A firm and a worker interact as follows. First, the firm can make 2 contract offers (wage, jobtype): (w, z=0) and (w, z=1) where z=0 denotes the "safe" job and z=1 denotes the "risky job. After observing the firm's contract offer(w,z) the worker accepts or rejects it. If the worker rejects the contract, then he gets a payoff of 100, which corresponds to his outside opportunities. If he accepts the job, then the worker cares about two things: his wage and his status. Then, the worker's payoff is [w+v(x)] where v(x) is the value of status x. The worker's status x depends on how he is rated by his peers, which is influenced by characteristics of his job as well as random events. Specifically his rating x can be either 1(poor), 2(good) or 3 (excellent). If the worker has the safe job, then x=2 for sure. On the other hand, if the worker has the risky job, then x=3 with probability q and x=1 with probability (1-q). That is with probability q, the worker's peers think of him as excellent. Assume that v(1)=0 and v(3)=100 and let v(2)=y. The worker searches to maximize his expected payoff. The firm obtains a return of (180-w) when the worker is employed in the safe job. The firm gets a return of (200-w) when the worker has the risky job. If the worker rejects the firm's offer then the firm obtains 0. You need to compute the subgame perfect equilibrium of this game by answering the following questions.
a) How large must the wage offer be in order for the worker to rationally accept the safe job? what is the firms' maximum payoff in this case? [hint: the parameter "y" must be included in the answer. Use e>0 to indicate a "positive small amount"]
b)How large must the wage offer be in order for the worker to rationally accept the risky job? What is the firm's maximum payoff in this case? [hint: the parameter q should be included in the answer. Use e>0 to indicate a "positive small amount"]
c) What is the firm's optimal contract offer for this case in which q=1/2 ? [hint: your answer should include an inequality describing conditions under which z=1 is optimal.]