Dynamics of the Distribution function

This chapter presents methods in order to compute the dynamics of an economy that is populated by heterogenous agents. In the first section, we show that this amounts to compute the law of motion for the distribution function

F

(ϵ, α) of wealth among agents. In the second section, we concentrate on an economy without aggregate uncertainty. The initial distribution is not stationary. For example, this might be the case after a change in policy, e.g. after a change in the income tax schedule, or during a demographic transition, as many modern industrialized countries experience it right now. Given this initial distribution, we compute the transition to the new stationary equilibrium. With the methods developed in this section we are able to answer questions as to how the concentration of wealth evolves following a change in capital taxation or how the income distribution evolves following a change in the unemployment compensation system. In the third section, we consider a model with aggregate risk. There are many ways to introduce aggregate risk, but we will focus on a simple case. We distinguish good and bad times which we identify with the boom and recession during the business cycle. In good times, employment probabilities increase and productivity rises. The opposite holds during a recession. As one application, we study the income and wealth distribution dynamics over the business cycle in the final section of this chapter. We will need to find an approximation to the law of motion

F

′ =

G(F)

and introduce you to the method developed by Krusell and Smith (1998).