The Rational Expectations-n(ϵ)ϵ-Equilibrium

In the rational expectations paradigm, one solves models of a large number of agents who optimize subject to a stochastic law of motion by assuming that all agents know that law of motion. If the agents do not know that law of motion perfectly, one needs a learning model. This paper follows the optimal learning literature by assuming that each agent constructs priors about the unknowns of the problem, and then updates those priors using the Bayes updating rule. The agents need to construct priors on the distribution of other agents' priors, and then on the distribution of priors on the distribution of priors, and so on, leading to an infinite hierarchy of beliefs. The existence of an optimal response given the current state vector and hierarchy of beliefs is proved. It is then shown that the resulting equilibrium, labeled the Rational Expectations-∞ equilibrium can be approximated by an ϵ-equilibrium where the infinite hierarchy is truncated at some level n(ϵ), and each agent believes that all of his higher level beliefs are concentrated at their true values