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

Yves Achdou,Benjamin Moll,Jiequn Han,Pierre-Louis Lions,Jean-Michel Lasry

doi:10.1093/restud/rdab002

Abstract

Abstract We recast the Aiyagari–Bewley–Huggett model of income and wealth distribution in continuous time. This workhorse model—as well as heterogeneous agent models more generally—then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (1) an analytic characterization of the consumption and saving behaviour of the poor, particularly their marginal propensities to consume; (2) a closed-form solution for the wealth distribution in a special case with two income types; (3) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including—but not limited to—the Aiyagari–Bewley–Huggett model.

Highlights

One of the key developments in macroeconomics research over the last three decades has been the incorporation of explicit heterogeneity into models of the macroeconomy
Iterating on r(t) until an equilibrium transition is found takes about 4 minutes (even though market clearing conditions like (4) that implicitly define prices are notoriously difficult to impose during transitions)
We prove a number of new theoretical results about the Aiyagari-Bewley-Huggett model, the workhorse theory of income and wealth distribution in macroeconomics: (i) an analytic characterization of the consumption and saving behavior of the poor, their marginal propensities to consume; (ii) a closed-form solution for the wealth distribution in a special case with two income types; (iii) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one; (iv) a characterization of “soft” borrowing constraints

Summary

Introduction

One of the key developments in macroeconomics research over the last three decades has been the incorporation of explicit heterogeneity into models of the macroeconomy. Fueled by the increasing availability of high-quality micro data, the advent of more powerful computing methods as well as rising inequality in many advanced economies, such heterogeneous agent models have proliferated and are ubiquitous. This is a welcome development for a number of reasons. Macroeconomists often want to analyze the welfare implications of particular shocks or policies. This is impossible without asking “who gains and who loses?”, that is, distributional considerations often cannot be ignored. Models with heterogeneity often deliver strikingly different aggregate implications than do representative agent models, for example with respect to monetary and fiscal policies.

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Results

Conclusion