2 is possible that some firms do not produce in a PE allocation. What can be said about which firms they are? c. Compare the levels of food consumption by different consumers in a given PE allocation. Compare the levels of food consumption that any given consumer gets in different PE allocations for the same economy. Explain how the different PE allocations differ from each other. d. Define a competitive (Walrasian) equilibrium (CE) for this economy. Characterize a CE in which money is numeraire (assuming that such a CE exists). Be as specific as possible. For each firm : that produces in a CE, find an inequality that relates K; to the output level and the marginal and total cost of firm i in equilibrium. e. Is a CE allocation necessarily Pareto efficient in this economy? f. Show that a CE allocation does not necessarily exist in this economy. Explain why CE might not exist. Use the notation above to specify an allocation that might plausibly arise if no CE exists. What can be said about how this allocation compares to PE allocations? 3. Consider the pricing problem faced by a monopolistic seller. There is a continuum of potential buyers of size 1. Each buyer demands at most one unit of the good. Buyers are of three types, i = 0, 1, 2. Each type i buyer values one unit of the good at V, = 0;VI, where r 2 0 is the quality of the good and @, is a parameter that describes the buyer's taste for quality. Assume 0 0 is constant, to produce one unit of the good of quality z. The monopolist's payoff is its expected profit. The buyers get payoffs equal to the value to them of what they buy minus what they pay. a. Suppose the monopolist can directly observe buyer type and can offer contracts contin- gent on type. Characterize the profit maximizing set of contracts for the monopolist. For the rest of the problem, suppose that types are not observable to the monopolist. The monopolist offers a menu of contracts of the form (po, a;) where a type i contract is meant for type i buyers. b. Formulate the monopolist's pricing problem with incentive and participation constraints, assuming each buyer has a reservation payoff equal to zero. c. Consider the relaxed monopoly pricing problem (RP) in which only the following down- ward adjacent incentive constraints (DAIC) and a participation constraint (PO) for type i = 0 are imposed. even - p2 2 02V/21 - Pi. (DAIC2) 01val - P1 2 01V/20 - po, (DAICI) GovID - Po 2 0 (PO) Show that all these constraints bind in a solution to this relaxed problem. d. Show that if the solution obtained in the relaxed problem (RP) satisfies monotonicity, i.e. r; 2 71 2 26, then all of the incentive constraints and participation constraints in the original problem are satisfied and the solution to the relaxed problem is also a solution to the original problem. e. Solve the relaxed problem (RP). Compare the optimal quality levels (25, 21, a;) to the quality levels the monopolist would choose in part a. Discuss any differences. f. Based on the solution to (RP) in part e, provide a sufficient condition on buyers' preferences such that the solution to (R.P) in part e is indeed a solution to the original problem in part b. Interpret this condition. Is the monopoly better off when this condition holds than when a solution to (RP) is not a solution to the original problem in part b?3. Five medical students i = 1,...,5 are candidates to be placed in residencies with four hospitals, j = 1, ...,4. No student wants more than one residency position and no hospital wants to hire more than one student. The tables below show the students' rankings of hospital residencies and the hospitals rankings of the students. Each student i [respectively, hospital ] ranks the hospitals [resp. students] as listed in the row beginning with i [resp. /], with more preferred listed first, reading left to right. For example, student 3 ranks hospital 4 first, then hospital 3, then hospital 1, then 2. Student 5 ranks 1 first, then hospital 2, then 4, and prefers no residency to a residency at hospital 3. Each hospital would rather fill its residency with any student instead of leaving the position empty. 2 3 Students: 2 231 4 Hospitals: 09 13 P 3 31 2 4 5 4 3 5 2 4123 4 452 3 2 4 a. Define a matching of students to hospital residencies. Note that no hospital can be matched with more than one student, so there must be at least one student without a residency. b. Define a stable matching in this environment. c. Find the outcome of the student-proposing deferred acceptance algorithm (SDA) applied to the preference rankings above. Show all your work. d. Is the outcome matching in part c stable? Show that your answer is correct. e. Is there a group of students who could each get a more preferred outcome than the one in part c if, for the SDA, they all reported different from their true preferences listed above? Assume that the agents outside the group report their preferences listed above. Justify your answer. You do not need to prove it is correct. f. Is there a hospital that could get a more