Rational Routes to Randomness

We introduce the concept of Adaptively Rational Equilibrium (A.R.E.) where agents base decisions upon predictions of future values of endogenous variables whose actual values are determined by equilibration. Predictors are chosen from a finite set. Each predictor is a function of past observations and has a performance measure attached to it which is publically available. Agents use a discrete choice model and make a rational choice concerning the predictor based upon the performance measure. This results in a dynamics across predictor choice which is coupled to the dynamics of the endogenous variables. When there is at least one “stabilizing” predictor (e.g., rational or long memory expectations) driving the endogenous variable toward its steady state value and at least one ‘destabilizing’ predictor (e.g., adaptive or short memory expectations) driving the endogenous variable away from its steady state value, then the adaptive rational equilibrium dynamics can be very complicated and cycles and chaos can arise. The irregularity of the equilibrium time paths is explained by the existence of a homoclinic orbit and its associated complicated dynamical phenomena, when the intensity of choice between predictors is high. Thus local instability and global complicated dynamics may be a feature of a fully rational notion of equilibrium.