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(1) Suppose there are two large grain elevators (consolidators) in Iowa that buy all of the wheat from many small farmers and sell it on the world market at the price p > 100. The aggregate supply curve among the farmers is S(p) = p 100, where p is the local price of wheat (i.e., the price the elevators pay), and this is known by the elevators. Prior to the start of the planting season, the elevators (buyers-duopsonists) announce their demands (the quantities they wish to purchase), and the local price is determined to clear the market, that is, to elicit the aggregate quantity demanded from the farmers. The buyers thus act as middlemen, and their objective is to maximize the difference between the value of their sales and the value of their purchases. (a) Determine the equilibrium demands of the elevator operators. (b) Discuss the effect of an increase in p on the equilibrium quantities, local price, and profits of the elevators. (c) Suppose all of the wheat is sold abroad and foreign governments impose a tariff t on U.S. wheat imports. Assuming those countries can continue to buy wheat from other sources at the price p, how would this affect the equililbrium quantities and local price? (d) Now suppose there is a domestic market for Iowa's wheat with market demand given by D(p 0 ) = 200 p 0 , where p 0 is the price elevators receive. Assuming the elevators sell their entire stock domestically, they then act as duopsonists in the local market and duopolists in the domestic or national market for Iowa wheat. Determine the amounts they will purchase locally and sell nationally and the prices at which they would do so, i.e., p and p 0 . (e) Now suppose the elevators can sell in both the domestic market as in part d and internationally (without the tariff) as in part a. Formulate the (Nash) decision problem each operator would face in determining how much it would supply to each market? (You do not have to solve the problem.) (f) Alternatively, suppose the farmers could sell their wheat directly on the national market where demand is as in part d. What would be the equilibrium price and quantity, p and Q , respectively? (g) At the outcome determined in part f, there would still be room for a (single) middleman to operate in this market, providing p < p. In particular, it would 1 2 purchase wheat in Iowa at the price p and sell it internationally at p. Given the above supply curve, determine the middleman's optimal quantity and profit as a function of p and p.

(2) Two researchers with identical ability and preference try to complete scientific projects in a single period. The endowment of time is 1 for each researcher. Each researcher i has a von Neumann-Morgenstern utility function u(li , xi) = lixi , where li = 1 ei is the amount of leisure time i consumes and xi is the credit the researcher gets from his work. All variables are nonnegative. The researchers maximize their expected utilities. In the following, the probability of success is given by p(z) = ( z for 0 z 1 1 for z > 1 , where z is some effort level to be specified. (a) First consider the game (game A) in which both researchers work alone independently and simultaneously. If researcher i puts in effort ei , the probability of his/her success is p(ei). If one succeeds and the other fails during the period, the one who succeeds gets credit 1 and the one who fails gets credit 0. If both succeed, both get credit c, where 0 < c < 1. If both fail, they get credit 0. Draw the extensive form of this game. (b) Given that researcher 2 puts in e2 effort, what is the optimal level of e1 for researcher 1? (c) What are the Nash equilibrium effort levels in the game described in (a) and resulting payoffs? (d) Now, suppose that the researchers work jointly. When e1, e2 are effort levels of the researchers, the probability of success of the project is p(e1+e2). If the project succeeds, both get credit c where c is the same value as in (a). If the project fails, both get 0 credit. The researchers choose their efforts independently and effort levels are not directly observable and not contractible. Draw the extensive form of this game (game B). (e) Derive strictly positive Nash equilibrium levels of effort of the two researchers and the corresponding expected utilities of game B. Show why there cannot be a Nash equilibrium involving a researcher devoting either 0 or 1 of his time. (f) If the researchers expect to obtain Nash equilibrium payoffs in Game A and Game B, which would they prefer? Interpret the result. (3) A seller sells one indivisible object to two (potential) buyers. The buyers' valuations are distributed identically and independently as follows: Each has valuation 1 with probability 1 2 and valuation 2 with probability 1 2 . Each buyer knows his valuation only but the prior distribution of valuations is a common knowledge. (a) A seller tries the following Bayesian Nash direct mechanism: buyers report their valuations (either 1 or 2). If both report 1, the winner of the object is chosen 3 randomly with each winning with probability 1 2 and winner pays x. If both report 2, the winner of the object is chosen randomly with each winning with probability 1 2 and winner pays z. If one reports 1 and the other reports 2, the one who reports 2 wins and pays y. In all of the above, a loser pays or gets nothing. A buyer (after knowing his valuation) participates in sale if and only if his expected surplus is non-negative. Find the values of x, y, z that make truth telling a Bayesian Nash equilibrium and allow both buyers to participate. What are the values of such x, y, z that maximize the expected revenue of the seller? What is the maximum expected revenue for the seller?

