Economics economics

\f{1} Suppose there are two large grain elevators l[consolidatorsj in Iowa that buy all of the wheat from many small farmers and sell it on the world market at the price 13 :2:- me. The aggregate supply curve among the farmers is 31:32} = 31 lm, 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 season1 the elevators {buyersduopsonists} 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 quantityr demanded from the farmers. The buyers thus act as middlemen1 and their objective is to maximize the dilfenence 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 13 on the equilibrium quantities1 local price, and prots of the elevators. {c} Suppose all of the wheat is sold abroad and foreign governments impose a tariff t on [1.5. wheat imports. Assuming those countries can continue to buyr 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 Hui") = 2m 15", where ,p'r 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 so1 i.e_, p and p'. {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. Fonnulate 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.) {1"} tltematively1 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 112', respectively? {g} At the outcome determined in part f, there would still be room for a l[single]I middleman to operate in this market, providing 30* a: p. In particular, it would 'I purchase wheat in Iowa at the price p\" and sell it internationally at p. Given the above supply curve, determine the middlemanls optimal quantity and prot as a function of p' and s. {2} Two researchers with identical ability and preference try to complete scientic projects in a single period. The endowment of time is l for each researcher. Each researcher i has a von NeurnannMorgenstern utility function MIL-,1.) = LIE, where I; = l e; is the amount of leisure time 1E consumes and I.- is the credit the researcher gets from his work. All variables are nonnegative. The researchers maximize their expected utilities. .3 for I] E z 5 l l for 2}]. P In the following1 the probability of success is given by p{z} = { where a is some effort level to be specied. {a} First consider the game l[game A} in which both researchers work alone inde pendently and simultaneously. If researcher r'. puts in effort ei, the probability of hisfher 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 D. If both succeed, both get credit c, where D 4: c c 1. If both fail, they get credit . Draw the extensive form of this game. {b} lGiven that researcher '2 puts in e; effort, what is the optimal level of 1 for researcher I? {c} Iv'v'hat are the Nash equilibrium effort levels in the game described in {a} and resulting payoffs?I {d} Now, suppose that the researchers work jointly. 'W'hen e1, E2 are elfort levels of the researehers, the probability of success of the project is p{e1+ea}_ If the project succeeds, both get credit c where c is the same value as in {a} . If the project fails, both get [I 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 E}. {e} Derive strictly positive Nash equilibrium levels of effort of the two researchers and the corresponding expected utilities of game E. Show why there cannot be a Nash equilibrium involving a researcher devoting either I] or 1 of his time