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.]

To find the subgame perfect equilibrium of this game, let's go step by step to answer the questions.

a) To find the wage offer that would make the worker rationally accept the safe job, we need to compare the payoffs of accepting and rejecting the job. If the worker accepts the safe job, he receives the wage offer plus the value of status, v(x=2), which is denoted by y in this case. So the payoff would be (w + y).

If the worker rejects the job, his outside opportunity payoff is 100. Comparing the two options, the worker will accept the job if the payoff from accepting is greater than the payoff from rejecting, i.e., (w + y) > 100.

To find the firm's maximum payoff in this case, we need to consider the return on investment for the firm. If the worker is employed in the safe job, the firm obtains a return of (180 - w). So the firm's maximum payoff would be (180 - w) if the worker accepts the safe job.

b) To find the wage offer that would make the worker rationally accept the risky job, we again compare the payoffs of accepting and rejecting the job. If the worker accepts the risky job, the payoff would be (w + v(x)), where x can be 1 or 3.

If the worker rejects the job, his outside opportunity payoff is 100. Comparing the two options, the worker will accept the job if the maximum possible payoff from accepting is greater than the payoff from rejecting, i.e., max{w + v(1), w + v(3)} > 100.

Given that v(1) = 0 and v(3) = 100, the worker will accept the risky job if the wage offer is such that w + v(3) > 100, which simplifies to w > -100.

To find the firm's maximum payoff in this case, if the worker accepts the risky job, the firm obtains a return of (200 - w). So the firm's maximum payoff would be (200 - w) if the worker accepts the risky job.

c) In this case, when q = 1/2, we need to determine the firm's optimal contract offer. The firm's goal is to maximize its payoff, which is given by the return on investment.

If the firm offers the safe job (z = 0), it obtains a return of (180 - w). If the firm offers the risky job (z = 1), it obtains a return of (200 - w). To find the optimal contract offer, we need to determine under which conditions the return on the risky job is greater than the return on the safe job.

Comparing the returns, we have:

(200 - w) > (180 - w)

The firm's optimal contract offer would be the risky job (z = 1) if this inequality holds true. Otherwise, the optimal contract offer would be the safe job (z = 0).

To summarize:
a) The wage offer for the worker to accept the safe job is (w + y), with y being the value of status.
b) The wage offer for the worker to accept the risky job is w > -100.
c) The firm's optimal contract offer is the risky job (z = 1) if (200 - w) > (180 - w), otherwise, it is the safe job (z = 0).