Robust chaos in dynamic optimization models

Abstract

The purpose of this paper is to investigate the (theoretical) importance of chaos as a phenomenon occurring in dynamic optimization problems. The intertemporal models we focus on are specified by a standard aggregative production function, an immediate return function depending on current consumption, capital input and a taste parameter, and a discount factor.

We interpret “chaos” as a situation in which the Liapounov exponent of the relevant dynamical system is positive. This notion of chaos is related to the concept of “unpredictability” as measured by the Kolmogorov-Sinai entropy.

In the family of intertemporal models, indexed by the taste parameter (with values lying in a closed interval), chaos is considered to be an “unimportant” phenomenon, if the set of parameter values for which chaos occurs is of Lebesgue measure zero.

We identify a family of dynamic optimization models, for which the optimal transition functions are represented by the quadratic family of maps. Relying on the mathematical literature on the robustness of chaos for this family of maps, we conclude that chaos cannot be considered to be an unimportant phenomenon in dynamic optimization models.