ISyE6230 Spring 2006 Homework 5Due Thurs April 27Grading: up to 12.5 points per problem for completing each problem; the remaining 50 points will come from grading one randomly selected problem for correctness. In all of the following questions, you are required to show your work and demonstrateclearly the logic of your method. Your final answer should be indicated by a box around itor two lines underneath. Ambiguity will always be penalized. Please write Last Name, First Name on the assignment you turn in. 1. Consider a problem in which a principal contracts an agent, whose effort determines the result, and that uncertainty is represented by three states of nature. The agent can choose between two effort levels. The results are shown in the table below. The principal and the agent both believe that the probability of each state is one third. Theobjective functions of the principal and the agent are, respectively:B(x,w) = x-wU(w,e) = √w-e2where x=(e,p) is the monetary result of the relationship and w=w(x) is the monetary pay-off that the agent receives. Assume that the agent will only accept the contract if he obtains an expected utility level of at least 112 (his reservation utility level). states of NatureEfforts p1 p2 p3e=8 60,000 60,000 30,000e=6 30,000 60,000 30,000a. What would be the effort and the wage in a situation of symmetric information? What would happen if the principal were not risk-neutral?b. What happens in a situation of asymmetric information? What pay-off scheme allows an effort level of e=6 to obtained? What pay-off scheme allows the effort level of e=8 to be obtained? Which effort level does the principal prefer? Discuss the result. 2. A local college requires the services of an expert in Semitic philology, and there is only one expert with whom they can contract. The college wants the expert to translate a part of a large manuscript written in a Semitic language that has fallen into their hands by chance. Since the translation is very complicated, the only person capable of doing it is the graduate, who can tackle it in his free time. The graduate could be one of two possible types: he is either a fast translator, capable of doing two pages per hour, or he is a slow translator, only capable of doing one page per hour. An hour of translating implies a disutility to the graduate (independent of his type) that must be compensated with $10. a. The college director has calculated how many pages of the manuscript he wants translated and how much to pay the translator. These calculations were done under the assumption that he is able to recognize whether the translator is fast or 1slow. His decision is to translate 60 pages of manuscript if the translator is slow, paying $500 for the job, and 90 pages if the translator is fast, paying only $450. Determine whether this contract is reasonable and explain.b. Once the translator arrives at the college, the director realizes that there is no way of knowing if he is fast or slow. Given the two contracts, the translator assures the director that he is slow, but the director does not believe him. Why not?c. The director decides to redo his calculations, and to offer a new contract menu to the translator. Explain the form that the menu will take. Will the director ask for more or less pages from each type? How will he calculate the payments? d. Verify your results of part (c) analytically, using the function Pi(n) for the value that the college places on a translation of n pages, with Pi’(n) > 0, Pi”(n) < 0. 3. Describe all the pure-strategy pooling and separating perfect Bayesian equilibria in the following signaling game. 4. Goliath Corporation is the manufacturer of devices and software programs for industrial espionage. The firm has noticed that there are two customer groups (e-commerce and government organizations) that have different demand characteristics N [0.5] [0.5]LRLRSendert1Sendert2UDUDReceiverUDReceiverUD2(1,1)(2,0)(2,2)(0,0)(0,0)(0,1)(1,0)(1,1) [p] [1-p] [q] [1-q]2The e-commerce group is more sensitive to price, perhaps because they can develop their own algorithms if the product is too expensive. Goliath has estimated the two groups’ demand functions as.Qe(p) = 80 – 4pQg(p) = 60-0.6p.Goliath faces a production cost of C(q) = 0.25q2, where q denotes the number of units produced. Compare and contrast Goliath’s optimal pricing decision(s) under several market structures. To make these comparisons, you should determine the following for each market structure: a. optimal price b. quantity purchased c. consumer surplus d. Goliath’s profit e. social welfare.First consider (i) perfect competition and (ii)single-price monopoly as we did in class. The final market structure is (iii) 3rd-degree price discrimination. For this one, first note that the production cost depends on the total quantity produced. Secondly, as a hint, set up an optimization problem to maximize Goliath’s profit when there are two pricing decisions and they can determine each type of customer.
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