on the demand side or on the supply side. The deviation of yt from yn depends not only on the level of the random elements, but also on: (a) The parameters of the economic system (both AD and AS). (b) The correctness of forecasting government monetary policy. Assuming no shocks (εt = νt = 0) then income can still be above/below the natural level if forecasters underestimate/overestimate the money supply. A positive monetary surprise (i.e. mt > Et−1 mt ) means a rise in yt , pt and π t. Although pt includes forecast errors these are random in nature and so there are no systematic forecast errors. To see this, note a1 (mt − Et−1 mt ) εt − νt + a 1 + b1 a1 + b1 If the money stock is constant (i.e. mt = Et−1 mt ), or if the money stock is forecasted correctly (Et−1 mt = mt ), or if the money stock itself is subject pt − Et−1 pt =

The dynamics of inﬂation and unemployment

493

to random shocks (which then means mt − Et−1 mt is a random variable), then pt − Et−1 pt is purely a random variable. Thus, the (mathematical conditional) expectation is E( pt − Et−1 pt ) = (5)

(6) (7)

(8)

E(εt ) − E(νt ) =0 a1 + b1

It can be shown (see exercise 5) that any systematic component of a money supply rule has no bearing on the solution value of output, i.e., systematic elements of policy which are known have no impact on real output. A systematic component of a money supply rule can have a bearing on the solution of pt , and hence on π t (see exercise 6). The procedure here adopted for deriving the rational expectations is possible only if the reduced form equations can be derived. Where this cannot be done, then other procedures are necessary. (See Holden, Peel and Thompson 1985; Leslie 1993). Most attention has been on the result derived in (1) indicating policy impotence with regard to inﬂuencing the level of real income. So long as the money supply is correctly forecasted (i.e. there are no monetary surprises), then income can deviate from the natural level only as a result of random shocks to either aggregate demand or aggregate supply.

11.5 Policy rules In the previous section we pointed out the possibility of policy impotence. Let us make this more precise. Consider some policy rule for the money supply. A variety has been considered – some active and some passive. An active policy rule is one in which the policy in period t depends on the performance of the economy in the previous periods. A passive policy rule is completely independent of recent economic performance. For our present analysis we shall consider policy rules based only on variables in the previous period. This in no way invalidates the conclusions. Let x denote the policy instrument used for monetary control12 and let q denote a vector of economy variables. Then an active policy rule takes the form mt = f (xt−1 , qt−1 )

(11.28)

where f (x, q) is nonstochastic and can be linear or nonlinear. A passive policy rule, on the other hand, can take the form mt = g(xt−1 ) where g(x) is nonstochastic which can be linear or nonlinear. Return now to the result in the previous section for income, given in equation (11.25) yt = yn + 12

a1 b1 (mt − Et−1 mt ) b1 εt + a1 νt + a1 + b1 a 1 + b1

A typical choice for x is either the money base or the rate of interest.

(11.29)

494

Economic Dynamics This result suggests that income will deviate from its natural level for two basic reasons: (1) (2)

deviation of mt from Et−1 mt random occurrences to either aggregate demand or aggregate supply.

Here our concentration is on the ﬁrst. Given either of the two rules above, so long as they are nonstochastic then (11.30)

Et−1 mt = Et−1 f (xt−1 , q) = f (xt−1 , q) Et−1 mt = Et−1 g(xt−1 ) = g(xt−1 ) which implies that deviations mt − Et−1 mt = 0 for both the active and the passive policy rule. It does not matter therefore whether the rule is active or passive nor whether it is simple or complex, the result is still the same. Deviations will be nonnegative only when forecasters get the government’s policy rule wrong. This would especially be true when the government ‘changed’ the rule without announcing it. Under rational expectations theory, however, market participants would soon come to know the rule as they attempted to minimise their errors. If the policy rule involved a random component, wt , which we again assume is normally distributed with zero mean and constant variance, then the two rules can take the form mt = f (xt−1 , q) + wt mt = g(xt−1 ) + wt where wt ∼ N(0, σw2 ). Taking expectations at time t − 1, i.e., Et−1 , we immediately arrive at the nonstochastic component since Et−1 wt = 0. Hence mt − Et−1 mt = wt in both cases. The result on income is the same, namely yt = yn +

a1 b1 wt + b1 εt + a1 νt a1 + b1

The only reason why income deviates from its natural level is because of random shocks, including random elements to the policy rule. Before leaving this topic a warning is in order. The impotence of policy may appear to be solely because of the assumption of rational expectations. But this is not true. It also depends on the model chosen to illustrate the problem. In particular the result crucially depends on the assumption of completely ﬂexible prices and a vertical long-run Phillips curve. (See Attﬁeld, Demery and Duck 1985, chapter 4).

11.6 Money, growth and inﬂation In this section we turn to yet another model where inﬂation takes place in a growing economy. In such models it is common to establish that along the equilibrium growth path, the expected rate of inﬂation equals the rate of monetary expansion

The dynamics of inﬂation and unemployment

495

minus the warranted rate of growth.13 Furthermore, in models involving rational expectations (which amount to perfect foresight models) then expected inﬂation equals actual inﬂation. The model we use is that given by Burmeister and Dobell (1970, chapter 6) and taken up by George and Oxley (1991). In this model agents have perfect foresight and all markets are assumed to clear continuously.14 The model assumes a ﬁxed money growth rule of the type advocated frequently by Milton Friedman. The goods market is captured by the following set of equations Y = F(K, L) y = f (k) C = cY

(11.31)

Y = cY + K˙ + δK where Y = output K = capital stock L = labour y = Y/L k = K/L C = consumption c = marginal propensity to consume K˙ = dK/dt = net investment δ = depreciation The ﬁnal equation in (11.31) is no more than income equals consumption plus investment, and is the condition for equilibrium in the goods market. This condition is assumed to hold continuously. In line with our discussion of the Solow growth model in section 2.7, we can derive the following differential equation k˙ = sf (k) − (n + δ)k

(11.32)

where s = 1−c ˙ (the rate of growth of the labour force) n = L/L Turning now to the money market we assume a constant monetary growth rule ˙ M ˙ = λM = λ or M M The real demand for money per capita, m = M/L, is given by m=

13 14

M = PG( y, r) L

For an analysis of the production function and the resulting differential equation, see section 2.7. As we have pointed out elsewhere, these are two quite separate assumptions.

(11.33)

(11.34)

496

Economic Dynamics where m = per capita nominal money balances P = price level r = nominal interest rate Gy = ∂G/∂y > 0 Gr = ∂G/∂r < 0 The model is more easily analysed in terms of per capita real money balances, namely x = m/P, where x=

(11.35)

m = G(y, r) P

Equation (11.35) is assumed to hold continuously because the money market is assumed to be always in equilibrium. From equation (11.35) we assume we can derive the implicit function r = H( y, x) In this model there are only two assets: money and physical capital. The real rate of interest is the nominal rate, r, minus the rate of inﬂation, π (where π e = π). In equilibrium this will be equated with the marginal product of capital, f (k), adjusted for the rate of depreciation, δ, i.e. r − π = f (k) − δ

(11.36)

or r = f (k) − δ + π We now require to obtain a differential equation for x. Since x = m/P then x˙ is P˙ m˙ m˙ x˙ = − x or x˙ = − πx P P P m˙ i.e. x˙ = + ( f (k) − r − δ)x P But

M L˙ L L M = (λ − n) = (λ − n)m = (λ − n)Px L

˙ M − m˙ = L

and so (λ − n)Px + ( f (k) − r − δ)x P = ( f (k) + λ − δ − n − r)x

x˙ =

i.e. (11.37)

x˙ = ( f (k) + λ − δ − n − H( f (k), x))x

The dynamics of inﬂation and unemployment

497

We can establish the ﬁrst isocline quite easily. In equilibrium k = k∗ and x = x∗ with the result that k˙ = 0 and x˙ = 0. Consider k˙ = 0, then sf (k∗ ) = (n + δ)k∗ and for positive k this is unique, as illustrated in ﬁgure 11.18. It is also independent of x. Hence, in (x, k)-space this gives rise to a vertical isocline at k∗ , as shown in ﬁgure 11.19. For k < k∗ then k is rising while for k > k∗ , k is falling, leading to the vector forces shown in Figure 11.19. The isocline x˙ = 0 is less straightforward, and in general is nonlinear. We shall pursue this isocline by means of a numerical example, using the ﬁgures in George and Oxley (1991, p. 218). Figure 11.18.

Figure 11.19.

498

Economic Dynamics Example 11.5 Let y = 2k0.25 ln x = ln y − 0.25 ln r s = 0.2, δ = 0.03, λ = 0.05,

n = 0.02

Then k˙ = 0.4k0.25 − 0.05k If k˙ = 0 then k(0.4k−0.75 − 0.05) = 0 i.e. k∗ = 0 or k∗ = 16. Before considering x˙ = 0, we note that x = yr−0.25 ... r = y4 x−4 = (2k0.25 )4 x−4 = 16kx−4 and f (k) = 0.5k−0.75 Hence x˙ = (0.5k−0.75 − 16kx−4 )x If x˙ = 0 then (0.5k−1.75 − 16x−4 )kx = 0 So x˙ = 0 if x=0

or

(0.5k−1.75 − 16x−4 ) = 0

The second term leads to the isocline x = 2.3784k0.4375 which is nonlinear, and is shown in ﬁgure 11.20. Figure 11.20.

The dynamics of inﬂation and unemployment

499 Figure 11.21.

Figure 11.22.

Since x˙ > 0 when x > 2.3784 k0.4375 then above the isocline vector forces are pushing x up; while below the isocline they are pushing x down. These forces are also illustrated in ﬁgure 11.20. The system’s dynamics are shown in ﬁgure 11.21. The two isoclines intersect at the equilibrium point E, where (k∗ , x∗ ) = (16, 8). What ﬁgure 11.21 shows is not only a nonlinear isocline, but that the equilibrium is a saddle-point solution whose stable arm is given by the equation satisfying x˙ = 0. The full dynamics of the system are shown in ﬁgure 11.22, which uses Maple’s phaseportrait command to

500

Economic Dynamics plot the vector forces and the x˙ = 0 isocline, and which has then been annotated to complete the ﬁgure.

11.7 Cagan model of hyperinﬂation 11.7.1

Original model

Cagan argued that during periods of hyperinﬂation the main determinant of the demand to hold money balances was the expected rate of inﬂation – the higher the rate of expected inﬂation the lower the demand to hold real money balances. Income and interest rates could be thought of as constant during periods of hyperinﬂation relative to the impact of expected inﬂation. The Cagan (1956) model consists of two equations, a demand for money equation (where nominal demand is equated with nominal supply) and an equation for adaptive expectations m(t) − p(t) = −απ e (t)

(11.38)

α>0

π˙ (t) = γ [π (t) − π (t)] γ > 0 e

e

where m = ln M = logarithm of nominal money stock p = ln P = logarithm of prices π e = expected inﬂation π = inﬂation ˙ = π (t). In order to appreciate Also note that since p(t) is a logarithm then p(t) the dynamics of this model, differentiate the ﬁrst equation in (11.38) with respect to time, holding the money stock constant at some level, then ˙ = −α π˙ e (t) −p(t) = −αγ [π(t) − π e (t)] ˙ = π(t) so But p(t) −π(t) = −αγ [π (t) − π e (t)] γ [m(t) − p(t)] −αγ π e (t) = i.e. π(t) = 1 − αγ 1 − αγ or (11.39)

˙ = p(t)

γ [m(t) − p(t)] 1 − αγ

which is a ﬁrst-order differential equation. ˙ = 0, which means p(t) = m(t). The ﬁxed point of equation (11.39) satisﬁes p(t) Differentiating this result with respect to time t gives the typical monetarist result ˙ = π (t) = m(t), i.e., inﬂation is equal to the growth of the money supply. The p(t)

The dynamics of inﬂation and unemployment

501 Figure 11.23.

ﬁxed point is stable only if the coefﬁcient of p(t) is negative, which requires 1 − αγ > 0

or

αγ < 1

The situation is shown in ﬁgure 11.23(a). Cagan’s stability condition emphasises that with a highly sensitive demand for money function (α high), then stability requires inﬂationary expectations to adapt slowly to past inﬂation rates (γ = 1/α small). If this is not the case, then the system is unstable, as shown in ﬁgure 11.23(b), and the economy will exhibit either accelerating inﬂation or accelerating deﬂation depending on the initial price level. Now consider the Cagan model with rational expectations as represented by perfect foresight. Then m(t) − p(t) = −απ e (t) π˙ e (t) = π (t)

α>0

(11.40)

Hence ˙ m(t) − p(t) = −απ (t) = −α p(t) i.e. 1 ˙ = − [m(t) − p(t)] p(t) α Since the coefﬁcient of p(t) is 1/α > 0, then this system is globally unstable. The dynamics is captured in terms of ﬁgure 11.24. Assume the system is in equilibrium with p0 = p∗ with money supply m0 . Now suppose there is a rise in the money supply from m0 to m1 . To restore equilibrium in the money market the demand for real money balances must also increase. In the present model this can occur only if expected inﬂation (equal to actual inﬂation) falls. But as the inﬂation rate falls the price level starts to fall (π < 0). With the money stock now ﬁxed at m1 , real money balances rise. To re-establish equilibrium in the money market means that

(11.41)

502

Economic Dynamics

Figure 11.24.

the demand for real money balances must fall, which means π e = π also falls. The result is a continuing fall in the price level. 11.7.2

Cagan model with sluggish wages15

This version of the model consists of the following equations, where we have suppressed the time variable m − p = ky − απ e y = c + (1 − θ)n w − p = a − θn w˙ = β(n − n) π e = π = p˙

(11.42)

α>0 0 0 then p > 10.1429 + 0.4286w, so above the p˙ = 0 isocline p is ˙ > 0 then p > 3 + w, and so rising while below p is falling. Similarly, when w above the w˙ = 0 isocline w is rising while below w is falling. Consider now a one-off rise in the money supply, so m = 25. This has an impact only on the p˙ = 0 isocline, which shifts up. The equation of this new isocline is p = 13 + 0.4286w resulting in a new equilibrium of (w∗ , p∗ ) = (17.5, 20.5). The situation is illustrated in ﬁgure 11.26. Notice that dp∗ = dw∗ = dm = 5. This readily follows from the equation of w˙ = 0. The new stable saddle path is S11 S11 which passes through the ﬁxed point E1 . But what trajectory does the economy follow? In this model we assume that prices are ﬂexible and can ‘jump’ to the new stable arm immediately. This is an implication of the assumption of rational expectations with perfect foresight. Wages are assumed to alter continuously but sluggishly owing to wage contracts. The path the economy follows, therefore, is E0 ->A->E1 as shown by the heavy line in ﬁgure 11.26.

506

Economic Dynamics

Figure 11.26.

11.8 Unemployment and job turnover In order to introduce the dynamics of unemployment (and employment) we consider in this section a very extreme model in which we assume that at the ruling wage there is full employment in the sense that the number of jobs is matched by the number of households seeking employment. The working population, N, is assumed ﬁxed and the number of jobs available is constant. At any instant of time a fraction s of individuals become unemployed and search over ﬁrms to ﬁnd a suitable job. Let f denote the probability of ﬁnding a job, i.e., the fraction ﬁnding a job. At any moment of time, if u is the fraction of the participating labour force unemployed, then s(1 − u)N = individuals entering the unemployment pool f uN = individuals exiting the unemployment pool. The change in the unemployment pool, uN, is therefore given by the differential equation (11.45)

(11.46)

d(uN) = s(1 − u)N − f uN dt Since N is constant then du = s(1 − u) − f u u˙ = dt or u˙ = s − (s + f )u

0 < s < 1, 0 < f < 1

Equilibrium requires that du/dt = 0, or s − (s + f )u∗ = 0 (11.47)

i.e.

u∗ =

s/f s = s+f 1 + (s/f )

where f ∂u∗ = > 0, ∂s (s + f )2

−s ∂u∗ = 0 and ∂m/∂v > 0. Following Diamond (1982) it is further assumed that the average return to each ‘input’ is diminishing, i.e., m/u and m/v diminishes with u and v, respectively. Finally, and purely for mathematical convenience, we assume that m(u, v) is homogeneous of degree k, so that m(u, v) = uk m(1, v/u) Using this analysis we can write the change in employment as the total match Nm(u, v) minus those losing a job s(1−u)N, i.e.

(11.48)

d(eN) dE = = Nm(u, v) − s(1 − u)N dt dt de or = e˙ = m(u, v) − se dt Although the time path of employment, e(t), must mirror the time path of the unemployment rate, u(t), since e = 1 − u, the present formulation directs attention to the matching rate m(u, v). In general (Mortensen 1990), the equilibrium hiring frequency m(u, v)/u will be a function of the present value of employment per worker to the ﬁrm, q, and the employment rate, e. This can be established by noting that

(11.49)

(11.50)

uk m(1, v/u) m(u, v) = = uk−1 m(1, v/u) u u = (1 − e)k−1 m(1, v/u) = h(q, e) The hiring function h(q, e) is a function of q since the value of v/u in (11.49) is that determined in equilibrium. In equilibrium, the return on ﬁlling a vacancy (mq/v) is equal to the cost of ﬁlling a vacancy, c, i.e. m(u, v) q=c v which gives (1 − e)k−1 q =

cv/u m(1, v/u)

which means that the hiring frequency is related to both q and e. Furthermore, we can establish from this last result that hq > 0 and he < 0 if k > 1 and he > 0 if

The dynamics of inﬂation and unemployment

509

k < 1. Hence

& he < 0 if k > 1 m(u, v) = h(q, e) hq > 0, he > 0 if k < 1 u .. . m(u, v) = uh(q, e) = (1 − e)h(q, e)

which in turn leads to the following equilibrium adjustment equation e˙ = (1 − e)h(q, e) − se

(11.51)

The proﬁt to the ﬁrm of hiring an additional worker is related to q and the employment rate, e. i.e., π(q, e), and will be different for different models of the labour market. This proﬁt arises from the difference in the marginal revenue product per worker, MRPL , less the wage paid, w. If we denote the MRPL by x(e), then π(q, e) = x(e) − w.16 However, the future proﬁt stream per worker to the ﬁrm is rq = x(e) − w − s(q − kv ) + q˙ where rq represents the opportunity interest in having a ﬁlled vacancy and kv is the capital value of a vacant job, i.e., the present value of employment to the ﬁrm is the proﬁt from hiring the worker less the loss from someone becoming unemployed plus any capital gain. Since in equilibrium no vacancies exist, then kv = 0 and so rq = π(q, e) − sq + q˙ or q˙ = (r + s)q − π(q, e)

(11.52)

To summarise, we have two differential equations in e and q, i.e. e˙ = (1 − e)h(q, e) − se q˙ = (r + s)q − π (q, e) Whether a unique equilibrium exists rests very much on the degree of homogeneity of the match function, i.e., the value of k in m(u, v) = uk m(1, v/u), and the productivity per worker x(e).

11.9 Wage determination models and the proﬁt function In order to establish the properties of this differential equation system we need to have information on the partial derivatives of the proﬁt function, i.e., πq and πe . But this in turn requires a statement about wage determination, and there are a variety of wage determination models. Here we shall consider just two: a market clearing model and a shirking model.17

16

17

In the case of the shirking model of wage determination MRPL = x(e) − aλ, where λ denotes the average number of times that the effort of each worker is checked and a the ﬁxed cost required to do the checking. This analysis draws heavily on Mortensen (1990) who also considers an insider–outsider model of wage determination.

(11.53)

510

Economic Dynamics For instance, in the simplest model, Diamond (1971), all workers are identical and all have the same reservation wage and so the wage must equal the value of leisure forgone when employed, denoted b. Thus, w = b and the proﬁt function is π (q, e) = x(e) − b, with πq = 0 and πe < 0 if x (e) < 0, i.e., if we have diminishing returns to labour employed. In the case of the shirking model, an individual can receive a wage w and if successful at shirking can receive a value b in leisure. If, however, the employer monitors the worker with a frequency λ and ﬁres them if they are found shirking, then the equilibrium wage must exceed b to ensure that the expected worker cost of shirking per period is no less than the beneﬁt b. If ye denotes the expected present value of a worker’s income when employed and yu the expected present value of a worker’s future income when unemployed, then in equilibrium λ( ye − yu ) = b Furthermore rye = w + s(yu − ye ) + y˙e m(u, v) ryu = b + ( ye − yu ) + y˙u u The ﬁrst equation states that the opportunity interest from holding a job must equal the wage received plus the income she receives when unemployed, which she faces with probability s, plus any capital gain. The second equation states that the opportunity interest on being unemployed must equal the return from not working (including any unemployment beneﬁt) plus the income she receives when employed, which she faces with an average match of m(u, v)/u, plus any capital gain. In equilibrium y˙e = 0 and sb λ m(u, v) b ryu = b + u λ rye = w −

Hence

(11.54)

m(u, v) b sb − r(ye − yu ) = w − b − λ u λ m(u, v) sb b rb =w−b− − λ λ u λ

In other words the wage rate is m(u, v) b w= b+ r+s+ u λ b = b + [r + s + h(q, e)] λ

The dynamics of inﬂation and unemployment

511

Using this result we can obtain the optimal values for λ, w and π (q, e) (see exercise 6), i.e., 1

1

λ = (b/a) 2 [r + s + h(q, e)] 2 1

1

w = b + (ab) 2 [r + s + h(q, e)] 2 π(q, e) = x(e) − b −

1 2(ab) 2 [r

(11.55)

+s+

1 h(q, e)] 2

Thus the optimal wage paid exceeds the market clearing wage and is an increasing function of h(q, e), so long as a > 0. We therefore have two alternative dynamic systems: Model 1 Market clearing e˙ = (1 − e)h(q, e) − se

(11.56)

q˙ = (r + s)q − x(e) + b Model 2 Shirking model e˙ = (1 − e)h(q, e) − se

1

1

q˙ = (r + s)q − x(e) + b + 2(ab) 2 [r + s + h(q, e)] 2 Both systems are nonlinear and the dynamics depend on the value of k, and hence on the properties of h(q, e), and on the properties of x(e). An equilibrium steady state requires e˙ = 0 and q˙ = 0. So in both models equilibrium satisﬁes (1 − e∗ )h(q∗ , e∗ ) = se∗ in other words, the hire ﬂow must equal the turnover ﬂow. The isocline e˙ = 0 is called by Mortensen (1990) the employment singular curve and for x (e) < 0 1 and k < 1 this curve is upward sloping. For instance, if m(u, v) = (uv) 4 so that 1 1 m(u, v) = u 2 (v/u) 4 with k = 1/2 we derive the following results (see exercise 7). v=

q4 3

c

1

u3 2

h(q, e) = (1 − e)− 3

q1 3

c

hq > 0, he > 0

For e˙ = 0 then q=

c(se)3 1−e

To pursue this analysis further, consider the following example in which we derive explicitly the isocline q˙ = 0, called the value singular curve by Mortensen.

(11.57)

512

Economic Dynamics

Figure 11.28.

Example 11.7 Let 1

m(u, v) = (uv) 4 x(e) = 3e−0.2 r = 0.05,

a = 0.1,

c = 1,

s = 0.2,

b = 3.2

then 2

1

h(q, e) = (1 − e)− 3 q 3 and for e˙ = 0 q=

0.008e3 1−e

Considering q˙ = 0 for each model we have Model 1 q˙ = 0.25q − 3e−0.2 + 3.2 = 0 i.e.

q = 12e−0.2 − 12.8

Hence the equilibrium level of employment is found from solving 0.008e3 = 12e−0.2 − 12.8 1−e which can be done by means of a software package.18 The solution is found to be e∗ = 0.7212. The stylised situation is shown in ﬁgure 11.28. 18

Recall that if you do not have a software package like Mathematica or Maple, you can use the Solver of Excel’s spreadsheet.

The dynamics of inﬂation and unemployment Model 2 In the case of model 2, the shirking model, the q˙ = 0 isocline is given by q˙ = 0.25q − 3e

i.e.

q˙ = 12e

−0.2

−0.2

+ 3.2 +

− 12.8 −

1 2[(0.1)(3.2)] 2

1 (2.56) 2

1 0.008e3 2 0.25 + 1−e

1 0.008e3 2 0.25 + 1−e

Qualitatively this leads to the same situation as in ﬁgure 11.28 except that the q˙ = 0 isocline is below that of model 1, so leading to a smaller level of equilibrium employment. In fact, given the parameter values this is found to be e∗ = 0.3207. These equilibrium values are consistent with the equilibrium wages in the two models, which are: Model 1

w = b = 3.2

Model 2

w = b + (ab) 2 [r + s + h(q, e)] 2 = 3.4836

1

1

Since the two models are qualitatively identical, we shall pursue here only the simple market clearing model illustrated in ﬁgure 11.28.

11.10 Labour market dynamics The situation we have developed so far for the simple market clearing model is a set of differential equations given by e˙ = (1 − e)h(q, e) − se q˙ = (r + s)q − x(e) + b which reproduces equations (11.56). The isocline e˙ = 0 is upward sloping and q˙ = 0 is downward sloping. Given the parameter values in example 11.7 we have the equilibrium point (e∗ , q∗ ) = (0.7212, 0.0108) which is unique. The isoclines are given by e˙ = 0

implying

q˙ = 0

implying

0.008e3 1−e q = 12e−0.2 − 12.8 q=

furthermore when e˙ > 0 then q >

0.008e3 1−e

so employment is rising above the e˙ = 0 isocline and falling below this isocline. Similarly when q˙ > 0 then q > 12e−0.2 − 12.8 so the present value of the future proﬁt stream of the marginal worker is rising above the q˙ = 0 isocline and falling below this isocline. These vector forces are

513

514

Economic Dynamics

Figure 11.29.

illustrated in ﬁgure 11.29 and indicate that the equilibrium point E is a saddle point solution. This property is also true for model 2 in which wages are determined within a shirking model.19 Given the saddle point nature of the equilibrium in all models the only solution trajectories are those which lie on the saddle path SS . Suppose the present level of employment is e0 , as shown in ﬁgure 11.30, then the only rational expectations trajectory must be the starting point (e0 , q0 ) at point A and the path along SS to point E. Any point below SS , such as point B, tends the system to zero present value proﬁt stream from the marginal worker; or, such as point C, to an ever expanding proﬁt, i.e., an unstable speculative bubble. The solution value so far is unique because we have assumed k < 1 and x (e) < 0. A number of labour economists, however, have been investigating the situation of increasing returns in the production exchange process, which allows various possibilities for x(e). Consider the situation shown in ﬁgure 11.31 in which the e˙ = 0 isocline is upward sloping while the isocline q˙ = 0 takes a variety of shapes. There are now three solutions: a low (L), medium (M) and high (H) (e, q)pair. The medium employment level is unstable. But for any level of employment such as e0 in ﬁgure 11.31, there are two values of q consistent with the rational expectations behaviour of the system: point A on S1 S1 and point B on S2 S2 . In the case of point A, the system will tend to solution point L; while for point B, the system will tend to solution point H. It is also possible that in the neighbourhood of point M a stable limit cycle can occur. 19

In fact, the insider–outsider model of wage determination also leads to the same qualitative model with a corresponding saddle–point solution (Mortensen 1990).

The dynamics of inﬂation and unemployment

515 Figure 11.30.

Figure 11.31.

This second version of the model, exhibiting as it does multiple equilibria arising from increasing returns, illustrates a point we made in chapter 1. Rational expectations alone is not sufﬁcient to determine outcomes. At e0 points A and B are equally likely and yet the solution points L and H, respectively, involve quite different welfare implications. This suggests quite strongly that some policy coordination is necessary to ﬁx the system on one or other of the solution paths.

516

Economic Dynamics

Exercises 1.

(i) Solve the nonhomogeneous differential equation dπ = βf (u) − β(1 − ξ )π dt for π(0) = π0 .

2.

(ii) Show that for β > 0 and 0 < ξ < 1 the equilibrium π ∗ is asymptotically stable. Given the model yt−1 = 9 + 0.2(mt−1 − pt−1 ) πt = α( yt−1 − yn )

3.

α>0

if mt = 5 and yn = 6, use a spreadsheet to investigate the dynamics of the system for different values of α. For the system y˙ = −1.85(y − y∗ ) − 10(π e − π e∗ ) π˙ e = 0.3( y − y∗ )

4.

establish that the characteristic roots are complex conjugate and that r, s = α ± βi has α < 0. Show that if Et Pt+1 − Et−1 Pt = (1 − λ)(Pt − Et−1 Pt ) then Pt =

5.

Mt a+b

+

0≤λ≤1

∞ b λk Pt−k (1 − λ) a+b k=0

Consider the model ydt = a0 + a1 (mt − pt ) yst = yn + b1 (pt − Et−1 pt ) ydt = yst = yt where expectations are formed rationally. (i) Show that if money supply follows a systematic component such that mt = µ0 which is correctly anticipated by market participants, then y = yn . (ii) Show that if mt = µ0 + zt where zt ∼ N(0, σz2 ), then yt = yn +

6.

a1 b1 zt a1 + b 1

Interpret this result. In the shirking model of wage determination the ﬁrm chooses the optimal value of λ. Given rq = max{x(e) − aλ − w − sq + q˙ } λ

The dynamics of inﬂation and unemployment (i) Show that 1 1 b 2 [r + s + h(q, e)] 2 λ= a (ii) Hence show that 1

1

w = b + (ab) 2 [r + s + h(q, e)] 2 1

π(q, e) = x(e) − b − 2(ab) 2 [r + s + h(q, e)] 1

7.

Given m(u, v) = (uv) 4 and [m(u, v)/v]q = c (i) Show that v=

q4 3

c

1

u3

(ii) Hence show that −

h(q, e) = (1 − e)

2 3

q1 3

c

(iii) Verify hq > 0 and he > 0. (iv) Show that the e˙ = 0 isocline is given by: c(se)3 1−e √ Given m(u, v) = uv and [m(u, v)/v]q = c (i) Show that q=

8.

v = (1 − e)(c/q)2 9.

10.

11.

(ii) Hence show that h(q, e) is independent of e. For the numerical model (11.4) establish the new steady-state equilibrium values for k and x for each of the following, and illustrate diagrammatically the trajectory the economy follows (i) A rise in s from 0.2 to 0.3 (ii) A rise in n from 0.05 to 0.06 (iii) A rise in technology such that y = 5 k0.25 . For the numerical model (11.4) establish the new steady-state equilibrium values for k and x for a rise in monetary growth from λ = 5% to λ = 6%. What trajectory does the economy traverse? Consider the model (1) c = a + b(1 − t)y a = 100 b = 0.8 t = 0.25 i0 = 600 h = 2.5 (2) i = i0 − h(r − π e ) (3) y = c + i + g g = 525 k = 0.25 u=5 (4) md = ky − ur m = 700 p=0 (5) ms = m − p (6) md = ms α = 0.2 (7) π = α(y − yn ) + π e yn = 3000 (8) π˙ e = β(π − π e ) β = 0.05

517

518

Economic Dynamics

12.

(i) What is the ﬁxed point of the system? (ii) Derive equations for the two isoclines (iii) Derive an equation for r∗ = 0 and hence establish the presence of a corridor. In the Cagan model with perfect foresight we have the model m − p = −απ e πe = π Given seigniorage is deﬁned as M P and money grows at a constant rate λ (i) Express ln S in terms of λ. (ii) Establish that the value of λ which maximises seigniorage is

S=

λm =

1 α

Additional reading Further material on the contents of this chapter can be found in Attﬁeld, Demery and Duck (1985), Azariades (1993), Burmeister and Dobell (1970), Cagan (1956), Carter and Maddock (1984), Diamond (1971, 1982), Frisch (1983), George and Oxley (1991), Groth (1993), Holden, Peel and Thompson (1989), Krugman (1999), McCafferty (1990), Mortensen (1990), Pissarides (1976, 1985), Scarth (1996), Sheffrin (1983), Shone (1989) and Turnovsky (1995).

CHAPTER 12

Open economy dynamics: sticky price models

In this chapter and chapter 13 we shall consider a number of open economy models that exhibit dynamic behaviour. We shall start with the very simplest – the income– expenditure model considered at the beginning of all courses on macroeconomics. This model assumes a ﬁxed exchange rate. Simple as it is, it will allow us to set the scene and illustrate, in the simplest possible terms, how instability may occur, but is less likely to occur in an open economy in comparison to a closed one. We then do the same in the context of the IS-LM model we discussed in chapter 10, extending it to the open economy, but considering the situation under both a ﬁxed and a ﬂexible exchange rate. This forms the basis of the Mundell–Fleming model. This model was originally concerned with the relative impact of monetary and ﬁscal policy under ﬁxed and ﬂoating exchange rate regimes, but with perfect capital mobility. It has become the standard model of open economy macroeconomics, and so we shall look into its dynamic properties in some detail – for models with some (but not perfect) capital mobility and for situations of perfect capital mobility. We shall ﬁnd that the assumption about the degree of capital mobility is quite important to the dynamic results. As in earlier chapters, we shall be particularly interested in what happens out of equilibrium, and hence in the dynamic forces in operation in an open economy.

12.1 The dynamics of a simple expenditure model The simplest macroeconomic model for an open economy is the one where prices are assumed constant, and so we need not distinguish between real and nominal variables. Expenditure, E, is the sum of consumption expenditure, C, investment expenditure, I, government expenditure G, and expenditure on net exports, NX – where net exports are simply the difference between exports, X, and imports, M. We make four behavioural assumptions with respect to consumption expenditure, net taxes, investment expenditure and imports. Consumption expenditure is assumed to be a linear function of disposable income, where disposal income, Y d , is deﬁned as the difference between income, Y, and net taxes, T, and we make a further behavioural assumption that net taxes is linearly related to income. Investment expenditure is assumed to be positively related to the level of income (we shall consider investment and interest rates more fully in the IS-LM dynamic model). Finally, we assume that imports are linearly related to the level of income. We treat

520

Economic Dynamics government spending and exports as exogenous. The deﬁnitions and behavioural equations of our model are, then E = C + I + G + NX

(12.1)

C = a + bY d Yd = Y − T

a > 0, 0 < b < 1

T = T0 − tY I = I0 + jY

0 AE0 ). There is a build up of stocks and so ﬁrms lay off workers. Because h is high, they lay off quite a number of workers. But the loss in income of the workers means that they in turn have less disposable income. With a high marginal propensity to spend, this means a major cut in consumer spending. But this will itself lead to a further excess supply of goods, and so ﬁrms will respond with further cuts. Hence, the economy goes into continuous decline. If income had begun above the equilibrium level, at Y = Y1, with stocks running down, then ﬁrms would expand their production, disposable income would rise and consumption expenditure would rise. The economy would expand. Of course, once it reached full employment, then this would eventually manifest itself in rising prices (which we have assumed constant so far).

Open economy dynamics: sticky price models Why is the open economy different? Again begin with income below the equilibrium level, as typiﬁed by the situation in ﬁgure 12.1 for Y = Y0 . At this level of income, if the economy were not importing then there would be a running down of stocks as expenditure is in excess of income. But with the economy open, part of this demand is directed abroad and so the run-down in stocks at home is not as great. Hence openness tends to dampen the multiplier. Furthermore, the greater the marginal propensity to import the more likely stability because the greater the stabilising inﬂuence.1 We shall state this in the form of a proposition: PROPOSITION 1 The higher the marginal propensity to import, the more likely the economy will exhibit a stable equilibrium. We can consider this proposition in more detail by considering a simple numerical discrete model that we can investigate by means of a spreadsheet. The model is an extension of that provided in table 10.2 (but here we ignore the money market). In line with our discussion in chapter 10, we introduce dynamics into this discrete model by assuming that income in period t adjusts according to total expenditure in the previous period. Our model is Et = Ct + It + G0 + NX Ct = 100 + 0.75Ytd Ytd = Yt − Taxt Taxt = −80 + 0.2Yt It = 320 + 0.1Yt Mt = 10 + 0.2Yt NXt = X0 − Mt Yt = Et−1 where we assume government spending remains constant at G0 = £330 million for all periods and exports remain constant at X0 = £440 million for all periods – unless either is shocked. Equilibrium income is readily found to be equal to Y ∗ = £2500 million, which can be found from the resulting difference equation Yt = 1250 + 0.5Yt−1 A rise in government spending from £330 million to £400 million results in a new equilibrium level of income of £2640 million. The movement of the economy over time in terms of the main variables is illustrated in table 12.1, which also includes the dynamic multiplier. What the table shows is that all variables gradually tend to their new levels as the multiplier impact comes closer to its ﬁnal value. A marginal propensity to import of m = 0.3 (and with autonomous exports at £690 million) also leads to an initial equilibrium level of income of £2500 million. For the same rise in government spending from £330 million to 1

Exactly the same argument holds for the tax rate. The higher the marginal rate of tax the greater the stabilising inﬂuence on the economy, and the more likely the equilibrium is stable.

