A firm and worker interact as follows. First, the firm offers the worker a wage w and a job z; 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. These are the only decisions made by the firm and worker in this game. 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. The worker's status depends on how he is rated by his peers, which is influenced by characteristics of his job as well as by random events.
Specifically, his rating is given by x, which is either 1 (poor), 2 (good), or 3 (excellent).
If the worker has the safe job, then x = 2 for sure. In contrast, 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. When employed by this firm, the worker's payoff is w + 
(x), where (x) is the value of status x. Assume that (1) = 0 and (3) = 100, and let y = (2). The worker maximizes 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 - ω when the worker has the risky job. If the worker rejects the firm's offer, then the firm obtains 0. 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 rationally to accept the safe job? What is the firm's maximal payoff in this case? The parameter y should be featured in your answer.
(b) How large must the wage offer be in order for the worker rationally to accept the risky job? What is the firm's maximal payoff in this case? The parameter q should be featured in your answer.
(c) What is the firm's optimal contract offer for the case in which q = 1>2? Your answer should include an inequality describing conditions under which z = 1 is optimal.