(Two effort levels) An unemployment insurance agency wants to insure unemployed workers in the most efficient way. An unemployed worker receives no income and chooses a sequence of search intensities at ∈ {0, a} to maximize the utility functional

Where u(c) is an increasing, strictly concave, and twice continuously differentiable function of consumption of a single good. There are two values of the search intensity, 0 and a. The probability of finding a job at the beginning of period t + 1 is

Where we assume that a > 0. Note that the worker exerts search effort in period t and possibly receives a job at the beginning of period t + 1. Once the worker finds a job, he receives a fixed wage w forever, sets a = 0, and has continuation utility . The consumption good is not storable and workers can neither borrow nor lend. The unemployment agency can borrow and lend at a constant one-period risk-free gross interest rate of R = β−1. The unemployment agency cannot observe the worker’s effort level.

a. Let V be the value of (1) that the unemployment agency has promised an unemployed worker at the start of a period (before he has made his search decision). Let C(V ) be the minimum cost to the unemployment insurance agency of delivering promised value V . Assume that the unemployment insurance agency wants the unemployed worker to set at = a for as long as he is unemployed (i.e., it wants to promote high search effort). Formulate a Bellman equation for C(V ), being careful to specify any promise keeping and incentive constraints. (Assume that there are no participation constraints: the unemployed worker must participate in the program.)

b. Show that if the incentive constraint binds, then the unemployment agency offers the worker benefits that decline as the duration of unemployment grows.

c. Now alter assumption (2) so that π(a) = π(0). Do benefits still decline with increases in the duration of unemployment? Explain.

d. Now assume that the unemployment insurance agency can tax the worker after he has found a job, so that his continuation utility upon entering a state of employment is , where τ is a tax that is permitted to depend on the duration of the unemployment spell. Defining V as above, formulate the Bellman equation for C(V ).

e. Show how the tax τ responds to the duration of unemployment.