Lecture 8: Heterogeneous-agent economies, Part I: Invariant measures

Suppose we are in a situation where something (say income or wealth) gets jumbled in a random but systematic way across individuals in each period. Random, because no one knows exactly who is going to get rich quick and who is going to end up on skid row. Systematic, because the same fraction of agents switch from a given level to another given level in each period. Under what circumstances can we be sure that the economy will converge to a unique distribu-tion of income or wealth? That is the topic of this set of lecture notes. Among economists, the treatment by Stokey and Lucas (1989) has become the standard introduction. They establish the following three conditions as sufficient: 1. The Feller property, for existence. 2. The mixing property, for uniqueness. 3. Monotonicity, for convergence. A bit later on, we will look at other sufficient conditions, including Theorem 11.12 in Stokey and Lucas (1989) and the result in Hairer (2006). In any case, we begin by describing formally what it means to jumble in a random but systematic way