Social optima in economies with heterogeneous agents

This paper analyzes the problem of computing the social optimum in models with heterogeneous agents subject to idiosyncratic shocks. This is equivalent to a deterministic optimal control problem in which the state variable is the infinite-dimensional cross-sectional distribution. We show how, in continuous time, the problem can be broken down into two finite-dimensional partial differential equations: a dynamic programming equation and the law of motion of the distribution, and we introduce a new numerical algorithm to solve it. We illustrate this methodology with two examples: social optima in an Aiyagari economy with stochastic lifetimes and in a model of on-the-job search with learning.