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Problem 1. Consider the Planner's problem in the Ramsey problem. Suppose there are only two periods and suppose further that the utility is logarithmic, that is, u(c) = log(c); output in period t is given by y, = Ack, for some a E (0, 1), and depreciation 6 = 0. The Ramsey problem is therefore given by max log(co) + Blog(ci) st : co + ki = Aoki C = Alki (i) Draw the set of feasible consumption bundles (co, ci). Next, draw the planner's indifference curves and identify the optimum. (ii) Consider an increase in Aj. What changes in the graph? How does the increase in A, affect co.ci, and by? Give an intuition and explain the relevant income and substitution effects. (iii) What is the effect of an increase in Ao? Problem 2. Consider the competitive equilibrium in the Ramsey model (where the saving rate is endogenous). The rates of technological progress and population growth are zero (n = g = 0). The technology is given by y = f(k) = Ak" for some a e (0,1). the depreciation rate is o E (0, 1), and the per-period utility is "() = 14 1-#, for some # > 0. The country under consideration features an inefficient law enforcement system and an extensive degree of corruption, fraud, theft, and the like. As a result, when an individual agent decides how much invest, he/she expects a fraction n of his/her capital returns to be confiscated by other agents in the economy. (Note that this fraction of capital returns does not disappear from the economy; it simply becomes part of the income and consumption of other agents.) Now suppose that, starting from the steady state corresponding to the above conditions, the country undertakes a reform of its legal and law enforcement system. The reform reduces the effective rate of confiscateon from , to new, where 0 0. The government finances this spending g with a distortionary tax r on capital income. (i) Find the conditions that characterize the steady state and e and & . Use the phase diagram to describe this economy. (ii) Now suppose that, starting from the steady state corresponding to the above con- ditions, the government undertakes a reform which makes his government spending more productive, i.e. f, increases. The reform is unanticipated and the aforementioned change happens immediately and is expect to last for ever. Use the phase diagram to describe the initial steady state (prior to reform) and the new steady state (after the reform). What is the impact of the reform on the long-run levels of capital, output, consumption wages, and the interest rate? Explain the intuition behind these effects.(60 points) Suppose an industry with inverse market demand P = 300-2Q is comprised of three firms which compete in quantities. Firm 1 is a Stackelberg leader and therefore acts first. Firms 2 and 3 observe Firm 1's decision, then act simultaneously. Firms produce output with constant marginal cost c = 60. (a) (10) Write the profit functions for the three firms as a function of choice variables (quantities for each firm) only. (b) (15) For any value of 91, find the best response functions for firms 2 and 3. (c) (10) Use your answers to part (b) to express qa and q, as functions of q only (i.e. not as functions of each other). (d) (10) Find the subgame perfect Nash equilibrium of this game (i.e. the optimal choices which characterize what all firms are doing in equilibrium), equilibrium profits, and market clearing price. (e) (15) Suppose that firms 1 and 2 act simultaneously in the first period, then Firm 3 observes their choices before making its decision. Does social welfare increase or decrease relative to the previous model? (Hint: which is closer to how a perfectly competitive market would behave?) (40 points) Consider a Hotelling model of the type discussed in class: there is a beach that is one mile long, with customers evenly distributed along the beach. Two dif- ferent ice cream shops are located at the endpoints of this beach (at points 0 and 1). Consumers dislike walking down the beach and pay transportation costs t = 2 per mile walked. All consumers have a value of V = 10 for an ice cream. Ice cream shops have constant marginal cost c = 2. (a) (5 points) Consider a consumer who is at the midpoint of the beach, z = . Write down their total cost (including transportation costs) of buying from each firm, assuming firm 1 (located at 0) sets price Pi and firm 2 (located at 1) sets price P2. b) (10 points) A consumer is indifferent between the two shops if the total cost the face at both shops is the same. Call this indifferent consumer's location rm . Fi i'm for any pair of prices Pi and P2. (15 points) All customers to the right of r' buy from firm 2, and all cust the left of r" buy from firm 1. Write down each firm's profits as a func6. General Equilibrium Exercise 1 Consider an initial endowment inside the Edgeworth box. Can the negociated contracts coincide to any point in the contract curve if the result is Pareto optimal? Comment. Exercise 2 The contract curve has this name because it is the set of all possible contracts. Comment. Exercise 3 To get a Parcto optimum it is necessary that the marginal rate of substitution between any two goods be equal for any two people that buy them, Comment, Exercise 4 Represent graphically, and justify, the optimal Pareto allocations for two goods- tea (X) and coffee (Y) - consumed by two people, Mr. Lipton and Mr. Delta, in the following situations. Assume that the endowment of coffee and tea are A and L . respectively. a) Mr. Delta only likes coffee and Mr. Lipton only likes tea. b) They both like coffee and tea. The two goods are perfect substitutes for Mr. Delta. Mr. Lipton likes consuming a certain quantity of tea that equals L/ A of the quantity of coffee. c) Mr. Delta and Mr. Lipton have preferences given by: Up =XD+Lp and UL=2X, +Y, d) Mr. Delta and Mr. Lipton have strictly concave to the origin indifference curves. Exercise 5 Assume that a certain consumer type A has X units of X and a certain consumer B has Y units of Y. The utility functions of these two consumers are given by the expressions: U. =XAY, and U. =X Y. where X and Y are the quantities consumed. a) What is the equilibrium relative price of X in a competitive market? b) Is this equilibrium unique? c) Define the contract curve. Exercise 6 Two tribes, the Ocidentals and the Orientals, inhabit a certain island. They live in the respective sides of the island and meet only once a year in the annual market. No contact to the rest of the world is established. Both tribes collect com (Y) and hunt deers (X). In the annual market, the deers are exchanged for com in competitive conditions. There are 1000 ocidental families and each produces 30 deers and 200 tons of corn. The utility function of each ocidental family is Ude = Xoe . You . where Xoc is the number of deers and Yo, is the number of tons of corn consumed in a year. There are 2000 oriental families and each produces 25 deers and 300 tons of corn. The utility function of each oriental family is Uo = Xor Yor , where Xor is the number of deers and Yo, is the number of tons of corn consumed in a year. The annual market achieves a competitive equilibrium
