10.6 Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let C denote the consumption of employed workers in state 1, and C," the consumption of un- employed workers The firm's profits in state are therefore A, F(L,)-[CEL, + C, (L-L,), where L is the number of workers. Similarly, workers' expected utility in state i is (L, LU(CF) -K]+[(IL)/IJU (C), where K > 0 is the disutility of working.

(a) Set up the Lagrangian for the firm's problem of choosing the L,'s, C,E's, and C's to maximize expected profits subject to the constraint that the representative worker's expected utility is u.33

(b) Find the first-order conditions for L., C, and C. How, if at all, do CE and C" depend on the state? What is the relation between C, and C,U?

(c) After A is realized and some workers are chosen to work and others are chosen to be unemployed, which workers are better off?