DeepHAM: A global solution method for heterogeneous agent models with aggregate shocks

Solving heterogeneous-agent (HA) models with aggregate shocks efficiently, reliably, interpretably, and generally has proven difficult: the Krusell-Smith (KS) method approximates the distribution with a small number of moments and is efficient for simple models but suffers a curse of dimensionality with multiple shocks or endogenous states, while the local-perturbation method of Reiter (2009) handles complex models but is unreliable wherever aggregate shocks generate nonlinear or nonlocal effects (a zero lower bound, large shocks, or a risky steady state that departs from the deterministic one). This paper’s method, DeepHAM, is designed to satisfy all four requirements – efficiency, reliability, interpretability, and generality – at once. It represents each agent’s value and policy functions with deep neural networks, and, rather than feeding these networks the entire cross-sectional distribution, first extracts a small number of “generalized moments”: neural-network-determined, permutation-invariant summary statistics of the distribution that play a role analogous to classical moments (such as the first moment of wealth) but are automatically optimized rather than fixed a priori. The networks are trained by directly optimizing the model’s objective over simulated economic paths, sidestepping the fixed-point iterations that both KS-style and Reiter-style methods require. In a calibrated Krusell-Smith benchmark, DeepHAM using only the first moment already reduces the Bellman equation error by 27.2% relative to the classic KS solution, and DeepHAM with one algorithmically-optimized generalized moment reduces the error by 40.3%; the resulting generalized moment reveals that the mapping from individual wealth to the moment is concave, implying that a purely redistributive, unanticipated policy shock changes the welfare of “middle” households who are not part of the redistribution program – a channel invisible to the standard KS solution, in which welfare depends only on the first moment. DeepHAM also efficiently solves more complex environments, including a model with a Brunnermeier-Sannikov-style financial sector, without suffering the curse of dimensionality that afflicts moment-matching or discretized state-space methods as the number of shocks or endogenous states grows. Finally, because DeepHAM’s neural networks are trained directly against a stated objective rather than derived from a decentralized equilibrium concept, the same framework solves the planner’s constrained-efficiency problem as easily as the competitive equilibrium, which the authors note “opens up new possibilities for studying optimal monetary and fiscal policies in heterogeneous agent models with aggregate shocks.” The paper’s stated scope excludes models in which aggregate variables are determined recursively as a function of expected future aggregate variables (such as inflation in a forward-looking New Keynesian Phillips curve), which the authors flag as requiring an additional price function and leave for companion work.

Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

This paper recasts the workhorse Aiyagari-Bewley-Huggett model of income and wealth distribution – in which households facing uninsurable idiosyncratic income risk save in a single asset – in continuous time, and shows that doing so reduces the model to a coupled system of two partial differential equations: a Hamilton-Jacobi-Bellman (HJB) equation describing an individual’s optimal consumption and saving given the evolution of prices, and a Kolmogorov Forward (KF) equation describing how the cross-sectional distribution of income and wealth evolves given individuals’ choices, a structure the mathematics literature calls a “Mean Field Game.” This reformulation supports two distinct contributions. First, a set of new analytic results: households near the borrowing constraint see their consumption and saving behave according to an explicit square-root law, implying they reach the constraint in finite time and generating clean, parameter-based formulas for their marginal propensity to consume; the resulting stationary wealth distribution has a point mass exactly at the borrowing constraint rather than smoothly vanishing there; a closed-form solution for the wealth distribution is available with two income types; and the stationary equilibrium is proven to be unique whenever the intertemporal elasticity of substitution is weakly at least one, ruling out poverty traps that would otherwise be theoretically possible. Second, the same HJB-KF structure underlies a simple, efficient, and portable finite-difference numerical algorithm – built around the fact that in continuous time a borrowing constraint appears only as a boundary condition rather than distorting first-order conditions throughout an interior region, unlike in discrete time – that the paper shows generalizes to a much wider class of heterogeneous-agent models, including ones with non-convexities and multiple assets that standard discrete-time methods find difficult to handle, and which the paper’s authors and others subsequently built on to solve heterogeneous-agent models with aggregate shocks, multiple assets, and other extensions.

Jiequn Han | Macro Paper Warehouse

DeepHAM: A global solution method for heterogeneous agent models with aggregate shocks

Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach