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.

The New Keynesian Transmission Mechanism: A Heterogeneous-Agent Perspective

This paper studies how the simplest possible form of household heterogeneity – splitting the representative agent of the textbook New Keynesian model into a “worker,” who receives only labor income, and a “capitalist,” who receives only firm profits – changes the model’s monetary transmission mechanism. Under the standard assumption that only goods prices are sticky and wages are flexible, the authors show that this 2-agent model behaves very differently from its representative-agent counterpart: the real interest rate, inflation, real wages, and profits all respond similarly to a monetary policy shock, but output and employment do not respond at all in the worker-capitalist model, whereas they fall sharply in the standard model. The reason is that with the balanced-growth (King-Plosser-Rebelo) preferences standard in macroeconomics, income and substitution effects on labor supply exactly cancel; once profit income is removed from a worker’s budget (because a worker earns only wages), this cancellation makes hours completely unresponsive to wage movements, so monetary policy only redistributes consumption between workers and capitalists – it does not move aggregate output. The authors then show that the representative-agent model’s own ability to generate an output response rests on an empirically fragile mechanism: profits move countercyclically with the policy rate, making the representative household poorer and inducing it, via a wealth effect, to work more – a channel undermined both by household balance-sheet data showing few households hold much non-labor income, and by the fact that profits are procyclical, not countercyclical, in the data. When wage stickiness is introduced instead of (or alongside) price stickiness, however, workers are pushed off their static labor-supply curve and simply supply whatever hours are demanded; in this case the worker-capitalist model’s impulse responses become nearly indistinguishable from the representative-agent model’s, and the authors show this equivalence strengthens as the degree of wage rigidity increases. The authors confirm these results are robust to allowing limited financial trade between workers and capitalists via a bond market with adjustment costs.

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

The New Keynesian Transmission Mechanism: A Heterogeneous-Agent Perspective