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Presented By: Department of Economics

Solving Heterogeneous Agent Models with the Master Equation

Adrien Bilal, Harvard University

Solving Heterogeneous Agent Models with the Master Equation Solving Heterogeneous Agent Models with the Master Equation
Solving Heterogeneous Agent Models with the Master Equation
This paper proposes an analytic representation of 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 representation of the econ- omy: 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 sim- plifications. 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 five main advantages: (i) finite dimension; (ii) closed-form mapping to steady-state objects; (iii) applicability when many distributional moments or prices en- ter individuals’ decision such as dynamic trade, urban or job ladder settings; (iv) block-recursivity bypassing further fixed points; (v) fast implementation using standard numerical methods. The Second-order Approximation to the Master Equation (SAME) shares these properties, making it 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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