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GT ISYE 6230 - Homework # 5 - ISyE 6230 – Economic Decision Analysis II

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Homework # 5 - ISyE 6230 – Economic Decision Analysis II – Spring 2010 Due Thursday April 29, 2010 Grading: Up to 10 points per problem for completing each problem; the remaining 50 points will come from grading one randomly selected problem for correctness. 1. Consider the following game played by two parties, A and B. First, “nature chooses either C or D. C is chosen with probability 0.7, and D is chosen with probability 0.3. Second, party A chooses either E or F. Party A does not observe nature’s choice when it makes this choice. Next, party B chooses either G or H. Prior to making this choice, party B observes the choice of party A; party B also observes nature’s choice if A has chosen E, but does not observe nature’s choice if A has chosen F. Payoffs are determined as follows: A and B always receive the same payoff; the payoff is 0 if A chooses E and B chooses G, regardless of nature’s choice; the payoff is 5 if A chooses E and B chooses H, regardless of nature’s choice; the payoff is 0 if nature chooses C, A chooses F, and B chooses G; the payoff is 10 if nature chooses C, A chooses F, and B chooses H; the payoff is 10 if nature chooses D, A chooses F, and B chooses G; and the payoff is 0 if nature chooses D, A chooses F, and B chooses H. Draw the extensive form of this game (You do not need to solve the problem). 2. Describe all the pure-strategy pooling and separating perfect Bayesian equilibria in the following signaling game:3. Find a real-life example that you have seen which relates to the topic of “Principal-Agent Problem” (try to use different examples than what has been discussed in class). Write a short paragraph explaining how the situation is related to the Principal-Agent Problem. Identify the principal(s), the agent(s), possible actions and outcomes, the contract (incentive scheme), and the payoffs. Also, discuss potential conflict in the motives of the principals(s) and the agent(s) if they exist. 4. Consider the following Principal-Agent problem: The principal is risk-neutral, but the agent is risk-averse with the utility function of U(w, e) = √2– e2, where w is the wage and e is the effort. In this problem, the agent’s effort impacts the work result, but the effort is not observable by the principal. The agent’s reservation utility is 18 and he can choose between putting low effort e=1, or high effort e=3. There are three results; they can value either V=0 or V=2000 or V=3000 to the principal. If the agent put effort e=1, the probability of having V=0 is 0.6; V=2000 is 0.2; and V=3000 is 0.2. If he puts e=3, the probability of having V=0 is 0.2; V=2000 is 0.3; and V=3000 is 0.5. (i) Indentify the optimal symmetric information contracts. What effort will the principal demand from the agent? (ii) When the effort is not observable, what is the constrained maximization problem that determines the optimal contract in the case that the principal would like the agent to exert an effort of e=3? (iii) Which contract will the principal offer to the agent under the moral hazard problem? 5. A hard working high-school student is having difficulty in understanding a concept of his main coursework, so he wants to hire a senior tutor. However, since he is new in his school, there are some aspects unknown to him, e.g., who can be considered to be a “good” or “bad” tutor. Assume that a tutor has disutility, ne2 (where n = 1 for a “good” and n=2 for a “bad” tutor, since a “bad” tutor should suffer greater disutility to effort). The probability that the tutor is “good” is p. The student pays tutoring fee w and both tutor types’ reservation utility is 0. Everyone is risk neutral and the student’s benefit of each session can be quantify as βe, where β is a sufficiently large and β is independent with the tutor type. In other words, for each unit of effort put in the session, the student receives β units of benefit. (i) What type of Principal-Agent problem is this case? (ii) Suppose the student had perfect information about the tutor’s type, formulate and solve this problem; i.e., what effort levels (e) are demanded, and what is the tutoring payment the student will pay (w)? (Hint: the tutor’s utility function is either UG (w,e) = w - e2 or UB (w,e) = w - 2e2, depending on his type.) (iii) If the information is asymmetry, formulate and solve for the optimal effort of each type (eG, eB). How are they compared to the symmetric information


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GT ISYE 6230 - Homework # 5 - ISyE 6230 – Economic Decision Analysis II

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