Optimal Consumption Decisions and Value Functions

We begin by defining Bellman values for employed, unemployed, and nonparticipating workers. We focus on steady state equilibria in which wages, capital stocks, queue lengths, and unemployment rates are constant. Let E(A,w) be the lifetime utility of a worker who is employed at wage w with assets A at the start of the time period. Similarly, let J(At w,q) be the expected value of an unemployed worker if he applies for wage w with queue length q in this period, and then follows an optimal consumption rule and application strategy thereafter. Let U(A) denote the Bellman value of an unemployed participant with assets A. Finally, define N(A,x) as the utility of a nonparticipating agent with asset level A} receiving home-produced income x.

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