523

524

Economic Dynamics Table 12.1 Impact of a rise in government spending of £70 million t

Et

Yt

Taxt

Ydt

Ct

It

Mt

NXt

kt

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2500.00 2570.00 2605.00 2622.50 2631.25 2635.63 2637.81 2638.91 2639.45 2639.73 2639.86 2639.93 2639.97 2639.98 2639.99 2640.00 2640.00 2640.00 2640.00 2640.00 2640.00

2500.00 2500.00 2570.00 2605.00 2622.50 2631.25 2635.63 2637.81 2638.91 2639.45 2639.73 2639.86 2639.93 2639.97 2639.98 2639.99 2640.00 2640.00 2640.00 2640.00 2640.00

420.00 434.00 441.00 444.50 446.25 447.13 447.56 447.78 447.89 447.95 447.97 447.99 447.99 448.00 448.00 448.00 448.00 448.00 448.00 448.00

2080.00 2136.00 2164.00 2178.00 2185.00 2188.50 2190.25 2191.13 2191.56 2191.78 2191.89 2191.95 2191.97 2191.99 2191.99 2192.00 2192.00 2192.00 2192.00 2192.00

1670.00 1712.00 1733.00 1743.50 1748.75 1751.38 1752.69 1753.34 1753.67 1753.84 1753.92 1753.96 1753.98 1753.99 1753.99 1754.00 1754.00 1754.00 1754.00 1754.00

570.00 577.00 580.50 582.25 583.13 583.56 583.78 583.89 583.95 583.97 583.99 583.99 584.00 584.00 584.00 584.00 584.00 584.00 584.00 584.00

512.00 524.00 531.00 534.50 536.25 537.13 537.56 537.78 537.89 537.95 537.97 537.99 537.99 538.00 538.00 538.00 538.00 538.00 538.00 538.00

−70.00 −84.00 −91.00 −94.50 −96.25 −97.13 −97.56 −97.78 −97.89 −97.95 −97.97 −97.99 −97.99 −98.00 −98.00 −98.00 −98.00 −98.00 −98.00 −98.00

0.00 1.00 1.50 1.75 1.88 1.94 1.97 1.98 1.99 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00 2.00

£400 million, the economy also gradually approaches its new equilibrium level of income (namely £2617 million) marginally sooner than with a lower marginal propensity to import (see exercise 4). This can also be seen in terms of ﬁgure 12.3, which captures the movement of the economy in the ﬁrst few periods. It is clear that the new level of income is lower. The economy is inherently more stable the higher the marginal propensity to import. Of course, the corollary of this is that government spending has less inﬂuence on the domestic economy. Or, put another way, the more open an economy the greater the change in government spending necessary to achieve a given change in national income.

12.2 The balance of payments and the money supply We precede our discussion of open economy models under ﬁxed and ﬂexible exchange rates with a consideration of two interrelated variables: the balance of payments and the money supply. Both of these play a prominent role in the modelling to follow, and it is important that they are fully understood. This is quite important because we shall be setting up the models in real terms, even though we shall be assuming that prices at home and abroad are constant. This assumption of constant prices will be relaxed in chapter 13. 12.2.1

The balance of payments

We deﬁne the balance of payments in real terms, bp, as the sum of real net exports, nx, and real net capital ﬂows, cf, i.e. (12.5)

bp = nx + cf Consider ﬁrst net exports.

Open economy dynamics: sticky price models

525 Figure 12.3.

Net exports in nominal terms, NX, is the value of exports minus the value of imports. Since we do not distinguish between goods in our modelling, then exports and imports in domestic currency are priced in terms of the domestic price level P. Making a clear distinction between real and nominal variables we have NX = Px − Pz where x is real exports and z real imports. Deﬁning S as the spot exchange rate expressed as domestic currency per unit of foreign currency,2 and letting P∗ denote the price level abroad, then NX = Px − SP∗ z Dividing throughout by P to bring everything into real terms ∗ SP NX =x− z P P or SP∗ nx = x − Rz where R = P and where R deﬁne the real exchange rate, a variable which features prominently in later models.3 2 3

This means that a rise is S denotes a devaluation of the domestic currency while a fall in S indicates a revaluation of the domestic currency. This variable denotes the competitiveness of home goods relative to those abroad.

(12.6)

526

Economic Dynamics Real exports depend on income abroad and the real exchange rate (competitiveness). We assume a simple linear function x = x0 + f R

(12.7)

f >0

The constant x0 can be considered as relating to income abroad, but we shall be holding this constant throughout. The second term captures competitiveness. Suppose the home currency depreciates, so S rises and hence so does R. Then domestic prices fall relative to those abroad and hence exports are stimulated. There is then a positive relationship between real exports and the real exchange rate.4 In the case of real imports we assume Rz = z0 + my − gR 0 < m < 1, g > 0

(12.8)

where m is the marginal propensity to import, and real imports decline with a devaluation of the domestic currency (a rise in S). Combining the results we can now express real net exports as nx = (x0 + f R) − (z0 + my − gR) = (x0 − z0 ) + ( f + g)R − my = nx0 + ( f + g)R − my

(12.9)

where nx0 = x0 − z0 . Turning now to real net capital ﬂows, cf, we assume that cf = cf0 + v(r − r∗ ) v > 0

(12.10)

where cf0 is real net capital ﬂows independent of interest rates, and r and r∗ are the nominal interest rates at home and abroad (which are here equal to the real rates since we shall be holding prices at home and abroad constant). At this point we do not need to consider expected changes in the exchange rate.5 The modelling is for a ﬁxed exchange rate with no expected devaluation or revaluation, i.e., dSe /dt = 0. In chapter 13 we shall relax this assumption. Under this assumption, capital ﬂows 4

We are assuming here that the Marshall–Lerner condition is satisﬁed. Differentiating net exports with respect to S we have dz dx dNX =P − SP∗ − P∗ z dS dS dS z S dz x S dx =P − SP∗ − P∗ z S x dS S z dS = P∗ xEx − P∗ zEz − P∗ z where Ex and Ez are the export and import price elasticities, respectively. Assuming initially x = z, then dNX = P∗ x(Ex − Ez − 1) dS dNX or > 0 if Ex + Ez > 1 dS i.e. |Ex | + |Ez | > 1

5

Had we done so then the net capital ﬂow equation would become e cf = cf0 + v(r − r∗ − S˙ )

Open economy dynamics: sticky price models

527

in real terms according to the uncovered interest differential, r − r∗ , with an inﬂow if r > r∗ and vice versa. Combining net exports and the capital ﬂow equation, we arrive at an expression for the balance of payments bp = nx + cf = nx0 + ( f + g)R − my + cf0 + v(r − r∗ ) = bp0 + ( f + g)R − my + v(r − r∗ ) where bp0 = nx0 + cf0 . Balance of payments equilibrium occurs when bp = 0, a deﬁcit when bp < 0 and a surplus when bp > 0. Recall that in chapter 10 we discussed the IS-LM model. We can now introduce a third relationship into the framework. Under the assumption of ﬁxed exchange rates, the BP curve denotes combinations of income and interest rates for which the balance of payments is in equilibrium. Setting bp = 0 and expressing the result as r a function of y, we have m bp0 + ( f + g)R ∗ + y r= r − v v hence the BP curve is, in general, positively sloped. But also note that if bp < 0 then bp0 + ( f + g)R − my + v(r − r∗ ) < 0 bp0 + ( f + g)R m ∗ y i.e. r < r − + v v In other words, below the BP curve the balance of payments is in deﬁcit, while above the BP curve the balance of payments is in surplus. This information is displayed in ﬁgure 12.4. One way to account for the situations off the BP curve is to take a point on the BP curve, such as point A in ﬁgure 12.4, and move horizontally across to point B, moving to a point below the BP curve. Since the rate of interest remains constant, then so too do net capital ﬂows. On the other hand, the rise in the level of income raises the level of imports, and hence worsens the current account. Hence, if initially the balance of payments was zero, then it must now be negative as a result of the worsening current account balance. Note also that this helps to explain why the BP curve is positively sloped. Given the deﬁcit at point B, then this can be eliminated by raising the rate of interest. This will increase the net capital inﬂow, so bringing the balance of payments back into equilibrium (at point C). A similar argument applies to points above the BP curve. Care must be exercised in interpreting the BP curve, and points off it. The BP curve denotes combinations of income and interest rates for which bp = 0. In other words, we interpret external equilibrium as a situation where the current account is matched by the capital account but with opposite sign (i.e. nx = − cf or bp = 0). This is not the only deﬁnition of external equilibrium, but it is the one we shall use throughout this chapter. But what about the vectors of force either side of the BP curve? It is here we must be especially careful. If the exchange rate is ﬁxed, then although there is some force acting on the market exchange rate, there is no change in the parity rate, and it is the parity rate that determines the

(12.11)

(12.12)

528

Economic Dynamics

Figure 12.4.

intercept of the BP curve.6 If the economy is below the BP curve, then the balance of payments is in deﬁcit. What is occurring in this situation is a running down of the country’s reserves. Such a situation can persist in the short term, but not necessarily in the medium and long term. A similar situation arises in the case of a surplus, which occurs above the BP curve. Here the economy is adding to its reserves. The implication of the change in the reserve position of the economy depends on a number of factors. These include: (i) (ii)

(iii)

The change in the money supply resulting from a change in the reserves. (We shall take this up in the next sub-section.) The extent to which the authorities sterilise the impact on the money supply. (We shall also take this up in the next sub-section.) Whether a change in the parity rate is considered a possibility.

What we observe here are asset market forces that act on the economy in the medium and long term. We shall return to these where appropriate. It is also worth noting some special cases: (i)

If v = 0 then the BP curve is vertical at income level y=

6

bp0 + ( f + g)R m

The market exchange rate is determined by the demand and supply of foreign exchange, but the parity rate is set by the authorities. Under the Bretton Woods system, where exchange rates were ﬁxed vis-`a-vis the dollar, the market exchange rate could ﬂuctuate either side of the parity rate by ± 1 per cent.

Open economy dynamics: sticky price models (ii) (iii)

529

If v = ∞ there is perfect capital mobility and the BP curve is horizontal at r = r∗ . With some, but not perfect, capital mobility then the BP curve is positively sloped. However, there are two further categories which can be distinguished here, depending on the relative slopes of the BP and LM curves, which are both positively sloped: (a) the BP curve is steeper than the LM curve (b) the BP curve is less steep than the LM curve.

Situations (i) and (ii) are illustrated in ﬁgure 12.5. Before we consider the IS-LM-BP model we need to take note of the fact that the expenditure function has now altered, since it includes net exports, and hence

Figure 12.5.

530

Economic Dynamics so too has the IS-curve that we developed in chapter 10. To be speciﬁc e = a + [b(1 − t) + j]y − hr + nx0 + ( f + g)R − my = [a + nx0 + ( f + g)R] + [b(1 − t) + j − m]y − hr

(12.13)

which leads to an IS curve of a + nx0 + ( f + g)R [1 − b(1 − t) − j + m]y r= − h h which indicates a change in the position of the IS curve and in its slope relative to that in the closed economy. 12.2.2

The money supply in an open economy

In the model developed in chapter 10 the money supply was exogenous and ﬁxed. In an open economy with a ﬁxed exchange rate this is no longer the case. To see why this is so, we need to be clear on the deﬁnition of money for an open economy. Here we shall consider just the narrow deﬁnition of money, the money base, and denoted M0, and a broader deﬁnition of money supply, namely M1. Speciﬁcally M0 = Cp + CBR Ms = Cp + D

(12.14)

where M0 = money base Cp = cash held by the public CBR = commercial bank reserves at the Central Bank Ms = money supply (here M1) D = sight deposits We shall further assume a simple money multiplier relationship between Ms and M0,7 i.e. Ms = qM0

(12.15)

Return to the money base M0 = Cp + CBR. This is the money base from the point of view of Central Bank liabilities. It is possible to consider a consolidated banking system from the point of view of the asset side.8 The money base from the asset side denotes Central Bank Credit, CBC, and international reserves, IR.9 Thus M0 = Cp + CBR = CBC + IR

(12.16)

Hence Ms = qM0 = q(CBC + IR)

(12.17)

7 8 9

If Cp = cD and CBR = rD then Ms = cD + D = (1+ c)D and M0 = cD + rD = (c + r)D. Hence, Ms/M0 = (1 + c)/(c + r) or Ms = qM0. See Shone (1989, pp.147–51). See Copeland (2000, pp. 120–8). International reserves, IR, should not be confused with commercial bank reserves at the Central Bank, CBR.

Open economy dynamics: sticky price models

531

Looking at the money base from the point of view of assets means that any change in the money base can occur from two sources: (i) (ii)

Open market operations (including sterilisation) which operates through changes in CBC. Changes in the foreign exchange reserves that, under a ﬁxed exchange rate, is equal to the balance of payments.

Open market operations, CBC, can usefully be thought of in terms of two components. (a) Open market operations which have nothing to do with the balance of payments, denoted µ, and which we shall refer to as autonomous open market operations. (b) A component that is responding to the change in the reserves. Let, then CBC = µ − λIR

0≤λ≤1

(12.18)

where λ denotes the sterilisation coefﬁcient. If λ = 0 then regardless of the change in reserves, no sterilisation occurs; if λ = 1, then we have perfect sterilisation. Thus, for a surplus on the balance of payments and a rise in the money base of IR, the Central Bank reduces the money base by an equal amount. If the country has a deﬁcit, leading to a reduction in the money base, then the Central Bank increases the money base by an equal amount. Where some, but not perfect, sterilisation occurs, then 0 < λ < 1. We are now in a position to consider the money supply in more detail. Ms = q(CBC + IR) Ms = q(CBC + IR) = q(µ − λIR + IR) = q[µ + (1 − λ)IR] Hence µq q(1 − λ)IR Ms = + P P P

(12.19)

Consider the two extreme cases: (i)

µ = 0 and λ = 0, no autonomous open market operations and no sterilisation qIR Ms = = q.bp P P

(ii)

where bp =

IR P

i.e. real money balances change by a multiple of the balance of payments (in real terms). A deﬁcit leads to a fall in the money supply, while a surplus leads to a rise in the money supply. µ = 0 and λ = 1 no autonomous open market operations and perfect sterilisation Ms =0 P i.e. under no autonomous open market operations and perfect sterilisation there is no change in the money supply regardless of the balance of payments situation.

532

Economic Dynamics

Figure 12.6.

It should be noted that retaining the assumption of an exogenous and constant money supply for an open economy is equivalent to assuming no autonomous open market operations and perfect sterilisation (i.e. case (ii)). In general, this is not true, and so for an open economy the money supply should be treated as endogenous. With no sterilisation, a deﬁcit on the balance of payments under a ﬁxed exchange rate leads to a shift left in the LM curve, while a surplus on the balance of payments leads to a shift right in the LM curve, as illustrated in ﬁgure 12.6. We are now in a position to consider the dynamics of monetary and ﬁscal policy under a ﬁxed exchange rate.

12.3 Fiscal and monetary expansion under ﬁxed exchange rates 12.3.1

Fiscal expansion

In chapter 10 we have already established the vectors of force either side of the IS curve and the LM curve. Even with the IS curve re-speciﬁed for an open economy, as outlined in the previous section, the forces either side remain the same. In subsection 12.2.1 we established the deﬁcit/surplus situation either side of the BP curve. In a dynamic context, the BP curve is the condition for which bp = 0. There is no equivalent to the adjustment functions in the goods market or the money market. Why is this? The exchange rate, S, is assumed to be ﬁxed. Prices at home, P, and abroad, P∗ , are assumed constant. Hence the real exchange rate, R = SP∗/P, is constant. Once income and interest rates are determined by the dynamics of the

Open economy dynamics: sticky price models Table 12.2 Parameter values and equilibrium points for ﬁgure 12.8 Equations: Parameter values: e = a + ( f + g)R + b(1 − t)y − hr + jy − my a = 43.5 f =5 md = Md/P = ky − ur ms = Ms/P = q(CBC0 + IR0 ) + q[µ + (1 − λ)IR] g = 2 R = SP∗ /P = S R = SP∗ /P b = 0.75 nx = (x0 − z0 ) + ( f + g)R − my cf = cf0 + v(r − r∗ ) t = 0.3 bp = nx + cf h=2 dy/dt = y˙ = α(e − y) α>0 j=0 dr/dt = r˙ = β(md − ms ) β > 0 Intercepts and slopes: IS intercept = 27.924 IS slope = −0.3375 LM intercept = −6 LM slope = 0.5 BP intercept = 6.152 BP slope = 0.2 IS1 intercept = 32.924 IS1 slope = −0.3375 LM1 intercept = −8.791 LM1 slope = 0.5

m = 0.2 P=1 k = 0.25 u = 0.5 CBC = 0 IR0 = 3 q=1 λ = 0, µ = 0

S = 1.764 P∗ = 1 x0 = 0 z0 = 24 cf0 = 20.5 v=1 r∗ = 15 α = 0.05 β = 0.8

Solutions for point E0 : y = 40.506 r = 14.253 nx = −19.753 cf = 19.753 bp = 0 Solutions for point E1 : y = 46.476 Ms = 3 r = 17.238 bp = 1.791 Solutions for point E2 : y = 49.809 Ms = 4.395 r = 16.114 bp = 0

goods market and the money market, the balance of payments is automatically determined from bp = bp0 + ( f + g)R − my + v(r − r∗ ) But this is a short-run result. Why? Because a deﬁcit leads to a fall in the reserves and hence to a reduction in the money supply, while a surplus leads to a rise in the reserves and hence to an expansion in the money supply, as explained in subsection 12.2.2. In the long run, with no sterilisation, interest rates and income will change until the deﬁcit/surplus is eliminated. Geometrically, the LM curve will shift until it intersects the IS curve on the BP curve. To see this adjustment consider the following numerical model outlined in table 12.2, where CBC0 and IR0 denote the initial level for Central Bank credit and international reserves, respectively. In this model all three curves intersect at the same point, namely (y, r) = (40.506, 14.253). The situation is shown in ﬁgure 12.7, in which it should be noted that the BP curve is less steep than the LM curve. Consider a rise in autonomous spending by 10, e.g., because of a rise in government spending. The situation is shown in ﬁgure 12.8. In the short run the economy moves from equilibrium point E0 to E1 . Since the money market always clears, or is very quick to clear, then the economy moves along either the LM curve or close to it. At E1 the economy is in surplus. This follows from the new IS curve (whose

533

534 Figure 12.7.

Figure 12.8.

Economic Dynamics

Open economy dynamics: sticky price models intercept and slope are indicated in the table 12.2), intersects the LM curve above the BP curve. This must be a short-run result. Assuming no sterilisation, then the money supply must rise as the balance of payments surplus leads to a rise in international reserves. Of course, equilibrium E1 will persist into the medium term if perfect sterilisation occurs and the money supply remains constant.10 In the case of no sterilisation, then the money supply will rise, the LM curve will shift right, and this will continue until the balance of payments becomes zero once again. This requires the ﬁnal LM curve to cut the BP curve and the IS curve on the BP curve. This is shown by LM1 in ﬁgure 12.8, where all three curves (IS1 , LM1 and BP0 ) all intersect at point E2 . Given the fact that the money supply under these circumstances is endogenous, it is possible to establish that it must increase from Ms = 3 to Ms = 4.395. We have so far concentrated on the comparative statics. But what type of trajectory will such an economy follow? From our analysis so far we know that the money market is quick to adjust and the initial movement will be close to the initial LM curve. But as the economy goes into surplus the money supply will rise so shifting the LM curve right, income will adjust and the interest rate will be brought down because of the monetary expansion. The expected trajectory, therefore, is shown by the path indicated in ﬁgure 12.8 on which the arrow heads are marked. To the extent that any sterilisation takes place, then the actual path the economy follows will deviate from the trajectory shown. For instance, with perfect sterilisation and instantaneous clearing in the money market, then the path will be along LM0 , between E0 and E1 . Under perfect capital mobility the BP curve is horizontal at r = r∗ (v = ∞). The qualitative results are similar. The ﬁscal expansion leads to a rise in interest rates, which in turn leads to an immediate capital inﬂow. This will continue until the interest rate is brought into line with the interest rate abroad. During this process the balance of payments is in surplus because of the favourable capital account. The resulting surplus on the balance of payments leads to a rise in the money supply. However, since adjustment is quite quick the trajectory is either along the BP curve or close to it, as shown in ﬁgure 12.9. This will occur, however, so long as no sterilisation takes place. 12.3.2

Monetary expansion

Consider next a monetary expansion for the model outlined in table 12.2. Suppose Central Bank credit is raised from zero to CBC = 2, raising the money supply from Ms = 3 to Ms = 5. This results in a new LM curve given by LM1

r = −10 + 0.5y

This cuts IS0 with solution values y = 45.282 10

r = 12.641

bp = −2.567

The situation is more serious where the BP curve is steeper than the LM curve, then the rise in autonomous spending leads to a deﬁcit. Perfect sterilisation will eventually lead to a running out of gold and overseas currency.

535

536

Economic Dynamics

Figure 12.9.

where the deﬁcit on the balance of payments results because LM1 cuts IS0 below the BP curve, as shown in ﬁgure 12.10. In the long run, however, the deﬁcit leads to a fall in international reserves and a fall in the money supply, shifting the LM curve back to LM0 . The ﬁnal equilibrium, E2 , is the same as E0 . What about the dynamic path of this result? This is quite different from a situation of a ﬁscal expansion. To see this, consider a situation of instantaneous adjustment in the money market. The initial impact is a fall in the rate of interest to r = 10.253 (point A). This not only overshoots the short run equilibrium point E1 , but leads to a greater deﬁcit because of the larger capital outﬂow. Two forces now come into operation. With a fall in the rate of interest investment rises which, through the multiplier, raises the level of income. Simultaneously, however, the deﬁcit leads to a fall in international reserves and a fall in the money supply. The economy moves along a shifting LM curve, with a trajectory shown by the arrows pointing from position A to E2 . How ‘bowed out’ the trajectory is depends on the extent to which the money supply is slow to fall as a result of the deﬁcit (i.e. as a consequence of the fall in the level of reserves). Also, the trajectory will be more bowed out the more the Central Bank engages in any sterilisation in an attempt to move (or keep) the economy at point E1 . Perfect capital mobility does not change the qualitative nature of the results just discussed. The major difference is that the fall in the rate of interest below the world level r∗ will lead to a rapid outﬂow of capital and a more immediate reduction in the money supply. The economy is more likely to return to E0 more quickly, with a less ‘bowed out’ return trajectory.

Open economy dynamics: sticky price models

537 Figure 12.10.

This section veriﬁes the Mundell–Fleming results: PROPOSITION 2 Under ﬁxed exchange rates, ﬁscal policy is effective at changing the level of income but monetary policy is totally ineffective. Also, our analysis indicates three dynamic forces in operation: (1) (2) (3)

pressure on income to change whenever expenditure differs from income pressure on interest rates to change whenever the demand and supply of money are not equal pressure on the money supply to adjust whenever there is a balance of payments disequilibrium, and where the extent of this change depends on the degree of sterilisation being undertaken by the Central Bank.

The change in income is likely to be slow since the goods market takes time to adjust to any disequilibrium. On the other hand, interest rates are likely to adjust quite quickly. This supposition, however, assumes that the Central Bank is not attempting to control the rate of interest. The speed of the change in the money supply arising from changes in the balance of payments (the level of reserves) is likely to lie between that of the change in the rate of interest arising from capital

538

Economic Dynamics ﬂows and that of the level of income. What is being hinted at here is the need for another adjustment function: namely, the rate at which the authorities change the money supply in response to a change in the level of reserves. This is a much more sophisticated analysis than we propose to investigate in this book. 12.3.3

Rise in the foreign interest rate

A less frequently discussed shock, but an important one, arises from a change in the foreign interest rate. Consider a rise in the foreign interest rate, a rise in r∗ . Such a rise shifts only the BP curve in the ﬁrst instance. From equation (12.12) we note that a rise in r∗ raises the intercept of the BP curve – in fact by exactly the rise in r∗ . The situation is shown in ﬁgure 12.11, where we start from the same initial position. The rise in the foreign interest rate from r∗ = 15 to r∗ = 18 shifts the BP curve up to BP1 . Given this situation, the economy is still at E0 and so experiences a deﬁcit on the balance of payments, bp = −3. Under a ﬁxed exchange rate and no sterilisation, the deﬁcit leads to a capital outﬂow and a fall in the money supply. LM shifts left to LM1 and the economy settles down at E1 . But what trajectory does the economy follow on its path to E1 ? The immediate impact of the deﬁcit is a fall in the money supply. If the money market adjusts instantaneously to this fall in the money supply, then the economy moves vertically up to point B on LM1 . Thereafter, as income falls in response to the rise in the rate of interest, money demand falls putting pressure on the interest rate to fall until point E1 is reached. In this scenario the trajectory of the economy is E0 ->B->E1 , and labelled trajectory T1 . What we observe is an overshoot of the domestic interest rate. If the shift in the LM curve is not complete or not so immediate, and depending on income adjustment, another trajectory is possible, shown by T2 . Also note one other feature. The change in interest rate abroad is shown by the vertical distance between BP0 and BP1 while the rise in the domestic interest rate is less. Why is this? The fact that income is falling means a fall in imports and so net exports are rising. Hence the size of the capital outﬂow does not have to be as great. Even with perfect capital mobility, the same basic logic holds. The only difference is that eventually the domestic interest rate will rise in line with the foreign interest rate. With instantaneous adjustment of money supply to the resulting deﬁcit and instantaneous adjustment in the money market, the interest rate will once again overshoot the ﬁnal rise. In this section we have concentrated on the impact of changes in ﬁscal and monetary policy under a ﬁxed exchange rate system – typical of the situation under Bretton Woods. However, since 1973 the exchange rate in most countries has been ﬂoating.11 In the next section we consider the IS-LM-BP model under the assumption that the exchange rate is allowed to ﬂoat. However, we retain the assumption that prices at home and abroad are constant. This reminder is important. A change in the exchange rate is most likely to lead to a change in the price level in the medium and longer term. We shall take up this question of the link between

11

Britain ﬂoated its exchange rate in June 1972.

Open economy dynamics: sticky price models

539 Figure 12.11.

the exchange rate and changes in the price level in chapter 13. Even so, what it does mean is that the real exchange rate, R = SP∗ /P is changing because of the change in S.

12.4 Fiscal and monetary expansion under ﬂexible exchange rates 12.4.1

Fiscal expansion

In this section we shall consider monetary and ﬁscal policy under ﬂoating exchange rates. In doing this we need to be clear on the implications of ﬂoating. With the spot exchange rate ﬂoating, S variable, and with ﬁxed prices at home and abroad (P and P∗ constant), then the real exchange rate, R, will vary directly with S. Whatever is happening in the economy, the exchange rate will vary so that the balance of payments is always in equilibrium, bp = 0. If we assume instantaneous adjustment in the foreign exchange market and the money market, then the full impact of any change in the economy will initially fall on interest rates and the exchange rate. Only over time will the economy adjust to the situation as income changes. In terms of the diagrammatic treatment we have been using, the BP curve will shift continuously so that it always passes through the intersection between the IS and LM curves. Consider the initial situation depicted in table 12.2. Once again let autonomous spending rise by 10. This shifts the IS curve to IS1 , as shown in ﬁgure 12.12, which

540

Economic Dynamics

Figure 12.12.

Table 12.3 Exchange rate values and equilibrium points for ﬁgure 12.12 S = 1.764 r = 27.924 − 0.3375y IS0 LM0 r = −6 + 0.5y BP0 r = 6.152 + 0.2y IS1

S = 1.764 Point E0 y = 40.506 r = 14.253

r = 32.924 − 0.3375y

S = 1.547 r = 32.165 − 0.3375y IS2 LM0 r = −6 + 0.5y BP2 r = 7.671 + 0.2y

Point E1 y = 46.476 r = 17.238 bp = 1.791

S = 1.547 Point E2 y = 45.57 r = 16.785

is the same IS1 curve indicated in table 12.2. The resulting surplus on the balance of payments leads to an immediate appreciation of the domestic currency. The BP curve shifts up, and the resulting appreciation results in the IS curve shifting left to IS2 . The ﬁnal results are set out in table 12.3 and illustrated in ﬁgure 12.12. Our discussion, however, concentrates on the comparative statics. Let us for a moment turn to the dynamics of adjustment. In doing this we shall, as already indicated, assume instantaneous adjustment in all asset markets (money and foreign exchange), but slow adjustment in the goods market. The initial impact of the ﬁscal expansion is to move the economy to point E1 , with a trajectory moving along LM0 from E0 to E1 , as shown in ﬁgure 12.12. Because of the resulting surplus, the domestic currency appreciates shifting the BP curve up to BP1 . It should be noted

Open economy dynamics: sticky price models

541 Figure 12.13.

that BP1 passes through point E1 , which it must do to eliminate any surplus on the balance of payments. The appreciation, however, leads to an appreciation of the real exchange rate, a fall in R, which leads over time to a reduction in net exports. As net exports decline, so too does income through the multiplier impact. As income falls, so too does the demand for money, and this leads to a fall in the rate of interest. What we observe, since the money market is continuously in equilibrium, is a movement along LM0 from E1 to E2 . As the interest rate falls, however, the amount of net capital inﬂows declines and so the exchange rate must depreciate. This shifts the BP curve down from BP1 to BP2 which occurs as the IS curve shifts from IS1 to IS2 . In other words, in the (y, r)-plane, the economy gradually moves down LM0 from E1 to E2 , establishing itself at the ﬁnal equilibrium point E2 . The most likely trajectory, therefore, is a movement along LM0 from E0 to E2 as all these forces take effect. There is some difference in the results for the situation of perfect capital mobility. This is illustrated in ﬁgure 12.13. We can be brief because the formal analysis is similar. The ﬁscal expansion shifts IS to IS1 and the economy from E0 to E1 . The domestic currency appreciates as a result of the balance of payments surplus, shifting the BP line up to BP1 . The resulting appreciation leads to a fall in net exports shifting IS left. As this takes place, income falls, interest rates fall, and the exchange rate depreciates, with the economy moving down the LM curve from E1 to E2 = E0 , and with the situation returning to its initial position, with no impact on the level of income. Two observations are worth noting about all these results, which we shall put in the form of two propositions. The ﬁrst is the typical Mundell–Fleming result concerning a ﬁscal expansion under the assumption of a ﬂoating exchange rate; the second proposition is in the spirit of Dornbusch and overshooting.

542

Economic Dynamics PROPOSITION 3 Under ﬂexible exchange rates, ﬁscal policy is effective in changing the level of income where there is some degree of capital mobility, but totally ineffective where there is perfect capital mobility. PROPOSITION 4 Under ﬂexible exchange rates, a ﬁscal expansion leads to an overshooting of interest rates and an overshooting of the exchange rate, and this result holds with some degree of capital mobility or with perfect capital mobility. The important results are those with regard to some degree of capital mobility, since this is likely to capture the real world. A ﬁscal expansion would lead to a rise in income, a rise in interest rates and an appreciation of the domestic currency. This would be followed by interest rates falling, income falling and the exchange rate depreciating – but all such that the initial impact outweighs the secondary impacts. 12.4.2

Monetary expansion

A monetary expansion under imperfect capital mobility and under perfect capital mobility is illustrated in ﬁgures 12.14 and 12.15, respectively. The adjustment is similar in both cases. However, ﬁgure 12.14 illustrates a numerical example. Figure 12.14.

Open economy dynamics: sticky price models

543 Figure 12.15.

Table 12.4 Equilibrium points for ﬁgure 12.14 IS, LM and BP curves r = 27.924 − 0.3375y IS0 r = −6 + 0.5y LM0 r = 6.152 + 0.2y BP0 LM1 IS1 BP2

r = −10 + 0.5y r = 29.013 −0.3375y r = 3.975 + 0.2y

Solution values E0 y = 40.506 r = 14.253 S = 1.764 E2

E1

y = 45.282 r = 12.641 bp = −2.567

y = 46.582 r = 13.291 S = 2.075

Which illustrates the result of increasing the money supply by 2. The relevant information is given in table 12.4 A rise in the money supply shifts the LM curve right to LM1 and moves the economy from point E0 to point E1 . At E1 the balance of payments is in deﬁcit. Since the exchange rate is ﬂexible and adjusts instantaneously, it will depreciate, shifting the BP curve down from BP0 to BP1 , where it intersects both the IS curve and LM curve at point E1 . The depreciation leads to a rise in the real exchange rate and hence to a stimulus to net exports. This leads to a shift right in the IS curve. In terms of ﬁgure 12.14, this will be to IS1 , and the resulting rise in the rate of interest leads to an appreciation of the exchange rate, but not enough to swamp the original depreciation. The economy accordingly moves to point E2 (the intersection point between IS1 , LM1 and BP2 ). The situation in ﬁgure 12.15 is somewhat similar. The depreciation leads to a shift right in the IS curve to IS1 , but this will cut the LM1 curve on the original BP curve because interest rates will have to be brought back into line with world interest rates, which is accomplished by an expected appreciation of the currency, which returns BP1 to BP0 . Again we have concentrated on the comparative statics. But what will the trajectory of the economy look like in the short run and in the long run? The analysis is similar for both ﬁgure 12.14 and ﬁgure 12.15. The immediate impact of the monetary expansion is a sharp drop in the rate of interest, to point A on LM1 . This

544

Economic Dynamics is because the money market adjusts immediately, while the goods market is yet to alter. But there is another immediate result. The sharp fall in the rate of interest leads to a major depreciation of the exchange rate. There will be another BP curve (not shown) that passes through point A. As the goods market adjusts to the lower interest rate, stimulating investment, and through the multiplier stimulating the level of income, the economy will move to point E1 , the movement taking place along LM1 and the BP curve continuously adjusting upwards until BP1 is reached. In the longer run, however, the depreciation which originally occurred will begin to shift the IS curve because of the stimulus to net exports. This will lead to a further movement along LM1 and a further shift up in the BP curve until the economy moves to point E2 . Again we arrive at two propositions: PROPOSITION 5 Under ﬂexible exchange rates, monetary policy is effective in changing the level of income, and the effect is greater the greater the degree of capital mobility. PROPOSITION 6 Under ﬂexible exchange rates, a monetary expansion leads to overshooting of interest rates and overshooting of exchange rates, and the less the degree of capital mobility the greater the overshooting of the exchange rate. The important results are those with regard to some degree of capital mobility, since this is likely to capture the real world. A monetary expansion would lead to a rise in income, a fall in interest rates and a depreciation of the domestic currency. This would be followed by interest rates rising, income rising and the exchange rate appreciating – but all such that the initial impact outweighs the secondary impacts. 12.4.3

A rise in the foreign interest rate

Finally, consider the situation where the foreign interest rate is increased under ﬂoating. As earlier, the initial impact is to raise the BP curve by the amount of the increase. The initial situation, point E0 in ﬁgure 12.16, now represents a deﬁcit on the balance of payments. This leads to a depreciation of the domestic currency, which improves competitiveness. The improvement in competitiveness stimulates net exports, so shifting IS to the right (to IS1 ) and BP down to BP2 in ﬁgure 12.16. The ﬁnal equilibrium is at E1 . But what is the trajectory of the economy over the adjustment period? As the depreciation stimulates net exports shifting IS right and BP down, the economy will move along LM0 , since the money market clears in every period. The economy will have a trajectory along LM0 between E0 and E1 , shown by the arrows. We arrive, then, at the important conclusion that under a ﬂoating exchange rate the economy’s adjustment exhibits no overshooting. The same basic conclusion holds with perfect capital mobility, except that the ﬁnal equilibrium must have the domestic interest rate equal to the new (higher) foreign interest rate. the trajectory remains along LM with no overshooting.

Open economy dynamics: sticky price models

545 Figure 12.16.

12.5 Open economy dynamics under ﬁxed prices and ﬂoating So far we have concentrated on the comparative statics with some reference to the dynamics. Keeping within the simple linear model, let us consider the dynamics in more detail. We begin by stating the adjustment functions for the three markets explicitly, namely Goods market Money market Foreign market

y˙ = α(e − y) α>0 r˙ = β(md − ms ) β > 0 S˙ = γ (bp) γ >0

(12.20)

To simplify we set P = P∗ = 1 so that R = S and hence e = (a + nx0 ) + [b(1 − t) + j − m]y − hr + ( f + g)S giving y˙ = α(a + nx0 ) + α[b(1 − t) + j − m − 1]y − αhr + α( f + g)S or y˙ = A0 + A1 y + A2 r + A3 S where A0 A1 A2 A3

= α(a + nx0 ) = α[b(1 − t) + j − m − 1] = −αh = α( f + g)

Equilibrium in the goods market is where y˙ = 0 which no more than traces out the IS curve in ( y, r)-space. If y˙ > 0 then e > y and r is below the value on the line y˙ = 0. Hence, below and to the left of y˙ = 0 then y is rising while above and to the right income is falling, as shown in ﬁgure 12.17(a).

(12.21)

546

Economic Dynamics

Figure 12.17.

The money market is unchanged. Money market equilibrium is where r˙ = 0 which traces out the LM curve in (y, r)-space. If r˙ > 0 then Md/P > Ms/P and r is below and to the right of the value on the r˙ = 0 line. Hence, below and to the right of r˙ = 0 then r is rising, while above and to the left r is falling, as shown in ﬁgure 12.17(b). The greater the value of β the faster interest rates rise or fall to clear the money market. Finally, the foreign exchange market is in equilibrium when bp = 0 or S˙ = 0, which simply traces out the BP curve in ( y, r)-space. Here we need to be careful.

Open economy dynamics: sticky price models

547

Below S˙ = 0 then bp < 0 and S is rising representing a depreciating of the domestic currency, resulting in the S˙ = 0 line shifting down, as shown in ﬁgure 12.17(c). The fact that the S˙ = 0 line shifts should be clear. The BP curve is drawn in ( y, r)-space for a ﬁxed exchange rate. When the exchange rate varies, which it will do for either a deﬁcit or a surplus, then the result is a shift in the BP curve. However, there is also a shift in the IS curve. A depreciation of the domestic currency shifts IS right while an appreciation shifts it left – assuming the Marshall–Lerner conditions are satisﬁed.12 The higher the value of γ , the more the BP curve shifts for any given deﬁcit or surplus. Turning to the money market we have r˙ = β(md − ms )

β>0

= β(ky − ur − m0 ) i.e.

r˙ = −βm0 + βky − βur

or r˙ = B0 + B1 y + B2 r

(12.22)

where B0 = −βm0 B1 = βk B2 = −βu Finally, for the foreign exchange market S˙ = γ (bp) = γ [bp0 + ( f + g)S − my + v(r − r∗ )] = γ (bp0 − vr∗ ) − γ my + γ vr + γ ( f + g)S or S˙ = C0 + C1 y + C2 r + C3 S

(12.23)

where C0 = γ (bp0 − vr∗ ) C1 = −γ m C2 = γ v C3 = γ ( f + g) Our model, then, amounts to three differential equations y˙ = A0 + A1 y + A2 r + A3 S r˙ = B0 + B1 y + B2 r S˙ = C0 + C1 y + C2 r + C3 S The ﬁxed point is where y˙ = 0, r˙ = 0 and S˙ = 0 and can be solved for the equilibrium values y∗ , r∗ (not the interest rate abroad)13 and S∗. 12 13

See n. 4, p. 526. There should be no confusion between the foreign interest rate and the equilibrium interest rate both being referred to as r∗ .

