Solving heterogeneous-agent models by projection and perturbation

This paper proposes a numerical method for solving stochastic general-equilibrium models with incomplete markets and a continuum of heterogeneous agents – a class of problems where, as the paper puts it, “the state vector includes the whole cross-sectional distribution of wealth,” an infinite-dimensional object in principle. The dominant approach at the time, pioneered by Krusell and Smith (1998), represents that distribution with only a small number of statistics (typically the mean) and works very well for the models it was built for; but the paper argues this cannot serve as a general solution, since in models where the shape of the distribution itself drives the dynamics – such as (S,s) pricing or inventory models, or the redistributive-shock example this paper constructs – a low-dimensional summary can miss essential dynamics. The proposed method instead computes a solution that is fully nonlinear in the idiosyncratic (individual) shocks but only linear in the aggregate shocks: it first solves precisely for the steady-state cross-sectional distribution and consumption function (with no aggregate shocks but the full idiosyncratic shock process), using cubic splines for the consumption function and a fine histogram (up to 1000, and as a robustness check 5000, intervals) for the wealth distribution; it then computes a first-order perturbation of that whole high-dimensional representation with respect to small aggregate shocks, using Sims (2001)’s solver for linear rational-expectations systems. Applied to a test model of household saving with uninsurable income risk, liquidity constraints, an aggregate technology shock, and an i.i.d. redistributive capital-tax shock, the method reproduces the Krusell-Smith “approximate aggregation” finding when only the technology shock is active (a one-moment forecast of future aggregate capital is nearly exact), but shows that this breaks down once the tax shock is introduced, in which case even a four-moment forecast leaves sizable error while the paper’s high-dimensional, spline-based solution remains accurate to roughly 10^-6 in absolute forecast error. The method is explicitly a linear approximation in the aggregate dimension – suited to cases “where individual shocks are much bigger than aggregate shocks” – and the paper proposes it as a first step that can be combined with state-space reduction (via spline or principal-component bases) before, in future work, attempting higher-order perturbations in a reduced aggregate state space.

Michael Reiter | Macro Paper Warehouse

Solving heterogeneous-agent models by projection and perturbation