Detailed solu8tions.

Problem 1 Slutsky equation Tomas is trying to figure out how to supplement the study allowances of 500 kr a week. He is considering a part-time job at a gas station. The wage is 50 kr per hour. His utility function is U(C, L) = C*L where C is his consumption measured in SEK and L his leisure measured in hours. The amount of leisure time that he has left after allowing for necessary activities is 50 hours a week. a. What is the monetary value of Tomas' endowment? b. Draw Tomas' budget set (horizontal axis: leisure and vertical axis: consumption). c. Set up the maximisation problem and decide optimal consumption and leisure. d. Let Y = study allowance and T = total amount of leisure time. Express his demand for consumption as a function of study allowance and wage. e. Express his supply function for labour as a function of study allowance and wage. f. How many hours would Tomas work if he did not receive any study allowance? Problem 2 Slutsky equation Assume that the function U(x, y) = x0 yois the utility function of a person who consumes two goods in quantities x and y, respectively. The price of x is p, = 5 and the price of y is py = 8 This person's income is m = 160. a) Find the optimal consumption choice of this person. b) Verify that at the optimum that you found the marginal rate of substitution equals the price ratio. Explain in terms of economic theory why this should be the case! c) Assume that the price of x falls to p, = 4. i. Draw the old and the new budget constraints in a diagram, (Indicate at what values they intersect the axes). ii. Calculate the person's demand for x and y at the new price. ili. Calculate the compensated income, m'. wv. Decompose the change in demand for good x into a substitution and an income effect. Problem 1. Consumer's surplus Mattias has quasilinear preferences and his demand function for books is B = 15 -0.5p. a) Write the inverse demand function b) Mattias is currently consuming 10 books at a price of 10 kr. How much money would he be willing to pay to have this amount, rather than no books at all? What is his level of consumer's surplus? Problem 2. Consumer's surplus 0.1 Suppose Birgitta has the utility function U = x x, . She has an income of 100 and P = 1 and P = 1. Calculate compensating and equivalent variation when the price of x1 increases to 2. Also, try to estimate the change in consumer's surplus measured by the area below the demand function. Problem 3. Consumer's surplus Explain the concept of "consumer surplus" in words and illustrate by a diagram. Problem 4. Consumer's surplus The inverse demand curve (the demand curve but with p instead of q on the left hand side) is given by p(q)=100-10q. The consumer consumes five units of the good (q).All problems are from Mankiw's Macroeconomics (6" Ed.) unless otherwise noted. 1. (Chap 12, problem 1) Use the open economy IS/LM model to predict what would happen to aggregate income (Y), the exchange rate (e), and the trade balance (NX) under both floating and fixed exchange rates in response to each of the following shocks: a. A fall in consumer confidence about the future induces consumers to spend less and save more. (The MPC falls) b. The introduction of a stylish line of Toyotas makes some consumers prefer foreign cars over domestic cars (NX demand falls) c. The introduction of ATMs reduces the demand for money (L falls) 2. (Chap 12, problem 3) The open economy IS/LM model takes the world interest rate (r) as given. Consider what happens when this world rate changes. a. What might cause world interest rates to rise? b. In the open economy IS/LM model with a floating exchange rate, what happens to aggregate income (Y), the exchange rate (e), and the trade balance (NX) when the world interest rate rises? C. Answer part b again, except now with a fixed exchange rate. 3. (Chap 12, problem 4) Business executives and policymakers are often concerned about the "competitiveness" of American industry (the ability of U.S. industries to sell their goods profitably in world markets). a. How would a change in the exchange rate affect competitiveness? b. Suppose you wanted to make domestic industry more competitive but did not want to alter aggregate income (Y). According to the open economy IS/LM model, what combination of monetary and fiscal policies should you pursue? 