There are two tasks, one in period t = 1 and one in period t = 2, to be performed by a worker in a firm. In each period, the worker chooses whether to work (e, = 1) or shirk (e = 0). Working costs c for the worker in each period. If the worker shirks in period t, he is caught with probability 0 q1. If caught shirking, the worker is fired immediately and receives zero wages in that period. Otherwise, he receives a bonus for that period. At the end of the first period, the worker retires with fixed probability 0 < r < 1. If the worker retires, the firm hires a replacement worker for period 2. The worker seeks to maximize his wages minus his effort costs in both periods: E[(w - ce) + (w-ce)], where w = 0 and e = 0 for him if he retires at the end of period 1. (Correspondingly, the replacement worker seeks to maximize his wages minus his effort costs in the second period: E[w-ce]). The firm seeks to minimize wages w + w, while inducing the worker/replacement worker to choose e = 1 and e = 1. The timing is as follows: Step 1: The firm chooses bonus levels by and b. Step 2: In the first period: (a) the worker chooses effort e = {0, 1} and incurs cost ce. (b) If the worker shirked (e = 0), he is caught with probability q and fired, so he receives w = 0. Otherwise, he is paid bonus w = b. (c) The worker retires with probability r, in which case a replacement worker is hired. Step 3: In the second period: (a) the worker/replacement chooses effort e = {0, 1} and incurs cost ce. (b) If the worker / replacement shirked (e2 = 0), he is caught with probability q and fired, so he receives w = 0. Otherwise, he is paid bonus w = b. We'll proceed step-by-step. a) Write down the condition on b that ensures the worker/replacement will choose e = 1. b) Suppose that the condition you specified in (a) is satisfied. Write down the condition on b and b that ensures the worker will choose e = 1. Your answer should take the following form: b + (1 - r)b (...). c) Add rb, to both sides of your answer to (b). You should now have an inequality that takes the following form: b + b (...). This expression replaces your answer to (b); combined with your answer to (a), it specifies conditions which b and b must satisfy to ensure e = 1 and e = 1. d) What is the firm's optimal choice of b and b? What is the corresponding total wage E[w + w]? How does this answer change with the retirement rate r? e) Explain, in words, why total wages increase as the retirement rate r increases.