(12.24)

548

Economic Dynamics It is clear from the speciﬁcation of the initial adjustment functions that the three parameters α, β and γ have no bearing on the existence of a ﬁxed point. What they do have a bearing on is the dynamic path or trajectory of the system from some initial value. Such a trajectory is less straightforward than any we have encountered so far because in (y, r)-space the trajectory is also governed by the movement of the exchange rate. The ﬁscal expansion we outlined in the previous section illustrates the problem. Suppose we start from an equilibrium. A ﬁscal expansion will shift the y˙ = 0 line to the right. Interest rates will be pushed up and there will be a capital inﬂow resulting in a balance of payments surplus. The extent of the interest rate rise depends on the value of β. The resulting surplus on the balance of payments leads to an appreciation of the domestic currency. The extent of the appreciation depends on the value of γ , which in turn will inﬂuence the trajectory arising from changes in the rate of interest and changes in the level of income. Finally, the changes in income will be governed by the parameter α. The difﬁculty, of course, is that we are attempting to reduce a three-variable problem into a two-dimensional plane. To appreciate some of the difﬁculties suppose we have the parameter values given in table 12.2 along with α = 0.05, β = 0.8 and γ = 0.0001. The ﬁxed point is (y∗ , r∗ , S∗ ) = (40.506, 14.253, 1.764) Now let autonomous spending rise by 10 as before. We have already established (see table 12.3) that the new ﬁxed point is (y∗ , r∗ , S∗ ) = (45.570, 16.785, 1.547) But is this new ﬁxed point attained? The ﬁrst thing we note is that E0 becomes our initial position and the dynamics of the system are governed by point E2 . The differential equation system associated with point E2 is y˙ = 2.675 − 0.03375y − 0.1r + 0.35S r˙ = −2.4 + 0.2y − 0.4r S˙ = −0.00185 − 0.00002y + 0.0001r + 0.0007S whose trajectory we can establish using a software package. One possible trajectory is shown in ﬁgure 12.18. The trajectory must begin in the shaded triangle and move anticlockwise. But there is nothing in the qualitative analysis precluding the system overshooting and spiralling towards E2 (or even away from E2 !). Such a possibility is very dependent on the reaction coefﬁcients α, β and γ . For instance, with the same ﬁscal expansion but now α = 0.1 (higher than before) then the trajectory lies outside that for α = 0.05, as shown in ﬁgure 12.19. This should not be surprising because a higher α is indicating a greater response on income in the goods market for any given level of excess demand. On the other hand, a higher value for β (say β = 1.5 rather than 0.8) leads to a trajectory inside that for β = 0.8, as shown in ﬁgure 12.20. Again this should not be surprising

Open economy dynamics: sticky price models

549 Figure 12.18.

Figure 12.19.

since a higher value for β will push the trajectory towards the LM curve.14 Notice, however, that the trajectory in ﬁgure 12.19 overshoots the equilibrium E2 while it is difﬁcult to see whether this is the case in ﬁgure 12.20. We have not considered the reaction coefﬁcient γ . The greater γ the more the BP curve and IS curve shift for any given level of balance of payments disequilibrium. The greater the value of γ the more likely the system over-reacts and equilibrium E2 not attained. By way of example, compare the following two situations for a ﬁscal change. The ﬁrst situation is as before, while situation II has a higher 14

Recall that with instantaneous adjustment in the money market the system would move along the LM curve.

550

Economic Dynamics

Figure 12.20.

Figure 12.21.

level of γ , with the exchange rate reacting signiﬁcantly to balance of payments disequilibria. Situation I α = 0.05, β = 0.8, γ = 0.0001 y˙ = 2.675 − 0.03375y − 0.1r + 0.35S r˙ = −2.4 + 0.2y − 0.4r S˙ = −0.00185 − 0.00002y + 0.0001r + 0.0007S Situation II α = 0.05, β = 0.8, γ = 0.05 y˙ = 2.675 − 0.03375y − 0.1r + 0.35S r˙ = −2.4 + 0.2y − 0.4r S˙ = −0.925 − 0.01y + 0.05r + 0.35S As can be seen from ﬁgure 12.21, the trajectories for these two situations are quite different. More signiﬁcantly, although point E2 exists, it is not attained in situation II. Why is this? We noted that the ﬁscal expansion led to a surplus and to an appreciation. The resulting rise in income led to a subsequent depreciation. But this is swamped in the present situation and the appreciation begins to dominate the dynamics pushing the system along the dotted line trajectory in ﬁgure 12.21. Of course, these parameter values are purely illustrative. But they act as a warning in the present example that the comparative statics is not sufﬁcient to establish

Open economy dynamics: sticky price models the likely trajectory of the system. It also suggests that the larger γ the greater the possibility that the system is dynamically unstable.

Exercises 1.

2.

3.

Set up the same numerical model as in section 12.1, but have the adjustment lag Yt = Et−2 . Compare your results for the time path of Yt with that in table 12.1. Consider the simple dynamics of section 12.1 with exactly the same parameter values but with the following alternative lags for the import function (take each one separately): (i) Mt = 10 + 0.2Yt−1 (ii) Mt = 10 + 0.2Et−1 Show that NXt = −mkt G and that lim mkt = mk

t→∞

4.

5.

6.

where Xt = X0 and Mt = M0 + mYt . Consider the numerical model in section 12.1 for three alternative marginal propensities to import: m = 0.2, m = 0.3 and m = 0.4, all other parameters constant. (i) Show that the three total expenditure lines emanate from the same point on the vertical axis but that their slopes differ by deriving the equation for each total expenditure curve. (ii) Obtain the equilibrium income in each case. (iii) Assuming income begins at Y0 = £2000 million, plot on the same graph the path of income over 10 periods for each marginal propensity. (iv) What conclusions do you draw from your analysis? Using the data in table 12.2, consider the new equilibrium income and interest rate for the following changes, assuming no sterilisation, constant prices and a ﬁxed exchange rate. Draw each situation and the likely path to the new equilibrium. (i) Rise in a from 43.5 to 50.0. (ii) Fall in IR (hence fall in the money supply) from 3 to 2. (iii) Devaluation of the exchange rate from 1.764 to 2. Using the data in table 12.3, consider the new equilibrium income and interest rate for the following changes, assuming no sterilisation and that P and P∗ are ﬁxed. Draw each situation and the likely path to the new equilibrium. (i) autonomous spending rises by 20 and under ﬂoating S = 1.33. (ii) Ms rises from 3 to 4, i.e., LM becomes r = −8 + 0.5y and under ﬂoating S = 1.92.

551

552

Economic Dynamics 7.

Consider the following discrete version of the numerical model of section 12.5, where again the ﬁgures refer to point E2 . yt+1 − yt = 2.675 − 0.03375yt − 0.1rt + 0.35St rt+1 − rt = −2.4 + 0.2yt − 0.4rt St+1 − St = −0.00185 − 0.00002yt + 0.0001rt + 0.0007St (i) Set the system up on a spreadsheet and show that ( y∗ , r∗ , S∗ ) = (45.570, 16.785, 1.547) is an equilibrium. (ii) Show that this equilibrium is not attained from the initial position ( y0 , r0 , S0 ) = (40.506, 14.253, 1.764).

8.

(iii) Is the equilibrium attained for initial values very close to the equilibrium? For situation II in section 12.5, the equivalent discrete model is yt+1 − yt = 2.675 − 0.03375yt − 0.1rt + 0.35St rt+1 − rt = −2.4 + 0.2yt − 0.4rt St+1 − St = −0.925 − 0.01yt + 0.05rt + 0.35St Given (y0 , r0 , S0 ) = (40.506, 14.253, 1.764) show that point E2 represented by (y∗ , r∗ , S∗ ) = (45.570, 16.785, 1.547) is not attained and that the system is explosive. Additional reading

Additional material on the contents of this chapter can be obtained from Copeland (2000), Dernburg (1989), Gapinski (1982), G¨artner (1993), Karakitsos (1992), McCafferty (1990), Pilbeam (1998) and Shone (1989).

CHAPTER 13

Open economy dynamics: ﬂexible price models

Since the advent of generalised ﬂoating in 1973 there have been a number of exchange rate models, most of which are dynamic. In this chapter we shall extend our discussion of the open economy to such models. Besides having the characteristic of a ﬂexible exchange rate they also have the essential feature that the price level is also ﬂexible, at least in the long run. This is in marked contrast to chapter 12 in which the price level was ﬁxed. The models are often referred to, therefore, as ﬁx-price models and ﬂex-price models, respectively. The majority of the ﬂex-price models begin with the model presented by Dornbusch (1976). Although the model emphasised overshooting, what it did do was provide an alternative modelling procedure from the Mundell–Fleming model that had dominated international macroeconomic discourse for many years. It must be stressed, however, that the model and its variants are very monetarist in nature. Although the Mundell–Fleming model assumed prices ﬁxed, which some saw as totally inappropriate, the models in the present chapter assume full employment, and hence a constant level of real income. This too may seem quite inappropriate. Looked at from a modelling perspective, it allows us to concentrate on the relationship between the price level and the exchange rate. Of particular importance, therefore, in such models is purchasing power parity. It does, of course, keep the analysis to just two main variables. Purchasing power parity (PPP) indicates that prices in one country are equal to those in another after translating through the exchange market. There is a vast literature on this topic that we shall not go into here. All we shall do is stipulate that this is supposed to hold at the aggregate level. Hence, if P is the price level at home, P∗ the price level abroad, and S the exchange rate (quoted in terms of domestic currency), then SP∗ = P Taking natural logarithms, and setting the foreign price to unity (i.e. P∗ = 1), which throughout is held constant, then ln S − ln P∗ = ln P i.e.

s=p

since ln P∗ = ln 1 = 0

where lower case letters denote natural logarithms. A rise in S (or s) is an

554

Economic Dynamics appreciation of the foreign currency, i.e., a depreciation of the domestic currency.1 For purchasing power parity to hold, therefore, we require s = p. So long as s differs from p, then purchasing power parity does not hold.2 It is the purchasing power parity condition that drives the long-run result in the models to be discussed in this chapter. In other words, in the short run it is possible for the economy to deviate from purchasing power parity but in the long run purchasing power parity must hold.3 One of the essential differences between the present models and those of chapter 12 is that they are presented in terms of natural logarithms. Accordingly we shall denote all variables in natural logarithms with lower case letters. The exception is interest rates. These are percentages and the home interest rate will be denoted r and the foreign rate r∗ , as in chapter 12. In section 13.1 we consider a simpliﬁed Dornbusch model in which the goods market is independent of the rate of interest. This captures most of the characteristics of the original Dornbusch model but is easier to follow. In section 13.2 we consider Dornbusch’s (1976) model. Both these models assume perfect capital mobility. In section 13.3 we consider what happens when capital is immobile (but not perfectly immobile). Next we consider the Dornbusch model under the assumption of perfect foresight, which gives a rational expectations solution (section 13.4). One of the main features of rational expectations modelling is the possibility of considering the impacts of government announcements. This topic we consider in section 13.5. The discovery of gas and then oil in the North Sea led to major impacts on the exchange rate, which in turn inﬂuenced adversely the non-oil sector. Section 13.6 presents a popular model for considering any resource discovery and its impact on the exchange rate. The ﬁnal section 13.7 considers the dynamics of a simple monetarist model. Throughout we concentrate on the economic dynamics, illustrating this with many numerical examples.

13.1 A simpliﬁed Dornbusch model4 We begin with a simpliﬁed Dornbusch model that captures nearly all the features of the original but is more manageable. We can then go on to further complications once this is fully understood. All the Dornbusch models begin with three markets. There is the goods market, the money market and the foreign exchange market (or the balance of payments). The goods market reduces down to two simple relationships, a total expenditure equation and a price adjustment equation, where income is assumed constant at the full employed level. The money market is a 1

2 3

4

The reader needs to be vigilant concerning which currency is appreciating and which depreciating. Since S (or s) is the price of overseas currency in terms of domestic currency (the European convention of quoting exchange rates, other than the UK), then a rise in the price is an appreciation of the foreign currency. However, most discussion takes place in terms of the price of domestic currency. In terms of the analysis of chapter 12 purchasing power parity requires the real exchange rate, R, to equal unity. There is something wholly unsatisfactory in this modelling. Although in the short run deviation from PPP is possible, but not in the long run, income cannot deviate from its full employment level either in the short run or in the long run, which is quite unrealistic. Based on a model presented in G¨artner (1993).

Open economy dynamics: ﬂexible price models

555

Table 13.1 Model 13.1 Goods market e = cy + g + h(s − p) p˙ = a(e − y)

Money market md = p + ky − ur ms = md = m

0 < c < 1, h > 0 a>0

k > 0, u > 0

International asset market r = r∗ + s˙e s˙e = v(s − s) v > 0

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = domestic price level p˙ = inﬂation rate (since p = ln P) md = demand for money r = domestic interest rate ms = supply of money m = exogenous money balances r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s = purchasing power parity rate (equilibrium rate)

straightforward demand for money and a constant level of real money balances. The international asset market varies considerably from one model to another. Here we assume perfect capital mobility and therefore the domestic interest rate is equal to the foreign interest rate adjusted for any expected change in the exchange rate. The expected change in the exchange rate, in turn, depends on the extent of the deviation of the exchange rate from its purchasing power parity level. The model, then, is captured by the set of equations in table 13.1. The model can be captured diagrammatically by deriving two equilibrium lines in (s,p)-space. A goods market equilibrium line, which denotes combinations of p and s for which the price level is not changing, i.e., p˙ = 0, which we shall denote GM; and an asset market line which denotes combinations of p and s which maintains equilibrium in the money market and satisﬁes the condition on the expected change in the exchange rate, which we shall denote AM. Substituting the expenditure function into the price adjustment relation p˙ = a(e − y), and setting p˙ equal to zero, gives the following relationship between the price level and the exchange rate p=s−

(1 − c)y g + h h

Equation (13.1) is a positive relationship between p and s with a slope of unity. If we impose the condition of purchasing power parity, which we shall do, then in the long run p = s, and so the intercept of the GM line must be zero. Hence, in ﬁgure 13.1 we have drawn the GM line through the origin with a slope of unity. Furthermore, p˙ > 0 if e > y, i.e. p 0, h > 0 p˙ = a(e − y) a>0

Money market md = p + ky − ur ms = md = m

k > 0, u > 0

International asset market r = r∗ + s˙e s˙e = v(s − s) v > 0

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = domestic price level p˙ = inﬂation rate (since p = ln P) md = demand for money r = domestic interest rate ms = supply of money m = exogenous money balances r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s = purchasing power parity rate (equilibrium rate)

different, this does add a signiﬁcant complication. The change is to the expenditure function, which now assumes that investment (a component of expenditure) is inversely related to the rate of interest; hence a component −dr (d > 0) is added to the expenditure function. This has the immediate implication that the goods market and the asset market are interdependent, and this interdependence arises through the rate of interest. The asset market line remains unaffected and therefore can be expressed as before, i.e., the AM line is (13.6)

(13.7)

m = p + ky − u[r∗ + v(s − s)] i.e. p = (m − ky + ur∗ + uvs) − uvs However, the goods market line, GM, now takes the form * $ g + (dm/u) hs (1 − c) + (dk/u) y+ + p= − h + (d/u) h + (d/u) h + (d/u) Notice in particular that the slope of the GM line (where p is on the vertical axis and s on the horizontal axis) is now

(13.8)

slope GM =

1 h = 0, b > 0

(13.9)

Equation (13.9) says no more than the balance of payments is the sum of the current account and the net capital ﬂow. The current account element is the same as that in the expenditure function,5 while the second element denotes net capital ﬂows which is responding to the difference between the two interest rates, adjusted for any expected change in the exchange rate. Perfect capital mobility implies b = ∞, while a value of b close to zero implies more extreme capital immobility. We need to make one further observation concerning this equation. Given a perfectly ﬂoating exchange rate, then the balance of payments is always in balance and so bp = 0. We retain the assumption about the expected change in the exchange rate, namely that it adjusts to the difference between the purchasing power parity level and the actual level. Since nothing else in the model is different, then there is no change in the goods market line. Only the speciﬁcation of the asset market line is changed. Consider example 13.1 again, which excludes any interest rate impact on the goods market, and so the GM line is the same as the purchasing power parity line, and is a 45◦ -line through the origin. The model is set out in detail below in table 13.3. 5

G¨artner (1993) has a different coefﬁcient on (s − p) in the expenditure function and the balance of payments equation. There is no real need for this. Both arise from net exports, which occurs identically in both equations.

Open economy dynamics: ﬂexible price models

565

Table 13.3 Model 13.3 Goods market e = cy + g + h(s − p) p˙ = a(e − y)

Money market md = p + ky − ur ms = md = m

0 < c < 1, h > 0 a>0

k > 0, u > 0

International asset market bp = h(s − p) + b(r − r∗ − s˙e ) h > 0, b > 0 s˙e = v(s − s) v > 0

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = domestic price level p˙ = inﬂation rate (since p = ln P) md = demand for money r = domestic interest rate ms = supply of money m = exogenous money balances bp = balance of payments r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s = purchasing power parity rate (equilibrium rate)

Figure 13.8.

As we have just indicated, the essential change is to the asset market line. This now takes the form uvs [uv + (uh/b)]s m − ky + ur∗ + − p= 1 − (uh/b) 1 − (uh/b) 1 − (uh/b) Notice that this is consistent with model 13.1. If b → ∞ then this equation reduces to the asset market equation of section 13.1. Of particular importance in this model is the slope of the asset market line, which is uv + (uh/b) slope of AM = − 1 − (uh/b) A very high value of b, a high degree of capital mobility, will mean the typical negatively sloped asset market line, with analysis identical to that in section 13.1. However, with a very low degree of capital mobility, a value of b close to zero, can mean a positively sloped asset market line. The situation is illustrated in ﬁgure 13.8.

(13.10)

(13.11)

566

Economic Dynamics A rise in the money supply will shift the asset market line to the right, from AM0 to AM1 , and the equilibrium will move from E0 to E1 on the goods market line, which coincides with the purchasing power parity condition. The movement of the economy now, however, is quite different. Although the trajectory is still along the new asset market line, with sticky prices initially the economy moves to point C on AM1 . Since again there is excess expenditure over income, prices will rise. The rise in the price level, although reducing real money balances and raising the rate of interest at home, will have only a small effect on capital inﬂows. In order, therefore, to maintain balance of payments equilibrium the domestic currency must also depreciate (s must rise). Hence, the economy moves along AM1 from point C to point E1 . In this version of the model, therefore, the exchange rate undershoots its long-run equilibrium level. There is initially a rapid depreciation of the domestic currency (a movement from point E0 to point C), followed by a further gradual depreciation in response to the price rise (a movement from point C to point E1 ). In this version of the model the rate of interest both before and after the change in the money supply will equal the interest rate abroad. Since purchasing power parity implies s = p, and since in long-run equilibrium s˙e = 0, then it follows that r = r∗ in long-run equilibrium. Once again price movements and exchange rate movements can be expressed by the equations p(t) = p + ( p0 − p)e−λt

(13.12)

s(t) = s + (s0 − s)e−λt but now

(13.13)

1 − (uh/b) +1 λ = ah uv + (uh/b)

Notice that for b → ∞ this reduces to the adjustment coefﬁcient of model 13.1 (see exercise 5 for a numerical example illustrating this model). The analysis is very similar in the original Dornbusch model, but with capital immobility. This model, model 13.4, is presented in table 13.4. Table 13.4 Model 13.4 Goods market e = cy − dr + g + h(s − p) 0 < c < 1, d > 0, h > 0 p˙ = a(e − y) a>0

Money market md = p + ky − ur ms = md = m

k > 0, u > 0

International asset market bp = h(s − p) + b(r − r∗ − s˙e ) h > 0, b > 0 v>0 s˙e = v(s − s)

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = domestic price level p˙ = inﬂation rate (since p = ln P) md = demand for money r = domestic interest rate ms = supply of money m = exogenous money balances bp = balance of payments r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s =purchasing power parity rate (equilibrium rate)

Open economy dynamics: ﬂexible price models

567 Figure 13.9.

The situation is illustrated in ﬁgure 13.9. A rise in the money supply shifts both the asset market line and the goods market line. But because capital is very immobile, the exchange rate initially undershoots its long-run equilibrium value, s2 < s1 . In this model we have (see exercise 6) (h + dv)(1 − (uh/b)) + h (p − p) p˙ = −a uv + (uh/b) so prices and the exchange rate have the same adjustment coefﬁcient (h + dv)(1 − (uh/b)) +h λ=a uv + (uh/b) This is consistent with all our previous results. If d → 0 the model reduces to model 13.3; if b → ∞ the model reduces to model 13.2; and if d → 0 and b → ∞ the model reduces to model 13.1.

13.4 The Dornbusch model under perfect foresight One of the advantages of the Dornbusch model is that it readily lends itself to different speciﬁcations of exchange rate expectations. One such speciﬁcation is perfect foresight. This model has a number of formal advantages. It can be shown that rational expectations is formally the same as expectations under perfect foresight, and since it is easier to handle models under the assumption of perfect foresight, then all the features of modelling rational expectations can be captured by this version. Second, the assumption that s˙e = v(s − s) with v > 0, is the same as the assumption of perfect foresight – so long as v is correctly chosen (see exercise 7). Again we shall begin with the simpliﬁed Dornbusch model in which expenditure is independent of the rate of interest. The model is captured in model 13.5, and set out in table 13.5, where we have replaced the assumption about exchange

(13.14)

(13.15)

568

Economic Dynamics Table 13.5 Model 13.5 Goods market e = cy + g + h(s − p) p˙ = a(e − y)

Money market md = p + ky − ur ms = md = m

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = domestic price level p˙ = inﬂation rate (since p = ln P) md = demand for money r = domestic interest rate ms = supply of money m = exogenous money balances r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s˙ = change in spot exchange rate

0 < c < 1, h > 0 a>0

k > 0, u > 0

International asset market r = r∗ + s˙e s˙e = s˙

rate expectations of model 13.1 with that of perfect foresight. This is the only difference from model 13.1, but it will be seen that it has signiﬁcant implications for the dynamic behaviour of prices and exchange rates. Since the formal algebraic manipulations are the same in deriving the goods market line and the asset market line, we shall be brief. There has been no change to the goods market, this remains the same as model 13.1, and under the assumption of purchasing power parity is a 45◦ -line through the origin. The dynamics of the goods market is still speciﬁed by the relationship (13.16)

p˙ = a[h(s − p) − (1 − c)y + g]

a > 0, h > 0, 0 < c < 1

The major change is in the foreign exchange market. Substituting the perfect foresight assumption into the interest rate condition, which retains the assumption of perfect capital mobility, and substituting this into the money market equilibrium, we obtain (13.17)

(13.18)

m = p + ky − u(r∗ + s˙) 1 ... s˙ = (p + ky − m) − r∗ u We therefore have the following dynamic system p˙ = a[g − (1 − c)y] − ahp + ahs 1 1 ∗ (ky − m) − r + p s˙ = u u which is a differential equation system which can be solved for the two variables p(t) and s(t). The critical point, stationary point, or equilibrium point of the system is where p˙ = 0 and s˙ = 0. Consider the second condition ﬁrst. We immediately have no change in the exchange rate if

(13.19)

p = p = m − ky + ur∗ which is a horizontal line in the phase plane, as shown in ﬁgure 13.10. Turning to the goods market, p˙ = 0 implies a straight line through the origin with slope 45◦ , as shown in ﬁgure 13.10.

Open economy dynamics: ﬂexible price models

569 Figure 13.10.

We are now in a position to consider the model’s dynamics. Consider points either side of the s˙ = 0 line and p˙ = 0 line. s˙ > 0 if p > m − ky + ur∗ s˙ < 0 if p < m − ky + ur∗ and p˙ > 0 if p < s + (g − (1 − c)y)/h p˙ < 0 if p > s + (g − (1 − c)y)/h In other words, below the horizontal line the exchange rate is falling (the domestic currency is appreciating), while above the horizontal line the exchange rate is rising (the domestic currency is depreciating). On the other hand, below the goods market line prices are rising while above it prices are falling, consistent with our earlier analysis. The vector forces in ﬁgure 13.10 illustrate all this information. What is quite clear from ﬁgure 13.10 is that we have a saddle point equilibrium. This can be established as follows. Consider the system in terms of deviations from equilibrium, which is particularly useful. p˙ = a[h(s − p) − (1 − c)y + g] 0 = a[h(s − p) − (1 − c)y + g] ... p˙ = −ah( p − p) + ah(s − s) and 1 ( p + ky − m) − r∗ u 1 0 = ( p + ky − m) − r∗ u .. . s˙ = 1 (p − p) u s˙ =

570

(13.20)

Economic Dynamics Hence, the system can be written in matrix form as follows. p˙ −ah ah p − p = s−s s˙ 1/u 0 Letting A denote the matrix of the system, then we immediately have −ah −ah ah 0) to the expenditure equation, where xp denotes the permanent income stream from the new wealth and f is a positive coefﬁcient. Thus, our expenditure equation now takes the form (13.25)

e = cy + g + h(s − p) + f xp On the other hand, x is the current income from oil that adds additional demand for money balances, which is captured by a term j x( j > 0) in the demand for money equation. But we need to make an additional change to the demand for money equation. To understand this, return to prices and exchange rates in unlogged form. We assume that the domestic price level (the RPI) is a weighted average of domestically produced goods, P, and imported goods, Pf = SP∗ , i.e.

(13.26)

Q = Pα (SP∗ )1−α taking natural logarithms and denoting these by lower case letters, then

(13.27)

q = αp + (1 − α)(s + p∗ ) But if we set P∗ = 1, then p∗ = 0, hence

(13.28)

q = αp + (1 − α)s

Open economy dynamics: ﬂexible price models

583

Table 13.8 Model 13.7 Goods market e = cy + g + h(s − p) + f xp 0 < c < 1, h > 0, f > 0 p˙ = a(e − y) Money market md = q + ky − ur + j x k > 0, u > 0, j > 0 q = αp + (1 − α)s 0 < α < 1

International asset market r = r∗ + s˙e s˙e = s˙

e = total expenditure y = real income (exogenous) g = government spending s = spot exchange rate p = price of domestic goods p˙ = inﬂation rate (since p = ln P) md = demand for money ms = supply of money r = domestic interest rate m = exogenous money balances q = domestic price level α =weight of domestic goods in q r∗ = interest rate abroad s˙e = change in expected spot rate (expected depreciation/appreciation) s˙ = change in spot exchange rate

Finally, we deﬂate money balances by the domestic price level Q. Thus the demand for money equation becomes md = q + ky − ur + j x

(13.29)

where q = αp + (1 − α)s. The complete model (model 13.7), under the assumption of perfect foresight and no interest rate effect on expenditure, is given in table 13.8. Carrying out the same manipulations as for model 13.5 we obtain p˙ = a[g − (1 − c)y + f xp ] − ahp + ahs α 1−α ky + j x − m − r∗ p+ s˙ = s+ u u u

(13.30)

In equilibrium p˙ = 0 and s˙ = 0, hence 0 = a[g − (1 − c)y + f xp ] − ahp + ahs α 1−α ky + j x − m − r∗ 0= p+ s+ u u u So taking deviations from the equilibrium we have p˙ = −ah(p − p) + ah(s − s) α 1−α ( p − p) + s˙ = (s − s) u u Or in matrix notation −ah ah p−p p˙ = 1−α α s−s s˙ u u Hence, the matrix of this system is −ah ah A= α 1−α u

u

(13.31)

(13.32)

584

Economic Dynamics

Figure 13.22.

(13.33)

(13.34)

with det(A) = −ah/u < 0. Since det(A) is negative the equilibrium point is a saddle point. From the conditions p˙ = 0 and s˙ = 0 given above, we can solve for p and s using Cramer’s rule. These are (1 − c)(1 − α) (1 − α)g f (1 − α)xp +k y+ + + ur∗ − jx p=m− h h h α(1 − c) αg α f xp −k y− + + ur∗ − jx s=m+ h h h It is apparent, therefore, that the discovery of a major resource leading to terms xp and x will inﬂuence the equilibrium price and exchange rate.10 To see this more clearly we need to consider the model in more detail. To do this we need to consider the equilibrium lines associated with p˙ = 0 and s˙ = 0. With some algebraic manipulation these are g − (1 − c)y + f xp p=s+ for p˙ = 0 h m − ky − jx + ur∗ 1−α p= − s for s˙ = 0 α α Consequently, the goods market line, the line associated with p˙ = 0, is a 45◦ -line. The second equilibrium line, that associated with s˙ = 0, is negatively sloped. Initially we assume that purchasing power parity is satisﬁed. This is best considered as the situation before any resource is discovered. Hence, the goods market line 10

These results are consistent with those of model 13.5. If α = 1 and x = xp = 0, then we have the same equilibrium results as model 13.5.

Open economy dynamics: ﬂexible price models

585 Figure 13.23.

passes through the origin, as shown in ﬁgure 13.22. This ﬁgure also shows the vector forces and the saddle paths associated with the equilibrium point E, and denoted SP1 and SP2 . Note in particular that SP2 denotes the asset market equilibrium. In this model, as in previous models, we assume asset markets are always clearing while the goods market takes time. Now consider the discovery of a natural resource, such as oil, as shown in ﬁgure 13.23. The change in xp (from zero to some positive amount) shifts the p˙ = 0 line up (i.e. leads to a rise in the intercept) to p˙1 = 0. But this will generate an income stream and so raise x (from zero to some positive amount). This in turn will shift the s˙ = 0 line left (i.e. will reduce the intercept) to s˙1 = 0. The economy will move from equilibrium point E0 to equilibrium point E1 . But what trajectory will such an economy take? Initially prices do not change, and so the economy moves horizontally from point E0 to point C, point C being on the new saddle path SP21 through E1 . The domestic currency has accordingly appreciated taking the full impact of the adjustment in the short run. Point C is on the new asset market line. The resulting increase in permanent income raises consumers’ expenditure. Since we have full employment this results in excess demand and hence to a rise in the price level. Accordingly the economy moves up the new asset market line from point C to point E1 . In this example there is no overshooting. This occurs, however, only where the goods market impact is greater than the money market impact. To see this consider ﬁgure 13.24, which shows the situation where the money market impact exceeds that of the goods market. Again the economy moves horizontally from point E0 to point C on SP21 , but then moves down the new asset market line until point E1 is reached. This is because the signiﬁcant effect of the current income stream on the demand for money leads to a signiﬁcant rise in the rate of interest. In order to maintain the condition r = r∗ + s˙e , the domestic

586

Economic Dynamics

Figure 13.24.

currency must depreciate, moving the economy along the path C to E1 on SP21 . Although prices gradually fall, the home currency ﬁrst appreciates, overshooting its long-run equilibrium value, and then depreciates but leading to an eventual appreciation of the exchange rate. The difference in the two results is determined by the rate of resource depletion. In ﬁgure 13.23 the rate of depletion is slow and so the permanent income stream outweighs the current income from the oil extraction. Figure 13.24, however, indicates that resource depletion is quick and there is a relatively large current income from resource sales. In either case, the discovery of a resource, although having ambiguous results on the price level, does lead to an appreciation of the home currency, with the possibility of overshooting the quicker the resource depletion. Given the discovery of North Sea Oil, we may hypothesise that market participants with perfect foresight would know that the domestic currency would appreciate in the long run and would act accordingly. The situation is similar to the analysis in the previous section, and the result is shown in ﬁgure 13.25. Because of the anticipated appreciation, the economy moves to point F. We can think of this as the situation the moment the discovery is made. The economy then moves along the trajectory F to G, which is governed by the unstable arm SP11 , and where point G is determined by the point in time that the oil comes on-stream. The economy then moves along the asset market line, SP21 , from point G to point E1 .

13.7 The monetarist model An early ﬂex-price model of exchange rate determination was the simple monetarist model. The model is set out in table 13.9. All variables are in natural logarithms except for interest rates.11 11

Since s is ln S, then s˙ is the percentage change in the exchange rate.

Open economy dynamics: ﬂexible price models

587 Figure 13.25.

Table 13.9 Monetarist model m − p = ky − ur r = r∗ + s˙e p = s + p∗ s˙e = s˙

k > 0, u > 0

m = nominal money supply y = real income r = nominal interest rate at home r∗ = nominal interest rate abroad p = domestic price level p∗ = foreign price level s = exchange rate

The ﬁrst equation is no more than real money balances is equal to real demand for money balances, where we assume a simple demand for money equation. The second equation is the interest parity condition under perfect capital mobility, while the third equation is purchasing power parity. The ﬁnal equation is rational expectations under perfect foresight. Real income is assumed constant at the natural level. Also m, r∗ and p∗ are assumed constant. Substituting, we have m − s − p∗ = ky − u(r∗ + s˙e ) or s˙ =

1 k 1 y − r∗ − (m − p∗ ) + s u u u

which is a ﬁrst-order differential equation. The dynamics of the model are illustrated in ﬁgure 13.26. Since m, y, r∗ and p∗ are constant, then so is k 1 y − r∗ − (m − p∗ ) u u which is the intercept on the vertical axis. The slope is 1/u. We have labelled the line A to denote asset market. The ﬁxed point is readily established by setting

(13.35)

588

Economic Dynamics

Figure 13.26.

Figure 13.27.

s˙ = 0, hence 0=

k 1 1 s y − r∗ − (m − p∗ ) + u u u

or (13.36)

s = (m − p∗ ) + ur∗ − ky Since the slope of the asset market line is positive, the system is dynamically unstable. Also we have a linear differential equation, so the system is globally unstable. To solve differential equation (13.35) we normally require an initial condition, say s(0) = s0 . But any s0 < s leads the system to a continual appreciation of the home currency (fall in s); while for s0 > s the home currency continually

Open economy dynamics: ﬂexible price models depreciates (s rises). Since in this rational expectations model market participants have perfect foresight, then s jumps immediately to s. The effect of a rise in nominal money balances is shown in ﬁgure 13.27. The asset market line shifts down (by (1/u)m) and the new equilibrium exchange rate increases to s2 . From equation (13.36) it immediately follows that s = m, i.e., the domestic currency depreciates by exactly the same percentage as the rise in nominal money balances. Under perfect foresight, expected depreciation and actual depreciation are identical and the system immediately jumps from s1 to s2 .

Exercises 1.

For the model outlined in table 13.1 we have the result p(t) = p + (p0 − p)e−λt (i) Show that s(t) = s + (s0 − s)e−λt where s0 is the initial exchange rate after the shock, but associated with the price level p0 . (ii) In terms of example 13.1, show that point C is represented by (s,p) = (155,100) and that s(t) = 105 + 50e−0.011t (iii) Plot on the same graph p(t) and s(t) for λ = 0.011 and λ = 0.02.

2.

3.

Given the model in section 13.1 (example 13.1), establish the comparative static and dynamics of a rise in the foreign interest rate from r∗ = 10 to r∗ = 12. For the Dornbusch model given in table 13.2 (model 13.2), show that (i) s − s = −

1 (p − p) uv

(ii) hence show d h + ( p − p) p˙ = −a h + uv u and

d h + λ=a h+ uv u 4.

Consider the following discrete version of the Dornbusch model of table 13.2 et = 0.8y − 0.1rt + 5 + 0.01(st − pt ) pt+1 − pt = 0.2(et − y) mdt = pt + 0.5y − 0.25rt mdt = mst = 105 rt = r∗ + set+1 − set

589

590

Economic Dynamics set+1 − set = 0.25(s − st ) y = 20

r∗ = 10

(i) Derive an expression for the GM line and the AM line of the form pt = φ(st ) and establish the ﬁxed point of the model. (ii) Set up the model on a spreadsheet and establish its ﬁxed point, starting from the initial value pt = 100. (iii) Let ms rise from 105 to 110 establish the new equilibrium and demonstrate that dmt = dst = dpt . 5.

Given the numerical model based on table 13.3 e = 0.8y + 4 + 0.01(s − p) p˙ = 0.2(e − y) md = p + 0.5y − 0.5r md = ms = 105 bp = 0.01(s − p) + b(r − r∗ − s˙e ) s˙e = 0.2(s − s) y = 20

r∗ = 10

(i) If b = 0.0045 establish that the initial equilibrium is (s,p) = (100,100). (ii) For a rise in the money supply to ms = 110, establish that point C is represented by (s,p) = (104.54128, 100). (iii) Conﬁrm that the new equilibrium satisﬁes dp = ds = dm. 6.

For the model in table 13.4 (i) Show that (h + dv)(1 − (uh/b)) + h ( p − p) p˙ = −a uv + (uh/b) and

(h + dv)(1 − (uh/b)) +h λ=a uv + (uh/b)

7.

(ii) If d → 0 then p˙ and λ reduce to the values of model 13.3 (table 13.3). (iii) If b → ∞ then p˙ and λ reduce to the values of model 13.2 (table 13.2). (iv) If d → 0 and b → ∞ then p˙ and λ reduce to the values of model 13.1 (table 13.1). A numerical version of model 13.4 (table 13.4) is e = 0.8y − 0.1r + 5 + 0.01(s − p) p˙ = 0.1(e − y) md = p + 0.5y − 0.5r md = ms = 105 bp = 0.01(s − p) + 0.004(r − r∗ − s˙e )

Open economy dynamics: ﬂexible price models s˙e = 0.2(s − s) y = 20

r∗ = 10

(i) Establish the following at the initial equilibrium. (a) Equilibrium is (s,p)= (100,100). p = 95.238095 + 0.0476190s (b) GM0 : p = −440 + 5.4s (c) AM0 : (ii) Let ms rise to 110. Find point C on AM1 and establish that for the new equilibrium dm = ds = dp. 8.

