No More Excuses! A Toolbox for Solving Heterogeneous Agent Models with Aggregate Shocks

How does the joint distribution of income and wealth interact with macroeconomic aggregates? We develop a new continuous-time method that improves the theoretical characterization and computation of incomplete-market models designed to study this question, taking advantage of the fact that such models can be conveniently solved as systems of partial differential equations. We provide a complete analytic characterization of individual consumption and saving behavior as well as a closed-form solution for the wealth distribution in a special case with two income states. We show that aggregate consumption and saving are approximately independent of the distribution of wealth only in the special case of homothetic (CRRA) preferences and hence approximately constant marginal propensities to consume for rich and poor households. With non-homothetic preferences, an increase in income inequality can instead lead to a sizable increase in aggregate savings. Conversely, changes in macroeconomic aggregates may affect the distribution of wealth in unexpected ways. For instance, an increase in aggregate productivity may lead to higher wealth inequality at the bottom of the distribution. Our efficient and flexible computational algorithm easily handles non-differentiable and non-convex problems as well as transition dynamics.