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1. Consider two islands, A and B. On each island there are 3 units of input L which can be used to produce goods r and y. On island A, x and y are produced according to the technologies x = VL and y = L, respectively, whereas on island B they are produced according to r = L and y = VL. On each island, there is one individual with preferences for x and y represented by u(x, y) = ry. Hint: Use the symmetry of the problem. a. Determine the respective production possibility sets on islands A and B. b. Assuming each island is isolated and there is no trade between them, determine a Pareto optimal allocation on each. c. Suppose markets are used to solve the allocation problem on each island. Assume the r and y firms maximize profits and the consumer maximizes utility, all taking prices as given. The consumer is endowed with the input and owns the firms. In this case determine the prices that would occur in equilibrium on each island. d. Next, suppose the inhabitants of the islands discover that it is possible to trade r and y; L is nontradable. Determine the competitive equilibrium prices and quantities that would occur in this case. e. What would be the pattern of trade in equilibrium? That is, which country would import and export each good? Explain why this pattern emerges. f. Show that the inhabitants of both islands would be better off with trade across islands than without it. g. Suppose island A were to specialize entirely in the production of y and B were to specialize in the production of x, after which they would each trade half of their output with the other island. Would this be better or worse than the outcome in part d. Explain why.2. Consider a rm that uses input, L, to produce output, 3:, in each of two periods. The prices of L and y are u: and p, respectively, and are the same in each period. a. First, assume the technology is the same in each period and is given by y = E. Assuming there is no discounting, determine the rms per period supply function, input demand function and its total prot function. b. Next, suppose the rm can set aside part of its prots in period 1 to improve the technology in period 2. Specically, if it sets aside r in period 1, the new technology would be p = (1 + x/Fli/TJ. Determine conditions on w and p under which the rm would choose to invest '1" :2 U. c. If is = p = 1, determine the rm's optimal level of investment. d. Next, suppose the outcome of R351] expenditure is uncertain. 1With probability p E (0,1), the eort will succeed and the technology will be [1+ Jail/E, but with probability {1 p], it will fail and the technology will remain y = E, despite spending 1'. Assume the outcome of the R851] expenditure is known at the start of the second period. In this case, determine the rm's total prot function if it were to invest r 32+ I}. e. Show that when w = p = 1, the rm's optimal level of investment is increasing in p. f. Returning to the case in which the outcome of REED is certain? suppose the rm did discount future prots at the rate 5 E {I}? 1}. Compare the role of discounting verse uncertainty [part d] in deciding the optimal level of investment. g. Is it possible to 5s},r whether the rm is more or less likely to invest when it discounts future prots versus when it does not discount?I 3. Two researchers try to complete scientic projects in a single period. Each is endowed with 1 unit of time. They can either work alone or together but they cannot split their time and do both. Each researcher can accomplish at most one project whether alone or jointly during the period. Researcher 1 working alone for e1 units of time [which represents his 'effort'} has probability el} of successfully completing his work. If researcher 1 completes his work alone? he consumes the credit (monetary or not} of 3:1 2 1. If he fails to complete the project.H he consumes the credit 3:1 2 0. Similarly? researcher 2 is successful working alone with probability q[e2] if he puts in time :23, and he gets credit :52 = 1 if successful and 3:2 = 0 otherwise. If the researchers work together and researcher i spends r; units of time on the joint work? then the probability of success for their joint work is q[r1 +13], but each i = 1:. 2 gets the credit 3.1,; = c} U a: c a: 1 if they succeed. Each gets I} if they fail. Each researcher i has a von NeumannMorgenstern utility function aflg? rag) = lira? where l3: 2 163: is the amount of leisure time i consumes. All variables are nonnegative. The researchers maximize their expected utilities, with (]_ 2:2+2z for Dizl d [}_ z for Dizl T\": 1 for 23:1 5\"\" '12: 1a]: 2:1. a. Find the researchers1 optimal efforts and expected utilities if they work alone. b. Now, suppose that efforts of the researchers are not directly observable and not con tractible. The researchers choose their labor inputs independently? but they can only do joint work and this can be enforced. Derive strictly positive Nash equilibrium levels of effort of the two researchers and the corresponding expected utilities. Show why there cannot be a Nash equilibrium involving a researcher devoting either [I or 1 of his time. c. Now consider the following game: Reseacher 1 either decides to work alone [5} or proposes to Reseacher 2 to work together {T}. If S is chosen} the researchers work alone. If T is chosen} then researcher 2 either decides to work alone {a} or to work together {t}. If researcher 2 chooses 5, both researchers work alone. If t is chosen? the workers work together but their efforts are chosen simultaneously and the efforts are neither observable nor contractible. Draw an extensive form game tree describing this game and nd pure strategy subgame perfect Nash equilibria for each possible value of c. [When 1. Consider a monopolist who produces output 11'} under constant returns to scale, with marginal cost of It). The monopolist produces during T :e 2 periods. Demand for its product in each period is D[P] = 100 2P, for P g 50. Hmvever, the monopolist does not know the demand at the outset. It only knows that the demand is linear and stationary. _ _ a. One recourse is for the monopolist to sample the market by announcing prices P1 and P3 in periods 1 and 2 and producing whatever quantities Q1 and Q3 consumers demand at these prices. It would then determine demand and behave optimally in subsequent periods. Show how this procedure works for the sample prices P1 = 20 and P2 = all] and determine the optimal supply and pricing strategy for each period t :1- 2. (Assume there is no discounting of future prots.) ' b. An alternative recourse would be for the monopolist to hire a market research rm at a cost of s to determine the exact demand it faces. What is the value of knowing the exact demand prior to the commencement of operations [i.e., prior to period 1) rather than having to announce the sample prices P1 = 20 and P2 = all] in periods 1 and 2? Would the rm be willing to pay an amwnt a up to this value? Is it capable of making this determination ex ante? Discuss. c. Instead of announcing specic prices P1 = 20 and P2 = All], suppose the monopo- list were to draw prices randomly from a uniform distribution on the interval [10,50]. Explain how to determine the rm's expected prot in the single period when it sets this randomly chosen price. Is the rm capable of performing this calculation ex ante? Explain. d. Alternatively, suppose the rm knows that its demand is of the form DU?) = mP+ it. The rm believes that m and a are independently distributed; or has density function f on {1, 4] and b has density p on [8G, EDD]. In this case, write an expression for the rm's expected single-period prot Erom announcing price P. Write an expression for its expected prot if it were to announce an optimal P. Is it capable of evaluating these? a. Discuss how the sampling procedure described above involving one observation per period might be generalized beyond the linear case to allow for an arbitrary demand curve. 'What considerations might be relevant in deciding whether to hire a market research rm to determine the rm's demand