Solving Heterogeneous Agent Models with the Master Equation; Adrien Bilal (Harvard University)

Abstract

This paper proposes a general method to analyze and compute equilibria in heterogeneous agent economies with aggregate shocks in continuous time. Treating the underlying distribution as an explicit state variable, a single value function defined on an infinite-dimensional state space provides a fully recursive representation of the economy: the 'master equation' recently introduced in the mathematics mean field games literature. I show that analytic local perturbations around the steady-state drastically reduce dimensionality. The First-order Approximation to the Master Equation (FAME) reduces to an explicit value function with only twice the number of idiosyncratic states. The FAME has five main advantages: (1) block-recursive structure bypassing price or distributional fixed points; (2) explicit stability and convergence speed results; (3) applicability when many distributional moments or prices enter individuals' decision such as trade, spatial or wage ladder settings; (4) fast numerical implementation using standard finite difference methods bypassing dimension reduction; (5) amenability to higher-order perturbations necessary in settings such as asset pricing. I apply the method to two business cycle economies: a precautionary savings model with a wage ladder, and a multi-industry-location model with migration and trade in intermediate inputs.