Inverse limits and statistical properties for chaotic implicitly defined economic models

TL;DR

This paper explores the dynamics, stability, and ergodic properties of implicitly defined chaotic economic models using inverse limit spaces, providing new methods to rank utility functions based on invariant measures.

Contribution

It introduces the analysis of inverse limits in economic models, demonstrating their stability, expansiveness, and specification properties, and develops novel utility ranking methods on these spaces.

Findings

Inverse limits of models are stable under perturbations.

Inverse limits are expansive and have the specification property.

Utility functions can be ranked by average values and equilibrium measures.

Abstract

In this paper we study the dynamics and ergodic theory of certain economic models which are implicitly defined. We consider 1-dimensional and 2-dimensional overlapping generations models, a cash-in-advance model, heterogeneous markets and a cobweb model with adaptive adjustment. We consider the inverse limit spaces of certain chaotic invariant fractal sets and their metric, ergodic and stability properties. The inverse limits give the set of intertemporal perfect foresight equilibria for the economic problem considered. First we show that the inverse limits of these models are stable under perturbations. We prove that the inverse limits are expansive and have specification property. We then employ utility functions on inverse limits in our case. We give two ways to rank such utility functions. First, when perturbing certain dynamical systems, we rank utility functions in terms of their \textit{average values} with respect to invariant probability measures on inverse limits, especially with respect to measures of maximal entropy. For families of certain unimodal maps we can adjust both the discount factor and the system parameters in order to obtain maximal average value of the utility. The second way to rank utility functions (for more general maps on hyperbolic sets) will be to use equilibrium measures of these utility functions on inverse limits; they optimize average values of utility functions while \textit{at the same time} keeping the disorder in the system as low as possible in the long run.