Implicit contracts without variable hours. Suppose that each worker must either work a fixed number of hours or be unemployed. Let CE i denote the consumption of employed workers in state i and CU i the consumption of unemployed workers. The firm’s profits in state i are therefore Ai F (Li ) − [CE i Li +

CU i (L − Li )], where L is the number of workers. Similarly, workers’ expected utility in state i is (Li /L )[U(CE i ) − K] + [(L − Li )/L ]U(CU i ), where K > 0 is the disutility of working.

(a) Set up the Lagrangian for the firm’s problem of choosing the Li ’s, CE i ’s, and CU i ’s to maximize expected profits subject to the constraint that the representative worker’s expected utility is u 0.

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(b) Find the first-order conditions for Li , CE i , and CU i . How, if at all, do CE and CU depend on the state? What is the relation between CE i and CU i ?

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