Benjamin Moll | Macro Paper Warehouse

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.

Monetary Policy According to HANK

This paper revisits the transmission mechanism from monetary policy to household consumption using a quantitative Heterogeneous Agent New Keynesian (HANK) model built to match the empirical distribution of household income, liquid wealth, and illiquid wealth. On the household side, the model extends the standard Aiyagari-Huggett-Imrohoroglu incomplete-markets framework, following Kaplan and Violante (2014), to let households save in a low-return liquid asset and a high-return illiquid asset subject to a transaction cost – a structure that, unlike one-asset incomplete-markets models, can simultaneously match a high aggregate wealth-to-output ratio and a realistically large marginal propensity to consume out of small windfalls. The paper’s central finding decomposes the aggregate consumption response to an interest rate cut into a “direct effect” (intertemporal substitution, operating even absent any income change) and an “indirect effect” (the general-equilibrium rise in labor demand and income that follows from the direct impulse): in representative-agent New Keynesian (RANK) models, direct effects account for nearly the entire response, but in the calibrated HANK model, indirect effects account for about 80 percent of the response and direct effects for only about 20 percent – a result the authors show is highly robust across specifications. The reversal is driven by the coexistence of poor and wealthy hand-to-mouth households (insensitive to interest rates but highly sensitive to income) and by dampened intertemporal substitution even among non-hand-to-mouth households, due to liquidity-constraint risk and portfolio rebalancing toward illiquid assets. A second major finding is that, because the government is a large issuer of liquid assets, Ricardian equivalence fails in this environment, so the specific fiscal response accompanying a monetary shock (transfers, taxes, spending, or government debt absorbing the change in interest payments) materially changes the overall size and timing of monetary policy’s effect on the economy – a dependence entirely absent from RANK models.

Partial differential equation models in macroeconomics

Written explicitly to get mathematicians interested in macroeconomics, this review collects the systems of coupled nonlinear partial differential equations that arise once a macro model tracks a whole population of heterogeneous households or firms in continuous time – a Hamilton-Jacobi-Bellman equation for one atomistic agent’s optimal control, paired with an equation for how the cross-sectional distribution evolves – and states, family by family, which of their basic properties are proved and which remain open. The authors are candid that this is the paper’s purpose: they “present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research,” and they call the pairing a “mean field game” after Lasry and Lions, noting that while each equation type is individually well understood, “our understanding of the coupled system is much more limited.” Five families are covered. The continuous-time Huggett-Aiyagari-Bewley model of income and wealth distribution (§2) yields a stationary HJB/Fokker-Planck pair in which the borrowing constraint, treated as a state constraint, makes the optimal saving drift behave like the square root of distance to the floor, so the stationary wealth density is unbounded and carries a Dirac mass exactly at the constraint for all incomes below a threshold; existence of a stationary equilibrium is proved in the companion Achdou-Lasry-Lions-Moll work, but uniqueness, and both existence and uniqueness of the time-dependent equilibrium, are listed as open. Models of power laws (§3) run on the Gabaix mechanism — geometric Brownian motion plus a small friction gives a stationary density that is exactly a power law with exponent ζ = 1 − 2μ̄/σ̄² — and become genuinely hard once an optimal-stopping exit decision, in the form of a variational inequality of the obstacle type, makes the minimum size endogenous. Knowledge-diffusion growth models (§4) replace the local Fokker-Planck law of motion with non-local Fisher-KPP or Boltzmann-type equations whose travelling-wave solutions deliver the closed-form pairing growth = σ√(2α) and tail inequality 1/ζ = σ/√(2α), implying a growth-inequality trade-off in the experimentation parameter σ but not in the diffusion parameter α, where higher diffusion raises growth and lowers inequality simultaneously. Business-cycle models with aggregate shocks (§5) are the hardest: the cross-sectional distribution becomes a random variable that must enter each individual’s own state space, producing an “HJB equation in the space of density functions” whose existence, uniqueness and numerical approximation are all open, and which the authors sidestep in practice by allowing shocks only at finitely many dates (ten shocks giving 2¹⁰ = 1024 finite-dimensional paths). Finally §6 notes that oligopoly applications with a finite number of strategic firms take the form of a differential game rather than a mean field game. The scope condition on the whole exercise is stated in the conclusion: this is a research agenda, an area the authors see “large ‘gains from trade’” in, not a set of settled economic findings — and the paper itself notes two places where these calibrated models fail quantitatively against data.