9.

Using the model in example 13.3, and assuming r∗ = 18, all other parameters the same, then establish that (i) the initial equilibrium is (s,p) = (104,104) (ii) the characteristic roots are r = 0.04422415 and s = −0.0452242 (iii) the saddle path equations are: (a) unstable arm: p = 101.70034 + 0.02211208s (b) stable arm: p = 106.35166 − 0.0226121s (iv) for a rise in the money supply to 110 the intercepts of the saddle paths only alter to 106.58978 for the unstable arm and to 111.46472 for the stable arm, respectively. In the model outlined in table 13.8 suppose we have the following numerical version of the model e = 0.8y + 4 + 0.01(s − p) + 2xp p˙ = 0.1(e − y) m = q + 0.5y − 0.5r + x q = αp + (1 − α)s md = ms = 105 r = r∗ + s˙e s˙e = s˙ y = 20 r∗ = 10 (i) If initially xp = x = 0 and α = 0.8, show that the initial equilibrium, E0 , is given by (s, p) = (100, 100). (ii) Show that the stable and unstable arms of the saddle point E0 are: stable arm

p = 125.31 − 0.253s

unstable arm p = 99.7531 + 0.002469s (iii) Now assume a resource discovery which leads to xp = 0.5 and x = 0.3. With α = 0.8, (a) show that equilibrium (s, p) = (19.7, 119.7) (b) the unstable arm is given by p = 119.65135 + 0.00246943s (c) the stable arm is given by p = 124.68608 − 0.2531005s

591

592

Economic Dynamics 10.

Use the model surrounding ﬁgure 13.25 to analyse the UK’s position in 1979 when the Conservative government under Mrs Thatcher took ofﬁce. The basic information at the time was as follows. (a) Oil had been discovered in the North Sea, was being drilled around 1975 and was known to come on-stream in 1979. (b) The Conservatives won the General Election in 1979 with Mrs Thatcher indicating: – removal of all UK exchange controls; and – a reduction in monetary growth to combat inﬂation.

11.

Take as your starting date 1975 when oil was being drilled. For the monetarist model in table 13.8 let y = 20,

12.

m = 106,

p∗ = 1,

k = 0.5,

u = 0.5,

r∗ = 10

(i) Derive the differential equation for this model. (ii) Solve for equilibrium s. Suppose nominal money supply grows at a constant rate λ and inﬂation abroad is constant at π ∗ , i.e. m˙ = λ

and

p˙ ∗ = π ∗

Derive an expression for equilibrium s under the assumption that a stationary equilibrium is one in which real money balances are constant. Additional reading Additional material on the contents of this chapter can be obtained from Buiter and Miller (1981), Copeland (2000), Dornbusch (1976), Dernburg (1989), Ford (1990), Frenkel and Rodriguez (1982), G¨artner (1993), MacDonald (1988), Niehans (1984), Obstfeld and Rogoff (1999), Pilbeam (1998), Rødseth (2000), Shone (1989, 2001).

CHAPTER 14

Population models

14.1 Malthusian population growth Population growth is frequently considered by means of differential equations, where the growth can be of persons, animal species, or bacteria. Although the increase in population is discontinuous, if the population is very large, then the additions to its size will be very small and so it can be considered as changing continuously. Hence, we assume population size, p, changes continuously over time and that p(t) is differentiable. The simplest population growth model is to assume that population grows/declines at a constant rate. Thus dp 1 =k dt p this means that the change in the population is proportional to the size of the population dp = kp dt where k is positive for a growth in the population and negative for a decline. The initial condition is that if at time t0 the population is p0 then p(t0 ) = p0 Although (14.1) is a simple equation to solve, let us investigate its qualitative properties by means of phase-space. For positive k the growth curve is linear, positively sloped, and passes through the origin, as shown in ﬁgure 14.1. It is clear, then, that the only equilibrium for this population is a population of zero, since this is the only value of p for which dp/dt = 0. Furthermore, for any population of size greater than zero, e.g., p0 , then dp/dt is positive and so population will be increasing over time. In other words, the arrows along the phase line indicate a continuously growing population. If, on the other hand, k is negative then equilibrium population size is still zero, but now for any population greater than zero means dp/dt is negative and so population will decrease over time until it is extinguished. Although not wholly realistic, let us solve for the population size explicitly. To do this integrate both sides of the differential equation

(14.1)

594

Economic Dynamics

Figure 14.1.

dp = kp dt dp = k dt dt ln p = kt + c p = c0 ekt where c is the constant of integration. Applying the initial condition p = p0 for t = t0 we have p = p0 ekt0 implying c0 = p0 e−kt0 Which leads to the result p = p0 e−kt0 ekt = p0 ek(t−t0 )

(14.2)

and which clearly satisﬁes the initial condition. In this model, population grows/declines exponentially, and is referred to as the Malthusian model of population growth. Of interest in rapidly growing populations is the time necessary for the population to double in size.1 It is readily shown that for the Malthusian model this 1

Biologists refer to this as the mean generation time, i.e., the time necessary for a population to reproduce itself.

Population models period depends only on the rate of growth, k. To show this let the population be p0 initially at time t0 . Let the time period when the population has doubled be denoted t1 . Then the length of time for the population to double is t1 − t0 . Furthermore, p1 = 2p0 , hence 2p0 = p0 ek(t1 −t0 ) ln 2 = k(t1 − t0 ) 0.6931 ln 2 = ... t1 − t0 = k k For example, if a population is growing at 2% per annum, then it will double approximately every 0.6931/0.02 = 35 years regardless of the initial population size. Example 14.1 Table 14.1 gives the population of the UK from 1781 to 1931. Our ﬁrst problem is to estimate the parameter k. Suppose we set p0 = 13 million for the initial year 1781. Further, take the population in year 1791 to be as in the table, namely 14.5 million. This allows us to estimate the value of k. Letting t0 = 0 to represent 1781, then t1 = 10 for 1791, i.e., t1 − t0 = 10 p(0) = p0 = 13 p(10) = p0 e10k = 13e10k = 14.5 2.6741 − 2.5649 ln 14.5 − ln 13 = k= 10 10 k = 0.01092 Using this estimate of k we compute the Malthusian estimate of population growth, as shown in column (3) of table 14.1.

Table 14.1 UK Population, 1781–1931 (million) Year

Actual

Malthusian

Logistic

1781 1791 1801 1811 1821 1831 1841 1851 1861 1871 1881 1891 1901 1911 1921 1931

13.000 14.500 15.902 18.103 21.007 24.135 26.751 27.393 28.977 31.556 34.934 37.802 41.538 45.299 47.168 49.007

13.000 14.500 16.173 18.039 20.121 22.442 25.032 27.920 31.142 34.735 38.743 43.213 48.200 53.761 59.964 66.883

13.000 14.996 17.143 19.410 21.756 24.135 26.498 28.799 30.993 33.046 34.934 36.641 38.162 39.500 40.662 41.662

Source: Deane and Cole (1962, table 3, p. 8).

595

596

(14.3)

Economic Dynamics A discrete version of the model may appear more appropriate. This takes the form pt+1 = kpt i.e. pt+1 = pt + kpt = (1 + k)pt Using the analysis of chapter 3, we have the general solution

(14.4)

pt = (1 + k)t p0 Again, using p0 = 13 and p10 = (1 + k)10 (13) = 14.5, we obtain k=

14.5 13

101

− 1 = 0.0109798

Using this estimate of k, and the discrete solution, we compute an alternative series based on the Malthusian assumption. However, it is readily established that this gives exactly the same ﬁgures (to three places of decimal) as the continuous model. The model is reasonably accurate up to 1851 but thereafter the error becomes not only quite large but increasing. This should not be surprising. In the ﬁrst instance, k was estimated from the ﬁrst two observations. Second, the population increases at an ever-increasing rate, which is unrealistic. Third, for distant population there is no account taken of competition of the population for the limited resources available. It may be thought that the model is inappropriate because it does not take account of births and deaths. But this is not strictly true. If births are assumed to follow the Malthusian law as well as deaths, i.e., both grow at constant rates b and d, respectively, then

(14.5)

dpd dpb = p0 ebt = p0 edt and dt dt dpb dpd dp = − = p0 e(b−d)t = p0 ekt dt dt dt Hence, the k we estimated using data from 1781 and 1791 would account for both births and deaths. This means that the problem lies elsewhere. Although we have considered births and deaths we have taken no account of immigration or emigration. Migration (immigration minus emigration), however, is usually fairly small relative to the total size of the population, or occurs only at speciﬁc times (most especially in human populations). This would suggest, therefore, that the exponential growth curve might not be the most appropriate speciﬁcation of the growth process.

14.2 The logistic curve An alternative approach is to assume that not only does population grow with population size, but that as it grows its members come into competition with each other for the food or limited resources. In order to capture this ‘competition’ it is assumed that there are p( p − 1)/2 interactions for a given population of size p. Assuming such interactions lead to additional deaths, for example because of disease or war, then we can assume that the growth in the population will also

Population models

597

diminish in proportion to this element of interaction. In other words, population now changes by k1 p( p − 1) dp = kp − dt 2 k1 p k1 p2 − = kp + 2 2 k1 p2 k1 = k+ p− 2 2 Therefore dp = ap − bp2 = p(a − bp) a > 0, b > 0 dt which is referred to as the logistic growth equation. In general the parameter b is small relative to the parameter a, so that the second term is often negligible. However, as the population size grows and competition becomes greater, the second term −bp2 becomes more signiﬁcant. This is especially true as time moves further away from the initial level. As the second term becomes more signiﬁcant, this dampens the growth in the population. This second formulation is referred to as the logistic law of growth. Before solving for population explicitly, let us investigate the qualitative properties of the population by considering the phase-space. The logistic growth equation

(14.6)

p˙ = p(a − bp) a > 0, b > 0 is an autonomous ﬁrst-order differential equation. The qualitative properties of this equation are shown in the phase diagram in ﬁgure 14.2. Figure 14.2.

598

Economic Dynamics The equilibrium population is where p˙ = 0, i.e., zero population growth, which occurs at a and p∗2 = p∗1 = 0 b Since we are interested only in positive populations we can ignore p∗1 = 0 and so just refer to equilibrium rate p∗ . For p0 < a/b, where p0 is the initial population, then dp/dt > 0, and so p rises over time. For p0 > a/b then dp/dt < 0, and p falls over time. The arrows in ﬁgure 14.2 show these properties. It is clear that p∗ = a/b is a (locally) stable equilibrium.2 Although population approaches the limit a/b, this is never in fact achieved (see exercise 5). We can solve for p explicitly as follows dp = ap − bp2 = p(a − bp) dt p p dp = dt p0 p(a − bp) p0 But

1 1 −b 1 = − p(a − bp) a p a − bp t p dp 1 p −bdp .. . 1 − = dt a p0 p a p0 (a − bp) t0

Solving we have p 1 1 1 ln p − (−b) ln(a − bp) = t − t0 a a −b p0 p a − bp 1 1 ln − ln = t − t0 a p0 a a − bp0 p(a − bp0 ) ln = a(t − t0 ) p0 (a − bp) ... p (a − bp)ea(t−t0 ) = p(a − bp ) 0

We can now solve for p(t) p0 aea(t−t0 ) = pbp0 ea(t−t0 ) + p(a − bp0 ) = p[bp0 ea(t−t0 ) + (a − bp0 )] i.e. p(t) =

(14.7)

(14.8)

ap0 bp0 + (a − bp0 )e−a(t−t0 )

This represents the logistic function, which is sketched in ﬁgure 14.3, and shows the logistic curve. This curve depends on the three parameters a, b and p0 . It has an upper limit of a lim p(t) = t→∞ b 2

∗ Expanding

p˙ = f ( p) in a Taylor series around p = a/b we obtain the following linear approximation p˙ = −a p − ab , which has a positive intercept (a2 /b) and a negative slope (−a).

Population models

599 Figure 14.3.

The zero population growth is, however, never reached. (This result is also established in exercise 5 using a linear approximation around the equilibrium.) A second property of the logistic function is that it has an inﬂexion point at p=

a 2b

This is readily established from the logistic growth equation, since the inﬂexion point occurs where the logistic growth equation is at a maximum. Thus, if f (p) = ap − bp2 f ( p) = a − 2bp = 0 a p= 2b The shape of the logistic curve depends on whether the initial population is below or above the inﬂexion value of p, or even above the limit value a/b. Figure 14.3 illustrates three different paths. We can use the logistic function and the data provided in table 14.1 to compute the values of a and b for the logistic growth equation. Using ﬁgures for 1781, 1831 and 1881, respectively, for t(0), t(50) and t(100), we have the following two equations 13a 13b + (a − 13b)e−50a 13a 34.934 = 13b + (a − 13b)e−100a 24.135 =

(14.9)

600

Economic Dynamics

Figure 14.4.

which provide two nonlinear equations in two unknowns. Using a mathematical software package for solving equations3 (and using the Malthusian value of k for a ﬁrst approximation for a), it can be established that a = 0.02038302

b = 0.0004605

As indicated above, the value of b is very small and the population has to be large before this second term becomes signiﬁcant. Even so, it implies an upper limit for the population of the UK of a/b = 46.745186 million. Using these values for a and b, we have the logistic results shown in column (4) of table 14.1. It is clear that these give signiﬁcantly different results than those of the Malthusian growth law and that towards the end of the period they under-estimate the growth in the population of the UK. The different growth processors relative to the actual observations are illustrated in ﬁgure 14.4. This shows quite clearly that the Malthusian law grossly over-estimates the UK population in 1931, while the logistic growth equation under-estimates it. Of course, a possible reason for the under-estimate of the logistic growth equation is the choice of years to estimate the parameters a and b. We quite arbitrarily chose t1 to be ﬁfty years on from t0 and t2 to be 100 years on. A different choice of years would give different computed values of a and b, and hence different values in column (4) of table 14.1. It is even possible to estimate a and b using nonlinear statistical estimation, which would use all the available data in table 14.1. However, the point being emphasised is that the logistic calculations are sensitive to the computed/estimated values of a and b, and most especially the limit in the growth of the population. We might, however, approach the logistic equation in terms of its discrete approximation we developed in chapter 3, section 3.7. It is assumed that the change in the population, pt+1 conforms to the rule pt+1 = apt − bp2t

(14.10) 3

After deﬁning the equations, Mathematica can solve these equations using the FindRoot command and using initial guesses for a and b. Maple can do the same using the fsolve command and giving ranges for a and b. The two programmes give the same results (see appendices 14.1 and 14.2). The same results can be established using TK Solver. With all programmes, care must be exercised in providing initial guesses.

Population models

601

which has the approximate solution pt =

ap0 bp0 + (1 + a)−t (a − bp0 )

This too has the limit a/b. Again using the ﬁgures for 1781, 1831 and 1881 we obtain two equations 13a 13b + (1 + 10)−50 (a − 13b) 13a 34.934 = 13b + (1 + 10)−100 (a − 13b) 24.135 =

which gives two slightly different estimates for a and b, namely a = 0.0205922

b = 0.00044052

However, once again using these estimates for a and b along with the discrete form for the population, we obtain exactly the same estimates as column (4) of table 14.1. Although the discrete approximation is good for forecasting population, care must be exercised in its use. The original model is nonlinear. As we showed in chapter 3, for certain values of the parameters a and b the model leads to cyclical behaviour. This is not true of the discrete approximation. Regardless of the values of a and b the discrete approximation leads to an equilibrium value of a/b in the limit for some arbitrary population size which is nonzero. For instance if we consider the two formulations4 : pt+1 = apt − bp2t = 3.2pt − 2.2p2t (1 + a)pt 4.2pt pt+1 = = 1 + bpt 1 + 2.2pt i.e. a = 3.2 and b = 2.2, then system (i) goes to a 2-cycle with values oscillating between 0.74625 and 1.16284. On the other hand, system (ii) converges very quickly on the limiting value of 1.45455. These quite different stability characteristics of the two systems are a warning about the use of approximations when dealing with nonlinear systems.

14.3 An alternative interpretation In modelling population change it is useful to consider the process from a different perspective. Population at a point in time is a stock. This stock level will change depending on the difference between the inﬂow and the outﬂow. Depending on the population under investigation there will be different factors contributing to each of these ﬂows. For example, a typical inﬂow will consist of births and immigration; while a typical outﬂow will consist of deaths and emigration. In the case of ﬁsh populations, however, there is also the extent of the harvesting over the period. We 4

See chapter 3, section 3.9 for a derivation of the second equation above.

(14.11)

602

Economic Dynamics shall consider ﬁsheries in chapter 15, and here we shall concentrate on ‘natural’ changes to population. We have then: Net change in population = inﬂow − outﬂow = (births + immigration) − (deaths + emigration) But births and deaths can be considered as ‘internal’ to the population, while immigration and emigration can be considered as coming from outside the system, as ‘external’ inﬂuences on the population. We can, therefore, redeﬁne the net change in the population as composed of internal change plus external change as follows: Net change in population = internal change + external change = (births − deaths) + migration where, of course, migration is immigration less emigration. Notice that this interpretation is particularly useful for open systems, for it is only in such systems that migration can take place. For example, when considering the population of the UK we can consider the internal change in terms of births and deaths of UK citizens, and we can consider the external change in terms of the migration of the population in and out of the UK. On the other hand, if we are considering world population, then this is a closed system (at least until planetary movements of population take place!). There can be only births and deaths in a closed system. Abstracting from the many characteristics that make up a population, like age, sex, density, fertility, etc., we can think of a representative unit that contributes a net amount to the internal change in the population, which we shall label n. The population size at a point in time is p(t), and denotes the number of individuals at time t. Hence, the internal change in the population is np(t). Letting m(t) denote the migration (immigration less emigration) over the same interval of time as we are measuring the internal change, and measured at time t, then m(t) denotes the external change. Accordingly, the change in the population, dp(t)/dt is given by

(14.12)

dp = np(t) + m(t) dt Example 14.2 (Malthusian population growth) In the case of the Malthusian population growth we considered earlier, there is no migration (m(t) = 0 for all t) and population is assumed to grow at a constant rate r. In other words, the net contribution of each member is assumed to be equal to r (i.e. n = r). Hence for n = r and m(t) = 0 for all t dp(t) = rp(t) dt with population at time t given by p(t) = p0 ert

Population models

603

Example 14.3 (Logistic growth curve) Again there is assumed to be no migration and m(t) = 0 for all t. Assume, as in the Malthusian case, that a population which is not inﬂuenced by other factors grows at a constant rate r. But now further assume that there is a restraint on the growth process that is proportional to the size of the population. In other words, the growth process r is reduced by a factor r1 p(t). The net internal contribution is therefore given by n(t) = r − r1 p(t) Notice in particular that the internal net contribution is a function of time since it is related to the stock size of the population. Under these two assumptions about migration and net internal change, we have for the growth of the population dp = (r − r1 p(t))p(t) dt p(t) p(t) = r 1− k

where k =

r r1

(14.13)

which is the logistic growth equation we discussed earlier. Notice ﬁrst that r is the Malthusian growth of population and k denotes the carrying capacity of the population. This version of the logistic equation will be found particularly useful when we discuss ﬁsheries in chapter 15. For this population its size at time t is given by

p(t) = 1+

k

k − 1 e−rt p0

As we shall see in the next section, this alternative view of population change will be found very useful when considering multispecies populations that interact with each other in complex ways.

14.4 Multispecies population models: geometric analysis Consider some closed system, a habitat, in which there are just two species. These two species can interact with each other in a variety of ways. They may be: (1) (2) (3) (4)

independent of each other, in competition with each other, one a predator and the other a prey, both mutually supportive of each other.

If both are independent of each other then the populations will grow according to the type of laws we have already considered. In this section we are more concerned with interacting species. But before we consider each of the possible interactions in turn, we need to model the problem.

(14.14)

604

Economic Dynamics Let the two species be denoted x(t) and y(t), respectively. Then we can posit that the growth of the two species, with no migration for each species, as

(14.15)

x˙ = Rx(t) y˙ = Qy(t) where R denotes the net contribution of each individual in the x-population and Q the net contribution of each individual in the y-population. The extent to which a typical member of the x-population contributes to the stock depends not only on births and deaths, but also on its interaction with the y-population. The same holds for the y-population. Consider a very general interaction speciﬁcation, namely

(14.16)

R = α + βx(t) + γ y(t) Q = δ + εy(t) + ζ x(t) For each population, α and δ denote the natural growth coefﬁcient of the species. The second term denotes the over-crowding (or self-limiting) coefﬁcient of the species. As with the logistic growth equation, if β and ε are negative, then overcrowding will occur and the species come into competition with themselves. On the other hand, if β and ε are positive, then growth expands as the population size increases, i.e., there is an increase in fertility as population expands. This we refer to as mutualism. If γ and ζ are both zero then the two species are independent of each other. If γ and ζ are both negative, then each is in competition for the limited resources of the habitat. The growth of one species is at the expense of the other. On the other hand, if γ and ζ are both positive, then we have a mutually supportive closed system: the growth of each species is mutually beneﬁcial. Finally we have a predatory–prey relationship. If γ is positive and ζ is negative then x is the predator and y is the prey; if γ is negative and ζ is positive, then x is the prey and y is the predator. The predatory–prey model has been discussed in some detail in the literature, and much of it is the model of Lotka and Volterra or its extension. Given the general speciﬁcations here, then it is possible, for example, to consider models that combine over-crowding and have predatory–prey features or only predatory–prey characteristics. We now turn to each of the various models to consider them in some detail. In doing this we shall employ Mathematica to illustrate, in particular, the numerical examples in the phase plane. Some of the basic instructions for doing this are provided in appendix 14.3, which also includes instructions for using Maple. Here we concentrate on the geometric features of the modelling, leaving the mathematical analysis of such models to the next section. 14.4.1

Competition with no over-crowding

Consider the following model (14.17)

x˙ = [a − by]x x(0) = x0 y˙ = [c − dx]y y(0) = y0

a > 0, b > 0 c > 0, d > 0

The terms −by and −dx (where we suppress the time variable) show that each species is in competition for the limited resources of the habitat. We assume the habitat represents a closed system so there is no migration. Does such a system

Population models

605

have an equilibrium? Stationary values occur when x˙ = 0 and y˙ = 0, i.e. x˙ = [a − by(t)]x(t) = 0 implying y = a/b or x = 0 y˙ = [c − dx(t)]y(t) = 0 implying x = c/d or y = 0 Hence, there are two stationary points (x1∗ , y∗1 ) = (0, 0) and (x2∗ , y∗2 ) = (c/d, a/b), as shown by points E0 and E1 , respectively, in ﬁgure 14.5. Figure 14.5 also illustrates the qualitative nature of the trajectories. In this problem only nonnegative values of x and y are meaningful. Consider ﬁrst the trajectories in the neighbourhood of the origin. Since y < a/b, then 0 < a − by, and so x˙ > 0 and hence x is increasing. Similarly, x < c/d means 0 < c − dx, and so y˙ > 0 and hence y is increasing. In fact, this speciﬁes the nature of trajectories in quadrant I in ﬁgure 14.5. Using the same reasoning, we can summarise the properties of the four quadrants as shown in table 14.2. The trajectories are looking complex. For some trajectories in quadrant I the system seems to tend towards the equilibrium point E1 . However, if it passes into quadrant II then it moves away from the equilibrium point E1 . This is because x dominates the habitat and fertility of y is now so low that it begins to decline. A similar problem occurs if the

Figure 14.5.

Table 14.2 Vector properties for competition with no over-crowding Quadrant I For x < c/d then c−dx < 0, hence y˙ > 0 For y < a/b then 0 < a−by, hence x˙ > 0 Quadrant III For x > c/d then 0 < c−dx, hence y˙ < 0 For y > a/b then 0 > a−by, hence x˙ < 0

Quadrant II For x > c/d then 0 > c−dx, hence y˙ < 0 For y < a/b then 0 < a−by, hence x˙ > 0 Quadrant IV For x < c/d then 0 < c−dx, hence y˙ > 0 For y > a/b then 0 > a−by, hence x˙ < 0

606

Economic Dynamics trajectory moves from quadrant I into quadrant IV. In this instance, however, species y dominates the habitat and x declines to extinction. A similar logic holds if the system begins in quadrant III. An initial situation in either quadrant II or IV simply moves the system away from the equilibrium point E1 . Example 14.4 We can try to see what is happening to this system by considering a numerical example. Consider the following competitive model x˙ = [4 − 3y]x y˙ = [3 − x]y Equilibrium points can readily be found by setting x˙ = 0 and y˙ = 0, which gives two equilibrium points E0 : (x0∗ , y∗0 ) = (0, 0)

E1: (x1∗ , y∗1 ) = (3, 4/3)

Point E1 , in particular, is the solution to the two equations 4 x˙ = 0 3 x = 3 y˙ = 0

y=

To highlight the stability/instability properties of equilibrium E1 (here we ignore E0 ), we can consider the direction ﬁeld, which is illustrated in ﬁgure 14.6. This diagram illustrates a number of features. First, equilibrium E1 appears to be a saddle path solution. Second, the possible trajectories of the system conform to those highlighted by the qualitative discussion of ﬁgure 14.5, in particular the movement of the system in the various quadrants, and the likely paths as trajectories move from one quadrant into another. Third, the movement of the system is from quadrant I into quadrants II and IV; and from quadrant III into quadrants II and IV. Fourth, it is not obvious whether any path will lead to the equilibrium point E1 . Figure 14.6.

Population models

607

Although it was not possible to solve the nonlinear system given in the general speciﬁcation of the system, we can obtain more detailed information on the properties of the trajectories in the phase-plane by noting5 dy dy/dt (c − dx)y = = dx dx/dt (a − by)x which uses the chain rule. We can re-arrange this expression as follows c a − b dy = − d dx y x Integrating both sides we have c a − b dy = − d dx y x a ln y − by = c ln x − dx + k1 a ln y − c ln x = by − dx + k1 ya x−c = keby−dx

k = ek1

where k1 is the constant of integration. Hence ya x−c eby−dx where k is a constant. For a given value of k this solution gives the solution trajectory in the phase-plane. k=

Example 14. 4 (cont.) Returning to our numerical example, we can use Mathematica or Maple, to plot the trajectories for various values of k. We do this using Mathematica’s ContourPlot command or Maple’s contourplot command (see appendix 14.3). Figure 14.7 shows a number of trajectories for different values of k. The trajectories in ﬁgure 14.7 verify the general features outlined in ﬁgures 14.5 and 14.6, most especially the saddle path nature of equilibrium E1 . 14.4.2

Predatory–prey model with no over-crowding (Lotka–Volterra model)

Consider the following model x˙ = (a − by)x = ax − bxy a > 0, b > 0 y˙ = (−c + dx)y = −cy + dxy c > 0, d > 0 In this model y is the predator and x is the prey. Notice that if the stock of x is zero, then the predator has no food and is assumed to die out, as indicated by −c. The greater the food stock, the greater the x-population, and hence the greater the growth in the predator. On the other hand, the natural growth of the x-stock does not depend on the predator for food and so a is positive, but it is subject to prey, 5

This is possible only for autonomous systems, see chapter 4.

(14.18)

608

Economic Dynamics

Figure 14.7.

and so the greater the y-population, the more the x-population will be subject to prey, as indicated by −b. Our ﬁrst task is to establish the equilibrium of the system, to ﬁnd the stationary points. We do this by setting x˙ = 0 and y˙ = 0 and solving for x and y. Thus x˙ = (a − by)x = 0 y˙ = (−c + dx)y = 0

implying y = a/b or x = 0 implying x = c/d or y = 0

Hence, there are two stationary points: (x1∗ , y∗1 ) = (0, 0) and (x2∗ , y∗2 ) = (c/d, a/b), represented by points E0 and E1 , respectively, in ﬁgure 14.8. Figure 14.8 also illustrates the qualitative nature of the trajectories. We can summarise the properties of the four quadrants as shown in table 14.3. It would appear, then, that the trajectories follow some sort of anticlockwise spiral. Example 14.5 To see whether this is so, consider a numerical example at this stage, namely y x x˙ = 2 − 100 x y˙ = −2 + y 50 The equilibrium (other than the origin) is readily found to be (x∗ , y∗ ) = (100, 200). But the much more interesting question is what is happening to the species out of equilibrium. To obtain some initial insight into this obtain the direction ﬁeld for this system. This is illustrated in ﬁgure 14.9. What is apparent from ﬁgure 14.9 is that the system has a cyclical pattern around the equilibrium point E1 , and that the movement of the system is anticlockwise.

Population models

609

Table 14.3 Vector properties for predatory–prey model Quadrant I For x < c/d then −c + dx < 0, hence y˙ < 0 For y < a/b then 0 < a − by, hence x˙ > 0

Quadrant II For x > c/d then −c + dx > 0, hence y˙ > 0 For y < a/b then 0 < a − by, hence x˙ > 0

Quadrant III 2 For x > c/d then −c + dx > 0, hence y˙ > 0 For y > a/b then 0 > a−by, hence x˙ < 0

Quadrant IV For x < c/d then −c + dx < 0, hence y˙ < 0 For y > a/b then 0 > a−by, hence x˙ < 0

Figure 14.8.

Figure 14.9.

610

Economic Dynamics However, we can go further into the trajectories by noting that the predator must be a function of the prey, i.e., y = f (x). By the chain rule we have dy dy/dt = dx dx/dt Substituting the speciﬁc general equations, we have (−c + dx)y dy = dx (a − by)x a −c or − b dy = + d dx y x Integrating both sides, we have a −c − b dy = + d dx y x a ln y − by = −c ln x + dx + k1 a ln y + c ln x = by + dx + k1 ya xc = keby+dx

k = ek1

where k1 is the constant of integration. Hence, k=

ya xc eby+dx

where k is a constant. For a given value k this solution gives the solution trajectory in the phase-plane. Once again, using Mathematica’s ContourPlot command or Maple’s contourplot command, we obtain typical trajectories shown in ﬁgure 14.10, which clearly illustrates the cyclical pattern of the solution. Using the information in ﬁgure 14.9 we further note that the system moves in an anticlockwise direction. Suppose, however, we concentrate on just one trajectory with the initial situation shown by point P0 in ﬁgure 14.11, where P0 denotes the initial point (x0 , y0 ) = (50, 300). Point P0 is in the northwest quadrant. In this situation the predator is in excess of its equilibrium level while the prey is below its equilibrium level. But because the number of predators is contracting, the number of prey will soon begin to rise as the system moves into the southwest quadrant. Once into the southwest quadrant, the number of prey begins to rise since the number of predators is too small to be a major threat. Eventually, this moves the system into the southeast quadrant, allowing sufﬁcient prey for the predator once again to expand towards its equilibrium. However, too great an expansion in the predatory population diminishes the prey as the system moves into the northeast quadrant. From ﬁgures 14.10 and 14.11 it is clear that the trajectories form closed curves. This means that neither the predator nor the prey becomes extinct. Each species cycles between its minimum and maximum level, as illustrated in ﬁgure 14.12. This ﬁgure plots the time path of the predator, y, and the prey, x. The starting point is represented by point P0 (i.e. x = 50, y = 300), the point shown in ﬁgure 14.11.

Population models

611 Figure 14.10.

Figure 14.11.

What is also clear from ﬁgure 14.12 is that the predator lags behind the prey in a cyclic pattern, and because of this the stationary state is never attained. 14.4.3

Competitive model with over-crowding

In section 14.4.1 we considered a competitive model in which two species were in competition for the limited resources. But suppose there is also competition within each species as well; in other words, there is the possibility of over-crowding. We

612

Economic Dynamics

Figure 14.12.

can capture this situation in the following model x˙ = (a − by − ux)x y˙ = (c − dx − vy)y where the terms −ux2 and −vy2 denote the over-crowding in the x-species, and y-species, respectively; while −by and −dx denote the interactive competition between the two species. This system is nonlinear and much more complex than our earlier models. But we can still readily obtain the stationary points of the system by setting x˙ and y˙ equal to zero. This is certainly satisﬁed for x = 0 and y = 0, and so the origin denotes an equilibrium of the system, and the axes represent isoclines. Once again we can use the chain rule to specify the situation in the phase-plane, dy/dt (c − dx − vy)y dy = = dx dx/dt (a − by − ux)x We cannot solve this because the expression is not separable. We can, however, derive expressions for two further isoclines dy = 0 when (c − dx − vy) = 0 dx dy = 0 when (a − by − ux) = 0 dx These represent two straight lines in the phase-plane, of which there are four conﬁgurations depending on the values of the six parameters, a, b, c, d, u, and v, as illustrated in ﬁgure 14.13. The markings along the isoclines indicate that x˙ = 0 y˙ = 0

when a − by − ux = 0 implying dy/dx = ∞ and y = (a/b) − (u/b)x when c − bx − vy = 0 implying dy/dx = 0 and y = (c/v) − (d/v)x

Population models

613 Figure 14.13.

while above and below the isoclines we have the properties x˙ > 0 x˙ < 0 y˙ > 0 y˙ < 0

when a − by − ux > 0 implying y < (a/b) − (u/b)x (below x˙ = 0) when a − by − ux < 0 implying y > (a/b) − (u/b)x (above x˙ = 0) when c − dx − vy > 0 implying y < (c/v) − (d/v)x (below y˙ = 0) when c − dx − vy < 0 implying y > (c/v) − (d/v)x (above y˙ = 0)

which are indicated by the vectors of force in ﬁgure 14.13. In the upper diagrams in ﬁgure 14.13 extinction will occur in one of the species. So long as the system does not begin at the origin, then the system will either move to equilibrium point E1 , in which the y-species dies out, or to equilibrium point E2 , in which the x-species dies out. In the lower diagrams it is also possible for the two species to coexist. Such a situation occurs where the two isoclines intersect, and is given by the solution (x∗ , y∗ ) =

av − bc uv − bd

But an important question is whether such a coexisting equilibrium is a stable solution of the model. Figure 14.13(c) would suggest that E3 is not a stable equilibrium,

614

Economic Dynamics

Figure 14.14.

while in ﬁgure 14.13(d) E3 appears a stable equilibrium. In order to verify these results we shall continue our discussion with two numerical examples. This not only allows us to compare the two diagrams in the lower part of ﬁgure 14.13 but also to consider some trajectories in the phase-plane. In order to show such trajectories, however, we need to solve the nonlinear system using numerical solutions. We do this within Mathematica, using the NDSolve command and the ParametricPlot command. Similar plots can be derived with Maple.6 Example 14.6 x˙ = (3 − y − x)x y˙ = (4 − 2x − y)y The basic properties of this system are illustrated in ﬁgure 14.14, which displays the isoclines and the vectors of force in the various quadrants. These forces are based on the following observations x˙ = 0 y˙ = 0 x˙ > 0 x˙ < 0 y˙ > 0 y˙ < 0

6

when 3 − y − x = 0 implying y = 3 − x and dy/dx = ∞ when 4 − 2x − y = 0 implying y = 4 − 2x and dy/dx = 0 ˙ when 3 − y − x > 0 implying y < 3 − x (below x) ˙ when 3 − y − x < 0 implying y > 3 − x (above x) ˙ when 4 − 2x − y > 0 implying y < 4 − 2x (below y) ˙ when 4 − 2x − y < 0 implying y > 4 − 2x (above y)

See Lynch (2001) for plotting multispecies models with Maple.

Population models

615 Figure 14.15.

This system in general leads to the extinction of one of the species, depending on the initial situation. This is clearly illustrated in ﬁgure 14.15. The trajectories in this ﬁgure required the use of a software package to solve numerically the nonlinear system of equations. What ﬁgure 14.15 clearly shows is that if the initial situation is not on the saddle path solution, then the system will tend either towards equilibrium E1 , where only the x-species survives, or equilibrium E2 , where only the y-species survives. Only in the unlikely event of the initial condition of the system being on the saddle path will the system converge to the coexistent equilibrium point E3 . Example 14.7 x˙ = (4 − y − x)x y˙ = (6 − x − 2y)y The basic properties of this system are illustrated in ﬁgure 14.16, which displays the isoclines and the vectors of force in the various quadrants. These forces are based on the following observations x˙ = 0

when 4 − y − x = 0 implying y = 4 − x and dy/dx = ∞

y˙ = 0

when 6 − x − 2y = 0 implying y = 3 − 12 x and dy/dx = 0

x˙ > 0

˙ when 4 − y − x > 0 implying y < 4 − x (below x)

x˙ < 0

˙ when 4 − y − x < 0 implying y > 4 − x (above x)

y˙ > 0

˙ when 6 − x − 2y > 0 implying y < 3 − 12 x (below y)

y˙ < 0

˙ when 6 − x − 2y < 0 implying y > 3 − 12 x (above y)

In this example, unlike the previous example, the system converges on the coexistent equilibrium point E3 , so long as the system does not have an initial point equal to the other stationary values. This is illustrated quite clearly in ﬁgure 14.17, which shows a number of trajectories for this nonlinear system. It is also quite clear from ﬁgure 14.17 that this system does not have a saddle path, except for the axes,

616

Economic Dynamics

Figure 14.16.

