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 < < Elw]. 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 w - nC, 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 w ~- nC 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 w > V, and rejects it if w < 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.)

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

(c) Show that an increase in reduces V.

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

10.13 In the setup described in Problem 10.12, suppose that w is distributed uniformly on In - a,n -i-a], and that < fj..

(a) Find V in terms of p, a, and C.

(b) How does an increase in a affect V? Explain intuitively.

10.14 Describe how each of the following affects equilibrium employment in the model of Section 10.8:

{a) An increase in the job breakup rate, b.

{b) An increase in the interest rate, r.

(c) An increase in the effectiveness of matching, K.

10.15 Suppose we replace the assumption in equation (10.76) that the worker and the firm divide the surplus from their relationship equally with the assumption that fraction f of the surplus goes to the worker and fraction 1- f goes

•i to the firm: (1 - - Vv) = fiVf - Vv)-

(a) How does this change in the model affect the equation implicitly defining £, (10.85)?

ih) How does a change in f affect the equilibrium level of £?

10.16 Consider the model of Section 10.8. Suppose the economy is initially in equilibrium, and that A then falls permanently. Suppose, however, that entry and exit are ruled out; thus the total number of jobs, F -y- V, remains constant. How do unemployment and vacancies behave over time in response to the fall in A?

10.17 The efficiency of the decentralized equilibrium in a search economy. Consider the model of Section 10.8. Let the interest rate, r, approach zero, and assume that the firms are owned by the households; thus welfare can be measured as the sum of utility and profits per unit time, which equals AE -(f -I- V)C. Letting JV denote the total number of jobs, we can therefore write welfare as W(N) = AE(N) - JVC, where E(N) gives equilibrium employment as a function of N.

(a) Use the matching function, (10.68), and the steady-state condition, (10.69), to derive an expression for the impact of a change in the number of jobs on employment, E{N), in terms of N,L,E(N), y, and j8.

(b) Substitute your result in part (a) into the expression for W{N) to find W{N) in terms of N,L,E(N), y.p, and A.

ic) Use (10.82) and the facts that a = bE/{L - E) and a = bE/V to find an expression for in terms of Neq,I,1;(Neq), and A, where Neq is the number of jobs in the decentralized equilibrium.

(d) Use your results in parts (h) and (c) to show that if j8 -i- = 1, W{Neq) > 0 if > I and W{Neq) < 0 if < i.

(e) If is 7 but p + is not necessarily 1, what determines the sign of

W(Neq)?

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