Present Bias Amplifies the Household Balance-Sheet Channels of Macroeconomic Policy

Maxted, Laibson, and Moll study fiscal and monetary policy in a partial-equilibrium heterogeneous-agent model in which homeowners have present-biased time preferences (Instantaneous Gratification preferences, the continuous-time limit of quasi-hyperbolic discounting) and naive beliefs, alongside a liquid savings account, an illiquid home, and access to credit card and mortgage debt. Because present bias substantially increases households’ marginal propensity to consume — in the calibrated model the quarterly MPC rises from 4% under exponential discounting to 14% under present bias, and the quarterly marginal propensity for expenditure (MPX) rises from 13% to 30% — present bias powerfully increases the effect of fiscal stimulus. Present bias also amplifies the overall effect of expansionary monetary policy, but at the same time slows down the speed of monetary transmission: interest rate cuts incentivize households to conduct cash-out refinances, which become targeted liquidity injections to households near the liquidity constraint who have especially high MPCs, but present bias with naive beliefs also introduces a motive for households to procrastinate on refinancing their mortgage, which substantially slows the speed at which this channel operates. A noteworthy feature of the model is that present bias amplifies the direct effect of monetary policy on household consumption while simultaneously delivering larger MPCs — a combination that is in contrast to standard heterogeneous-agent models, where modeling choices that amplify MPCs typically deliver smaller consumption responses to interest rate changes. The calibrated present-biased economy also replicates several empirical regularities that are difficult to match with exponential discounting: high-cost credit card borrowing by homeowners, empirically plausible cash-out behavior and loan-to-value ratios, and refinancing inertia.

Structural Reinforcement Learning for Heterogeneous Agent Macroeconomics

Standard recursive formulations of heterogeneous-agent models with aggregate risk force the entire cross-sectional distribution of agents into the Bellman equation characterizing individual decisions – the “Master equation” – purely because low-dimensional equilibrium prices, unlike the distribution itself, do not follow a Markov process, so rational agents forecasting prices end up needing to forecast the whole distribution. This extreme curse of dimensionality remains the central computational bottleneck for global solutions of heterogeneous-agent models, so severe that even a Huggett (1993) model with aggregate risk – despite looking simple – proved impossible for any team to solve in an influential benchmarking exercise and was dropped from the project altogether. This paper sidesteps the Master equation entirely using ideas from reinforcement learning (RL): agents learn equilibrium price dynamics directly from simulated paths, as standard RL would, but the paper’s “structural reinforcement learning” (SRL) approach departs from standard RL by assuming agents have structural knowledge of their own individual-state dynamics (their budget constraint and idiosyncratic income process), letting the authors compute exact policy gradients by differentiating through these known dynamics rather than relying on the noisy, approximate policy gradients standard RL methods estimate; only the equilibrium price process itself is treated as unknown and learned from simulation. By further restricting agents to condition their policies only on current (or briefly lagged) prices rather than the full price history or the distribution, the paper solves for a low-dimensional “restricted perceptions equilibrium” in the sense of Sargent (1991) rather than the full rational-expectations equilibrium – expectations are restricted in functional form but remain statistically consistent with actual outcomes. Because policy functions depend only on prices, they double as individual supply/demand schedules that can be integrated across the distribution and market-cleared period-by-period along a simulation, treating market clearing as part of the “environment” (in RL parlance) rather than something solved inside an optimization loop – which is what lets the method efficiently handle nontrivial market-clearing conditions that have historically been very hard. Implemented in JAX on a single GPU, the resulting structural policy gradient (SPG) algorithm solves the Krusell and Smith (1998) model in about 55 seconds, the previously-unsolved Huggett (1993) model with aggregate risk in around one minute, and a one-asset HANK model with a forward-looking New Keynesian Phillips curve in around three minutes – with the Krusell-Smith solution closely matching alternative global solutions of the rational-expectations equilibrium, and allowing agents a longer history of lagged prices barely moving the solution, indicating most of the information relevant for forecasting prices is already contained in current prices. The paper is explicit that its algorithm, as presented, is not itself intended as an empirically realistic theory of how real economic agents form expectations, though it suggests the “sampling”-based logic behind SRL could in principle be developed into one.

