# Saddle path Stability – Economics For Good (2023)

I have always found that tabulating information, cases and sub-cases, is very useful in providing a coherent picture of the situation, making interlinks visible, as well as general rules.

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In the context of Dynamic Economics, stability of differential equations that describe laws of motion of economic variables are of primary interest. Since in Economics we are almost exclusively concerned with the existence of “saddle-path stability“, we usually do not provide the whole picture and possibilities, of which saddle path stability is just one case.

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So I wrote and I attach here a summary of what can happen (and when) in a 2 by 2 system of differential equations, in terms of the system’s stability. While writing it, I hit upon an interesting way to show how important is the actual arrangement of equations when one wants to use matrix algebra -arrangements that are equivalent in the “plain” formulation of a system of equations, become totally different if viewed through the lenses of matrix algebra, and lead to different results.

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Stability of Systems of Diferrential Equations

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Maybe in the future I will do the same for Difference Equations, since they are more convenient when one wants to work in a stochastic framework.

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## FAQs

### What is saddle path in economics? ›

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. Dynamic Stability for economic models.

What is the equation of the saddle path? ›

(1) q(T - ε)/q(T + ε) = (1 - f).

Can a saddle point be stable or unstable? ›

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.

Why is a saddle point unstable? ›

As the eigenvalues are real and of opposite signs, we get a saddle point, which is an unstable equilibrium point.

What is the saddle point answer? ›

Definition of Saddle Points

Saddle points of a multivariable function are those points in its domain where the tangent is parallel to the horizontal axis, but this point tends to be neither a local maximum nor a local minimum.

What is the saddle path in the Ramsey model? ›

Phase space graph (or phase diagram) of the Ramsey model. The blue line represents the dynamic adjustment (or saddle) path of the economy in which all the constraints present in the model are satisfied. It is a stable path of the dynamic system.

What is saddle point example? ›

Surfaces can also have saddle points, which the second derivative test can sometimes be used to identify. Examples of surfaces with a saddle point include the handkerchief surface and monkey saddle.

What is a saddle point in statistics? ›

Well, mathematicians thought so, and they had one of those rare moments of deciding on a good name for something: Saddle points. By definition, these are stable points where the function has a local maximum in one direction, but a local minimum in another direction.

What is a saddle point calculus? ›

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function.

How do you tell if a point is maximum or minimum or saddle? ›

If D < 0, then our critical point, (a,b), is a saddle point. If D > 0 and A > 0, then our critical point, (a,b) is a minimum. If D > 0 and A <0, then our critical point, (a,b) is a maximum.

### How do you know if a point is stable or unstable? ›

A fixed point is said to be stable if a small perturbation of the solution from the fixed point decays in time; it is said to be unstable if a small perturbation grows in time. We can determine stability by a linear analysis.

Does saddle point always exist? ›

This is in a way assigning probabilities to his strategies. Where as in mixed strategies, as we will see, there always exists a saddle point. This implies we can always find a Nash equilibrium in mixed strategies.

What happens when there is no saddle point? ›

Ans: If a Rectangular game does not have a Saddle point, the two players cannot use Pure Strategies i.e. Maximin and Minimax criterion of optimality.

Does a saddle point exist when maximum? ›

A necessary and sufficient condition for a saddle point to exist is the presence of a payoff matrix element which is both a minimum of its row and a maximum of its column. A game may have more than one saddle point, but all must have the same value.

What is true about a saddle point? ›

A saddle point is a point (x0,y0) where fx(x0,y0)=fy(x0,y0)=0, but f(x0,y0) is neither a maximum nor a minimum at that point.

How do you escape the saddle point? ›

To escape from saddle points and find local minima in a general setting, we move both the assumptions and guar- antees in Theorem 2 one order higher. In particular, we require the Hessian to be Lipschitz: Definition 5.

What is strategic saddle point? ›

Accordingly a game with saddle point is that in which both the players use pure strategies i.e. both players use the same course of action throughout the game. Thus a saddle point is the point of intersection of the optimal pure strategies of the two players.

What is the Ramsey rule in economics? ›

The Ramsey rule states (approximately) that the optimal taxes cause every good to have the same proportional reduction in compensated demand. See also inverse elasticity rule.

What is saddle point in classical mechanics? ›

If a great circle path terminates beyond the kinetic focus of its initial point, the length of the great circle path is a saddle point.

How do you classify a saddle point? ›

If ∆(h, k) > 0 for all (h, k) = (0,0) sufficiently close to (0,0), (a, b) is a local minimum. If ∆(h, k) > 0 then (a, b) is a local maximum, otherwise, (a, b) is a saddle point. To classify a critical point we first use the second derivative test and if D = 0 then we use first principals and look at ∆(h, k).