Figure 14.17.

corresponding to the equilibrium point (x∗ , y∗ ) = (0, 0), which is an uninteresting case. We can give another interpretation to our results. First we note that a denotes the natural growth of the x-species and c denotes the natural growth of the y-species. We can then make the following deﬁnitions: u/a the competitive effect of x on itself relative to the natural growth of x b/a the competitive effect of y on x relative to the natural growth of x v/c the competitive effect of y on itself relative to the natural growth of y d/c the competitive effect of x on y relative to the natural growth of y

Population models Consider, then, ﬁgure 14.13(a). Here we have c/v > a/b and c/d > a/u or b/a > v/c and u/a > d/c, i.e., the relative competitive effect of the y-species on x is greater than its relative competitive effect on itself; while the relative competitive effect of the x-species on itself is greater than its relative effect on the y-species. Depending, therefore, on the starting position, either the x-species will die out or the y-species will. Turning to ﬁgure 14.13(b) we have a/b > c/v and a/u > c/d or v/c > b/a and d/c > u/a, i.e., the relative competitive effect of the y-species on itself is greater than its effect on the x-species; while the relative impact of the x-species on the y-species is greater than the relative competitive effect on itself. Hence, depending on the starting position, one of the species will die out. Next consider ﬁgure 14.3(c) where we have c/v > a/b and a/u > c/d or b/a > v/c and d/c > u/a. In this instance the relative competitive effect of the y-species on x is greater than its relative competitive effect on itself; while the relative competitive effect of the x-species on y is greater than on itself. This is unstable, and which species wins out depends on the initial conditions, but E3 cannot be attained unless the starting point lies on a saddle path.7 Finally, in ﬁgure 14.13(d) we have a/b > c/v and c/d > a/u or v/c > b/a and u/a > d/c, i.e., the relative competitive effect of the y-species on itself is greater than the relative impact of the y-species on x; while the relative competitive effect of the x-species on itself is greater than the relative impact of the x-species on y. Accordingly, the species will settle to some mutually coexistent level – namely at E3 . 14.4.4

Predatory–prey model with over-crowding

In sub-section 14.4.2 we considered the predatory–prey model (often referred to as the Lotka–Volterra model), which involved no competition from within the species. But suppose there are many predators and so they are in competition with themselves for the prey. Suppose, too, that the prey, besides being under attack from the predator is also in competition for the resources of the habitat from members of its own species. Consider then the most general situation of predatory–prey with over-crowding of both species in the model x˙ = (a − by − ux)x y˙ = (−c + dx − vy)y where the terms −ux2 and −vy2 denote the over-crowding in the prey (x-species), and the predator (y-species), respectively. The stationary points of the system are, once again, obtained by setting x˙ = 0 and y˙ = 0. This is certainly satisﬁed at the point (x∗ , y∗ ) = (0, 0). Hence the origin denotes one equilibrium solution, but an uninteresting one. The other solution is found by setting the terms in brackets to zero, which provides two isoclines. x˙ = 0 y˙ = 0 7

when a − by − ux = 0 implying y = (a/b) − (u/b)x when − c + dx − vy = 0 implying y = −(c/v) + (d/v)x

We shall illustrate this in the next section.

617

618

Economic Dynamics which gives the nontrivial equilibrium a d u a a c − + b v b v x∗ = b d y∗ = u d u d + + b v b v Furthermore, we can use the chain rule to express the slope of the trajectory in the phase plane, i.e. dy/dt (−c + dx − vy)y dy = = dx dx/dt (a − by − ux)x which is nonlinear and cannot be solved. Using the isoclines, however, we can get some insight into the possible trajectories. However, this is a much more complex system than the straight predatory–prey model, and so we shall continue our discussion with a numerical example. Example 14.8 Let

y x x˙ = 2 − − x 100 75 x y y˙ = −2 + − y 50 200

Then the nontrivial equilibrium point is (x∗ , y∗ ) = (112.5, 50). The question arises, however, as to whether, like the Lotka–Volterra model, a closed cycle occurs around the equilibrium point. In fact, this is not the case in the present model. The fact that there is competition from within each of the species leads the system towards the equilibrium point in the limit. This is illustrated in ﬁgure 14.18, which portrays the direction ﬁeld along with a number of typical trajectories in the phase plane. It is quite clear that, given the parameter values, this system will always converge on the equilibrium in the limit. Hence, point (x∗ , y∗ ) = (112.5, 50) is asymptotically stable. Figure 14.18.

Population models Of course, it is not always the situation that the equilibrium is asymptotically stable. For different parameter values the ﬁxed point can be asymptotically unstable (see exercise 12), but once again does not converge on a closed orbit around the equilibrium, as in the Lotka–Volterra model. This section has illustrated quite a variety of solution paths to systems involving the interaction between two species depending on whether the interaction is competitive, mutual or of the predatory–prey variety. With more than two species the variety of interactions becomes even more complex, but the nature of the solutions is basically similar. In the next section we shall consider the mathematical properties of these systems.

14.5 Multispecies population models: mathematical analysis8 In this section we shall set out the general approach, look at just two of the examples in the previous section in detail, and summarise all remaining examples. We shall then conclude with some general comments about such linear approximations to nonlinear systems. Suppose we have a general nonlinear system denoting the interaction between two species of the form x˙ = f (x, y) y˙ = g(x, y) Suppose further that this system has at least one ﬁxed point, denoted (x∗ , y∗ ), at which x˙ = 0 and y˙ = 0. If we wish to consider the stability of the system in the neighbourhood of the ﬁxed point then, following our treatment in chapter 4, we can expand the system in a Taylor expansion around the ﬁxed point. Thus ∂ f (x∗ , y∗ ) ∂ f (x∗ , y∗ ) (x − x∗ ) + ( y − y∗ ) ∂x ∂y ∂g(x∗ , y∗ ) ∂g(x∗ , y∗ ) (x − x∗ ) + ( y − y∗ ) y˙ − y∗ = ∂x ∂y

x˙ − x∗ =

Let fx , fy , gx and gy denote the partial derivatives, each evaluated at the ﬁxed point (x∗ , y∗ ). Then x˙ − x∗ = fx (x − x∗ ) + fy (y − y∗ ) y˙ − y∗ = gx (x − x∗ ) + gy (y − y∗ ) or, in matrix notation f x˙ − x∗ = x gx y˙ − y∗

fy gy

x − x∗ y − y∗

The matrix composed of elements fx , fy , gx and gy are evaluated at the ﬁxed point and this is a square matrix9 and the system is a linear approximation of the original nonlinear system. 8 9

This section requires knowledge of chapter 4. It is a Jacobian matrix.

619

620

Economic Dynamics Let A=

fx gx

fy gy

We have already shown in chapter 4 that all the stability properties of this linear system can be established from the eigenvalues and eigenvectors of A, along with the trace and determinant of A, where tr(A) = fx + gy det(A) = fx gy − fy gx These properties, however, are only local and apply only for the neighbourhood of the ﬁxed point under investigation. For nonlinear systems with more than one ﬁxed point, as in all the examples in the previous section, then the neighbourhood of each ﬁxed point must be investigated individually. Example 14.4 (cont.) Example 14.4 has the nonlinear system x˙ = f (x, y) = 4x − 3xy y˙ = g(x, y) = 3y − xy Taking an arbitrary equilibrium point (x∗ , y∗ ), then we can expand this system in a Taylor expansion around this value x˙ − x∗ = fx (x − x∗ ) + fy ( y − y∗ ) y˙ − y∗ = gx (x − x∗ ) + gy ( y − y∗ ) where fx = 4 − 3y evaluated at (x∗ , y∗ ) fy = −3x evaluated at (x∗ , y∗ ) gx = −y evaluated at (x∗ , y∗ ) gy = 3 − x evaluated at (x∗ , y∗ ) We have already established two ﬁxed points E0 = (0,0)

and

E1 = (3, 4/3)

and we need to consider the system’s behaviour in the neighbourhood of each. Take the point E0 = (0, 0). Then fx = 4, fy = 0, gx = 0 and gy = 3. Hence 4 0 A= 0 3 and our system has the linear approximation x˙ 4 0 x = y˙ 0 3 y

Population models in the neighbourhood of the origin. The eigenvalues and eigenvectors are found from 4−λ 0 A − λI = 0 3−λ where det(A − λI) = (4 − λ)(3 − λ) = 0 with the two eigenvalues r = 4 and s = 3. Furthermore, tr(A) = 7 and det(A) = 12. (Note that tr(A)2 > 4 det(A)). For r = 4 then 0 0 x 0 = 0 −1 y 0 Hence it does not matter what values x and y take in forming the eigenvector vr Let 1 vr = 0 Next consider s = 3, then 1 0 x 0 = 0 0 y 0 and again it does not matter what values x and y take in forming the eigenvector vs . Since vr must be linearly independent of vs , then let 0 s v = 1 The general solution is, therefore, 1 4t 0 3t x(t) e + c2 e = c1 0 1 y(t) and it is quite clear that this is asymptotically unstable. The situation is shown in ﬁgure 14.19, at the point E0 . For any value not the origin and in the positive quadrant will move the system away from the origin over time. (Also notice that the two independent eigenvectors form part of the axes.) Next consider the point E1 = (3, 4/3). Then fx = 0, fy = −9, gx = −4/3 and gy = 0. Hence 0 −9 A= −4/3 0 and our system has the linear approximation x˙ − x∗ 0 −9 x − x∗ = y − y∗ y˙ − y∗ −4/3 0 where (x∗ , y∗ ) = (3, 4/3). The eigenvalues and eigenvectors are found from −λ −9 A − λI = −4/3 −λ √ √ where √ det(A − λI)√= λ2 − 12 = 0, with the two eigenvalues r = 12 = 2 3 and s = − 12 = −2 3. Furthermore, tr(A) = 0 and det(A) = −12. From our analysis

621

622

Economic Dynamics

Figure 14.19.

in chapter 4 we already know that these results identify a saddle point, since the eigenvalues √ are real and of opposite sign.10 For r = 2 3 then √ −2 3 −9 x 0 √ = 0 −4/3 −2 3 y giving

√ −2 3x − 9y = 0 √ Let x = 3 then y = −2/3, hence √ 3 vr = −2/3 √ For s = −2 3 then √ 2 3 −9 x 0 √ = 0 −4/3 2 3 y

giving

√ 2 3x − 9y = 0 √ Let x = 3 then y = 2/3, hence √ 3 s v = 2/3

10

Another identifying feature of the saddle point is that det(A) < 0.

Population models The general solution is, therefore, √ √ √ x − x∗ 3 3 (−2√3)t (2 3)t + c2 e e ∗ = c1 y−y −2/3 2/3 Suppose c2 = 0 then √ √ x − x∗ 3 = c1 e(2 3)t y − y∗ −2/3 and so vr represents an unstable arm because the system if perturbed will move over time away from (x∗ , y∗ ) along the vector vr . On the other hand, if c1 = 0 then √ x − x∗ 3 (−2√3)t e ∗ = c2 y−y 2/3 which converges on (x∗ , y∗ ) over time. Hence, vs is a stable arm of the saddle point E1 = (3, 4/3). The behaviour of the system, therefore, in the neighbourhood of E1 is illustrated in ﬁgure 14.19. Unlike our analysis in the previous section, this present analysis indicates that if the system begins on the stable arm of the saddle point in the neighbourhood of the ﬁxed point, then it will converge on the ﬁxed point over time. However, for all other perturbations in the neighbourhood of the critical point, the system will diverge away from it. In which direction depends on how the system is disturbed, i.e., which of the four quadrants the system is moved into (but not along the arm through vs ). The next example has more equilibrium points to consider but the formal analysis is the same. Accordingly we shall be more succinct in our presentation. Example 14.6 (cont.) The system is x˙ = f (x, y) = 3x − xy − x2 y˙ = g(x, y) = 4y − 2xy − y2 which has four equilibrium points: E0 = (0, 0),

E1 = (3, 0),

E2 = (0, 4),

E3 = (1, 2)

In each case we shall consider a linear approximation of the system in that neighbourhood. The matrix of the linear system has elements fx = 3 − y − 2x

fy = −x

gx = −2y

gy = 4 − 2x − 2y

E0 = (0,0)

3 A= 0

0 4

3−λ A − λI = 0

0 4−λ

623

624

Economic Dynamics

Figure 14.20.

Hence det(A − λI) = (3 − λ)(4 − λ) = 0 with eigenvalues r = 4 and s = 3. The eigenvectors are 1 0 vr = and vs = 0 1 and the general solution is x 1 4t 0 3t = c1 e + c2 e y 0 1 and is asymptotically unstable. The behaviour of the system in the neighbourhood of E0 is shown in ﬁgure 14.20. In particular, since both x and y are positive then the system will move away from the origin. E1 = (3,0) −3 −3 −(3 + λ) −3 A= A − λI = 0 −2 0 −(2 + λ) Hence, det(A − λI) = (3 + λ)(2 + λ) = 0, with eigenvalues r = −3 and s = −2. Using r = −3 the associated eigenvector is 1 vr = 0 while for s = −2 the associated eigenvector is 3 s v = −1

Population models and so the general solution in the neighbourhood of E1 is x − x∗ 1 −3t 3 + c = c e e−2t 1 2 y − y∗ 0 −1 Since both r and s are negative, then the system is asymptotically stable in the neighbourhood of E1 . E2 = (0,4) −1 0 −(1 + λ) 0 A= A − λI = −8 −4 −8 −(4 + λ) Hence, det(A − λI) = (1 + λ)(4 + λ) = 0, with eigenvalues r = −4 and s = −1. The associated eigenvectors are, respectively, 0 1 r s v = and v = 1 −8/3 and the general solution in the neighbourhood of E2 is x − x∗ 0 −4t 1 + c = c e e−t 1 2 y − y∗ 1 −8/3 Since both r and s are negative, then the system is asymptotically stable in the neighbourhood of E2 . E3 = (1,2) −1 −1 −(1 + λ) −1 A= A − λI = −4 −2 −4 −(2 + λ) Hence, det(A − λI) = (1 + λ)(2 + λ) − 4 = λ2 + 3λ − 2 = 0, with eigenvalues √ √ −3 − 17 −3 + 17 = 0.56155 and s= = −3.56155 r= 2 2 The fact that the eigenvalues are of opposite sign indicates that E3 is a local saddle point. The eigenvector associated with r = 0.56155 is 1 r v = −1.5616 while that associated with s = −3.56155 is 1 vs = 2.5616 and the general solution in the neighbourhood of E3 is x − x∗ 1 1 0.56155t = c1 e e−3.56155t + c2 y − y∗ −1.5616 2.5616 It readily follows, therefore, that vr is the unstable arm of the saddle point and vs is the stable arm. Again the situation is illustrated in ﬁgure 14.20. Although this second example involves more critical points, the linearisation of the nonlinear system enables us to investigate some useful properties of the system. Furthermore, it supports the analysis of the previous section.

625

626

Economic Dynamics Table 14.4 Eigenvalues and eigenvectors for examples 14.4–14.8 Ex

Points

Eigenvalues

14.4

E0 = (0,0)

r=4 s=3

E1 = (3,4/3)

√ r = +2√3 s = −2 3

E0 = (0,0)

r=2 s = −2

E1 = (100,200)

r = 2i s = −2i

E0 = (0,0)

r=4 s=3

E1 = (3,0)

r = −3 s = −2

E2 = (0,4)

r = −4 s = −1

E3 = (1,2)

r = 0.5616 s = −3.5616

E0 = (0,0)

r=6 s=4

E1 = (4,0)

r = −4 s=2

E2 = (0,3)

r=1 s = −6

E3 = (2,2)

r = −0.7639 s = −5.2361

E1 = (0,0)

r=2 s = −2

E2 = (112.5,50)

r = 1.4375 + 1.0588i s = −1.4375 − 1.0588i

14.5

14.6

14.7

14.8

Eigenvectors 1 0 vr = , vs = 0 1 √ √ 3 3 vr = , vs = −2/3 2/3 1 0 vr = , vs = 0 1 1 1 vr = , vs = 2i 2i 1 0 vr = , vs = 0 1 1 3 vr = , vs = 0 −1 0 1 vr = , vs = 1 −8/3 1 1 vr = , vs = −1.5616 2.5616 0 1 vr = , vs = 1 0 1 2/3 vr = , vs = 0 1 1 0 vr = , vs = 3/7 1 1 1 vr = , vs = 0.61805 1.61805 1 0 vr = , vs = 0 1 Complex conjugate vectors∗

∗ If required they can be obtained using Mathematica or Maple. Obtain the matrix A and deﬁne it in Mathematica or Maple as a matrix, say m. Then use the Eigenvectors[m] command in Mathematica or eigenvects(m) command in Maple.

Before we comment generally on the linearisation of nonlinear systems, table 14.4 provides a summary of all the mathematical properties of examples 14.4–14.8 of the previous section. 14.5.1

Some general remarks

For linear systems there are general formulae for solutions. These general formulae include all solutions. Even where discontinuities exist, these can be located. Unfortunately, for nonlinear systems no such general formulae exist. This means that it is very difﬁcult, or not even possible, to establish properties of solutions. Another difﬁculty with nonlinear systems is that of determining the interval in which a solution exist which satisﬁes an initial value.

Population models

627 Figure 14.21.

14.6 Age classes and projection matrices Since it is females that give birth, then a number of population models consider only the number of females. However, the probability of giving birth varies throughout the lifespan of the female – something that we have so far ignored. In any period there are two probabilities (events): (1) the probability of dying in that period, and (2) the probability of giving birth in that period. Put another way, there is a rate of survival and a reproduction rate for a given class. Many models, therefore, consider the population of women of childbearing age. In such models, equal class intervals are taken. If, for example, we assume women can bear children to age 45 and we have three class intervals, then we have the three classes: 0–15, 15–30 and 30–45. Let x denote the continuous variable ‘age’. In general, if the terminal age is N = 45, and we have n age-classes, then N/n is the duration of the age class. In our present example this is 45/3 = 15. Finally, let t denote the projection interval, which has the same duration as the age class (here 15 years). The population is observed at the end of each projection interval. Turning now to the characteristics of the female population, let bi denote the birth rate for the ith-class (i = 1, 2, 3) and sij the survival rate from class i into class j. In our present example we have only s12 and s23 . The model can be captured in terms of a state diagram, presented in ﬁgure 14.21. Let xi (t) denote the population of the ith-age class at the tth-time step. To be in class i = 1, age 0–15, then the female can be born from a woman of any of the three age classes. So x1 (t + 1) = b1 x1 (t) + b2 x2 (t) + b3 x3 (t) But account must be taken of women in the population surviving into the second and third age class. The number surviving to the second age class is x2 (t + 1) = s12 x1 (t); while the number surviving to the third age class is x3 (t + 1) = s23 x2 (t). We have, then, the set of equations x1 (t + 1) = b1 x1 (t) + b2 x2 (t) + b3 x3 (t) x2 (t + 1) = s12 x1 (t) x3 (t + 1) = s23 x2 (t) This can be written in the matrix form x1 (t + 1) b1 b2 b3 x1 (t) x2 (t + 1) = s12 0 0 x2 (t) x3 (t + 1) 0 s23 x3 (t) 0

(14.19)

628

Economic Dynamics or more succinctly x(t + 1) = Ax(t) In general

(14.21)

(14.20)

b1 s12 A= . ..

b2 0 .. .

b3 0 .. .

... ...

. . . sn−1,n

bn−1 0 .. .

bn 0 .. . 0

The matrix A is often called a Leslie matrix after P. Leslie who ﬁrst introduced them. Of course (14.20) is just a recursive equation with solution x(t) = At x(0)

(14.22)

where x(0) denotes the vector of females in each ith-class in period 0. Before continuing, consider the following simple numerical example. Example 14.9 Let b1 = 0.4,

b2 = 0.8,

s12 = 0.9,

s23 = 0.8

b3 = 0.2

Suppose a population has 10 million females in each of the three age classes, giving a total female population of 30 million. Using the recursive relations speciﬁed in (14.19), by means of a spreadsheet we can derive the time proﬁle of this population over, say, ten periods as shown in table 14.5(a). The ten periods cover 150 years, since the projection interval is 15 years. Figure 14.22 shows the time proﬁle of this population in terms of the three classes. Alternatively, using result (14.22) we could derive a particular row of the spreadsheet. For example, for t = 4 we have 10 19.2960 1.0048 .7680 .1568 x(4) = A4 x(0) = .7056 .6912 .1584 10 = 15.5520 .6336 .3456 .0576 10 10.3680 which is exactly the same as the row for t = 4 in table 14.5(a). In table 14.5(b) we have computed the proportion of the total female population in each class. It should be noticed that these proportions are settling down to 42.7% in class 1, 33.7% in class 2 and 23.6% in class 3 by period 10. What we shall now illustrate is that the dominant eigenvalue of the matrix A establishes the growth rate of the population, while the eigenvector associated with the dominant eigenvalue allows a computation of the proportion to which each class stabilises. Such results are highly signiﬁcant. Once we know the matrix A, it is relatively easy to establish with computer software the eigenvalues and eigenvectors. To show these properties we utilise the following theorem.11 11

This theorem itself utilises the Perron–Frobenious theorem.

Population models Table 14.5 Age class projections (a) Numbers t

Years

x1 (t)

x2 (t)

x3 (t)

Total

0 1 2 3 4 5 6 7 8 9 10

0 15 30 45 60 75 90 105 120 135 150

10 14 14.4 17.28 19.296 22.237 25.275 28.897 32.958 37.629 42.943

10 9 12.6 12.96 15.552 17.366 20.010 22.747 26.007 29.662 33.866

10 8 7.2 10.08 10.368 12.442 13.893 16.008 18.198 20.806 23.730

30 31 34.2 40.32 45.216 52.042 59.178 67.652 77.163 88.097 100.538

(b) Per cent t

Years

x1 (t)

x2 (t)

x3 (t)

0 1 2 3 4 5 6 7 8 9 10

0 15 30 45 60 75 90 105 120 135 150

33.3 45.2 42.1 42.9 42.7 42.7 42.7 42.7 42.7 42.7 42.7

33.3 29.0 36.8 32.1 34.4 33.4 33.8 33.6 33.7 33.7 33.7

33.3 25.8 21.1 25.0 22.9 23.9 23.5 23.7 23.6 23.6 23.6

THEOREM 14.1 If A is a Leslie matrix of the form b1 b2 b3 . . . bn−1 s12 0 0 . . . 0 A= . .. . . .. .. .. . 0

. . . sn−1,n

bn 0 .. .

bi ≥ 0 i = 1, . . . n 0 < si−1,i ≤ 1 i = 2, . . . n

then (1) (2) (3)

there exists a unique dominant eigenvalue, λd , which is positive, the eigenvector associated with the dominant eigenvalue has positive components, all other eigenvalues, λi = λd satisfy |λi | < λd .

Example 14.9 (cont.) The Leslie matrix for example 14.9 is 0.4 0.8 0.2 0 A = 0.9 0 0 0.8 0

629

630

Economic Dynamics

Figure 14.22.

with eigenvalues λ1 = 1.14136

λ2 = −0.4767

λ3 = −0.26466

We see that λ1 is the dominant eigenvalue and |λ2 | < λ1 and |λ3 | < λ1 . The eigenvector, v1 associated with the eigenvalue λ1 is 0.72033 v1 = 0.56800 0.39812 which clearly has all positive components. Given the three eigenvalues, the system has the general solution x(t + 1) = c1 λt1 v1 + c2 λt2 v2 + c3 λt3 v3 which will be governed in the limit by the dominant root, λ1 . Since λ1 > 1 then the system grows over time. This is clearly shown in ﬁgure 14.22. Furthermore, the growth of the system is given by λ1 − 1 = 0.14136, which means a growth rate of 14.1%. Since the eigenvector v1 has elements x1 (t) = 0.72033, x2 (t) = 0.56800 and x3 (t) = 0.39812 then their sum is 1.68645 and so normalising the eigenvector by dividing each term by this sum, we arrive at the values 0.42713 0.33680 0.23607 which in percentage terms are the values we obtained as the limiting values in table 14.5(b).

Appendix 14.1 Computing a and b for the logistic equation using Mathematica When solving for a and b in the logistic growth equation do not use Solve or Nsolve command, rather use FindRoot. It is important to have ‘good’ initial estimates of

Population models a and b. First deﬁne the two equations: 13a 13b + (a-13b) e -50 13a In[2]:= eq2 := 34.934 == 13b + (a-13b) e -100

In[1]:= eq1 := 24.135 ==

a

a

Then use the FindRoot command using initial estimates for a and b. In[3]:=FindRoot[{eq1, eq2}, {a, 0.02}, {b, 0.0004}] Out[3]= {a → 0.020383, b → 0.000436045}

We do the same for the linear approximation In[4]:= eq3 := 24.135 ==

13a 13b + a-13b50 (1+a)

13a In[5]:= eq4 := 34.934 == 13b + a-13b 100 (1+a)

In[6]:= FindRoot[{eq3, eq4}, {a, 0.02}, {b, 0.0004}] Out[6]= {a → 0.0205922, b → 0.00044052}

Appendix 14.2 Using Maple to compute a and b for the logistic equation When solving for a and b in the logistic growth equation do not use the solve command, rather use the fsolve command. Because solving can be problematic, include ranges for a and b, e.g. a = 0.02..0.03 and b = 0.0004..0.0005. First deﬁne the two equations: > eq1:=24.135=13*a/(13*b+(a-13*b)*exp(-50*a));

eq1 := 24.135 = 13

a 13b + (a − 13b)e(−50a)

> eq2:=34.934=13*a/(13*b+(a-13*b)*exp(-100*a));

eq2 := 34.934 = 13

a 13b + (a − 13b)e(−100a)

Then use the fsolve command using ranges for both a and b > fsolve ({eq1,eq2},{a,b}, {a=0.02..0.03,b=0.0004..0.0005});

{a = .02038301946, b = .0004360453198} We do the same for the linear approximation > eq3:=24.135=13*a/(13*b+((1+a)ˆ(-50))*(a-13*b));

eq3 := 24.135 = 13

a a − 13b 13b + (1 + a)50

631

632

Economic Dynamics > eq4:=34.934=13*a/(13*b+((1+a)ˆ(-100))*(a-13*b));

eq4 := 34.934 = 13

a a − 13b 13b + (1 + a)100

> fsolve ({eq3, eq4}, {a,b}, {a=0.02..0.03, b=0.0004..0.0005});

{a = .02059217184, b = .0004405196282}

Appendix 14.3 Multispecies modelling with Mathematica and Maple In this appendix we give detailed instructions for deriving direction ﬁelds and trajectories for example 14.4 employing both Mathematica and Maple. We also give some basic instructions for the linear approximation. All other problems in this chapter can be investigated in the same manner. Here we concentrate only on the input instructions. The equation system we are to investigate is x˙ = (4 − 3y)x y˙ = (3 − x)y This is a nonlinear system and cannot be solved directly by any known method. However, as we pointed out in sub-section 14.4.1, we can express the properties of x˙ = (a − by)x a > 0, b > 0 y˙ = (c − dx)y c > 0, d > 0 in the phase plane by plotting the solution trajectories ya x−c k a constant eby−dx Alternatively for the nonlinear system k=

x˙ = f (x, y) y˙ = g(x, y) we can investigate the linear approximation x − x∗ fy x˙ fx = gx gy y − y∗ y˙ in the neighbourhood of a particular ﬁxed point (x∗ , y∗ ), and where fx , fy , gx and gy are evaluated at a ﬁxed point.

14A.1 Mathematica To derive the contour plot in Mathematica input the following instructions: k[x-,y-]:=y^ a x^ (-c)/E^ (b y - d x) {a=4, b=3, c=3, d=1} graph1=ContourPlot[ k[x,y], {x,0.5,6}, {y,0.5,4}, ContourShading->False, PlotPoints->50]

Population models The contour plot, however, does not indicate in which direction the vector forces go. For this purpose we need to invoke the PlotVectorField command and then combine this with the contour plot. Thus graph2=PlotVectorField[ {(4-3y)x,(3-x)y}, {x,0.5,6}, {y,0.5,4}] Show[graph1,graph2]

There may be memory problems with showing the two graphs together. Turning to the linear approximation, the system can be investigated by means of the following input instructions roots=Solve[ {(4-3y)x==0, (3-x)y==0}, {x,y} ] eq3=(4-3y)x eq4=(3-x)y matrixA= { {D[eq3,x], D[eq3,y]}, {D[eq4,x], D[eq4,y]} }; MatrixForm[matrixA] matrixA1=matrixA /. roots[[1]] Eigenvalues[matrixA1] Eigenvectors[matrixA1] matrixA2=matrixA /. roots[[2]] Eigenvalues[matrixA2] Eigenvectors[matrixA2]

Although Mathematica gives the eigenvectors for matrixA2 as √ √ −3 3 3 3 r s v = and v = 2 2 1 1 these are, in fact, the same as those in the text.

14A.2 Maple The equivalent in Maple is not as satisfactory. The contour plots can be obtained using the following input instructions. with( plots): equ:=y^e*x^(-3)/exp(3*y-x); contourplot(equ, x=0.5..6, y=0.5..4, grid=[40,40]);

This plot has only contour lines to the left and right of the ﬁxed point (x∗ , y∗ ) = (3, 4/3) and not above or below this value. A better rendition of the phase portrait is to utilise the following instructions. with( plots): with(DEtools): seq1:=seq( [0,0.5,0.5+0.25*i], i=0..10); seq2:=seq( [0, 6, 1+0.25*j], j=1..10); seq3:=seq( [0,1,0.1+0.1*k], k=1..10)

633

634

Economic Dynamics inits:={seq1, seq2, seq3}; phaseportrait( equ, [x,y], 0..1, inits, x=0.5..6, y=0.5..4,arrows=THIN);

Turning to the linear approximation, the system can be investigated by means of the following input instructions with(linalg): sol1:=solve( {(4-3*y)*x=0, (3-x)*y=0} ); equ1:=(4-3*y)*x; equ2:=(3-x)*y; matrixA:=matrix( [ [diff(equ1,x), diff(equ1,y)], [diff(equ2,x), diff(equ2,y)] ] ); matrixA1:=matrix( [ [ subs(sol1[1],diff(equ1,x) ), subs(sol1[1], diff(equ1,y) ) ], [subs(sol1[1], diff(equ2,x)), subs(sol1[1], diff (equ2,y)) ] ] ); eigenvals(matrixA1); eigenvects(matrixA1); matrixA2:=matrix( [ [ subs(sol1[2],diff(equ1,x) ), subs(sol1[2], diff(equ1,y) ) ], [subs(sol1[2], diff(equ2,x)), subs(sol1[2], diff (equ2,y)) ] ] ); eigenvals(matrixA2); eigenvects(matrixA2);

These instructions produce the same results as with Mathematica, with the same eigenvectors that, as indicated above, are the same as those in the text – which can readily be veriﬁed.

Exercises 1.

Given the following data for population in England and Wales over the period 1701–91, obtain the estimated population using the continuous Malthusian population model and compare your results with those provided. Why do you think the estimated population under-estimates the actual population in 1791?

Year

1701

1711

1721

1731

1741

1751

1761

1771

1781

1791

Population (million)

5.8

6.0

6.0

6.1

6.2

6.5

6.7

7.2

7.5

8.3

Source: Tranter (1973, table 1).

2.

Two countries, A and B, have populations of equal size, p0 , and are growing at the same net rate of 2% per annum. However, population A has a birth rate of 3% per annum and a death rate of 1% per annum

Population models

3.

4.

5.

while country B has a birth and death rate of 5% and 3%, respectively. Unfortunately, country A suffers a major spread of AIDS and its death rate rises to 2% per annum. Assuming both populations conform to the Malthusian model, how long will it take for the population of country B to be twice the size of country A? A population has births b, deaths d and migration m each growing exponentially. If b < (d + m), how long before the population is half its original size? (i) If a population conforms to the Malthusian population model and is growing at 3% per annum, how long will it take for the population to treble in size? (ii) Derive a general formula for the time interval necessary for an increase in population to grow by λ times its initial size, assuming it is growing at some general rate k% per annum? For the logistic equation p˙ = p(a − bp)

6.

expand this as a Taylor series around the equilibrium a/b and hence show that the population in the neighbourhood of the equilibrium can be expressed a at a e p − = p0 − b b Show that as t → ∞ then p → a/b. What does this imply about the achievement of equilibrium? Suppose p˙ = p(a + cp)

7.

a > 0, b > 0

a > 0, c > 0

(i) Explain this equation. (ii) Draw the phase line for this population and show that the population tends to inﬁnity. (iii) Derive an explicit solution for the population and use this to show that an inﬁnite population is reached at a ﬁnite point in time. A population is thought to have the feature that if it falls below a minimum level, m, then it will die out and that there is a maximum carrying capacity of M for the same population. (i) Given an intrinsic growth rate of r, discuss the usefulness of p˙ = r(M − p)( p − m)

8.

to describe this population. (ii) Compare (a) p˙ = rp(M − p) (b) p˙ = r(M − p)( p − m) For the Gompertz equation p˙ = rp(a − ln p) a > 0 (i) Solve the equation subject to p(0) = p0 . (ii) Sketch this graph and its associated phase line.

635

636

Economic Dynamics

9.

(iii) Obtain the ﬁxed points and establish their stability/instability. (iv) What happens to p as t → ∞? Solve the following system for two competing species x and y x˙ = −3y(t) y˙ = −9x(t)

10.

11.

and derive explicitly the phase line. Trout, species T, and bass, species B, are assumed to conform to the following model T ˙ T − bTB T = a 1− k1 B B˙ = c 1 − B − dTB k2 Analyse this model in detail using a graphical analysis. Suppose N(t) denotes the biomass of halibut in the Paciﬁc Ocean. It has been estimated that for the equation N(t) =

12.

13.

N0 K N0 + (K − N0 )e−rt

Note: N(t)/K = N0 /(N0 + (K − N0 )e−rt ) = (N0 /K)/((N0 /K) + (1 − (N0 / K))e−rt ) r = 0.71 per year and K = 80.5 × 106 kg. If the initial biomass is one-quarter of the carrying capacity, (i) What is the biomass 2 years later? (ii) What is the time at which the biomass is (a) half the carrying capacity? (b) three-quarters of the carrying capacity? In each of the following systems which describes the interaction between two species of population x and y, (i) Find the stationary values. (ii) Linearise each system in the neighbourhood of all critical points. (iii) Find the eigenvalues and eigenvectors for each linearisation and describe the nature of the critical point. (iv) Try to establish the nature of the system by plotting sufﬁcient trajectories. 2xy xy y˙ = 2y − (a) x˙ = −x + 100 25 xy xy 1 (b) x˙ = − 2 x + 1 y˙ = y − y2 − 1 ( 4 + y) ( 4 + y) 2 2 (c) x˙ = x − x − xy y˙ = y − 2xy − 2y Consider the following discrete numerical predatory–prey model, where x is the prey and y is the predator xt+1 − xt = 1.4(1 − yt )xt yt+1 − yt = 0.6(1 − 4yt + xt )yt (i) Establish the critical points. (ii) Find the linearisation coefﬁcient matrix, A, for each critical point.

Population models

14.

(iii) Establish the eigenvalues and eigenvectors of A. (iv) Set up the system on a spreadsheet and establish the limit value of x and y as t → ∞. Investigate fully the discrete dynamical system xt+1 = 1.3xt − 0.3xt2 − 0.15xt yt yt+1 = 1.3yt − 0.3y2t − 0.15xt yt

15.

(i) Showing in particular that four critical points exit. (ii) Linearising the system about each critical point. (iii) Establishing the behaviour of the system by considering sufﬁcient trajectories in relation to (i)–(ii). The American bison population can be sub-divided into three categories (Cullen 1985): calves, yearlings and adults. These are denoted x1 (t), x2 (t) and x3 (t), respectively. In each year the number of newborns is 42% of the number of adults from the previous year. Each year 60% of the calves live to become yearlings, while 75% of yearlings become adults. Furthermore, 95% of adults survive to live to the following year. (i) Write out the system as a set of difference equations. (ii) Draw a state diagram for this population. (iii) Show that this system has one real eigenvalue and two conjugate complex eigenvalues. (iv) What is the eventual growth rate of the bison population? (v) What proportion of calves, yearlings and adults does this system settle down to? Additional reading

Additional material covered in this chapter can be found in Boyce and DiPrima (1997), Braun (1983), Caswell (2000), Deane and Cole (1962), Haberman (1977), Hoppensteadt (1992), Lynch (2001), Meyer (1985), Mooney and Swift (1999), Renshaw (1991), Sandefur (1990), Tranter (1973) and Vandermeer (1981).

637

CHAPTER 15

The dynamics of ﬁsheries

In this chapter we consider a renewable resource. Although we shall concentrate on ﬁshing, the same basic analysis applies to any biological species that involves births and deaths. A ﬁshery consists of a number of different characteristics and activities that are associated with ﬁshing. The type of ﬁsh to be harvested and the type of vessels used are the ﬁrst obvious characteristics and activities. Trawlers ﬁshing for herring are somewhat different from pelagic whaling.1 In order to capture the nature of the problem we shall assume that there is just one type of ﬁsh in the region to be harvested and that the vessels used for harvesting are homogeneous and that harvesters have the same objective function. Because ﬁsh reproduce, grow and die then they are a renewable resource. But one of the main characteristics of biological species is that for any given habitat there is a limit to what it can support. Of course, harvesting means removing ﬁsh from the stock of ﬁsh in the available habitat. Whether the stock is increasing, constant or decreasing, therefore, depends not only on the births and deaths but also on the quantity being harvested. The stock of ﬁsh at a moment of time denotes the total number of ﬁsh, and is referred to as the biomass. Although it is true that the biomass denotes ﬁsh of different sizes, different ages and different states of health, we ignore these facts and concentrate purely on the stock level of ﬁsh. But like any renewable resource, over an interval of time the stock level will change according to births, deaths and harvesting. We shall deal with harvesting later. For the moment we shall concentrate purely on the biological characteristics of the ﬁsh stock. Our ﬁrst aim is to represent the biological growth curve of a ﬁshery.

15.1 Biological growth curve of a ﬁshery We assume that the growth rate of the ﬁsh stock, denoted ds/dt, is related to the biomass (the stock level), denoted s. Although stock size and the growth in stock size are related to time, in what follows we shall suppress the time variable in the stock. Thus, the instantaneous growth process for ﬁsh can be represented by the equation ds = f (s) dt

(15.1)

1

See Shone (1981, application 11) for a review of pelagic whaling.

