Solving Heterogeneous Agent Models with the Master Equation - Bilal, Adrien (National Bureau of Economic Research, 2023)
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 representation 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
Hardcopy version available to institutional subscribers
Keywords: Arbeitspapier; Graue Literatur; Agentenbasierte Modellierung; Wirtschaftsmathematik; Mathematical Methods; Mathematical Methods; Programming Models; Mathematical and Simulation Modeling; General