10.12 Partial-equilibrium search. Consider a worker searching for a job. Wages, w, have a probability density function across jobs f(w) that is known to the worker; let F(w) be the associated cumulative distribution function. Each time the worker samples a job from this distribution, he or she incurs a cost of C, where 0 < C < E[w]. When the worker samples a job, he or she can either accept it (in which case the process ends) or sample another job. The worker maximizes the expected value of wnC, where w is the wage paid in the job the worker eventually accepts, and n is the number of jobs the worker ends up sampling. Let V denote the expected value of wn'C of a worker who has just rejected a job, where n' is the number of jobs the worker will sample from that point on.

(a) Explain why the worker accepts a job offering w if > V, and rejects it if < V. (A search problem where the worker accepts a job if and only if it pays above some cutoff level is said to exhibit the reservation-wage property.)

(b) Explain why V satisfies V = F(V)V + =v wf(w)dw - C.

(c) Show that an increase in C reduces V.

(d) In this model, does a searcher ever want to accept a job that he or she has previously rejected?