The dynamics of ﬁsheries which is a representation of the births and deaths of the species in the absence of harvesting. In order to take the analysis further we need to assume something about the biological growth curve. A reasonable representation is given by the logistic equation, which we discussed in chapter 14. Thus ds s = f (s) = rs 1 − dt k

639

(15.2)

The coefﬁcient r represents the intrinsic instantaneous growth rate of the biomass, i.e., it is equal to the rate of growth of the stock s when s is close to zero. More importantly, the coefﬁcient k represents the carrying capacity (or saturation level) of the biomass, i.e., it represents the maximum population that the habitat can support. This follows immediately from the fact that the stock size will be a maximum when ds/dt = 0, i.e., when s = k. In what follows we assume that both r and k are constant. These, and other features of the logistic equation representing ﬁsh growth, are illustrated in ﬁgure 15.1. In the upper section of the diagram we have the growth curve represented by the logistic equation, while in the lower section we draw the equation of the stock size against time, i.e., the solution equation

s= 1+

k

k − 1 e−rt s0

(15.3)

Figure 15.1.

640

Economic Dynamics In the lower section we also draw the curve denoting the biotic potential. This curve represents the growth of a species in which there is no negative feedback from overcrowding or environmental resistance. In other words, the biotic potential denotes the exponential growth curve

(15.4)

s = s0 ert The shaded region in the lower section of ﬁgure 15.1 shows the environmental resistance, which increases sharply after the inﬂexion point. The environmental resistance occurs because of the carrying capacity of the habitat. From the upper diagram in ﬁgure 15.1 it is clear that the growth function has a maximum point. At the stock size denoted smsy the growth of the ﬁsh stock is at a maximum, and is referred to as the maximum sustainable yield. The maximum sustainable yield is readily found. Since the growth curve is at a maximum, then we can establish this maximum by differentiating the growth curve and setting it equal to zero to solve for smsy and then substituting this value into f (s). Thus s 1 =0 +r 1− f (s) = rs − k k

(15.5)

−rs + r(k − s) = 0 k smsy = 2 k rk = f 2 4 i.e. the stock level of ﬁsh at the maximum sustainable yield of a particular species is exactly half of its carrying capacity. Furthermore, given the logistic equation, equation (15.2), the growth function is symmetrical about smsy . The importance of the maximum sustainable yield is in relation to harvesting. The situation is shown in ﬁgure 15.2. With no harvesting, the species will be in

Figure 15.2.

The dynamics of ﬁsheries

641

biological equilibrium when there is no growth, i.e., when ds/dt = 0. There are two biological equilibria, s∗ = 0 and s∗ = k, as shown in ﬁgure 15.2. Any stock size above zero and below the carrying capacity will lead to positive growth and hence an increase in the stock. Any stock level above the carrying capacity will lead to excessive environmental resistance, and hence to a decline in the stock size. (This naturally follows since above s∗ = k, ds/dt < 0, and hence the stock size must be declining.) Now consider three constant levels of harvesting, h1 ,h2 and h3 . Harvesting level h1 is above the growth curve, which means that the ﬁsh are being extracted faster than they can reproduce. The ﬁsh will accordingly be harvested to extinction. At a harvest level of h2 , the harvest line just touches the growth curve at the maximum sustainable yield. What does this imply? Suppose the ﬁsh stock begins at the level of the carrying capacity. There will be no natural growth in the ﬁsh stock, but there will be a level of harvesting equal to h2 , which will result in the ﬁsh stock declining. When the ﬁsh stock declines to the maximum sustainable yield, then the natural growth in the ﬁsh population is just matched by the level of harvesting, and so this level of ﬁsh stock can be sustained perpetually. However, if the ﬁsh stock should fall below the maximum sustainable yield, then the rate of harvesting will exceed the natural population growth, and the ﬁsh stock will decline, and extinction will eventually result. This suggests that a management policy to harvest at the level h2 is not necessarily a sensible one, especially with the uncertainty involved in estimating ﬁsh stocks. If harvesting were at a level of h3 , there are two possible equilibria, s∗1 and s∗2 , given by the stock levels where the harvesting line cuts the growth curve. Both s∗1 and s∗2 represent sustainable yields. This is because at each of these stock levels the growth rate equals the rate of harvesting, and so the ﬁsh stock will remain constant. There are a number of characteristics of the ﬁshery in this instance: (1) (2) (3) (4)

(5)

If 0 < s < s∗1 then harvesting exceeds natural ﬁsh growth, and the species will decline to extinction. If s∗1 < s < s∗2 then harvesting is less than the natural ﬁsh growth, and so the stock size will increase, and increase until s∗2 is reached. If s > s∗2 then again harvesting exceeds the natural ﬁsh growth and so the ﬁsh stock will decline until it reaches s∗2 . Any deviation of the stock size away from s∗1 will lead to a further movement of the ﬁsh stock away from this level, either to extinction or to the level s∗2 . Accordingly, s∗1 denotes a locally unstable equilibrium. Any deviation of the stock size around the level s∗2 will lead to the stock size changing until it reaches s∗2 . Hence, s∗2 denotes a stable equilibrium.

Consider the following discrete form of the model2 st − ht st+1 = st+1 − st = rst 1 − k which can readily be investigated by means of a spreadsheet. Let r = 0.2 and k = 1000; further assume h = 20 for all time periods. The results are shown 2

It is well known that the discrete form of the model can produce far more complex behaviour than the continuous model depending on the value of r. See Sandefur (1990).

(15.6)

642

Economic Dynamics

Figure 15.3.

in ﬁgure 15.3. An initial stock size equal to the carrying capacity leads to an equilibrium stock size of s = 887 approached from below (i.e. s falls). On the other hand, an initial stock level equal to s = 500 (the msy stock level), leads to the same equilibrium but approached from above (i.e. s increases). In fact any initial stock in excess of s = 113 will lead to the stable equilibrium. An initial stock level below s = 113 leads to extinction.3 Any constant level of harvesting in excess of 50 will automatically lead to extinction. We can generalise the problem by letting h(t) denote the harvesting function, then the net growth in the ﬁsh population is given by ds(t) = f (s(t)) − h(t) dt s(t) = rs(t) 1 − − h(t) k

(15.7)

Suppressing the time variable for convenience, this can be expressed more simply as s ds = rs 1 − −h dt k Equilibrium is established by setting equation (15.7) equal to zero, which gives the steady-state equilibrium. The steady-state equilibrium is at the maximum sustainable yield only if the harvest is at the level h2 . One of the simplest models, developed by Crutchﬁeld and Zellner (1962), is to assume that the harvest level is partly determined in a demand and supply market. Demand for ﬁsh is determined by price while the supply of ﬁsh is determined by price and by ﬁsh stocks. The market is assumed to clear, which determines the 3

Solving the quadratic rs(1−s/k) = h gives s∗1 = 112.702 and s∗2 = 887.298.

The dynamics of ﬁsheries harvest. Thus the model is ds s = rs 1 − −h dt k qd = a0 − a1 p a0 > 0, a1 > 0 s b1 > 0, b2 > 0 q = b1 p + b 2 s qd = qs = h

643

(15.8)

From the market equations we can eliminate the price and solve for the harvest function in terms of stock levels. This gives h = h(s) =

a0 b1 + a1 b2 s b1 + a1

(15.9)

which is linear with positive intercept and positive slope. As before, the model is captured by superimposing the harvest function on the biological growth curve, as shown in ﬁgure 15.4. Example 15.1 Using the following numerical discrete version of the model on a spreadsheet st st+1 = st + 0.2 1 − 1000 qdt = 45 − pt qst = 1.2pt + 0.05st qdt = qst = ht the two equilibrium values are readily found to be s∗1 = 172 and s∗2 = 715. At the stable equilibrium value the price is found to be approximately p∗ = 4.2, with Figure 15.4.

644

Economic Dynamics

Figure 15.5.

equilibrium harvest of h∗ = 41.4 The stable equilibrium of the model is illustrated in ﬁgure 15.5. Although this model relates the harvest to the ﬁsh stock, through the supply equation, no account is taken of the number of vessels employed. A consequence of this is that no account is taken of the proﬁtability of the ﬁshing to the ﬁshermen, and hence no account is taken of entry into and exit from the industry. In order to consider such possibilities we need to consider the harvesting function in more detail.

15.2 Harvesting function The harvesting function, or catch locus, is a form of production function of the ﬁshermen. Considered from this point of view, the harvest function is the catch at

4

Using the continuous form of the model it is readily established, employing a software programme such as Mathematica or Maple, that s∗1 = 171.736, s∗2 = 714.628, p(s∗2 ) = 4.213 and h(s∗2 ) = 40.787.

The dynamics of ﬁsheries

645 Figure 15.6.

time t, denoted h(t), as a function of the inputs. We assume the inputs are of two kinds. First, the stock of ﬁsh available for catching, s(t); and, second, the ‘effort’ expended by the ﬁshermen, e(t). Effort is here an index of all inputs commonly used for ﬁshing – such as man-hours, trawlers, time spent at sea, nets, etc. Again we suppress the time variable and simply write the harvesting function as h = h(e, s)

(15.10)

A common, and very simple, harvesting function used in the literature is h = aes

a>0

(15.11)

where a denotes the technical efﬁciency of the ﬁshing ﬂeet. This function is illustrated in ﬁgure 15.6, where we have drawn the harvesting (measured in terms of numbers of ﬁsh) against effort. For a given stock size, harvesting is a constant fraction of effort. It follows therefore that the marginal product to effort is constant and equal to the average product with respect to effort. Also shown in ﬁgure 15.6 is that for a higher stock size, the harvesting function is to the left, i.e., for given effort (e0 say) the catch size is greater the greater the stock of ﬁsh in the habitat (h2 > h1 if s2 > s1 ). Although a common harvesting function, this is but a special case of the Cobb– Douglas type harvesting function that allows for a diminishing marginal product to effort and to stock size. Such a function would be h = aeα sβ

a > 0,

0 < α ≤ 1,

0 h1 if e2 > e1 ). Second, as drawn in ﬁgure 15.7, we have a constant marginal product with respect to ﬁsh stock (in terms of the Cobb–Douglas function this means β = 1). Now that we have outlined the harvesting function we can return to consider equilibria. Steady-state equilibria requires that ds/dt = 0, hence it requires the condition (15.13)

f (s) = h(s) where we have suppressed the time variable. The situation is illustrated in ﬁgure 15.8. First consider effort at level e1 with a corresponding harvesting function h1 . There are two equilibria, an unstable equilibrium at s = 0 and a stable equilibrium at s = s1 . On the other hand, if effort is raised to the level e2 , with the corresponding harvesting function h2 , then the stable stock equilibrium falls to s2 . But another feature is illustrated in ﬁgure 15.8. As drawn the harvesting function h1 and h2 both yield the same level of harvest at the respective equilibrium stock sizes. Economic efﬁciency would imply that the same harvest level would always

The dynamics of ﬁsheries

647 Figure 15.8.

be undertaken at the lowest effort. This means that harvest function h1 would be chosen over harvest function h2 . In fact, on the grounds of economic efﬁciency, no effort would be employed which led to a harvesting function resulting in an equilibrium ﬁsh stock size less than the maximum sustainable yield. However, we have so far assumed open access to the ﬁshery by all companies. But are there any circumstances where an equilibrium in open access to the ﬁshery would be at a stock level below the maximum sustainable yield? To answer this question it must be recalled that open access involves no restrictions on companies harvesting in the locality under study or of new ﬁrms entering the industry (or ﬁrms leaving the industry). What is clearly missing from the analysis so far is any consideration of proﬁts to the industry.

15.3 Industry proﬁts and free access We simplify our analysis by assuming that the unit cost of effort expended is constant. Let this be denoted w, and can be considered as the ‘wage’ for effort. Then the total cost is given by TC = we

(15.14)

Turning to revenue, we assume that all ﬁsh are sold at the same price, denoted p. Hence, with the total ﬁsh caught being h, it follows that total revenue is TR = ph = paes

(15.15)

It follows, then, that proﬁts for the industry are π = TR − TC = paes − we = (pas − w)e

(15.16)

648

(15.17)

Economic Dynamics What is the shape of the TR and TC functions? We wish to construct TR and TC against effort. TC is linear since TC = we and w is assumed constant. TR is less straightforward. As effort rises we have already established that the stock size in equilibrium falls. Return to ﬁgure 15.7. At zero effort the harvesting function lies along the horizontal axis and the stock size is at the level k, but total revenue is zero. As effort rises the harvesting line shifts left, h rises and, with p constant, total revenue rises. Once effort has risen to a level such that s = smsy , then total revenue must be at a maximum since the harvest is at the maximum level. Effort beyond this means a fall in harvesting and a fall in total revenue. Hence, TR takes a similar shape to f (s), adjusted by the factor p. More formally TR = ph. But in equilibrium s − aes = 0 rs 1 − k ae or s = k 1 − r Hence we can express h = aes as a function of e ae h = aek 1 − r ak = (r − ae)e r Hence pak (r − ae)e TR = ph = r which is quadratic in e. It is readily established that for this TR function: (1) (2)

TR = 0 at e = 0 and e = r/a TR is a maximum at e = r/2a.

The situation is shown in ﬁgure 15.9. Figure 15.9 highlights two other features. A rise in the ‘wage’ to effort shifts the total cost function to the left. Second, a rise in the price of ﬁsh shifts the total revenue function up, but still passing through the points e = 0 and e = em . The results on the proﬁts function are illustrated in ﬁgure 15.10. In ﬁgure 15.11(a) assume effort is at the level e2 . At this level of effort TR exceeds total cost (TR2 > TC2 ). There are excess proﬁts in the industry and there will be entry by ﬁrms to take advantage of the excess proﬁts. The increase in ﬁrms is captured in this model by an increase in effort. Entry will continue in open access while total revenue exceeds total cost. As effort rises with entry, total revenue will rise initially beyond TR2 but will then fall. Furthermore, as effort rises we move along both TC and TR. Effort will rise (entry will continue) until effort level e1 is reached, where TR = TC. In ﬁgure 15.11(b) we note that at effort level e1 we have the harvest function h1 and the equilibrium stock is s1 which is less than smsy . What we have established here is that although for the same harvest lower effort would be the most efﬁcient, with open access and free entry, effort would be established at level e1 and the equilibrium stock size s1 < smsy . In other words, effort will always be adjusted until proﬁts reduce to zero because only then will

The dynamics of ﬁsheries

649 Figure 15.9.

Figure 15.10.

there be no further entry into the industry. But at effort e1 , MR is negative as shown by the slope of the TR curve at e = e1 , and MC is constant (and equal to w). Hence, at e = e1 MR < MC, which is economically inefﬁcient. It is also biologically inefﬁcient in that it leads to a stock size below the maximum sustainable yield. What we have illustrated is that with free access, entry will continue until proﬁts are reduced to zero. This leads to excess effort (over ﬁshing) and hence to a stock size below the maximum sustainable yield. It is, however, sustainable in the sense that the level of harvesting equals the level of natural population growth. It readily follows in the case of open access that depending on the price of ﬁsh and the wage rate to effort, it is possible that effort is such that the harvesting function in ﬁgure 15.11(b) lies wholly above the growth function f (s) and so the catch is to extinction. This has been argued to be the case for a number of species, including the blue whale (see Shone 1981, application 11).

650

Economic Dynamics

Figure 15.11.

15.4 The dynamics of open access ﬁshery So far we have concentrated on the equilibrium situation. However, it is necessary to consider what happens to the ﬁsh stock when it is out of equilibrium. To do this we need to consider the model in terms of phase space. In order to do this we shall assume that effort changes according to a fraction of the proﬁts. In other words, if proﬁts are positive then effort will rise, and the greater the proﬁts the more entry will occur and the more effort expended. Similarly, if proﬁts are negative, then some ﬁrms will leave the industry and effort will fall. The greater the loss, the greater the number of ﬁrms leaving the industry. We can capture this change in

The dynamics of ﬁsheries

651

effort over time quite simply as follows de = vπ v>0 dt = v(pas − w)e Our ﬁsheries model with open access can therefore be captured by means of two dynamic equations s − aes s˙ = rs 1 − k e˙ = v(pas − w)e

(15.18)

(15.19)

where we have used the dot notation to denote the derivative with respect to time. The two variables under consideration are the stock size, s, and the amount of effort, e. In equilibrium both variables must be jointly determined. When out of equilibrium, equations for s˙ and e˙ will determine the dynamic path taken. To this we now turn. The situation is shown in ﬁgure 15.12. We measure the stock size on the horizontal axis and effort on the vertical axis. Our ﬁrst problem is to determine the equilibrium paths. In equilibrium we know that s˙ = 0 for equilibrium stock size e˙ = 0 for equilibrium effort First consider the effort equilibrium. Entry will occur until proﬁts are zero, at which point effort is zero. There is only one stock size consistent with this result, Figure 15.12.

652

Economic Dynamics namely

(15.20)

(15.21)

(15.22)

s∗ =

w ap

This is shown by the vertical line in ﬁgure 15.12. Now consider a stock size less than w/ap. In this case pas < w (or pas − w < 0), and so losses are being made. Firms leave the industry and so effort is reduced. In other words, to the left of the vertical line there is a force on effort to fall. Similarly, to the right of the vertical line pas − w > 0 and so proﬁts are being made and ﬁrms enter the industry with a resulting increase in effort. Hence, to the right of the vertical line there is a force on effort to rise. These forces are shown by the vertical arrows in ﬁgure 15.12. Now consider the stock equilibrium. We derive this as follows s − aes = 0 s˙ = rs 1 − k r(k − s) = kae r r e= − s a ka Result (15.21) indicates that the equilibrium situation for stock size is linear with a negative slope. The intercept on the effort axis is given by (r/a), the slope is given by −(r/ka) and the intercept on the stock axis is k. Consider next points either side of this equilibrium line. Above the stock equilibrium line we have the condition r r s e> − a ka s aes > rs 1 − k which means the harvest exceeds the natural stock growth for a given stock size. This, in turn, means that the stock size will fall over time. Hence, above the stock equilibrium line the forces are shown by arrows pointing to the left. By similar reasoning, points below the stock equilibrium line lead to growth in excess of harvesting for any stock size, and so the stock size will increase over time. Hence, below the stock equilibrium line the forces are shown by arrows pointing to the right. All these vectors of forces are illustrated in ﬁgure 15.12. Finally, we can readily establish the equilibrium stock size and effort level by solving the two linear equations. The equilibrium stock size is given immediately as s∗ = w/pa and the equilibrium effort is readily found to be equal to r w ∗ e = 1− a kap One ﬁnal observation to make concerning the dynamics is the slope of the path when it crosses either equilibrium line. In the case of the vertical line at any point on this line effort is unchanging and so the trajectory must have a zero slope when crossing the vertical line. Similarly, a trajectory crossing the stock equilibrium line must have the stock size unchanging; hence the trajectory must have no slope when crossing this line. The trajectory over time must depend on the starting position of the species. Suppose, for illustrative purposes, that the species is at the level of its carrying

The dynamics of ﬁsheries

653 Figure 15.13.

capacity, i.e., s = k, as shown in ﬁgure 15.13. At this stock level proﬁts are to be had and entry will occur. This entry will raise the effort of the industry and will simultaneously reduce the ﬁsh stock. Because proﬁts are initially large there is a sizable entry into the industry that pushes effort beyond its eventual equilibrium. As a result, the path must cross the vertical line (which it does with a zero slope). In other words, the system moves from quadrant I into quadrant II. In this quadrant harvesting is still above the natural growth level and so the stock size is continuing to fall. On the other hand, proﬁts are now negative and some ﬁrms will be leaving the industry, resulting in a reduction in effort. However, the reduction in effort results in the system moving from quadrant II into quadrant III (cutting the stock equilibrium line with a zero slope). In quadrant III losses are still being made and so effort is falling, but the reduction in effort leads to less harvesting and a rise in the stock of ﬁsh. This pushes the system into quadrant IV. Now proﬁts are again positive and ﬁrms will enter the industry resulting in increased effort. Furthermore, the harvesting is less than the natural growth and so the stock size will be rising. It follows, then, that the trajectory of the system over time is shown by the heavy line, showing a counter-clockwise spiralling path. Although ﬁgure 15.13 illustrates a stable spiral we have implicitly assumed certain values on the parameters in the construction of the diagram. This can best be noted by considering the mathematical properties of the system (equations (15.19)). In order to do this, however, we need to make two adjustments to the dynamical system. First, we consider percentage changes rather than simply changes. Hence, we need to divide the ﬁrst equation by the stock size, s, and the second equation by the effort, e. Using hats to denote percentage changes, then our dynamic system

654

Economic Dynamics takes the form

s − ae sˆ = r 1 − k eˆ = v( pas − w)

(15.23)

Next we note that in equilibrium the percentage changes are zero, hence s 0 = r 1− − ae k 0 = v(pas − w)

(15.24)

Subtracting the equilibrium conditions from these equations gives r (s − s) − a(e − e) sˆ = − k eˆ = vpa(s − s) The matrix of this system is

r −a − A= k vpa 0 where the trace and determinant of A are r tr(A) = − k det(A) = a2 vp A stable spiral, as indicated in table 4.1 (p. 180), requires three conditions to be met (1) (2) (3)

tr(A) < 0 i.e. tr(A) = −(r/k) < 0 det(A) > 0 i.e. det(A) = a2 vp > 0 2 [tr(A)] < 4 det(A)

It is the third condition that we have implicitly assumed in graphing ﬁgure 15.15. This requires the condition r 2 < 4a2 vp k to be met. In order to see this issue, we shall now consider a numerical example.

15.5 The dynamics of open access ﬁshery: a numerical example Example 15.2 Consider the following numerical example of the open access ﬁshery s − 0.005es s˙ = 0.5s 1 − 200 p = 25, w = 4, v = 0.02

The dynamics of ﬁsheries The two dynamic equations are, then s − 0.005es s˙ = 0.5s 1 − 200 e˙ = 0.02(0.125es − 4e) Which gives the two equilibrium lines s = 32 e = 100 − 0.5s and the equilibrium solutions s∗ = 32 and e∗ = 84. The diagrams consistent with these results are illustrated in ﬁgure 15.14. The stable spiral in this example is readily established. The matrix A, its trace and determinant are −0.0025 −0.005 A= 0.0375 0 tr(A) = −0.0025 det(A) = 0.0001875 from which it readily follows that not only are the ﬁrst two conditions for a stable spiral met, but so is the third condition, since [tr(A)]2 < 4 det(A). A change in p or w The model readily illustrates the result of either a change in the wage rate or a change in the price level. Neither of these changes does anything to the stock equilibrium line. Only the effort equilibrium is altered. Thus, a rise in the price of ﬁsh will result in the effort equilibrium line shifting to the left. Assuming the system was initially in equilibrium, the result is shown in ﬁgure 15.15, where effort rises and the ﬁsh stock falls. The assumed trajectory is shown by T1 . For example, in our numerical example if the price of ﬁsh rises from p = 25 to p = 32, then the system will settle down at s∗ = 25 and e∗ = 87.5. On the other hand, a rise in the wage paid to effort will shift the effort equilibrium line to the right. The situation is shown in ﬁgure 15.16, where effort falls and the ﬁsh stock rises. The assumed trajectory is shown by T1 . For example, in our numerical example, if the wage rate rises from w = 4 to w = 10, then the system will settle down at s∗ = 80 and e∗ = 60. Figure 15.15 and 15.16 highlight a potential misleading result if concentration is paid only to equilibrium values. The equilibrium stock size will lead to extinction only if the price rises inﬁnitely. But this ignores the dynamic behaviour out of equilibrium. If the trajectory of the system is that shown by T2 in ﬁgure 15.15, then extinction occurs before some positive equilibrium stock size can occur. In the case of trajectory T2 , the rise in the price of ﬁsh leads to a glut of ﬁrms entering the industry. The rise in effort that results leads to excess harvesting and ﬁsh harvested faster than they can reproduce, so leading to extinction. The system never reaches its eventual equilibrium!

655

656

Economic Dynamics

Figure 15.14.

The same issue can arise with a rise in the wage rate. This is shown by trajectory T2 in ﬁgure 15.16. Effort is reduced to zero and the ﬁshing industry effectively collapses before the new equilibrium can be reached. One of the parameters of signiﬁcance in these last two results is v, which is a reaction coefﬁcient of the industry to proﬁts and losses. The larger v, then the more likely are the results shown by trajectory T2 in ﬁgures 15.15 and 15.16. This is because the larger v, then the more ﬁrms will enter the industry when proﬁts are positive. This means that effort rises more for any given size of ﬁsh stock, hence, the more steep the trajectory resulting from a rise in the price of ﬁsh. Similarly, if wages rise, then the resulting losses lead to a more rapid exit from the

The dynamics of ﬁsheries

657 Figure 15.15.

Figure 15.16.

industry and a relatively greater reduction in effort. The resulting trajectory is fairly steep. The converse of these results is that the smaller the value of the reaction coefﬁcient, v, the more likely the system will converge to its equilibrium without oscillations, shown by trajectory T3 in ﬁgures 15.15 and 15.16.

658

Economic Dynamics

15.6 The ﬁsheries control problem In the next section we shall discuss school ﬁsheries, i.e., ﬁsh populations that shoal in large numbers. However, before we can do this we need to consider the optimal control problem more closely. So far we have ignored the facts that proﬁts are spread over time, they need discounting, and ﬁsh left in the sea is forgone revenue for the ﬁsherman. In this section we shall consider only a continuous model formulation, leaving the discrete form as exercises. Since the aim of the ﬁsherman or agency is to maximise discounted proﬁts subject to the biological growth function, we have a typical control problem as outlined in chapter 6, sections 6.1–6.3. Knowledge of these sections is required for the present section and the next one. To recap before we consider ﬁsheries, a typical maximisation principle problem is to5 t1 max V(x, u, t)dt + F(x1 , t1 ) {u}

t0

s.t. x˙ = f (x, u, t)

(15.25)

x(t0 ) = x0 x(t1 ) = x1 by a suitable choice of the control variable u. Here V(x, u, t) is the objective function, F(x1 ) is the value of the terminal state, x˙ = f (x, u, t) denotes how the state variable changes over time, while x(t0 ) = x0 and x(t1 ) = x1 denote the initial and ﬁnal values of the state variable. In solving this maximisation principle problem, we form the dynamic Lagrangian. t1 t1 ˙ V(x, u, t)dt + F(x1 ) + λ[ f (x, u, t) − x]dt L= t0

t0 t1

=

˙ + F(x1 ) [V(x, u, t) + λ f (x, u, t) − λx]dt

t0

We further deﬁne the Hamiltonian function H(x, u, t) = V(x, u, t) + λ f (x, u, t) which implies L=

t1

˙ + F(x1 ) [H(x, u, t) − λx]dt

t0

and using

−

t1

˙ = λxdt

t0

t1

˙ − [λ(t1 )x(t1 ) − λ(t0 )x(t0 )] xλdt

t0

(see exercise 2, chapter 6) then t1 ˙ [H(x, u, t) + λx]dt + F(x1 ) − [λ(t1 )x(t1 ) − λ(t0 )x(t0 )] L= t0 5

In chapter 6 we assumed F(x1 , t1 ) = 0 and so λ(t1 ) = 0. Here we assume a nonzero terminal state x(t1 ) = x1 with value F(x1 ).

The dynamics of ﬁsheries

659

The necessary conditions for an (interior) solution are, then, (i) (ii) (iii) (iv) (v)

∂H =0 t 0 ≤ t ≤ t1 ∂u ∂H t 0 ≤ t ≤ t1 λ˙ = − ∂x ∂H = f (x, u, t) x˙ = ∂λ ∂F λ(t1 ) = 1 ∂x x(t0 ) = x0

(15.26)

Our ﬁrst task, therefore, is to appropriately deﬁne the objective function V(x, u, t). Returning to our ﬁsheries question, the proﬁt at time t is total revenue less total cost. We continue to assume the following relationships: (i) (ii) (iii)

s˙ = f (s) − h(e, s) TR = ph = ph(e, s) p is a constant TC = we w is a constant

where f (s) is the biological growth function and h = h(e, s) is the harvesting function; and where TR = total revenue and TC = total cost. The price of ﬁsh, p, and, w, the ‘wage’ to effort are both assumed constant. Since proﬁts are deﬁned as total revenue minus total cost, then π = TR − TC = ph(e, s) − we where we have suppressed the time variable. One possible objective function is to let V(s, e, t) be given by the proﬁts function π(e, s, t) which can be maximised over the interval t0 ≤ t ≤ t1 . But this is not appropriate for two reasons: (i) (ii)

It does not allow discounting either of proﬁts, or of the terminal state F(s1 ). It ignores the fact that stock that is left in the sea will lead to a capital gain or loss if the shadow price of ﬁsh should rise or fall, respectively.

This extreme problem with no discounting is left as an exercise (see exercise 8). Now consider the objective function when proﬁts are discounted between 0 ≤ t ≤ T. Letting E−δt denote the discount factor and δ the discount rate, then the objective is

T

max {e}

max {e}

E−δt V(s, e, t)dt + E−δt F(s1 ) = (15.27)

T

E 0

−δt

(ph(e, s) − we)dt + E

−δt

1

F(s )

660

Economic Dynamics subject to the growth function and the terminal condition. Our problem, then, is

T

E−δt ( ph(e, s) − we)dt + E−δt F(s1 )

max {e}

(15.28)

s˙ = f (s) − h(e, s)

s.t.

s(T) = s1 Hence L=

T

E−δt [ ph(e, s) − we]dt 0 T λ[ f (s) − h(e, s) − s˙]dt + E−δt F(s1 ) − λ(T)s(T) + 0

The Hamiltonian function associated with this is H(e, s, t) = E−δt [ ph(e, s) − we] + λ[ f (s) − h(e, s)] with ﬁrst-order necessary conditions (i)

∂H =0 ∂e i.e.

( p − λE−δt )

∂H λ˙ = − ∂s (ii)

(iii) (iv) (v)

E−δt p

or

∂h =w ∂e

∂h ∂h −w−λ =0 ∂e ∂e

∂h ∂h λ˙ = − E−δt p + λ f (s) − λ ∂s ∂s

or

∂h − λf (s) λ˙ = −(E−δt p − λ) ∂s ∂h − λf (s) i.e. λ˙ = (λ − E−δt p) ∂s ∂H s˙ = = f (s) − h(e, s) ∂λ ∂F λ(T) = T = 0 ∂s s(0) = 0

Now deﬁne µ(t) = Eδt λ(t)

or

Then λ = E−δt µ and ˙ − δE−δt µ λ˙ = E−δt µ Condition (i) then becomes (p − µ)

∂h =w ∂e

µ = Eδt λ

The dynamics of ﬁsheries

661

while condition (ii) becomes ˙ − δE−δt µ = (λ − E−δt p) E−δt µ

∂h − λf (s) ∂s

∂h ∂h −p − µf (s) ∂s ∂s ∂h i.e. µ ˙ = [δ − f (s)]µ − ( p − µ) ∂s or µ ˙ = δµ + µ

The three necessary conditions can now be summarised ∂h =w ∂e

(i)

( p − µ)

(ii)

µ ˙ = [δ − f (s)]µ − ( p − µ)

(iii)

s˙ = f (s) − h(e, s)

∂h ∂s

(15.29)

These three conditions are known, respectively, as the maximum principle, the portfolio balance condition and the dynamic constraint.6 The maximum principle indicates that the current value Hamiltonian is maximised if the marginal net revenue from effort equals the marginal cost of that effort – the typical marginal revenue equals marginal cost condition. Since p is the market price of ﬁsh sold and µ is the resource price of ﬁsh in the sea (the shadow price) then ( p − µ) can be considered as the net price of a caught ﬁsh. Hence, (p − µ)∂h/∂e is the marginal net revenue of ﬁsh caught. Now consider condition (ii). On the right-hand side of condition (ii), the ﬁrst term denotes the net interest from selling the ﬁsh and investing the proceeds. The second term denotes the net revenue from holding ﬁsh. Thus, the ﬁrst term denotes the net interest forgone, while the second term is the net beneﬁt from holding ﬁsh. If [δ − f (s)]µ − (p − µ)

∂h >0 ∂s

then

µ ˙ >0

i.e. there is need of a capital gain to compensate for the loss of interest forgone. The third equation is simply the constraint. A steady state requires e˙ = 0,

s˙ = 0

and

µ ˙ =0

15.7 Schooling ﬁshery It is well known that some ﬁsh move in shoals for purposes of migration, reproduction or to fend off predators. Although schooling activity is a defence against natural predators, it makes them especially vulnerable to human predation. Modern equipment means shoals are easy to locate and the ﬁsh easy to catch. This means that the stock size has little impact on the catch so ∂h/∂s = 0. We can, therefore, deﬁne the catch function as h = h(e, s) = h(e) and the three necessary conditions

6

See Neher (1990, chapter 9).

662

Economic Dynamics as

(15.30)

(i) (ii) (iii)

( p − µ)h (e) = w µ ˙ = [δ − f (s)]µ s˙ = f (s) − h(e)

where we retain the assumption that the ﬁsh sell at a constant price p and the ‘wage’ per unit of effort, w, is constant. The steady state requires three conditions to be met e˙ = 0,

s˙ = 0

and

µ ˙ =0

The dynamics of the problem is solved in stages. First, two variables are chosen whose dynamics are ‘solved’ in terms of the phase-plane. Both the steady state (equilibrium) and out-of-equilibrium situations can be depicted. The third variable is then considered in the light of what is occurring with these two. The common approach is to consider the phase-plane in terms of ﬁsh stock, s, and its shadow price µ.7 In continuing our analysis we shall assume some speciﬁc functional forms. The catch locus we shall assume is h = aeb

(15.31)

(15.32)

a > 0, 0 < b < 1

while the biological growth f (s) we shall assume is logistic s f (s) = rs 1 − k where k is the carrying capacity and r is the intrinsic growth. Using these speciﬁcations h (e) = abeb−1 2rs f (s) = r − k and the three necessary conditions are ( p − µ)abeb−1 = w 2rs (ii) µ ˙ = δ− r− µ k s − aeb (iii) s˙ = rs 1 − k If µ ˙ = 0 then 2rs δ− r− =0 k k(r − δ) s∗ = 2r which is shown by the vertical line in ﬁgure 15.17. If µ > 0 then s > k(r − δ)/2r, and µ is rising. Similarly, when s < k(r − δ)/2r then µ is falling. We can conclude (i)

(15.33)

(15.34)

7

This is the resource cost of the ﬁsh caught as distinct from the price at which the ﬁsh sells on the market.

The dynamics of ﬁsheries

663 Figure 15.17.

that for s < s∗ then µ is falling8 while for s > s∗ then µ is rising. These results are shown by the vector forces in ﬁgure 15.17. In order to derive the second isocline s˙ = 0, we ﬁrst need to eliminate effort, e. From (15.33)(i) we have b(p − µ)aeb−1 = w 1 b−1 w .. . e = ab(p − µ) Substituting this into (15.33)(iii) gives b b−1 w s −a s˙ = rs 1 − k ab(p − µ) If s˙ = 0, then the above expression is equal to zero, which is a quadratic in s. There is little gain in solving this for µ in terms of s explicitly. We can, however, express it implicitly. Deﬁne b b−1 w s +a φ(s, µ) = −rs 1 − k ab( p − µ) Then for a turning point 2rs ∂φ = −r + =0 ∂s k k i.e. s∗ = 2 2r ∂ 2φ >0 = ∂s2 k 8

For s* > 0 then δ must be less than r, i.e., the discount rate must be less than the intrinsic growth rate.

(15.35)

(15.36)

664

Economic Dynamics

Figure 15.18.

Figure 15.19.

hence it is a minimum. The relationship between µ and s for s˙ = 0 is shown in ﬁgure 15.18. If s˙ > 0 then φ(s, µ) is above the curve, i.e., above s˙ = 0 and s is rising; while below s˙ = 0, s is falling. These forces are represented by the arrows in ﬁgure 15.18. We can now combine all the results, as shown in ﬁgure 15.19. The ﬁxed point, the equilibrium point, is given by (s∗, µ∗ ), which occurs at the intersection of the two steady-state conditions µ ˙ = 0 and s˙ = 0. They lead to four quadrants with vector forces illustrated by the arrows. The typical trajectories arising from this problem are illustrated in ﬁgure 15.20, indicating the presence of a stable saddle path S1 S1 and an unstable saddle path S2 S2 .

The dynamics of ﬁsheries

665 Figure 15.20.

Example 15.3 Let

s f (s) = 0.2s 1 − 100 h(e) = 5e0.5 p = 10, w = 8,

i.e. r = 0.2, k = 100 i.e. a = 5, b = 0.5 δ = 0.1

Our three maximisation conditions are, then (i) (ii) (iii)

(10 − µ)(2.5)e−0.5 = 8 0.4s µ ˙ = 0.1 − 0.2 − µ 100 s − 5e0.5 s˙ = 0.2s 1 − 100

which can be solved for e, s and µ once the steady-state conditions µ ˙ = 0 and s˙ = 0 are imposed. For this example e∗ = 0.5625,

s∗ = 25,

µ∗ = 7.6

The curve denoting s˙ = 0 is given by s 0.2s 1 − − 1.5625(10 − µ) = 0 100 i.e. µ = 10 − 0.128s + 0.00128s2 with minimum point at sm = 50. This example is nonlinear but we can investigate the stability of the equilibrium point (s∗, µ∗ ) = (25, 7.6) using a linear approximation.9 Let µ ˙ = g(s, µ) and 9

See chapter 4.