preferred outcome than the one in part c by reporting different preferences to be used in the SDA when all other agents report their true preferences listed above? Show that your answer is correct. g. Is there a procedure that would take reported rankings by students and hospitals and determine a stable matching such that every hospital would have an incentive to report its true ranking listed above when the students report their true preferences? Explain. h What economic reason might explain why, in the absence of a formal organized matching procedure, hospitals typically offer residencies to students rather than students proposing to hospitals?3 4. A worker has to decide whether to finish her degree or not before applying for a job with a particular firm. The firm knows that the worker is of high or low ability and will know if the worker finishes her degree. Finishing the degree costs the worker 1 if she is of high ability and 2 otherwise. If the worker is of high ability, then she will produce 5 units of revenue for the firm if she finishes her degree and 3 units if she does not. If the worker is of low ability she will produce 2 units of revenue for the firm if she finishes her degree and 1 unit if she does not. The firm pays each of its workers with a degree W (0 0. The firms act as Cournot competitors facing uncertain (future) market demand described by the inverse demand function p(y, o), where y is aggregate output and o is a random variable distributed on the interval [o, a] with known density f(o). The firms must choose their outputs prior to the realization of market demand (a).2 a. In light of the above, set up the decision problem facing firm i assuming each firm seeks to maximize its expected profit taking the other firm's output as given. (Do not solve the problem.) b. For the special case in which p(y, a) = -y + o, solve for the Cournot equilibrium as a function of the expected value of a, denoted a. c. Argue that if a > 3k, then firm I's revenue will be greater and it's costs less than firm 2's in Cournot equilibrium, and hence I's profits will be greater. d In light of part c, suppose that the government decides to subsidize firm 2 in the event of a loss in order to ensure that there remain at least two competitors in the industry. This works as follows. As before, the firms first engage in Cournot competition, choosing their output levels to maximize their expected profits. Market demand is then realized. If the market price is below firm 2's break-even price, denoted p(yz), the government would raise the price for firm 2 (only) so that it would be able to break even. Assuming firm 2 knows it will be subsidized in the event of a loss, explain how the government intervention would affect its decision problem. e. Discuss the effects of a and & on the likelihood of the need for government subsidiza- tion. f. Next, suppose that even when subsidies are not necessary (i.e., when firm 2 would be profitable otherwise), the government decides to provide a price subsidy to firm 2 since firm 1 has a cost advantage. In this case, firm I would receive the market price p(y, a), but firm 2 would receive p(y, o) + 8 per unit. Prove that this would always have the desired effect of increasing firm 2's market share relative to firm l's. 3. Apple's market research shows that there are two types of consumers for its iMac computer and bMac computer (baby iMac). The bMac is physically the same as the iMac but has some functions disabled. Both iMac and bMac cost 300 dollars to produce per unit. Both consumer types, h and I, get utility m if they have m dollars and no computer: up (0, m) = w(0, m) = m. High type consumers with m dollars (after paying for a computer) get utility us(i, m) = 1500 + m and us(b, m) = 800 + m, where i denotes having one iMac and b having one bMac. Low types get utility up(i, m) = 600 + m and w(b, m) = 500 + m. Apple knows that there are one million consumers of high type and 2 million consumers of low type. Once Apple announces a price, it must serve all consumers who demand its product at that price. The consumers start with enough money to buy a computer if it raises their utility. They demand at most one computer during the period considered. a. Suppose Apple sells iMacs at 1500/unit and bMacs at 500/unit. Find the maximum sales and profits Apple could get. b Suppose Apple offers only iMacs. Find profit maximizing price, sales and profit. c. Suppose Apple offers only bMacs. Find profit maximizing price, sales and profit. d. Suppose Apple offers iMacs at price pa and bMacs at price pr. Show that no prices are such that high types buy bMacs and low types buy iMacs. e. Find profit maximizing prices pa and p that can induce high types to buy the iMac and low types to buy the bMac. What is the corresponding profit? f. Do part e assuming instead that the number of low types is 1 million. In this case, find profit maximizing prices and compare the resulting profit to what Apple receives if it offers only the iMac at a profit maximizing price