Matching and Saving in Continuous Time: Theory

Abstract

We analyse optimal saving of risk-averse households when labour income stochastically jumps between two states. The generalized Keynes-Ramsey rule includes a precautionary savings term. A phase diagram analysis illustrates consumption and wealth dynamics within and between states. There is an endogenous lower and upper limit for wealth. We derive the Fokker-Planck equations for the densities of individual wealth and employment status. These equations also characterize the aggregate distribution of wealth and allow us to describe general equilibrium. An optimal consumption path exists and distributions converge to a unique limiting distribution.

Keywords: matching model, optimal saving, incomplete markets, Poisson uncertainty, Fokker-Planck equations, general equilibrium

JEL Classification: D91, E24, J63, J64