Heterogeneous Agent Models in Continuous Time

Abstract

We study a class of continuous time heterogeneous agent models with idiosyncratic shocks and incomplete markets. This class can be boiled down to a system of two coupled partial differential equations: a Hamilton-Jacobi-Bellman equation and a Kolmogorov Forward equation, a system that Lasry and Lions (2007) have termed a “Mean Field Game.” We study two concrete model economies to show that continuous time allows for both tighter theoretical results and more precise and efficient computations as compared to traditional discrete time methods. The first one is an exact reformulation of Aiyagari (1994) and we obtain three theoretical results: a tight characterization of household savings behavior near the borrowing constraint, uniqueness of a stationary equilibrium (not yet in the current draft), and a tight link between the amount of capital “overaccumulation” and the number of borrowing constrained households. In our second economy, heterogeneous producers face collateral constraints and fixed costs in production, creating the possibility of a “poverty trap,” i.e. multiple stationary equilibria. We find that such “poverty traps” arise only in extreme special cases. Instead the economy typically features a unique but twin-peaked stationary distributions to which it converges extremely slowly. The precision of our algorithm is key for this finding, and coarse, simulation-based discrete-time algorithms may have obtained misleading results. We conclude by discussing an extension of our framework to the case with both idiosyncratic and aggregate shocks as in Krusell and Smith (1998).