A Perturbational Approach for Approximating Heterogeneous Agent Models

This paper develops a perturbational technique for approximating equilibria of a wide class of discrete-time heterogeneous-agent (HA) models with complex, possibly infinite-dimensional state spaces, including occasionally binding borrowing constraints that create kinks in policy functions. Its central insight is that traditional perturbation – which requires computing derivatives of policy functions with respect to every dimension of the state, an object whose size explodes with the dimension of that state – can instead be reformulated as finding a small number of scalar “directional derivatives”: values of policy-function derivatives evaluated along economically meaningful directions, such as the deterministic path traced out by a one-time (“MIT”) shock, or a direction capturing precautionary motives from anticipated risk. Because directional derivatives are always one-dimensional regardless of the underlying state’s dimensionality, and because the paper analytically derives the linear systems and recursive coefficient formulas that characterize them (using, among other inputs, generalized functions to handle kinks from borrowing constraints), the resulting method preserves the speed, simplicity and flexibility of classical representative-agent perturbation techniques (as in DYNARE) while remaining tractable at any approximation order in HA settings. Compared with the sequence-space-Jacobian method of Auclert, Bardóczy, Rognlie and Straub (2021), the paper’s approach is modestly faster at first order and, unlike that method, extends cleanly to second order and beyond, to models with stochastic volatility, to transition dynamics away from steady state, and to portfolio-choice problems in which risk premia and second-moment properties matter for even first-order dynamics. Illustrating the method on a Krusell-Smith-style economy with capital adjustment costs, the paper shows first-order approximation takes well under a second and second-order approximation only a few seconds more; that second-order terms materially change simulated aggregate paths relative to both the first-order and representative-agent counterparts, driven mostly by a precautionary-savings channel; that welfare from countercyclical labor-tax stabilization is exactly zero at first order but has an interior optimum at second order, with the popular “histogram method” for discretizing the distribution generating a quantitatively different (and, the paper shows analytically, generically incorrect) optimum at second order; that a five-times spike in aggregate uncertainty (calibrated to the VIX) generates aggregate and cross-sectional welfare losses substantially amplified relative to a representative-agent benchmark, concentrated among asset-poor households; and that letting households optimally choose between risky capital and safe debt materially changes the first-order response of aggregate capital relative to forcing households to hold a fixed portfolio.

David Evans | Macro Paper Warehouse

A Perturbational Approach for Approximating Heterogeneous Agent Models