Odd period cycles and ergodic properties in price dynamics for an exchange economy

TL;DR

This paper analyzes price dynamics in an exchange economy, identifying conditions for odd period cycles, chaos attraction, and ergodic properties, revealing that chaos can still allow for predictable average behavior.

Contribution

It provides necessary and sufficient conditions for odd period cycles and chaos attraction in price adjustment processes, and explores ergodic properties and sensitivity analysis in economic models.

Findings

Conditions for odd period cycles identified

Chaos can be associated with predictable average behavior

Sensitivity analysis links ergodic sums to price adjustment speed

Abstract

In the first part of this paper (Sections 1-4), we study a standard exchange economy model with Cobb-Douglas type consumers and give a necessary and sufficient condition for the existence of an odd period cycle in the Walras-Samuelson (tatonnement) price adjustment process. We also give a sufficient condition for a price to be eventually attracted to a chaotic region. In the second part (Sections 5 and 6), we investigate ergodic properties of the price dynamics showing that the existence of chaos is not necessarily bad. (The future is still predictable on average.) Moreover, supported by a celebrated work of Avila et al. (Invent. Math., 2003), we conduct a sensitivity analysis to investigate a relationship between the ergodic sum (of prices) and the speed of price adjustment. We believe that our methods in this paper can be used to analyse many other chaotic economic models.