Solving Heterogeneous Agent Models with the Master Equation

This paper proposes a new conceptual and computational framework for perturbing heterogeneous-agent economies with aggregate shocks, built around the “Master Equation” recently characterized in the mathematics mean-field-games literature. The core idea is to include the underlying cross-sectional distribution of agents as an explicit state variable in each individual’s decision problem – since knowledge of the distribution pins down prices, and households (who know the distribution’s law of motion) can forecast future prices from it just as they would forecast any other state variable – yielding a single Bellman-type equation, the Master Equation, that merges the usual forward-looking/backward-looking fixed point of heterogeneous-agent models into one Markovian object. The paper’s second core idea is to take analytic (rather than numerical) perturbations of this equation around a deterministic steady state, using Fréchet derivatives in the space of distributions while preserving the full nonlinearity of individual decisions in idiosyncratic states, and working in continuous time so that borrowing constraints can be handled without Lagrange multipliers. The resulting First-order Approximation to the Master Equation (FAME) reduces to an ordinary, finite-dimensional Bellman equation for an “Impulse Value” – the directional derivative of the value function with respect to a distributional impulse – whose dimension is only twice the number of idiosyncratic states (four in a standard Krusell-Smith example), which depends in closed form on interpretable steady-state objects, applies even when many prices or distributional moments enter decisions (as in dynamic spatial, job-ladder or search models), is block-recursive (the distribution’s evolution requires no further fixed point once the Impulse Value is known), delivers the first general stability criteria and characterization of the stochastic steady state for heterogeneous-agent economies, and can be computed with standard finite-difference methods as a modified Sylvester matrix equation, typically in a tenth of a second. The Second-order Approximation to the Master Equation (SAME) shares all these properties while capturing risk, nonlinearity, and asset pricing, at the cost of tracking three times the idiosyncratic states. Two applications illustrate the framework’s reach: in an incomplete-markets business-cycle model with countercyclical unemployment risk and a low-liquidity, high-MPC calibration, the paper finds the welfare cost of business cycles is 2.3% of steady-state consumption – 23 times Lucas’s (1987) classic representative-agent estimate – concentrated among low-wealth and unemployed households, who could gain over 10% from eliminating business cycles; in a dynamic spatial migration model spanning 381 U.S. metropolitan areas, the paper finds first- and second-order perturbation solutions are virtually indistinguishable for aggregate shocks up to 30%, suggesting first-order methods suffice for many applications of this kind. Both applications solve in a couple of seconds on a laptop.

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Solving Heterogeneous Agent Models with the Master Equation