Solving Heterogeneous Agent Models with the Master Equation

Title: Solving Heterogeneous Agent Models with the Master Equation

Abstract: This paper proposes an analytic representation of perturbations in heterogeneous agent economies with aggregate shocks. Treating the underlying distribution as an explicit state variable, a single value function defined on an infinite-dimensional state space provides a fully recursive epresentation of the economy: the ‘Master Equation’ introduced in the mathematics mean field games literature. I show that analytic local perturbations of the Master Equation around steady-state deliver dramatic simplifications. The First-order Approximation to the Master Equation (FAME) reduces to a standard Bellman equation for the directional derivatives of the value function with respect to the distribution and aggregate shocks. The FAME has six main advantages: (i) finite dimension; (ii) closed-form mapping to steady-state objects; (iii) applicability when many distributional moments or prices enter individuals’ decision such as dynamic trade, urban or job ladder settings; (iv) block-recursivity bypassing further fixed points; (v) mapping to analytic sequence-space derivatives; (vi) fast implementation using standard numerical methods. I develop the Second-order Approximation to the Master Equation (SAME) and show that it shares these properties, making the approach amenable to settings such as asset pricing. I apply the method to two economies: an incomplete market model with unemployment and a wage ladder, and a discrete choice spatial model with migration.

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