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.

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.

Pierre-Louis Lions | Macro Paper Warehouse

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

Partial differential equation models in macroeconomics