666

Economic Dynamics s˙ = v(s, µ), then ∂v(s∗ , µ∗ ) ∂v(s∗ , µ∗ ) (s − s∗ ) + (µ − µ∗ ) ∂s ∂µ ∂g(s∗ , µ∗ ) ∂g(s∗ , µ∗ ) (s − s∗ ) + (µ − µ∗ ) µ ˙ = ∂s ∂µ

s˙ =

Or s˙ = (0.2 − 0.004s∗ )(s − s∗ ) + 0.97652(10 − µ∗ )(µ − µ∗ ) µ ˙ = 0.004µ∗ (s − s∗ ) + (−0.1 + 0.004s∗ )(µ − µ∗ ) i.e. s˙ = 0.1(s − s∗ ) + 2.34375(µ − µ∗ ) µ ˙ = 0.0304(s − s∗ ) Writing the equations in matrix form, we have s˙ 0.1 2.34375 s − s∗ = µ ˙ 0.0304 0 µ − µ∗ where the matrix of the system, A, is given by 0.1 2.34375 A= 0.0304 0 from which we can readily compute tr(A) = 0.1 det(A) = −(0.0304)(2.34375) = −0.07125 Since det(A) < 0, we know from chapter 4 that we must have a saddle point. This is also veriﬁed if the characteristic roots are of opposite sign. The roots of the characteristic equation are √ tr(A) ± tr(A) − 4 det(A) λ 1 , λ2 = 2 0.1 ± (0.1)2 − 4(−0.07125) = 2 λ1 = 0.32157 λ2 = −0.22157 Using the ﬁrst solution we have s − s∗ 0.1 2.34375 s − s∗ = 0.32157 µ − µ∗ µ − µ∗ 0.0304 0 which leads to the relationship µ = 5.2375 + 0.0945s Using the second solution we have s − s∗ 0.1 2.34375 s − s∗ = −0.22157 µ − µ∗ µ − µ∗ 0.0304 0

The dynamics of ﬁsheries

667 Figure 15.21.

Figure 15.22.

which leads to the relationship µ = 11.03 − 0.1372s These saddle path solutions, along with the isoclines s˙ = 0 and µ ˙ = 0 are shown in ﬁgure 15.21. Finally Mathematica was used to produce a number of trajectories, as shown in ﬁgure 15.22. In doing this we employed the linear approximation results, i.e., we used the NDSolve command on the simultaneous linear differential equations s˙ = 0.1(s − s∗ ) + 2.34375(µ − µ∗ ) µ ˙ = 0.0304(s − s∗ ) where (s∗ , µ∗ ) = (25, 7.6). Notice that this linear approximation was reasonable. The stable and unstable saddle paths are given by µ = 11.03 − 0.1372s µ = 5.2375 + 0.0945s, respectively. Given s = 10, then the corresponding points on the saddle paths are µ0 = 9.658 and µ0 = 6.1825, respectively. Taking the trajectory through the point (10,9.658) did indeed lead to a trajectory straight towards the equilibrium (s∗ , µ∗ ). Similarly, for the point (10,6.1825) the trajectory moved directly away

668

Economic Dynamics

Figure 15.23.

from (s∗ , µ∗ ). The same is true for µ0 = 4.17 and µ0 = 9.9625 on the stable and unstable saddle paths for s = 50. All other initial points were taken off the saddle paths. It is quite clear from ﬁgure 15.22 that the numerical results support our earlier qualitative results. What meaning can we give to these results? All the trajectories in ﬁgure 15.22 represent solution paths, in the sense that each satisﬁes the three conditions listed above. The decision-maker, whether it be the ﬁshery manager or the manager of an agency, begins with a stock size s0 at time t = 0. Suppose this is below s∗ , as shown in ﬁgure 15.23. The manager has control over effort, e(t), and by implication over the shadow price µ. In particular, the manager can control the initial shadow price µ0 = µ(0). Given (s0 , µ0 ) then the particular trajectory starting at this point will move the system to reach sT . If the terminal time T is also a choice variable, then the manager has to choose both µ0 and T. Consider ﬁrst the choice of T. If T is ﬁnite, then sT would be a target. But this implies that net beneﬁts beyond sT are of no concern to the manager. Since, under free choice, this is unlikely, then the only logical possibility is for T to be inﬁnite. Now consider all inﬁnite planning horizons and the choice of µ0 . With an initial stock s0 , three possible choices are shown by µ10 , µ20 and µ30 . µ10 belongs to the stable arm of the saddle path S1 S1 . Hence, the trajectory is along this path tending to (s∗, µ∗ ) in the limit.10 A choice of µ20 (above S1 S1 ) implies a shadow price of ﬁsh in the sea higher than µ10 . Fishing effort is not so great and the ﬁsh stock rises until it reaches its own biological carrying capacity of k. On the other hand, an initial shadow price of µ30 leads to increased effort and to eventual extinction. The trajectories emanating from µ20 and µ30 cannot, therefore, be maximising paths. 10

Since the paths are inﬁnite, the trajectory never actually reaches µ∗ .

The dynamics of ﬁsheries

669

Only µ10 is optimal. The same logic holds if s0 is initially above s∗ , at s0 = s0 , µ40 is the only optimal path. The conclusion we arrive at is that the optimal path is to choose an initial point on the stable arm of the saddle path. Although this result is appealing, there is here a warning. If the shadow price is incorrectly estimated, then a divergent path is the most likely outcome – either raising the stock size to its carrying capacity or diminishing the species to extinction. This is certainly the problem facing a ﬁshing agency which is attempting to balance proﬁt and conservation.

15.8 Harvesting and age classes11 So far we have assumed the ﬁsh are homogeneous and in particular have not made any distinction between males and females or age composition. As we pointed out in chapter 14, section 14.6, we can model female populations in terms of age classes. In doing this here for ﬁsh populations, we shall simplify drastically and concentrate on sustainable harvesting. Let xi (t) denote the population of ﬁsh in the ith-age class just before harvesting and assume harvesting takes place at discrete time intervals. Throughout we will consider just three age classes. As in section 14.6, let bi denote the birth rate for the ith-class and sij the survival rate from class i to class j. This gives the Leslie matrix in the present example for i = 1, 2, 3 of b1 b2 b3 L = s12 0 0 0 s23 0 and without harvesting we have x1 (t + 1) b1 b2 x2 (t + 1) = s12 0 x3 (t + 1) 0 s23

x1 (t) b3 0 x2 (t) 0 x3 (t)

or x(t + 1) = Lx(t)

(15.37)

Now suppose hi are harvested (killed) for the age class i (i = 1, 2, 3). Then the number of females harvested is h1 x1 (t) h2 x2 (t) h3 x3 (t) where

h1 H=0 0

11

or

0 h2 0

Hx(t)

0 0 h3

This section is based on the analysis in Lynch (2001, chapter 13).

670

Economic Dynamics This means that HLx(t) will be the total harvested and the remaining population will be x(t + 1) = Lx(t) − HLx(t) = (I − H)Lx(t) If harvesting is to be sustainable across the age classes, then we require x(t + 1) = x(t), i.e. x(t) = (I − H)Lx(t)

(15.38)

Let M = (I − H)L, then M is also a Leslie matrix, and by theorem 14.1 (p. 629) there is a unique dominant eigenvalue λd which is positive. Also recall from section 14.6 that the population will then grow at a rate λd − 1. If we require the population to stabilise (no growth), then we require λd = 1 and there is a nonzero vector solution of (I − H)Lx(t) = λd x(t) = x(t)

(15.39)

where x(t) is the eigenvector associated with λd = 1. This means that we require the dominant eigenvalue of the matrix M = (I − H)L to equal unity. In our present example of i = 1, 2, 3 we have b1 b2 b3 0 0 1 − h1 0 s12 0 0 1 − h2 M = (I − H)L = 0 0 0 1 − h3 0 s23 0 (1 − h1 )b1 (1 − h1 )b2 (1 − h1 )b3 = (1 − h2 )s12 0 0 0 0 (1 − h3 )s23 Since λd = 1 is to be the dominant root, then we require |M − λd I| = |M − I| = 0 which will impose restrictions on the h-values. We have (1 − h1 )b1 − 1 (1 − h1 )b2 (1 − h1 )b3 =0 |M − I| = (1 − h2 )s12 −1 0 −1 0 (1 − h3 )s23 i.e. [(1 − h1 )b1 − 1] + (1 − h1 )b2 (1 − h2 )s12 + (1 − h1 )b3 [(1 − h2 )(1 − h3 )s12 s23 ] = 0 or

(15.40)

(1 − h1 )b1 + (1 − h1 )(1 − h2 )b2 s12 + (1 − h1 )(1 − h2 )(1 − h3 )b3 s12 s23 = 1 Only values of hi (i = 1, 2, 3) lying in the range 0 ≤ hi ≤ 1 will satisfy (15.40) in order to produce a sustainable policy. Once hi (i = 1, 2, 3) are found, the eigenvector of M associated with λd = 1 can be computed. Finally, from this the normalised vector can be obtained.

The dynamics of ﬁsheries Example 15.4 A ﬁsh species is divided into three age classes with a Leslie matrix 0.4 0.8 0.2 0 L = 0.9 0 0 0.8 0 We have already solved for the eigenvalues of this in example 14.9, where we found λ1 = 1.14136,

λ2 = −0.4767,

λ3 = −0.26466

with dominant root λ1 = 1.14136 and associated eigenvector 0.72033 v1 = 0.56800 0.39812 and associated normalised eigenvector 0.42713 x1n = 0.33680 0.23607 If no harvesting takes place, therefore, the ﬁsh population grows at λd − 1 = 0.14136 or about 14% every period. Furthermore, the female population will settle down at 42.7% in age group 1, 33.7% in age group 2 and 23.6% in age group 3. Now consider four harvesting policies: (i) (ii) (iii) (iv)

Uniform harvesting h1 = h2 = h3 = h Harvesting only the youngest age class h1 = 0, h2 = 0, h3 = 0 Harvesting only the middle age class h1 = 0, h2 = 0, h3 = 0 Harvesting only the oldest age class h1 = 0, h2 = 0, h3 = 0

(i) Uniform harvesting h1 = h2 = h3 = h Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1 − h)(0.4) + (1 − h)2 (0.8)(0.9) + (1 − h)3 (0.2)(0.9)(0.8) = 1 with solution h = 0.123855. Given this value for h, the matrix M takes the form 0.350458 0.700916 0.175229 0 0 Mu = 0.788531 0 0.700916 0 and the associated eigenvector is 0.720329 v1 = 0.568001 0.398121 while the normalised eigenvector is 0.427127 xu = 0.336803 0.236070

671

672

Economic Dynamics (ii) Harvesting the youngest age class h1 = 0, h2 = 0, h3 = 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1 − h1 )(0.4) + (1 − h1 )(0.8)(0.9) + (1 − h1 )(0.2)(0.9)(0.8) = 1 with solution h1 = 0.208861. Given this value for h1 , the matrix M takes the form 0.316456 0.632911 0.158228 0 0 M1 = 0.9 0 0.8 0 and the associated eigenvector is 0.655347 v1 = 0.589812 0.471850 while the normalised eigenvector is 0.381679 xn1 = 0.343511 0.274809 (iii) Harvesting the middle age class h1 = 0, h2 = 0, h3 = 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1)(0.4) + (1 − h2 )(0.8)(0.9) + (1 − h3 )(0.2)(0.9)(0.8) = 1 with solution h2 = 0.305556. Given this value for h2 , the matrix M takes the form 0.4 0.8 0.2 0 M2 = 0.625 0 0 0.8 0 and the associated eigenvector is 0.78072 v1 = 0.48795 0.39036 while the normalised eigenvector is 0.470588 xn2 = 0.294118 0.235294 (iv) Harvesting the oldest age class h1 = 0, h2 = 0, h3 = 0 Under this policy, and given the Leslie matrix above, equation (15.40) becomes (1)(0.4) + (1)(0.8)(0.9) + (1 − h3 )(0.2)(0.9)(0.8) = 1 There is no solution for h3 that lies in the range 0 ≤ h3 ≤ 1. The normalised vectors xu , xn1 and xn2 determine the long-term distribution of the ﬁsh in the different age categories. If, however, it was considered beneﬁcial to leave as many of the young age ﬁsh in the sea as possible, then pursuing a policy of catching middle age ﬁsh should be undertaken. In this case, the female population of the youngest age class would settle down at about 47%.

The dynamics of ﬁsheries

Exercises 1.

A discrete-time production function takes the form ht = st (1 − Eaet )

2.

where E is the exponential term. Show that this function exhibits diminishing returns to effort and constant returns with respect to stock size. For the Gompertz growth function k f (s) = rs ln s

3.

derive the sm level and the growth at this level. Given the following two growth functions, f (s) and g(s), respectively, and the same harvest function h(e, s), s f (s) = rs 1 − k k g(s) = rs ln s h(e, s) = aes

4.

(i) Show, after eliminating the stock size, that the yield to effort (h/e) is linear and log-linear, respectively. (ii) Demonstrate that ordinary least squares estimates of the parameters in (i) are not sufﬁcient to determine the three parameters a, r and k. In the following numerical model s˙ = 0.2s 1 −

s −h 1000

qd = 45 − p qs = 1.2p + 0.05s q d = qs = h

5.

establish whether it is possible for a regulatory authority to set the price to clear the market at a stable equilibrium which has a yield equal to the maximum sustainable yield. A typical ﬁsheries problem can be captured by the following three fundamental relationships, where the ﬁrst denotes the biological growth process of a ﬁsh stock; the second a harvesting function (or catch function); while the third denotes the proﬁts of a ﬁshing agency. g = ks(su − s) h = aes π = ph − we

673

674

Economic Dynamics where: g= k= s= su = h= a= e= p= w=

growth of ﬁsh stock stock-speciﬁc parameter ﬁsh stock maximum sustainable ﬁsh population harvest level technology efﬁciency parameter per unit of ‘effort’ expended in ﬁshing price per unit of ﬁsh “wage” rate

(i) Interpret these three equations in detail using sketches where possible. For the remaining questions assume: g = 0.5s(25 − s) h = 2.5es (ii) Given the level of effort is e = 2. (a) Establish the maximum sustainable yield (msy) and the stock size at this level. (b) Establish the steady-state value of the stock size. (c) Explain why the steady-state value of s exceeds the msy value. (iii) Now assume that ‘effort’ is not known and must be determined along with the stock size. Assume p = 0.6 and w = 12. (a) Establish the equilibrium values of stock size and effort that maximise proﬁts. What is this level of proﬁts? What is the growth rate? (b) Under the same price and wage levels, calculate the proﬁts under the condition e = 2 (i.e. use your answers in (ii)) and compare them with the present level of maximum proﬁts. (c) Explain why the stock size in this maximisation problem is greater than under (ii)(b). (iv) Undertake the calculations in question (iii) (a) with the following values for the respective parameters: (a) p = 0.8, w = 12 and a = 2.5 (b) p = 0.6, w = 15 and a = 2.5 (c) p = 0.6, w = 12 and a = 3 Discuss the implications of these results. (v) Suppose entry and exit occurs according to the following rule: e˙ = vπ = v( ph − we) where the parameter v denotes the speed of entry and exit in response to proﬁts. For v = 5 , p = 0.6 and w = 12, establish the stock level associated with equilibrium (i.e. no entry or exit). Establish the vector of forces each side of the s˙ = 0 phase line drawn with e on the vertical axis and s on the horizontal axis.

The dynamics of ﬁsheries

6.

(vi) For a steady-state solution s˙ = 0, and using p = 0.6 and w = 12, show that e is a linear function of s, and hence establish the vector of forces either side of this phase line. (vii) Using the results in (v) and (vi), establish the equilibrium for stock size and effort. Show that for a starting stock size of s0 = 25, the dynamic path gives a stable oscillation to equilibrium. Verify this result by establishing the characteristic roots of the dynamic system and the trace and determinant. (viii) Discuss this ﬁsheries model with entry and exit commenting, in particular, on the importance of the parameter v. (i) In the following model s −h s˙ = 0.2s 1 − 1000 h = 0.125es what is the level of effort that should be set by the regulatory authority to have a stable equilibrium with the maximum sustainable yield? (ii) Show that the level of effort obtained in (i) is the level that maximises 125 (0.2 − 0.125e)e 0.2

7.

8.

(iii) What is the stock size which maximises proﬁts in equilibrium if TR = ph and TC = we, with p = 0.4 and w = 10? Show that in the case of open access ﬁshery, a rise in either the price of ﬁsh or the productivity of the ﬁshing industry leads to an increase in effort and a reduction in ﬁsh stocks. Show that if the aim is to T max πdt {u}

s˙ = f (s) − h

9.

where π = ph − we, and p and w are constant and h = h(e, s). Then the ﬁrst-order conditions for an interior maximum are: ∂h =w (i) (p − λ) ∂e ∂h (ii) λ˙ = (λ − p) − λ f (s) ∂s (iii) s˙ = f (s) − h(e, s) where λ is the Lagrangian multiplier (the shadow price of the resource). In a ﬁshery the ﬁsh can be divided into three age groups, each one a year long. The Leslie matrix for this population is 0 3 40 L = 0.1 0 0 0 0.5 0 (i) What is the growth rate of the ﬁsh population if no harvesting takes place?

675

676

Economic Dynamics

10.

(ii) What is the long-run behaviour of the system if 25% of each age class is harvested? What is the optimal sustainable harvesting policy for the system in question 9, given the youngest age class is not harvested and groups two and three are harvested to the same extent? Additional reading

Additional material on the contents of this chapter can be obtained from Conrad (1999), Conrad and Clark (1987), Crutchﬁeld and Zellner (1962), Cunningham, Dunn and Whitmarsh (1985), Dasgupta and Heal (1979), Fisher (1981), Hamilton (1948), Hartwick and Olewiler (1986), Hilborn and Walters (1992), Lynch (2001), McVay (1966), Neher (1990), Peterson and Fisher (1987), Shone (1981) and Smith et al. (1977).

Answers to selected exercises

Chapter 2 2

−tk

p(t) = p0e e−a(e −1) p(0) = p0 and p → ea as t → ∞ tk

2

3

4 5

7 8 11 12 13 14 15 16 17 18

ce x − 1 ce x2 + 1 x (iii) y = 1 − cx

(ii) y =

(i) y =

x−1−c x−c

x3 − x2 + x + 1 (ii) y = 1 ± 4 + x3 + 2x2 + 2x 3 1 e−x (ii) y = (i) y = −x e +c −x + 1 + ce−x x e 1 = + ce−2x (iii) y 3

(i) y =

Table about 1220AD 1990BC (iii) £2,910 (i) r (ii) P(t) = P0 ert 14 years (i) x = 3 (repellor), x = −5 (attractor) (ii) k∗ = 0.908 (ii) Y(t) = Y0 e(s/v)t k D(t) = D0 + Y0 (ert − 1) r 7.77% (a) ‘Rich’ countries

Years to double

‘Poor’ countries

Years to double

France Japan West Germany UK USA

26 14 28 35 50

China India Uganda Zimbabwe

29 35 −347 347

(b) Years for the population to halve.

678

Answers to selected exercises 19

20

(a) 23.32% (b) 3 years (2.97) (c) 0.7506 × 1010 £292,507.52

Chapter 3 1

(i) (ii) (iii) (iv)

2

Pt = (1 + r)Pt−1 − R R R + Pn = (1 + r)n P0 − r r

3

Pn = (1 + r)n P0 − (1 + r)n−1 R1 − (1 + r)n−2 R2 − . . .

4

(i) (ii) (iii) (iv)

second-order, linear, autonomous, nonhomogeneous second-order, linear, autonomous, nonhomogeneous ﬁrst-order, linear, autonomous, homogeneous second-order, linear, non-autonomous, nonhomogeneous

− (1 + r)Rn−1 − Rn 1 n 1 n yn = y0 − −2 − + 2, stable 2 2 3 n 4 3 n 4 − − + , unstable yn = y0 − 2 5 2 5 n n yn = y0 (−1) − 3(−1) + 3, cyclical n n 1 1 −6 + 6, stable yn = y0 2 2

1 1 (−1)n + , cyclical 4 4 (i) a3 − a2 − a + 1 = (a + 1)(a − 1)2 8 1 (i) pt = − pt−1 , stable 3 3 (ii) pt = 5.5 − pt−1 , oscillatory (v) yn = y0 (−1)n −

5 7

8 10 16

(iii) pt = 16 − 3pt−1 , unstable a+I+G a+I+G n Yn = + b Y0 − , stable if 0 < b < 1 1−b 1−b d a−c de − (1 − e)pt−1 − pt−2 (i) pt = b b b λ(b + d) λ(a − c) pt−1 + (i) pt = 1 − b b (ii) p =

17 18

a−c , b+d

£248.85 22.25 minutes

q=

ad + bc b+d

Answers to selected exercises 19 20

xn =

x0 1 + nx0

√

√ 5)1+n + (1 + 5)1+n ] (ii) Mathematica xn = √ 5 n n √ 2√ 1 1 5 −2 5 −2 √ √ 2 5 1− 5 1+ 5 (ii) Maple xn = − + √ √ 5 1− 5 1+ 5 2−1−n [−(1 −

Chapter 4

1 5 7 8

9x 2 1 v1 = , 1

y=

v2 =

W=4 (i) λ1 = 1,

1 −1

λ2 = 2,

1 v1 = −3/2 , 1

λ3 = 5 0 v2 = 1 , −1

0 v3 = 1 2

x(t) = c1 et 3 (ii) y(t) = − c1 et + c2 e2t + c3 e5t 2 z(t) = c1 et − c2 e2t + 2c3 e5t (iii) W = 3 9

vr =

(i) (r, s) = (−2, −4), x(t) = c1 e−2t − c2 e−4t −2t

y(t) = c1 e

+ c2 e

vr =

x(t) = 4c1 et + c2 e−2t y(t) = c1 e + c2 e t

4 , 1

vr =

1 1

vs =

−i/2 , 1

x(t) = c1 sin(2t) + c2 cos(2t) y(t) = 2c1 cos(2t) − 2c2 sin(2t) (iv) (r, s) = (−1 + i, −1 − i), −t

vs =

i/2 1

centre

−i v = , 1 r

x(t) = c2 e−t sin(t) − c1 e−t cos(t) −t

−1 1

saddle point

−2t

(iii) (r, s) = (2i, −2i),

vs =

improper node

−4t

(ii) (r, s) = (1, −2),

1 , 1

y(t) = c1 e sin(t) + c2 e cos(t)

spiral

i v = 1 s

679

680

Answers to selected exercises 10

(i) P1 = (6,0) and P2 = (0.846,2.114). (ii) Fixed point P2 , exhibits a stable limit cycle

12

(i) (p∗ , Y ∗ ) = (7.055, 26.221) (ii) Yes.

13 14

System has limit cycles that shrinks as β rises. System has limit cycles that expand as α rises.

Chapter 5 7

(i) (a) (r, s) = (i, −i)

i 0 (c) D = 0 −i

(b) vr =

(ii) (a) (r, s) = (−3, −1)

−3 0 (c) D = 0 −1

2 (c) D = 0 11

12

0 −1

(b) vr =

(iii) (a) (r, s) = (2, −1)

−i , 1

(b) vr =

−1 , 1

4 , 1

vs =

i 1

vs =

vs =

1 1

1 1

(i) trace = 11, determinant = 14 3 5 (ii) 4 2 11 −5 1/2 −1 (iii) 1/2 −2 √ √ (iv) r = 7, s = − 7 for mA 3 1√ 3 1√ 17, s= − 17 for mB r= + 2 2 2 2 1 1 vr = 1 √ vs = 1 √ 2 , 2 for mA − 7− 7− 3 3 3 3 1 1 vr = 5 1 √ , vs = 5 1 √ for mB − + 17 17 4 4 4 4 (v) λ2 − 7 i i t t Mathematica xt = −9 + 5 + (−i) + 5 − i 2 2 i 1 + yt = ((−5 + 5i) + (10 + i)(−i)t − (1 + 10i)it ) 4 4

Answers to selected exercises Maple

13 14 15

J=

−i 0

1 1 xt = −9 + 5(−i)t + 5it + i(−i)t − i it 2 2 5 9 11 9 11 yt = − + (−i)t + i(−i)t + it − i it 2 4 4 4 4 0 1−i 1+i , V= i 1 1

5 (i) (x∗ , y∗ ) = (−9, − ), 4-period cycle results 2 −0.5 0 0 1 J= , V= 0 −0.4 −4 4

Chapter 6 1 8

9

10

(i) Minimum ABEHJ = 12, (ii) JHEBA, hence same. x(t) = 5 cos(1.11803t) + 8.68811 sin(1.11803t) y(t) = 3.2697 cos(1.11803t) + 0.688441 sin(1.11803t) 5e−t (−3e + 2e−2 − 2e2t + 3e1+2t ) e2 − 1 10e−t (−9e + 6e2 − 2e2t + 3e1+2t ) y(t) = 1 − e2 x(t) =

c˙ = (0.4k−0.7 − 0.1067)c k˙ = k 0.3 − 0.05k − c k∗ = 6.6047, c∗ = 1.4316

Chapter 7 λ2 = 3.5458 λ1 − λ0 1 1 λk = + k−1 + · · · − 1 + λ1 δ δ k−2 δ λ1 − λ 0 + λ1 lim λk = k→∞ δ

1 2

Chapter 8 2

3 4

(i) pt = 60 − 1.5pt−1 , (ii) pt = 15 + 0.5pt−1 , (iii) pt = 60 − pt−1 , 28 4 − pt−1 , (iv) pt = 3 3 (i) pt = 24 − 14(−0.25)t (ii) pt = 30 − 20(−0.9)t (i) pt = 4.5 − 0.2pt−1 , (ii) pt = 100 + 0.2pt−1 ,

oscillatory and divergent convergent cyclical oscillatory and divergent

cyclical and convergent convergent

681

682

Answers to selected exercises (i) p∗ = 4, pt = 4 − 0.5(pt−1 − 4), pt = 10 + 2.5(pt−1 − 10), p∗ = 10, pt = 6 + 0.5(pt−1 − 6), (ii) p∗ = 6, pt = 8 + 1.5(pt−1 − 8), p∗ = 8, ∗ q∗ = 20.5 (ii) 4 periods (i) p = 3.5, 14λ 49λ + 1− pt = pt−1 9 9 (i) 4 periods, (ii) 2 periods, (iii) 2 periods, (iv) 5 periods.

5

6 7

stable unstable stable unstable

(i) p(t) = 3 + c1 e−37.5t

(ii) stable $ 10 − 1.5w (i) w∗ = 4, (ii) Investigate f (w) = 3 t 2 stable pt = 1 + − (p0 − 1), 3 7λ 28λ + 1− pt = pt−1 5 5 (i) P˙ = r(P − P∗ ) − R (h∗ )(h − h∗ ) h˙ = g (P∗ )(P − P∗ ) − (d + n)(h − h∗ )

8 9 12 13 14

w ≤ 14/3 w > 14/3

Chapter 9 4

5

9 − 2a1 + a2 3 − 2a2 9 + a 1 q∗2 = 3

(ii) Cournot solution (q∗1 , q∗2 ) = 83 , 23 (i) q∗1 =

Firm 1 monopolist.

(i) (q∗1 , q∗2 , q∗3 ) = (1, 1, 1) (ii) q1,t = 2 − 12 q2,t−1 − 12 q3,t−1 q2,t = 2 − 12 q1,t−1 − 12 q3,t−1 q3,t = 2 − 12 q1,t−1 − 12 q2,t−1 (iii) Yes.

6

) (i) (q∗1 , q∗2 , q∗3 ) = ( 34 , 74 , 11 4 (ii) q1t =

3 7 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 4 4 3 t 1 1 + (2q10 − q20 − q30 ) 3 2

q2t =

7 7 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 4 4 3 t 1 1 + (−q10 + 2q20 − q30 ) 3 2

Answers to selected exercises 11 7 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 4 4 3 t 1 1 (−q10 − q20 + 2q30 ) + 3 2

, 17 (i) (q∗1 , q∗2 , q∗3 ) = 58 , 13 8 8 q3t =

7

(iii) q1t =

8 9 10

5 35 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 8 24 3 t 1 1 (2q10 − q20 − q30 ) + 3 2

q2t =

13 35 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 8 24 3 t 1 1 (−q10 + 2q20 − q30 ) + 3 2

q3t =

17 35 1 − (−1)t + (−1)t (q10 + q20 + q30 ) 8 24 3 t 1 1 (−q10 − q20 + 2q30 ) + 3 2

(a) (q∗1 , q∗2 ) = (1.778, 1.778) (b) (q∗1 , q∗2 , q∗3 ) = 43 , 43 , 43 (d) (q∗1 , q∗2 , q∗3 ) = (1, 1, 1) (c) (q∗1 , q∗2 ) = (1.176, 1.176)

(i) (q∗1 , q∗2 ) = 16 , 16 9 9 (ii) Yes.

, 16 Cournot solution (q∗1 , q∗2 ) = 16 . Yes, all paths converge on Cournot 9 9 solution. (i)

Chapter 10 2

yt = 1000 + 0.6yt−1 − 0.1yt−2

3

Smaller for any given time period.

4

(i) Yt = 110 + 4.75Yt−1 − 4Yt−2 , (ii) Yt = 110 + 3.75Yt−1 − 3Yt−2 ,

6

(i) Change in a, b or t. (ii) Change in k, u or m0 .

explosive explosive

12

No. Roots are (r, s) = (0.22984, −0.5715) with r − r∗ = 0.2825(y − y∗ ) associated with r r − r∗ = 0.2298(y − y∗ ) associated with s

14

(i) ( y∗ , r∗ ) = (27.907, 13.178) (ii) IS curve: r = 16.6667 − 0.125y LM curve: r = −33.3333 + 1.6667y (iii) trace = −0.129, determinant = 0.016, stable spiral.

15

(i) (y∗ , q∗ ) = (35.720, 0.720) (iii) q = 6.5498 − 0.1632y

683

684

Answers to selected exercises

Chapter 11 1 3 11

12

f (u) f (u) −β(1−δ)t +e 1− (i) π(t) = 1−δ 1−δ (r, s) = (−0.925 + 1.464i, −0.925 − 1.464i) (i) (m∗s , π e∗ ) = (700, 0) ˙ s = 0 gives π e = 68.29 − 0.098ms (ii) m π˙ e = 0 gives π e = 140 − 0.2ms (iii) π e = −488.125 + 0.625ms (i) ln S = ln λ − αλ

Chapter 12 1 2

Yt = 1320 − 0.5Yt−2 (i) Yt = 1320 − 0.7Yt−1 − 0.2Yt−2 (ii) Yt = 1320 + 0.5Yt−1

4

(i) Yt = 1250 + 0.5Yt−1 , Yt = 1250 + 0.3Yt−1 (ii) Y ∗ (m = 0.2) = 2500, Y ∗ (m = 0.4) = 1785.7

5

(i) ( y, r) = (44.39, 16.19) (ii) (y, r) = (38.12, 15.06) (iii) ( y, r) = (41.49, 14.75)

6

(i) ( y, r) = (52.447, 20.223) (y, r) = (50.663, 19.316) (ii) ( y, r) = (42.894, 13.447) (y, r) = (43.544, 13.772)

Yt = 1250 + 0.4Yt−1 , Y ∗ (m = 0.3) = 2083.3,

s ﬁxed, s = 1.7640145 s variable, s = 1.33 s ﬁxed, s = 1.7640145 s variable, s = 1.9195

Chapter 13 2 4

7 11 12

AM(r∗ = 12) p = 111.1 − 0.1s (i) GM0 : pt = 95.122 + 0.0244st AM0 : pt = 103.5938 − 0.0625st (s∗ , p∗ ) = (97.5, 97.5) (ii) GM1 : pt = 100 + 0.0244st AM1 : pt = 108.90625 − 0.0625st (s∗ , p∗ ) = (102.5, 102.5) (ii) C(s, p) = (104.07407, 100) (i) s˙ = −200 + 2s (ii) s¯ = 100 s = (m − p∗ ) − ky + u(r∗ + λ − π ∗ )

Chapter 14 2 3

69 years − ln 2 years b − (d + m)

Answers to selected exercises 4 5

6

8

ln λ years (i) 37 years (ii) k a a a −at p˙ −a p − p(t) = + p0 − e b b b a lim p(t) = , hence equilibrium never achieved in ﬁnite time period. t→∞ b −a (iii) p(t) = k = constant of integration c − kae−at ln(c/ak) which is ﬁnite p(t) = ∞ at t = −a −rt

(i) p(t) = pe0 e−a(e

−rt

−1)

(iii) p = 0, unstable; p = ea , stable (iv) lim p(t) = ea t→∞

10

√ y = 3x2 + c a a B= − T for T˙ = 0 b bk1 dk2 T for B˙ = 0 B = k2 − c

11

(i) 46.67 × 106 kg

12

(i) (a) E1 = (0, 0), (b) E1 = (0, 0), (c) E1 = (0, 0),

9

(ii) (a) 1.27 years

(b) 3.095 years

E2 = (25, 100) E2 = (0, 1), E3 = (0.375, 0.25) E2 = (1, 0), E3 = (0, 0.5)

(ii) (a) oscillations around E2 = (25, 100) (b) limit cycle around E3 = (0.375, 0.25) (c) competing predators at E2 = (1, 0) 13

(i) E1 = (0, 0) E2 = (3, 1) E3 = (0, 14 ) 1.4 0 0 −4.2 (ii) A1 = , A2 = , 0 0.6 0.6 −2.4 1.05 0 A3 = 0.15 −0.6 (iii) E1 = (0, 0), 0 r v = , 1

(r, s) = (0.6, 1.4) 1 s v = 0

E2 = (3, 1), (r, s) = (−1.2 + 1.039i, −1.2 − 1.039i) −1.732 + 2i −1.732 − 2i vr = , vs = i −i E3 = (0, 14 ), 0 r v = , 1

(r, s) = (−0.6, 1.5) 1 s v = 0.09

685

686

Answers to selected exercises 14

15

(i) E1 = (0, 0) 3 10 (ii) A1 = 0

E2 = (1, 0) E3 = (0, 1) E4 = −3 −3 0 10 20 , , A2 = 3 3 0 10 20 3 −1 −1 0 20 5 10 A4 = A3 = −3 −3 , −1 −1 20 10 10 5

2 3

, 23

(i) x1 (t + 1) = 0.4x3 (t) x2 (t + 1) = 0.6x1 (t) x3 (t + 1) = 0.95x2 (t) + 0.75x3 (t) (iv) 10.5% (v) 24%, 13% and 63%

Chapter 15 1

∂ht = st (aE−aet ) > 0, ∂et ∂ht = 1 − E−aet , ∂st

2 3

4 5

∂ 2 ht =0 ∂s2t

rk where E is the exponential E 2 a k h (i) = ak − e for f (s) e r a h = ln(ak) − e for g(s) ln e r ˆ and ln(h/e) = αˆ + βe. ˆ In each case αˆ and βˆ are (ii) Use h/e = αˆ + βe insufﬁcient to identify all parameters.

sm =

k , E

∂ 2 ht = −a2 st E−aet < 0 ∂e2t

f (sm ) =

No. (ii) (a) sm = 12.5, g(sm ) = 78.125, (b) s = 15 (iii) (a) sm = 16.5, e = 17, π = 21.675, g = 70.125 (b) π = 21 (iv)

s e π g

p = 0.8 w = 12 a = 2.5

p = 0.6 w = 15 a = 2.5

p = 0.6 w = 12 a=3

15.5 1.9 36.1 73.625

17.5 1.5 16.875 65.625

15.83 1.53 25.21 72.57

(v) s∗ = 8 (vi) e = 5 − 0.2s

Answers to selected exercises (vii) s∗ = 8, e∗ = 3.4 −0.5 −2.5 A= , (r, s) = (−0.25 + 4.32i, −0.25 − 4.32i) 7.5 0 tr(A) = −0.5, det(A) = 18.75 6 9 10

(i) e = 0.8 (iii) s = 600 (i) 34% (ii) group one, 90.7%; group two, 6.8%; group three, 2.5% h = 0.7192 with group 1, 94.5%; group 2, 4.5%; group 3, 1.0%

687

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695

Author index

Abell, M.L., 25 Allen, R.G.D., 114n, 141, 374 Arrowsmith, D.K., 84, 200 Attﬁeld, C.L.F., 494, 518 Azaraides, C., 4, 16, 25, 250, 483n, 518 Baker, G.L., 321 Barro, R.J., 275n, 330n, 374 Baumol, W.J., 121, 141, 298, 320, 321, 374 Beavis, B., 200, 285 Benhabib, J., 298, 312–314, 320, 321 Berry, J., 84 Blachman, N., 25 Blackburn, K., 285 Blanchard, O.J., 455, 458–459, 464n, 469 Blume, L., 217, 233, 250, 264n Borelli, R.L., 84, 200 Boyce, W.E., 37, 84, 162–163, 181, 200, 637 Braselton, J.P., 25 Braun, M., 49n, 84, 200, 637 Brock, W.A., 16, 25, 310n, 321 Brown, D.P., 25 Bryson, A.E. Jr., 285 Buchanan, N.S., 374 Buiter, W.H., 592 Bullard, J., 25 Burbulla, D.C.M., 25 Burmeister, E., 84, 275n, 285, 495, 518 Butler, A., 25 Cagan, P., 500–502, 505, 518 Carter, M., 518 Caswell, H., 637 Chiang, A.C., 114n, 141, 157n, 200, 250, 264n, 285, 355n, 374 Clark, C.W., 260, 261n Cole, W.A., 595, 637 Conrad, J.M., 260, 261n, 262n, 285

Coombes, K.R., 25 Copeland, L.S., 530n, 552, 592 Crandall, R.E., 25 Crooke, P.S., 25, 264n Crutchﬁeld, J.A., 642, 676 Cunningham, S., 676 Dasgupta, P.S., 676 Davies, S., 53, 84 Day, R.H., 16, 25, 31–34, 321 Deane, P., 595, 637 Demery, D., 494, 518 Dernburg, T.F., 552, 592 Devitt, J.S., 25 Diamond, P.A., 508, 510, 518 DiPrima, R.C., 37, 84, 162–163, 181, 200, 637 Dobbs, I., 200, 285 Dobell, A.R., 84, 275n, 285, 495, 518 Dodson, C.T.J., 25 Domar, E.D., 83, 141 Don, E., 25 Dornbusch, R., 541, 553–554, 579, 592 Duck, N.W., 494, 518 Dunn, M.R., 676 Eckalbar, J.C., 333, 374 Elaydi, S.N., 88–89, 118, 141, 217, 250 Ellis, W.J., 25 Ezekiel, M., 332, 346, 374 Farmer, R.E.A., 7, 25, 141 Fisher, C.A., 676 Flaschel, P., 182, 200 Ford, J.L., 592 Frenkel, R., 592 Friedman, J., 375, 423 Frisch, H., 518 Fryer, M.J., 285

