The efficiency of the decentralized equilibrium in a search economy. Consider the steady state of the model of Section 10.6. Let the discount 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  Letting N denote the total number of jobs, we can therefore write welfare as  where E(N) gives equilibrium employment as a function of N.

(a) Use the matching function, (10.53), and the steady-state condition, M(U,V ) = λE, to derive an expression for the impact of a change in the number of jobs on employment, E (N), in terms of E(N) and the parameters of the model.

(b) Substitute your result in part (a ) into the expression for W(N) to find W (N) in terms of E(N) and the parameters of the model.

(c) Use (10.66) and the facts that a = λE/(1 − E ) and α = λE/V to find an expression for c in terms of N EQ , E(N EQ ), and y, where N EQ is the number of jobs in the decentralized equilibrium.

(d) Use your results in parts (b) and (c) to show that