(b) In (a), what are the values of x, y, z that make truth telling a dominant strategy equilibrium and allow both buyers to participate?. Here, a buyer (after knowing his valuation) participates in sale if and only if his ex-post surplus is always nonnegative. What are the values of such x, y, z that maximize the expected revenue of the seller? What is the maximum expected revenue for the seller? (c) Suppose that the seller conducts the first price sealed bid auction to sell the object but specifies two values {b1, b2} where b2 > b1 0, from which each bid must come from. If there is a tie in the bids, each bidder wins with probability 1 2 and the winner pays the (highest) bid. Find the set of {b1, b2} that form a strictly increasing, symmetric Bayesian Nash equilibrium in which both bidders participate regardless of their valuations. What are the expected revenue maximizing {b1, b2} for the seller? What is the expected revenue? (d) Suppose the seller allows bids of b1 or b2 (only) 0 b1 < b2 but conducts the second price sealed bid auction. Again, in the case of a tie (say at b2), each bidder wins the object with probability 1 2 and pays the second highest bid if he wins (in this case, b2). Find all values of {b1, b2} that induce participation of both bidders and make it a dominant strategy for type 1 (a bidder with valuation 1) to bid b1 and type 2 (a bidder with valuation 2) to bid b2 . What are the expected revenue maximizing {b1, b2} and the maximum expected revenue? (e) Compare the answers to (a) (d) and interpret the results.

(4) Consider an economy with three consumers, three different inputs and three different outputs. Each consumer i is initially endowed only with one unit of input i (time for labor of type i) and consumes only output i (i = 1, 2, 3). The consumers live on a circle, facing inward, and each consumer can use its input to produce the output it consumes and/or to produce the output that is consumed by its neighbor to the left (clockwise along the circle). Formally, consumer i (i = 1, 2, 3) can use `i 0 units of input i to produce `i units of output i and can use Li 0 units of input i to produce 2Li units of output i + 1, where `i + Li 1. Here, "output 4" is another name for output 1 and "consumer i = 4" is another name for consumer 1. (Note that each 4 consumer is less productive in producing its own output than in producing the output consumed by its neighbor.) Consumer i gets utility xi by consuming xi 0 units of good i. (a) Specify the set of feasible allocations for this economy, assuming that there is free disposal of all goods. Use the notation above and any other notation you need, defining any notation you introduce. (b) Suppose that the consumers are not able to trade. They can only produce (using their own initial endowments) and consume. What are their optimal input, output and consumption choices? (c) Suppose, instead, that any two consumers can freely trade any goods with each other. Each consumer i is still the only consumer able to produce output using input i. Assume that trading does not use up any resources (either inputs or outputs). No consumer is forced to trade and no trade by a pair of consumers occurs if it reduces a consumer's utility. What trades might consumer 2 make with consumer 3 that both consumers would agree to, with no involvement of consumer 1? What trades might consumer 2 make with consumer 1 that both consumers would agree to, with no involvement of consumer 3? What conclusion can you draw about the final allocation if the only possible trades are bilateral (between two consumers, without involvement of the third)? Explain. (d) The economy can be viewed as a private ownership economy in which each consumer i owns firm i, which has the production possibilities of consumer i. Specify the production set of firm 2 in this private ownership economy. (e) Find a competitive (Walrasian) equilibrium of the private ownership economy of part d in which the consumers consume equal amounts of output. Let ri be the price of input i and let pi be the price of output i. Find the profit made by firm i in this equilibrium. Give a verbal description of the entire allocation and the equilibrium prices. Justify the claim that the equilibrium allocation is Pareto optimal. (f) Explain why the allocation in part e is not attainable through the unforced bilateral trading by pairs of consumers that is described in part c. What institutional arrangement for trading could make the competitive equilibrium allocation attainable? (g) Use the equilibrium prices in part e to compare the total value of equilibrium output in part e to the total value of output in a different Pareto optimal allocation.