Solving Heterogeneous Agent Models with the Master Equation

“The classic difficulty in characterizing this economy is that individual decisions are forward-looking in time, while the evolution of the infinite-dimensional distribution is backward-looking in time. Prices are the fixed point of this forward-backward system that clear the capital and labor market” (Introduction, p. 3). The paper resolves this by making the distribution an explicit state variable in household decisions – since the distribution fully determines prices and households know its law of motion, they can forecast prices exactly as they forecast any other state, “merg[ing] the fixed point on decisions, prices and the distribution into a single object”: the Master Equation (Introduction, pp. 3-4).

It “consists of a single Bellman equation that describes the entire behavior of a system of interacting agents”; in the Krusell and Smith (1998) example, the Master Equation defines a value function that depends on a given household’s own idiosyncratic states – assets and productivity – as well as the underlying distribution of assets and productivity of all other households in the economy (Introduction, p. 4). Because this includes as a state variable all the information needed to forecast the economy’s evolution, it is “a Markovian representation of the economy.”

The FAME’s solution, the Impulse Value, “consists of the directional derivatives of the value function with respect to the distribution” and “depends on only twice the number of idiosyncratic states, down from infinity in the fully nonlinear Master Equation”; in the Krusell-Smith example, “the Impulse Value has dimension four” (Introduction, p. 4). Concretely, individuals need only know their own idiosyncratic states and the idiosyncratic state at which a hypothetical distributional impulse is occurring – twice the underlying state count – “a drastic dimension reduction” that is “a feature of the local perturbation.”

Because the FAME is a perturbation with respect to the entire distribution rather than a small set of pre-selected moments, it “applies equally well to settings in which few or many prices summarize feedback between the general equilibrium and individual decisions” – for instance dynamic spatial models where households track several location-specific prices, or search-and-matching models where the entire distribution of wage offers matters directly to workers (Introduction, p. 5). The paper cites Bilal and Rossi-Hansberg (2023), which uses the FAME to evaluate the cost of climate change in a U.S. economy disaggregated into over 3,000 counties, as an example of this generality.

The FAME provides, “to the best of my knowledge, the first stability criteria and a description of the stochastic steady-state in heterogeneous agent economies,” showing that dynamic stability and exponential convergence to steady state obtain when steady-state transition probabilities satisfy either a mixing condition or a Lyapunov function condition (Introduction, pp. 5-6). Crucially, “if these conditions fail, checking the dominant eigenvalue in numerically discretized economies can be misleading about true convergence rates: the numerical dominant eigenvalue will converge to zero as the discretization becomes finer if the underlying law of motion exhibits a continuous spectrum that includes zero” – the mixing/Lyapunov conditions rule this failure mode out and guarantee a genuine spectral gap.

“Conceptually and practically, the Second-order Approximation to the Master Equation (SAME) is the same as the FAME,” again depending only on closed-form steady-state objects, but its solution “now depends on three times the number of idiosyncratic states because pairwise impulses in the distribution matter to second order,” computed via tensor Sylvester equations, and the paper further characterizes second moments of the stochastic steady-state distribution and a welfare formula, making the SAME “well-suited for applications that focus on non-linearities, aggregate risk or asset pricing which requires second-order perturbations to depart from certainty equivalence” (Introduction, pp. 6-7).

Using a low-liquidity, high-MPC calibration (targeting an average MPC of 0.2 and aggregate consumption volatility of 0.032, as in Lucas 1987) in an incomplete-markets economy with countercyclical income risk, the paper finds an aggregate cost of business cycles of 2.3% of steady-state consumption, “23 times larger than Lucas (1987)’s seminal calculation,” with losses “concentrated on the low-wealth and unemployed individuals, who can gain over 10% from the elimination of business cycles” (Introduction, p. 5). The gap from Lucas’s representative-agent estimate reflects the combination of incomplete markets and countercyclical idiosyncratic risk, which exposes low-wealth, borrowing-constrained households to business-cycle risk far more severely than a representative consumer.

Disaggregating the U.S. into 381 Metropolitan Statistical Areas, with bilateral migration costs, idiosyncratic preference shocks, locally supplied housing, and location-specific exposure to an aggregate productivity shock, the paper estimates the model on U.S. data and finds “the response of population and welfare across all locations is virtually identical in the FAME and the SAME for aggregate shocks up to 30%,” indicating “that first-order perturbations are sufficient for many applications of interest” (Introduction, pp. 5-6). Both the FAME and SAME solve in a couple of seconds on a laptop in this application.

The paper positions itself as reversing the usual order of operations in sequence-space methods like Boppart, Krusell and Mitman (2018) and Auclert, Bardóczy, Rognlie and Straub (2021), which “first discretize, then linearize to first order”: by “linearizing first, discretizing next, the FAME is the internally consistent foundation for this computational approach,” providing an economic interpretation of otherwise opaque numerical output (Introduction, p. 6). The paper also notes that, since its first circulation in 2021, “Bhandari et al. (2023) have developed complementary techniques to compute perturbations in discrete time heterogeneous agent economies,” and that the Master Equation approach “delivers a systematic approach to higher order perturbations such as the SAME,” with third and higher orders possible in principle though not derived explicitly in this paper.