# What is Saddle Point Stability?

What is saddle point stability and what sorts of functions or systems of equations give rise to such a thing?

For example, would we say that a two systems of stochastic difference equations that are interlinked give rise to an environment with saddle point stability?

• Do you mean a "system of two stochastic difference equations"? – luchonacho Sep 2 '17 at 20:42
• No. I mean two systems of two stochastic difference equations that are linked together. Think of two economies that are linked to one another where each economy is itself represented (approximately) by a system of stochastic difference equations that serve as laws of motion for two important economic indicators like inflation and output. – Macronogood Sep 2 '17 at 22:04

"Saddle point stability" refers to dynamical systems, (usually systems of difference or differential equations), where the system has a fixed point, and there exists a single trajectory that leads to the fixed point.

It follows that from a mathematical point of view these systems are in reality unstable.

A 2 X 2 system is the standard example because one can construct intuitive two-dimensional phase diagrams to understand the properties and the behavior of the system over time.

For natural sciences, unstable systems are useless as models -the tiniest deviation, if not corrected, would lead to corner solution (elimination or explosion of the system).

But saddle-path stable systems have found important uses in economics, because this feature of theirs accommodates purposeful behavior. Assume that an economic system described by a set of difference equations was properly stable in the full mathematical sense. That would imply that no matter where we started, automatically the system would tend to its fixed point/long-run equilibrium. But this means that no matter what the economic agents did and how they interact, the system would settle in equilibrium... this does not sound very convincing once we accept that agents try to optimize their situation (in terms of utility, wealth, etc).

But if the system is "saddle-path stable", then agents track the trajectory followed, and if it appears to deviate from the one that leads to the fixed point, they purposefully change their own behavior in order to go back to the desired trajectory, since corner solutions are not optimal.

The standard example here is the basic growth model.

• So if we have an open economy described by two interlinked sets of difference equations, where agents use a policy instrument to stabilize these economies, then we would say that this open economy is a saddle-path stable system since agents must act purposefully through the policy instrument to stabilize this economy? I am also unsure here because the system I have in mind features a multiplicity of equilibria by design but does have a clearly defined optimal equilibrium. – Macronogood Sep 3 '17 at 18:52
• @Macronogood Ultimately, saddle-point stability depends on the mathematical properties of the specific model. If you have multiple equilibria, then some may be stable, some unstable, some saddle-point stable. – Alecos Papadopoulos Sep 3 '17 at 19:34