697

698

Author index Gander, W., 25 Gandolfo, G., 337, 374, 375, 423 Gapinski, J.H., 141, 552 G¨artner, M., 552, 554n, 564n, 592 Gehrig, W., 423 George, D.A.R., 495, 497, 518 Giordano, F.R., 36n, 37, 84, 165n, 200 Gleick, J., 321 Glynn, J., 25, 301, 302, 319, 321, 338, 367n Goldberg, S., 141, 250 Gollub, J.P., 321 Goodwin, R.M., 336–337, 374 Gray, T.W., 25, 301, 302, 319, 321, 338, 367n Greenman, J.V., 285 Gregory, M., 455n, 469 Grifﬁths, H.B., 84, 118n, 141, 250 Grilli, V., 330n, 374 Groth, C., 484, 487, 518 Gulick, D., 308–310, 321 Haberman, R., 637 Hamilton, J.E., 676 Hartwick, J.M., 676 Heal, G.M., 676 Heck, A., 25 Henderson, J.M., 423 Hicks, J.R., 125–126, 141 Hilborn, R., 15, 321, 676 Ho, Y., 285 Holden, K., 493, 518 Holmgren, R.A., 19n, 141, 250, 341 Hommes, C.H., 293, 310, 315, 321, 363n, 374 Hoppensteadt, F.C., 637 Hrebicek, J., 25 Huang, C.J., 25, 264n

Kreyszig, E., 25 Krugman, P., 484, 489, 490, 518 L´eonard, D., 285 Leslie, D., 493, 628 Li, T.Y., 302n, 303 Long, N.V., 285 Ludsteck, J., 135n Lynch, S., 84, 301, 306, 306n, 321, 614n, 637, 669n, 676 MacDonald, R., 592 Maddock, R., 518 Mahajan,V., 53, 84 Malliaris, A.G., 16, 25 Mandelbrot, B., 286–287, 321 Mankiw, N.G., 358, 374 Mas-Colell, A., 182, 200 May, R.M., 288n, 321 McCafferty, S., 469, 478n, 518, 552 McMannus, M., 423 McVay, S., 676 Medio, A., 321 Miller, M.H., 592 Mirowski, P., 286, 321 Mizrach, B., 25 Mooney, D., 5n, 25 Mortensen, D.T., 508, 509n, 511, 514n, 518 Mullineux, A., 25 Mundell, R.A., 239n, 243 Muscatelli, V., 455n, 469 Neal, F., 8n, 25 Neher, P.A., 661n, 676 Nerlov, H., 374 Nicolaides, R., 25 Niehaus, J., 592 Norwinton, E.J., 25

Intriligator, M.D., 285 Jeffrey, A., 84, 141, 200 Jones, C.I., 83 Jong, F.J. de, 8n, 25 Judge, G., 262n

Obstfeld, M., 455n, 469, 592 Okuguchi, K., 423 Oldknow, A., 84, 118n, 141, 250 Olewiler, N.D., 676 Oxley, L., 495, 497, 518

Karakitsos, E., 552 Keiper, J.B., 25 Kelley, W.G., 141, 250 Kesley, D., 321 Keynes, J.M., 6 King, D.N., 330, 374 Kirk, D.E., 285 Koﬂer, M., 25

Parker, D., 405, 410, 423 Parkin, M., 330, 374 Parta, H., 25 Peel, D.A., 493, 518 Peng, W., 25 Percival, I., 84, 200 Peterson, A.C., 141, 250 Peterson, F.M., 676

Author index Peterson, R.A., 53, 84 Pilbeam, K., 552, 592 Pissarides, C.A., 518 Place, C.M., 84, 200 Pontryagin, L.S., 285 Pratt, J.W., 277n Quandt, R.E., 423 Ramsey, F.P., 275, 285 Renshaw, E., 637 Richards, D., 84, 200 Rodriguez, C.A., 592 Rødseth, A., 592 Rogoff, K., 455n, 469, 592 Romer, D., 275n, 285, 455n, 469 Ruskeepaa, H., 25 Sala-i-Martin, X., 275n Samuelson, P.A., 123, 141 Sandefur, J.T., 121n, 141, 212n, 250, 288n, 321, 637, 641n Scarth, W.M., 460n, 469, 518 Scheinkman, J.A., 25 Schwalbe, D., 25, 179n, 191, 200 Shafer, W., 16, 25 Shaw, W.T., 25 Sheffrin, S.M., 518 Shone, R., 8n, 19n, 21n, 25, 135, 141, 183n, 200, 212n, 239n, 240, 243, 250, 277n, 355n, 374, 375, 437n, 469, 472n, 473, 518, 552, 577n, 592, 638n, 649, 676 Simon, C.P., 217, 233, 250, 264n Skeel, R.D., 25 Smith, V.L., 676 Solow, R.M., 141

Stevenson, A., 455n, 469 Swift, R., 5n, 25 Szidarovsky, F., 423 Takayama, A., 84, 285 Teigen, R.L., 429n, 469 Theocharis, R.D., 423 Thompson, J.L., 493, 518 Tigg, J., 25 Tinbergen, J., 240 Tobias, A., 405, 423 Tobin, J., 455, 469 Tranter, N., 637 Tu, P.N.V., 84, 141, 200, 250, 321 Turnovsky, S.J., 518 Uhl, J., 25 Vandermeer, J., 637 Varian, J.R., 25 Wagon, S., 25, 179n, 191, 200 Walkington, N., 25 Walters, C.J., 676 Waugh, F.V., 345, 346, 374 Weil, D.N., 358, 374 Weir, M.D., 36n, 37, 84, 165n, 200 Whigham, D., 262n Whitby, S., 405, 423 Whitmarsh, D., 676 Wolff, E.N., 121, 141 Wolfram, S., 25 Yorke, J.A., 302n, 303 Zellner, A., 642, 676

699

Subject index

absolute adaptation mechanism, 415 absolute policy adaptation, 412–414 active policy rule, 493–494 adaptive expectations, 364–365, 470, 484 see also hyperinﬂation adjoint variable, 253 aggregate demand (AD), 473–475, 476 shocks, 490, 493 aggregate supply curve, long-run, 474–475 shocks, 490, 493 annuities, 108 aperiodic series, 293–294 arctan function, 363–365, 366, 407 art forgery, testing, 49–50 asset market Dornbusch model, 555–557, 560–567, 568–570 monetarist model, 587–589 resource discovery, 583–586 asymptotic stability, 402, 618, 624, 625 see also ﬁxed points asymptotically stable rest point, 57 attractors, 56–59, 60, 89–93, 95, 97, 127–128 chaotic, 310 Cobweb model, 341 see also strange attractor, H´enon map, Lorenz [strange] attractor auxiliary equation, 61–64 baby boom, housing market, 358, 361–363 baby bust, 359 balance, 239–245 balance of payments, 524–530, 532–535 external balance, 239–243 see also IS–LM–BP model Bank of England, interest rates, 489 Bank of Japan, 490 basin of attraction, 148

700

Beckham, D. and V. erratic behaviour, 314n Bernoulli equation, 36, 44, 51n, 81 bifurcation chaotic behaviour, 298, 304–307 demand and supply, 365–366 diagram, 288–289, 299, 300–301, 318–319 H´enon map, 308 period doubling, 296n, 297, 299n, 366 pitchfork, 292, 296, 299n: sub/supercritical 292n points, 288, 295, 299n saddle-node, 290, 299n transcritical, 291 value 287, 289–290, 295 see also Hopf bifurcation biological growth curve, see ﬁsheries biomass, see ﬁsheries biotic potential, 640 Bretton Woods, 528n, 538 Brock’s residual test, 310n butterﬂy effect, 16n, 310 Cagan model, see hyperinﬂation calculus of variations, 251 canonical form, 219–220, 224–228 complex root, 232–234 repeated root, 229–231 capital ﬂows, 6 capital mobility Dornbusch model, 558, 562, 564–567 IS–LM–BP model, 541–542, 544 carrying capacity, see ﬁsheries Central Limit Theory, 287 centre point, 177 chaos, 288 rational choice, 314 theory, 293–301

Subject index chaotic attractor see attractors chaotic behaviour of systems, 298–300 demand and supply, 363–367 chaotic series, 310 characteristic equation, 110, 205 characteristic polynomial Mathematica, 205 Maple, 207 characteristic roots, see roots Cobb–Douglas production function Cagan model, 502–503 Solow growth model, 36, 44, 56, 59, 67, 131 cobweb model, 12, 87–88, 92–93, 102, 332–337 interrelated markets, 346–349 using Mathematica and Maple, 338–339, 367–371 phase plane, 339–346 with stock behaviour, 352–353 competitive equilibrium, 353–358 complementary solution, 64, 65, 116, 117–118 complex conjugate roots, see roots compound interest, 39–40, 105–108 conjectural variation, 376 constant elasticity, 52 continuous control problem, 255–259 with discounting, 265–270 phase diagram, 270–283 Corel Draw, 22, 125 corn-hog cycle, 346–349 costate variable, 253 Cournot duopoly, 379 Cournot solution (Cournot–Nash solution), 375–377, 378, 381, 382, 384, 387, 389, 390 Cramer’s rule, 491, 584 critical point, 55, 167, 174, 177, 178–179 see also nodes Crutchﬁeld and Zellner model, see ﬁsheries De Moirre’s formula, 232 deﬂationary spiral, 489, see also dynamic liquidity trap demand and supply chaotic behaviour, 363–367 continuous time, 326–332 labour market, 329–332 short market, 327–329 stock behaviour, 349–353

stock levels, 329 see also Cobweb model; competitive equilibrium demand equation, 11–12, 587 see also hyperinﬂation demand pressure curve, 476–478, 480 Derive, 20 deterministic systems, 15–16 difference equations, 4 autonomous, 86 ﬁrst-order, 85, 99–105 ﬁrst-order linear, 37–39 linear, 85 non-autonomous, 86 non-linear, 85 recursive, 85, 87 second-order homogeneous, 110–115 second-order non-homogeneous, 116 second-order solutions, 110–118 second-order linear difference solutions, 61–66, 110–115 simple, 99–105 differential equations, 4, 26–27 autonomous, 27–28 approximation, 66, 68–70 complex roots, 164–166 constant-coefﬁcient nth-order, 27 ﬁrst-order 142: initial condition, 30, 39; with chaos, 298, 300, 309–310 ﬁrst-order linear, 37–39 general solutions, 27–37, 64, 65, 113, 114, 116–118, 124, 142–143 graphical solution, 29 implicit solution, 29, 41 homogeneous, 27 linear, 27, 142, 154–155 matrices, 156–161 non-homogeneous, 27 non-linear, 27, 67, 142 order classiﬁcation, 27 ordinary, 27 partial, 27, 30, 64, 65, 116, 117 particular solution, 30 qualitative properties, 41–45 repeating roots, 162–164 second-order linear homogeneous, 59–64 second-order linear non-homogeneous, 64–66 separable, 45–48, 81: logistic curves 50–52; radioactive decay, 48–50 time-invariant, 27 diffusion, 53–54

701

702

Subject index dimensionality, 8–12 diminishing returns, 645–646 direction ﬁelds, 42–45, 152 Mathematica, 77–79, 189–191 Maple, 79–80, 189–190, 192–194 discount rate, 108–109 discounting, and optimal control, 265–270 discrete time control model, 259–264 disequilibrium inventory model, 315–319 dominant solution, 110 Dornbusch model, 559–564 capital immobility, 564–567 original, 559–564 perfect foresight, 567–573, 574–581 duopoly incomplete and non-instantaneous adjustment, 389–392, 395–396 output adjusting, 377–380, 386–387 static, 375–377 dynamic adjustment process, 326 constraint, 661 liquidity trap, 484–487, 489 modelling, 3–4 multiplier, 426–427, 431 programming, 251 dynamics, computer software, 17–24 dynamical systems autonomous, 86, 142–145, 149 matrix speciﬁcation, 156–160 non-autonomous, 86, 142 random shocks, 286, 490, 492–493 effective interest rate, 105–106 eigenvalues, 158–161, 166, 168 ﬁsh harvests, 670–672 linear systems, 208–214, 226–228 Mathematica and Maple, 213–214, 626 phase plane, 237–239 population growth, 624–626, 628–630 eigenvectors, 158–162, 164–165, 166, 168, 170, 172 ﬁsh harvests, 670–672 independent, 172–173 linear systems, 208–214 Mathematica and Maple, 213–214, 626 population growth, 624–626, 628–630 employment rate, 506–509 singular curve, 511 sticky wage theory, 331 endogenous propagation mechanism, 16 environmental economics, 7–8

environmental resistance, 640, 641 equilibrium lines, 432 see also trajectories equilibrium point, 55–56, 58 autonomous systems, 145–147, 153 bifurcation theory, 288–290 in discrete dynamic systems, 88–89, 94 stability, 454 Euler’s approximation, 183–185, 437n Euler’s theorem, 355 European Central Bank (ECB), 489–490 Evans, C. and erratic behaviour, 314n Excel, 17, 18, 20, 25 discounting, 267 discrete optimisation problem, 262–264 employment level, 512n equilibrium value, 280n trajectories, 220–221, 309n, 441–442 exchange of stability, 291, 295, 299 exchange rate ﬁxed, 532–539 ﬂoating, 6, 538, 539–544 internal and external balance, 243–245 exchange technology, 508 exogenous shocks, 16 expectations, 6–7 inﬂationary, 470–472, 476, 477–478, 482, 501 expectations-augmented Phillips curve, 470, 484 expenditure model, see income-expenditure model explicit solution, 29, 41 exponential growth curve, 596 external equilibrium, 527 Federal Reserve, 489 Feigenbaum’s universal constant, 301–302, 309 Fibonacci series, 141 ﬁrst-order initial value problem, 30 ﬁscal expansion ﬁxed exchange rate, 532–535 Tobin–Blanchard model, 462–463, 464 ﬁscal shocks, 444–446 ﬁsheries biological growth curve, 638–644, 646 biomass, 638–639, 645, 649, 651–652, 653 carrying capacity, 639, 640, 641, 668 Crutchﬁeld and Zellner model, 642–644 entry and exit, 644, 648–649, 650–653

Subject index harvesting, 601, 648, 640, 641, 642, 653, 669–672 harvesting function, 644–646, 649, 659 proﬁts, 644, 647, 650, 658, 659: under open access, 647–657 optimal control problem, 658–661 schooling activity, 661–669 stable spiral, 653–654, 655 ﬁxed points, 55–58, 60 asymptotically stable, 12, 57, 147 autonomous systems, 145–148, 150–153 in discrete dynamic system, 88, 98–99, 101, 127, 224 ﬁsh stocks, 664 IS–LM model, 453 neutrally stable, 147n qualitative properties, 272–275 shunt, 56 stable, 56, 58, 8890, 93–94, 147 unstable, 147 see also attractor, repellor ﬂexible wage theory, 330 ﬂex-price model, 553 see also Dornbusch model, monetarist model ﬂoating exchange rate, see exchange rate ﬂow of solutions, 42 ﬂow variables, 10 foreign interest rate, 538–539 foreign market adjustment function, 545 see also balance of payments; Dornbusch model; IS–LM–BP model global stability, 58, 59, 68–69, 89–90, 148, 153 monetarist model, 588 Gompertz function, 53–54, 81, 635, 673 goods market, 473, 495 Dornbusch model, 554–557, 560–570 resource discovery, 583–586 see also IS–LM model; IS–LM–BP model goods market equilibrium condition, 424 Goodwin model, 337 government spending, internal balance, 240–245 half-life, 48–50 Hamiltonian function, 253–260, 266, 268, 272, 273, 658, 660–661 current value, 266, 268–269, 277

Harrod–Domar growth model, 34, 56, 59, 83, 101 using Mathematica and Maple, 133 H´enon map, 221, 301, 307–310 hiring frequency, 508–509 Holling-Tanner predatory–prey model, 199 Hopf bifurcation, 304–307, 311 housing market, 358–363 demographic change, 358–363 hyperinﬂation Cagan model, 500–505 hysteresis, 306–307 imitation process, 415–419 income-expenditure model, 519–524 indeterminism, and random shocks, 286 inﬂation growth, 494–500 Lucas model, 490–493 Phillips curve see IS–LM model see also hyperinﬂation initial value problem, 30, 39, 81, 86–87 Mathematica, 71–72 Maple, 74–76 insider–outsider model, 509n, 514n integral curves, 43–44 integrating factor, 38 intercepts, see IS–LM model interest parity condition, 587 interest rate, and external balance, 240–243 internal rate of return, 109–110 interpolating function, 437 invariant set, 180 inventory model, and rational expectations, 315–319 investment function IS–LM model, 447 q-theory, 455 IS curve, see IS–LM model, Tobin–Blanchard model IS–LM model, 24, 424–425, 465–467, 484–485, 487–488, 519 BP curve, 529–530, 532–535 continuous model, 431–437, 447–453 lags, 425–426, 429 income, 424–428, 447–448 intercepts, 448–449 interest rate, 424, 447–448 investment function, 447 monetary expansion, 535–538 non-linear, 453–455 Phillips curve, 472–483

703

704

Subject index IS–LM model (cont.) shift in IS curve, 442–443 shocks, 443–447 IS–LM–BP model, 529–530 ﬁscal expansion, 532–535 ﬁxed prices, 545–551 foreign interest rate rise, 538–539, 544–545 monetary expansion, 535–538, 542–544 isoclines, 42–44, 355–356, 358, 399, 456–457, 459, 462 Cagan model, 505 ﬁsh stocks, 663, 667 inﬂation, 478, 479, 480 liquidity trap, 485–486, 488–489 Mathematica, 77 Maple, 79–80 money growth, 497–498 population growth, 612–613, 615–616, 617–618 wage determination, 511, 513–514 isoproﬁt curves, 376–377 Jacobian matrix, 619n job-ﬁnding rate, 506–507 John, Sir E. and erratic behaviour, 314n Jordan blocks, 217–218 Jordan form xi, 216–217, 225, 230, 231 k-periodic point, 93–94 Kuhn–Tucker condition, 264n Lagrangian, 253–260, 266, 267, 658, 675 current value, 269 Leslie matrix xi, 628, 629, 669, 670–672, 675 Liapunov theorem, 68, 147 limit cycle, 147, 179–183, 514 Hopf bifurcation, 306 large-amplitude, 306–307 linear approximation, 127–130 dependence, 60–61 systems, 201–204 linearity, exception to norm, 287 Li–Yorke theorem, 303, 314 LM curve, see IS–LM model, Tobin–Blanchard model logistic equation, 118–123 chaos, 293–301 productivity growth, 121–123 Sarkovskii theorem, 302–304

logistic function, 598–599 logistic growth curves, 44, 50–52, 56–57, 299, 597, 598 direction ﬁeld, 46 ﬁxed point, 56 logistic growth equation, 597–601, 603, 630–632, 638–639 see also ﬁsheries long-run aggregate supply curve, 474–475 Lorenz [strange] attractor, 301, 307, 310–312 Lorenz curve, 186, 191, 193 Lotka–Volterra model, 604, 607–611, 617, 618, 619 Lotus, 1, 2, 3, 17, 20 Lucas model, 49–53 Lyapunov dimension, 310, 312 Malthusian population growth, 4, 33, 100–101, 593–596, 600, 602–603 direction ﬁelds, 44, 46 ﬁxed points, 55, 59 Maple xi, xii, 19–23, 25 basic matrices, 204–205, 206–207 Cobweb, 370–371 complex roots, 134 DEtools, 79–80 differential equations, 73–77, 149–150, 186–190, 192–194 direction ﬁelds, 44 discrete systems, 214–216 dsolve, 73–77, 187–189 eigenvalues/vectors, 212–214, 230 employment level, 512n equilibrium values, 280 Jordan form, 216–217 linalg, 204 LinearAlgebra, 204 logistic equation, 118–119, 134, 137–138, 631–632 optimal control problem, 258n oligopoly models, 378–379, 382, 383, 411 oscillations, 125 parametric plots, 195–196 phase plane in Cobweb, 342–343, 345–346 phaseportrait, 192–193 population growth, 600n, 604, 607, 610, 614: procedural function 76 multispecies model, 633–634 recursive equations, 105, 131–134: Cobweb 338–339, 340

Subject index rsolve, 131–134, 214–216, 378–379, 382 trajectories, 222–223, 439–441, 499 marginal revenue product per worker, 509 market clearing model, 330, 509–513 market share, and R&D, 406, 409–410, 416, 417–419 Marshall–Lerner condition, 526n, 547 MathCad, 20 Mathematica, xi, xii, 19–23, 25 basic matrices, 204–206 bifurcation diagrams, 301, 307 Cobweb model, 334, 342–343, 345–346, 367–369 complex roots, 134, 195–196 differential equations, 70–73, 149–150, 186–191 direction ﬁelds, 44 discrete systems, 214–216 DSolve, 70–71, 187–189, 195 eigenvalues/vectors, 212–214, 230 equilibrium values, 280 H´enon map, 308 ImplicitPlot, 125 Jordan form, 216–217 NDSolve, 70, 72–73 oligopoly model, 378, 382–383, 411 oscillations, 125 parametric plots, 195–196 piecewise function, 318n PlotField, 77–79 PlotVectorField, 190 population growth, 600n, 604, 607, 610, 614: logistic equation, 630–631; multispecies model, 632–633 recursive equations, 105, 131–134 recursive Cobweb, 338–339, 340 RSolve, 131–134, 214–216, 378, 382 trajectories, 222–223, 437–438, 452, 667 VisualDSolve, 179n, 181 MatLab, 20 maximisation principle problem, 658 maximum principle, 661 see also Pontryagin maximum principle maximum sustainable yield, 640, 641 mean generation time, 594n Microﬁt, 20 migration, 596, 602–603, 604 monetarist model, 586–589 monetary expansion Dornbusch model, 560–562, 568 ﬁxed exchange rates, 535–538 Tobin–Blanchard model, 463–465 monetary shocks, 443–447

money market, 473, 495, 501 adjustment function, 545 Dornbusch model, 554–555, 560, 565, 566 resource discovery, 58–64 see also IS–LM model; IS–LM–BP model money supply growth, 474, 493, 500 open economy, 530–532 policy rules, 493–494, 495 multiple equilibria, 12, 15 multiplier, 523 multiplier–accelerator model, 123–126, 427–428 Mundell–Fleming model, 519, 537, 541, 553 mutualism, 604 Nash solution, see Cournot solution natural growth coefﬁcient, 604 natural unemployment rate, 506–507 net present value, 109 nodes, 167–168 improper, 174, 402, 404 proper, 172 spiral, 176, 402, 404, 479, 481 Van der Pol equation, 305 non-accelerating inﬂation rate of unemployment (NAIRU), 471–472 non-linear discrete systems, 245–247 nonlinearity, 8, 12–15 chaotic behaviour, 15–17, 300 North Sea gas and oil discovery, 554, 581–586 North Sea herring ﬁsheries, 646 Occam’s razor, 20n Okun’s law, 472 oligopoly two-ﬁrm models, 375–380, 386–387, 389–392, 395–396 three-ﬁrm models, 380–383, 387–388, 393–394, 396–397, 400–401, 404 four-ﬁrm models, 384–386, 388–389, 394–395, 397–398 openness, 523–524 optimal control problem, 251–252 continuous model, 252–259 discounting, 265–270 maximum control, 259–264 see also ﬁsheries optimal growth model, 16

705

706

Subject index orbit, 55, 145, 179 orbital stability, 179 ordinary differential equation, 27 oscillations, 125, 129 over-crowding, 605, 605 competition, 611–617 ﬁsh stocks, 640 predatory–prey model, 617–619 overlapping generation model, 16 overshooting, 442–444, 447, 466, 538, 541, 544 ﬂex-price models, 553, 557, 574 parametric plot, 194–196 parity rate, 527–528 partial differential equation, 27 particular solution, 30, 64, 65, 116, 117 passive policy rule, 493–494 paths, 55, 145 counter-clockwise, 449 counter-clockwise spiral, 437 counter-clockwise stable spiral, 451 spiral, 435, 442–443 stable, 449 stable spiral, 451 unstable spiral, 449, 451 see also trajectories pelagic whaling, 638 perfect foresight, 15, 495 Dornbusch model, 567–573, 574–581 monetary model, 587, 589 rational expectations, 502–503 periodic solution, 93 Perron–Frobenious theorem, 628n Peruvian anchoveta ﬁsheries, 646 phase diagrams, 3, 4 control models, 270–283 single variable, 54–59 two-ﬁrm model, 400 phase line, 55, 56–57 phase plane, 45, 145, 166 discrete systems, 235–239 housing market, 360–363 internal and external balance, 239–245 optimal trajectory, 271 see also Cobweb model phase portrait, 145, 149, 189 direction ﬁelds, 190–194 discrete systems, 219 Phillips curve, 470–472, 477–478, 481, 494 and Cagan, 502–503 and Lucas, 490

Poincar´e–Bendixson theorem, 180 policy announcement, time periods, 579–581 Pontryagin maximum principle, 251–264 continuous model, 252–259, 272 discrete model, 259–264 population growth by age of women, 626–627 Malthusian, 593–596, 600, 603 multispecies analysis, 619–626, 632–634 natural changes, 601–602 see also ﬁsheries, logistic growth equation, migration, predatory–prey relationship portfolio balance condition, 661 predatory–prey relationship, 604, 607–611, 617–619 present value, 108–109 price inﬂation, see Phillips curve price-ceiling Cobwel model, 345 principle of effective market classiﬁcation, 243 product differentiation, 418 product innovation, 406 production function, 272n proﬁt function, 509–510 proportional policy adaptation, 412–414 purchasing power parity (PPP), 553–555, 557, 560, 562–564, 568, 584–585, 587 q-theory of investment, 425, 455 QuattroPro, 17, 20 radioactive decay, half-life, 33–34, 48–50 ﬁxed points, 59 Ramsey growth model, 275–283 rational expectations, 7, 15, 228, 494, 495 Cagan model, 501, 505 Dornbusch model, 567–573 Lucas model, 490–493 monetarist model, 587, 589 reaction coefﬁcients, 435–437, 451, 656–657 real income level, internal balance, 239–240 real wages, 502–503 recursive equation, 85, 87 dominant, 405 Mathematica/Maple, 134 multiplier–accelerator model, 123–126 solutions, 105–108

Subject index regression, spreadsheets, 19 relative risk aversion, 277 repellor, 56–57, 59, 60, 89–93, 95, 97, 128, 147 Cobweb model, 341 resource depletion rate, 586 resources, 9 rest point, 55 roots characteristic, 461, 481, 666 complex conjugate, 63–64, 65, 111, 114–115, 125, 159–160, 164–166, 174–177, 178–179, 217–218, 231–234, 428 real and distinct, 61–62, 64, 111–112, 124–125, 159–161, 167–169, 178, 217–218, 223–228, 405 real and equal, 62–63, 64, 111–113, 159–160, 172–174, 178, 228–231 R¨ossler attractor, 199, 200 R¨ossler equations, 320–321 saddle path, 15, 169, 282, 447, 452–453, 460, 461, 464, 572, 575–576, 585, 606–607, 615, 664, 667–668 saddle point, 169–170, 179, 2325, 274, 278, 281, 461, 622, 666 Cagan model, 503–505 housing market, 361–362 oligopoly model, 400–401, 402, 404 stable arm, 169–170, 172, 227–228, 237, 275, 281–282, 361–362, 461, 462–466, 499, 505, 570–572, 576–577, 623, 625, 668 unstable arm, 169–170, 172, 227, 237, 275, 281 wage determination, 514 Sarkovskii’s theorem xi, 302–304 Schumpeterian dynamics xi, 414–419 secondary dimensions, 9 selection process, 414–419 Shazam, 20 shirking model, 509–513, 514 slopes ﬁsh stocks, 652 IS–LM model, 430, 448–449, 451, 454–455, 527 Tobin–Blanchard model, 457–459 Solow growth model, 16, 34–37, 56, 67, 82, 495 direction ﬁelds, 44, 47 discrete time, 130–131: multiple equilibria, 59

speculative demand for money, 453 spiral point see nodes spreadsheets recursive systems, 19–20 SPSS, 20 stability problem, 97–99, 128–129 stability competitive equilibrium, 353–358 demand and supply models, 349–353 discrete systems, 223–234 expenditure model, 520–523 linear systems, 203–204, 219, 551, 619–620 local, 12–15, 59, 97, 127–128, 148, 454: Cobweb 344; ﬁsh stocks, 641; liquidity trap, 486–487 Lorenz system, 311 non-linear systems, 601, 620–626 oligopoly models, 385–386, 387–389, 392, 396, 397–400, 402, 403 see also asymptotic stability, global stability stable ﬁxed point see attractor, repellor, ﬁxed points state diagram, 627 Statgraphics, 20 sticky prices, 557, 566, 573 sticky wage theory, 330–332 stock behaviour demand and supply models, 349–353 stock market behaviour, see Tobin–Blanchard model stock variables, 9–10 stock-adjustment model, 32 stock-ﬂow, 6–7, 10 strange attractor, 186, 307–312 survival of the ﬁttest, see selection process Systat, 20 Taylor expansion, 67, 128, 460 non–linear discrete system, 246 Solow growth model, 131 tent function, 320 Thatcher, M., 592 time-independency, 144–145 time-series data discrete processes, 287 randomness, 286 Tobin–Blanchard model, 24, 424–425, 455–465, 467 trade-cycle model (Hicks), 125–126 trajectories, 55, 145, 149, 177–178 Cagan model, 505

707

708

Subject index trajectories (cont.) deﬂation, 489 discrete systems, 220–223 Dornbusch model, 559, 572–573, 577–579, 580–581 eigenvectors, 173–174, 390 using Excel, 441–442 IS–LM model, 433–435, 437, 442–445, 448 IS–LM–BP model, 54–50 using Maple, 439–441 using Mathematica, 437–438, 452 population growth, 605–607, 614–615: ﬁsh stocks, 652–653, 655–657, 664, 667–668; Lotka–Volterra model, 608, 610 rational expectations, 514, 535, 536, 541 see also paths transactions demand for money, 453 transient chaos, 293 TSP, 20 two-cycle result, 97 undershooting, 564, 566–567 undetermined coefﬁcients, 65–66 unemployment level, 506–509 see also Phillips curve

unit limit cycle, 181 utility, satisfaction, 9 vacancy rate, 507–508 value singular curve, 511 Van der Pol equation, 181–182, 190, 193 bifurcation features, 304–307 vector forces, 151–156 Cagan model, 505 discrete systems, 235–236 ﬂex-price models, 585 inﬂation, 478–480 IS–LM model, 433–435 liquidity trap, 486 market clearing model, 513–514 phase plane analysis, 239, 240, 356–357 population growth, 605–606, 615–616: ﬁsh stocks, 652, 66–74; predatory–prey model, 609 Tobin–Blanchard model, 459–460 Vermeer, 50 VisualDSolve, 179n wage determination, 509–513 Walrasian price and quantity adjustment, 182–183, 199, 354 warranted rate of growth (Harrod), 34 Wronksian, 198

## FAQs

### What is a phase diagram in economics? ›

• Phase diagrams are **graphical representations of a dynamic system**. **away from its steady state**: •Graphical: usually in two dimensions, so there are two variables. – the state variable (eg capital) and the choice variable (eg. consumption)

**What is the definition of economic dynamics? ›**

Econodynamics is **an empirical science that studies emergences, motion and disappearance of value**—a specific concept that is used for description of the processes of creation and distribution of wealth.

**What is a saddle path? ›**

Saddle-path stability is a central concept in dynamic economics, being **the mathematical concept that is consistent with dynamic adjustment that results from purposeful behavior, and can accommodate structural shifts**.

**What is phase diagram answers? ›**

Phase diagram is **a graphical representation of the physical states of a substance under different conditions of temperature and pressure**. A typical phase diagram has pressure on the y-axis and temperature on the x-axis. As we cross the lines or curves on the phase diagram, a phase change occurs.

**What are the 3 parts of the phase diagram? ›**

The diagram is divided into three areas, which represent the **solid, liquid, and gaseous states of the substance**. The best way to remember which area corresponds to each of these states is to remember the conditions of temperature and pressure that are most likely to be associated with a solid, a liquid, and a gas.

**What are the 4 types of economies in economics? ›**

Each economy functions based on a unique set of conditions and assumptions. Economic systems can be categorized into four main types: **traditional economies, command economies, mixed economies, and market economies**.

**What is an example of dynamic in economics? ›**

The models which are directly considering time factor are usually called dynamic. In such models all variables of economic processes and systems are functions of time. Examples of dynamic models are **equilibrium processes by Walras and the interaction of supply and demand by Marshall**.

**Why is economic dynamics important? ›**

Dynamic economics has an important place in economics because **many economic theories are based on it**. For example, saving and investment theory, theory of interest, effect of time element in price determination, etc. are based on dynamic economics.

**What is good saddle placement? ›**

Your saddle fits just right if it sits level on your horse's back and the bars of the tree do not pinch. **The front of your saddle should be positioned behind your horse's shoulder blade**, allowing him freedom of movement.

**What is the Wronskian method? ›**

In the mathematics of a square matrix, the Wronskian (or Wrońskian) is **a determinant introduced by Józef Hoene-Wroński (1812) and named by Thomas Muir (1882, Chapter XVIII)**. It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

### What does an unstable saddle mean? ›

Saddle when eigenvalues are real and of opposite signs. **The saddle is always unstable**; Focus (sometimes called spiral point) when eigenvalues are complex-conjugate; The focus is stable when the eigenvalues have negative real part and unstable when they have positive real part.

**What is the main purpose of phase diagram? ›**

A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used **to show conditions (pressure, temperature, volume, etc.)** **at which thermodynamically distinct phases (such as solid, liquid or gaseous states) occur and coexist at equilibrium**.

**Why is phase diagram important? ›**

Phase diagrams are important for a metallurgist, as they **provide relationship between phases in a system as a function of temperature, pressure and composition**. The development of microstructures of an alloy of a particular composition at different temperatures is clearly depicted by a phase diagram.

**What is a phase diagram give an example? ›**

A phase diagram **depicts the phase change of the substance at, for example, a certain pressure and temperature**. A phase diagram may represent a PT curve (pressure, temperature curve). The PT curve is formed by taking readings at various pressure and temperature conditions.

**What is the most common phase diagram? ›**

The **pressure-temperature phase diagram** is the most common and basic type. A phase diagram of the type shown above plots pressure (in atmospheres) versus temperature (in degrees Celsius or Kelvin).

**What is the rule of phase diagram? ›**

The phase rule states that **F = C − P + 2**. Thus, for a one-component system with one phase, the number of degrees of freedom is two, and any temperature and pressure, within limits, can be attained.

**What are the four components of a phase diagram? ›**

To understand the basics of a one-component phase diagram as a function of temperature and pressure in a closed system. To be able to identify the triple point, the critical point, and four regions: **solid, liquid, gas, and a supercritical fluid**.

**What are the 3 main economic systems? ›**

There are three main types of economies: **free market, command, and mixed**. The chart below compares free-market and command economies; mixed economies are a combination of the two. Individuals and businesses make their own economic decisions. The state's central government makes all of the country's economic decisions.

**What are the types of economy 3? ›**

There are 3 types of economic systems, namely **mixed economy, capitalist economy, and socialistic economy**. Here are some general characteristics of an economy: The type of economy is based on the means of production and ownership of resources.

**What are the 4 types of economic systems and their characteristics? ›**

In economics, four types of economic systems characterize most economies around the world: **traditional, command, market, and mixed economies**. A traditional economic system focuses exclusively on goods and services that are directly related to its beliefs and traditions.

### What is the most dynamic economy in the world? ›

**United States**. The United States of America is a North American nation that is the world's most dominant economic and military power. Likewise, its cultural imprint spans the world, led in large part by its popular culture expressed in music, movies and television.

**What are the characteristics of a dynamic economy? ›**

(i) In a dynamic economy, **population grows;** **(ii) Quantity of capital grows; (iii) Modes of production improve;** **(iv) Industrial institutions undergo changes**.

**What does dynamics economics deal with? ›**

Dynamic economics → deals with **relations and processes under the assumption of change in either the absolute or the relative economic quantities**. Static economics → study the relation of forces at the equilibrium level.

**Why are economic dimensions important? ›**

The economic dimension **handles all economic outputs/externalities of the event firm and the individual event**. Values measured range from the direct economic impacts on the host community and the world- to the more complex indirect impacts, both being of great interest to event researchers.

**Why is dynamics useful? ›**

**Without dynamics, objects would not interact with each other and cause changes in motion**. Changes in motion and other types of interactions are central to the entire field of physics.

**What degree should your saddle be? ›**

A flat saddle should only be set around **0 to 2 degrees of nose down tilt** to prevent the rider sliding forward on the saddle. A wave-shaped saddle with a kick up at the back is designed to be tilted down to some degree and can generally be tilted around 2 to 5 degrees.

**Is a higher saddle better? ›**

**A saddle that is too high will cause the hips to rock back and forth**. Not only does this detract from pedalling efficiency, but it can also be extremely uncomfortable. Discomfort can show up in your lower back or as knee pain (especially in the back of the knee).

**What does a good saddle fit look like? ›**

**The saddle should have 2-3 fingers clearance on the top and around the side of the withers**. The saddle must have be an opening (clearance) on the sides of his withers to accommodate the shoulder rotation upwards and backwards during movement. A horse whose saddle pinches his withers may be reluctant to go forward.

**How do you quickly check linear independence? ›**

The linear independence of a set of vectors can be determined by **calculating the determinant of a matrix with columns composed of the vectors in the set**. If the determinant is equal to zero, then the set of vectors is linearly dependent. If the determinant is non-zero, then the set of vectors is linearly independent.

**What is the linearly independent solution theorem? ›**

Theorem 5 **The columns of a matrix A are linearly independent if and only if the equation Ax = b has a unique solution for every b ∈ Col A**. This gives the connection between linear independence of vectors and uniqueness of solutions to linear systems.