### What are saddle points of function of two variables? ›

Critical points of a function of two variables are those points at which both partial derivatives of the function are zero. A critical point of a function of a single variable is either a local maximum, a local minimum, or neither.

Is every saddle point an inflection point? ›

Note: all turning points are stationary points, but not all stationary points are turning points. A point where the derivative of the function is zero but the derivative does not change sign is known as a point of inflection, or saddle point.

What is the difference between minimum and saddle point? ›

To be a local minimum, it has to be a local minimum in every direction. In contrast, for a saddle point, only 1 direction has to be different than others. It is much more likely that 1 or more have different behaviorthan the others, as compared to the same behavior in all directions.

How do you know if origin is saddle point? ›

If there are two real eigenvalues of opposite sign, the origin is a saddle point (and therefore unstable). If there are two real positive eigenvalues, the origin is an unstable node. If there are two real negative eigenvalues, the origin is an asymptotically stable node.

What do eigenvalues tell us about stability? ›

Eigenvalues can be used to determine whether a fixed point (also known as an equilibrium point) is stable or unstable. A stable fixed point is such that a system can be initially disturbed around its fixed point yet eventually return to its original location and remain there.

How do you determine stability? ›

In terms of the solution of a differential equation, a function f(x) is said to be stable if any other solution of the equation that starts out sufficiently close to it when x = 0 remains close to it for succeeding values of x.

How do you determine the stability of a steady state? ›

A steady state solution y⇤ is called (locally) asymptotically stable if any solution f (y⇤) = 0 and f 0(y⇤) < 0. A steady state solution y⇤ is called unstable if any solution f (y⇤) = 0 and f 0(y⇤) > 0.

What is an example of a saddle point? ›

Surfaces can also have saddle points, which the second derivative test can sometimes be used to identify. Examples of surfaces with a saddle point include the handkerchief surface and monkey saddle.

What are saddle point strategies? ›

The saddle point describes the solution of the game. The saddle point may not be unique. Maximin principle says that the player A always tries to maximize his minimum gains corresponding to the opponent strategies. A always tries to maximize his minimum gains corresponding to the opponent strategies.

What is saddle point in product life cycle? ›

The saddle is a sudden, sustained, and deep drop in sales of a new product, after a period of rapid growth following takeoff, followed by a gradual recovery to the former peak.

### What is the saddle point in a payoff matrix? ›

A saddle point is a payoff that is simultaneously the lowest entry in its row (row minimum) and the greatest entry in its column (column maximum).

How do you determine if a point is a saddle point? ›

Given the function z=f(x,y), the point (x0,y0,f(x0,y0)) is a saddle point if both fx(x0,y0)=0 and fy(x0,y0)=0, but f does not have a local extremum at (x0,y0).

What is a saddle point explanation? ›

1. : a point on a curved surface at which the curvatures in two mutually perpendicular planes are of opposite signs compare anticlastic. : a value of a function of two variables which is a maximum with respect to one and a minimum with respect to the other.

How do you prove a point is a saddle? ›

The standard test for extrema uses the discriminant D = AC − B2: f has a relative maximum at (a, b) if D > 0 and A < 0, and a minimum at (a, b) if D > 0 and A > 0. If D < 0, f is said to have a saddle point at (a, b). (If D = 0, the test is inconclusive.) F(x, y) = Ax2 + 2Bxy + Cy2.

How do you overcome saddle points? ›

A popular approach to overcoming saddle points is to incorporate second order information. However, the popular second order approach of Newton's method is not suitable since it converges to an arbitrary critical point, and does not distinguish between a local minimum 3 and a saddle point.

What is the difference between saddle point and turning point? ›

Note: all turning points are stationary points, but not all stationary points are turning points. A point where the derivative of the function is zero but the derivative does not change sign is known as a point of inflection, or saddle point.

Is saddle point the same as Nash equilibrium? ›

, then the Nash equilibrium is actually the same concept as a saddle point. It applies, however, to much more general games. , which is better for both players. For zero-sum games, the existence of multiple saddle points did not cause any problem; however, for nonzero-sum games, there are great troubles.

Is saddle point equal to Nash equilibrium? ›

7.1. 3 Saddle Point: A strategy profile (i∗, j∗) is said to be saddle point if ai∗j ≤ ai∗j∗ ≤ aij∗ , ∀i, j It turns out that this is also a Nash equilibrium.

What is an example of a saddle point in a matrix? ›

A saddle point of a matrix

The canonical example is f(x,y) = x2 - y2 at the point (x0, y0) = (0, 0). Along the horizontal line y=0, the point (0, 0) is a local minimum. Along the vertical line x=0, the point (0, 0) is a local maximum.

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