A Rational Route to Randomness

TLDR

Adaptively rational equilibrium (A.R.E.) describes markets where agents choose among predictor functions based on publicly available performance, and its dynamics can generate irregular price paths due to homoclinic orbits. Agents iteratively select predictor functions according to past performance, producing a coupled dynamics of predictor choice and endogenous variables, as illustrated in a cobweb demand–supply model with rational and naive expectations. In an unstable market with small information costs, frequent switching of predictors generates highly irregular prices that converge to a strange attractor, showing that local instability can coexist with globally complex dynamics in a fully rational equilibrium.

Abstract

The concept of adaptively rational equilibrium (A.R.E.) is introduced. Agents adapt their beliefs over time by choosing from a finite set of different predictor or expectations functions. Each predictor is a function of past observations and has a performance or fitness measure which is publicly available. Agents make a rational choice concerning the predictors based upon their past performance. This results in a dynamics across predictor choice which is coupled to the equilibrium dynamics of the endogenous variables. As a simple, but typical, example we consider a cobweb type demand-supply model where agents can choose between rational and naive expectations. In an unstable market with (small) positive information costs for rational expectations, a high intensity of choice to switch predictors leads to highly irregular equilibrium prices converging to a strange attractor. The irregularity of the equilibrium time paths is explained by the existence of a so-called homoclinic orbit and its associated complicated dynamical phenomena. Thus local instability and global complicated dynamics may be a feature of a fully rational notion of equilibrium.