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Texts: 8.16 Consider the following principal-agent problem. The owner of a firm (the principal) employs a worker (the agent). The worker can exert low effort, e=0, or high effort, e=1. The resulting revenue, r, to the owner is random, but is more likely to be high when the worker exerts high effort. Specifically, if the worker exerts low effort, e = 0, then: r=0, with probability 2/3 and r=4, with probability 1/3. If instead the worker exerts high effort, e = 1, then r=0, with probability 1/3 and r=4, with probability 2/3. The worker's von Neumann-Morgenstern utility from wage w and effort e is u(w, e) = √(w - e). The firm's profits are π = r - w when revenues are r and the worker's wage is w. A wage contract (w0, w4) specifies the wage, w ≥ 0, that the worker will receive if revenues are r ∈ {0, 4}. When working, the worker chooses effort to maximize expected utility and always has the option (his only other option) of quitting his job and obtaining (w, e) = (0, 0). Find the wage contract (w0, w4) ∈ [0, ∞)² that maximizes the firm's expected profits in each of the situations below. (a) The owner can observe the worker's effort and so the contract can also be conditioned on the effort level of the worker. How much effort does the worker exert in the expected profit-maximizing contract? (b) The owner cannot observe the worker's effort and so the contract cannot be conditioned on effort. How much effort does the worker exert in the expected profit-maximizing contract now?
Akash M.
A firm needs to hire an employee to complete a project. The project may be successful or not depending on how hard the employee works. Suppose there are two effort levels the employee can choose from, high effort (e = eH) and low effort (e = eL). If she chooses eH, the project is successful with a probability of 0.8, while if she chooses eL, the project is successful with a probability of 0.4. A successful project yields revenue of x = 2500 for the firm, and an unsuccessful project yields x = 0. The employer cannot observe the employee's effort choice but can only observe whether the project is successful or not. Therefore, the employer can only base the wage w on the success of the project, i.e. w = w(x). The firm (employer) is risk-neutral, with profit equal to x - w(x), and since the success of the project is random, the expected profit of the firm is E[x - w(x)]. The employee is risk-averse with utility equal to w(x) - c(e), and again since the success of the project is random, the expected utility of the employee is E[w(x)] - c(e). Exerting high effort is costly, in particular, suppose c(eH) = 16 and c(eL) = 0. (1) If the wage is w(x) = x, i.e. the employer pays the employee all the revenue the project brings along, what effort level would the employee choose? (2) Suppose the employee has no outside opportunity, so that her reservation utility is 0. What is the optimal incentive contract the firm will provide? (3) Suppose the employee has an outside offer that guarantees her a reservation utility of 20, what is the optimal incentive contract the firm will provide?
In a school machine shop, 60% of all machine breakdowns occur on lathes and 15% occur on drill presses. Let E denote the event that the next machine breakdown is on a lathe, and let F denote the event that a drill press is the next machine to break down. With P(E) = .60 and P(F) = .15, calculate: P(Eᶜ ∩ Fᶜ) 6.91 There are five faculty members in a certain academic department. These individuals have 3, 6, 7, 10, and 14 years of teaching experience. Two of these individuals are randomly selected to serve on a personnel review committee. What is the probability that the chosen representatives have a total of at least 15 years of teaching experience? (Hint: Consider all possible committees.) 6.92 The general addition rule for three events states that P(A or B or C) = P(A) + P(B) + P(C) - P(A and B) - P(A and C) - P(B and C) + P(A and B and C) A new magazine publishes columns entitled “Art” (A), “Books” (B), and “Cinema” (C). Suppose that 14% of all subscribers read A, 23% read B, 37% read C, 8% read A and B, 9% read A and C, 13% read B and C, and 5% read all three columns. What is the probability that a randomly selected subscriber reads at least one of these three columns? 6.93 A theater complex is currently showing four R-rated movies, three PG-13 movies, two PG movies, and one G movie. The following table gives the number of people at the first showing of each movie on a certain Saturday: Number of Theater Rating Viewers 1 R 600 2 PG-13 420 3 PG-13 323 4 R 196 5 G 254 6 PG 179 7 PG-13 114 8 R 205 9 R 139 10 PG 87
Lucas F.
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