The Trouble with Rational Expectations in Heterogeneous Agent Models: A Challenge for Macroeconomics

This essay – delivered as the Economic Journal Lecture at the Royal Economic Society’s 2024 Annual Conference – argues that the assumption of rational expectations about equilibrium prices should be abandoned in heterogeneous-agent macroeconomics. In these models, because equilibrium prices generically depend on the entire cross-sectional distribution of households’ idiosyncratic states, rational expectations forces decision makers to forecast prices by forecasting that distribution, producing a Bellman equation – the “Master equation,” nicknamed the “Monster equation” in the Mean Field Games literature – whose state variable is itself infinite-dimensional. The paper’s central claim is that this is not merely a computational nuisance but a conceptually implausible description of real behavior: “if even our most advanced computational tools struggle with the ‘Monster equations,’ how can we justify the assumption that real-world households and firms solve the associated decision problems?” Reviewing three broad classes of existing solution methods – MIT-shock and linearization approaches (which sidestep the problem via certainty equivalence but cannot address aggregate risk and nonlinearity), methods that solve the full Master equation directly (which the paper’s criticism targets), and Krusell-Smith-style moment-forecasting methods (which the paper judges similarly unrealistic, except in the variant where the forecasted moments are the prices themselves) – the paper argues for a different path: have decision makers forecast prices directly via subjective beliefs, without ever forecasting the distribution. It proposes three criteria such subjective-belief models should satisfy – computational tractability, consistency with empirical evidence on expectations formation, and endogeneity of beliefs to policy and other structural change (a form of the Lucas critique) – and surveys several candidate directions, including temporary equilibrium and “internal rationality,” disciplining beliefs with survey expectations, least-squares learning, and reinforcement learning, while explicitly cautioning that it offers a diagnosis rather than a settled solution: “I only know the problem, not the solution!”

When inequality matters for macro and macro matters for inequality

This paper argues that two standard excuses for relying on representative-agent macro models – that heterogeneous-agent models with aggregate shocks are computationally intractable, and that realistic household heterogeneity does not matter much for aggregate dynamics anyway – are both weaker than commonly believed. To address the first, the authors extend Michael Reiter’s discrete-time linearization approach to continuous time: they solve a model’s stationary equilibrium fully nonlinearly using the finite-difference methods of Achdou et al. (2015), take a first-order Taylor expansion of the full discretized equilibrium system around that steady state using automatic differentiation, and solve the resulting large linear system of stochastic differential equations by standard techniques, exploiting continuous time’s tendency to generate sparse transition matrices; a companion model-free dimensionality-reduction method, adapted from the engineering “model reduction” literature, lets the computer – rather than the researcher – identify the low-dimensional information in the cross-sectional distribution needed to forecast prices accurately, generalizing the “approximate aggregation” logic of Krusell and Smith (1998) beyond cases where a hand-picked set of moments happens to work. On the standard Krusell-Smith (1998) business-cycle model, the method is roughly 1,500 times faster and about three times more accurate (by Den Haan’s 2010 error metric) than the best-performing algorithm in the JEDC comparison project, though its accuracy – being a local, linear approximation – degrades as the size of aggregate shocks grows. To address the second excuse, the authors apply their (open-sourced) toolbox to a two-asset incomplete-markets model, calibrated to match the U.S. joint distribution of income, wealth, and marginal propensities to consume, in which “wealthy hand-to-mouth” households arise endogenously from a costly-to-access illiquid asset. This richer model jointly reproduces two features of aggregate consumption dynamics – sensitivity to predictable income changes and relative smoothness – that have long challenged representative-agent and simple spender-saver benchmarks, and, in an extension with capital-skill complementarity, shows that aggregate productivity shocks can generate substantial, shock-specific swings in income and consumption inequality, providing what the authors call “a striking counterexample to the main result of Krusell and Smith (1998).”