4. (Based on Chap 11, problem 3 - with my own extensions) The economy is a described by a closed economy IS/LM model. a. Consumption is C = 200 + 0.75(Y - T) and investment is I = 200 - 25r. G = T = 0. Solve for the IS curve (i.e. an equation for Y in terms of r). b. Money demand is given by L = Y- 100r, the money supply is 1000 and the price level is P. Solve for the LM curve (i.e. an equation for Y in terms of r and P). c. Find the equilibrium interest rate of r and the equilibrium level of income (i.e. solve for where the IS and LM curves cross) d. The LRAS curve is Y* = 975. What is the price level at which output is exactly equal to this? e. Now the money supply increases to 1200. What is the new equilibrium level of income (i.e. solve the IS/LM problem again with M = 1200). f. If prices remain at exactly the level you found in d), what is the new level of income? Does the increase in money supply cause an expansion or contraction in income? g. What would prices have to be so that income is exactly equal to Y* = 975 again?Problem 1 (Uncertainty) Rick is considering whether to spend 5 dollars betting on Republicans winning the next election. If Republicans were to win the election, Rick would be paid 4 dollars for any dollar that he has bet. The utility that Rick derives from a (positive or negative) cash transfer of a dollars is determined by the following utility function, u(r) = (475 + 75x)1/2. Rick believes that the probability of republicans winning the next election is 1/3. 1. Find the expected value of such a lottery. Intermediate Microeconomics F. Nava 2. Find Rick's expected utility of taking such a gamble. Would he accept it? Or would he reject it and get r = 0? 3. What's the certainty equivalent of such a lottery. Problem 2 (Static Games) Consider the following static two-player game: 1\\2 A B A 3, 7 7,3 2,2 Player 1'is the row player, and his payoff is the first to appear in each entry. Player 2 is the column player and his payoff is the second to appear in each entry. 1. Find the pure strategy Nash equilibria of the game, and show that they are equilibria. 2. Find the mixed strategy Nash equilibrium of the game. 3. Derive the mixed strategy best responses. Problem 3 (Bayesian Games) Consider the following Bayesian game played by two players (1 and 2) who are deciding whether to cooperate, C, or defect, D. Two states are possible, Good and Bad. Suppose that Player 2 knows the state, while Player 1 thinks that the state is Good with probability p. Payoffs in each state respectively satisfy 12 C D 1\\2 C D State Good: 0 0.0 1, 1 State Bad: C 0.0 0. 1 . D | 1,1 0.0 D 1,0 3,3 Player 1 is the row player, and his payoff is the first to appear in each entry. Player 2 is the column player and his payoff is the second to appear in each entry. 1. What is the set of possible strategies for the two players in this game? 2. Find the pure strategy Bayes-Nash equilibria for all values of pe (0, 1). Problem 4 (Cournot Uncertainty) Two firms compete to sell a good. Firm I has total costs of production Ci(ni) = (qi)" + 21 and its costs are known to Firm 2. The total costs of Firm 2 depends on its type. If Firm 2 is of type L, its costs are Cr(q1) = 2qr. If Firm 2 is of type H, its costs are Cu(qn) = 2 (qu). Firm 2 knows its type. But Firm I only knows that Firm 2 can have either cost structure with equal probability. The inverse demand for the output produced by the two firms in this market satisfies: p(gi + 92) = 10 - 2(q1 + 42) Firms choose how much output to produce in order to maximize their profits. Find the Bayes-Nash equilibrium of this game. Characterize the equilibrium output strategies for both firms. Find the market price for each of the two possible cost configurations. Problem 5 (Repeated Games) Consider the following asymmetric Prisoner's Dilemma: 1\\2 C D 3.4 1,6 4,0 2.2 1. Find the minmax values of this game. 2. Then, consider the following "trigger" strategy: any player chooses C provided that no player ever played D; otherwise any player chooses D. Write the two incentive constraints that if satisfied would make such a strategy a NE. Then, write the two additional incentive constraints that if satisfied would make such a strategy a SPE. What is the lowest discount rate for which such strategy satisfies all the constraints.Exercise 12.1.1 If you throw a die, you will get a number between I and 6 with equal probability. What is the expected value? Exercise 12.1.2 In Figure E.12.1, we have drawn the amount of utility a certain individual gets from different levels of wealth. Figure E. 12.1 U 10 W 500 1 00 non Does this person have diminishing, increasing, or constant marginal utility of wealth? b) Is she risk-averse, risk-neutral, or a risk-lover? Why? Suppose she has 500,000 and is invited to participating in a lottery that with a probability of 50% increases her wealth to 1,000,000 and with equal probability causes her to lose everything. c) What is the expected value of the lottery? d) What level of utility does she achieve if she does not participate in the lottery? Indicate hat point in the graph. e) What is her expected utility if she does participate in the lottery? Indicate that point as well. f) Indicate in Figure E.12.1, what represents the risk premium.10.2 The Demand for Labor Exercise 10.2.1 When a firm is to determine how much labor it needs, it is often interested in how revenue is affected by, say, one more hour of labor. This is called "the marginal revenue product of labor," MRPL, and can be defined as ATR MRP = AL Exercise 10.2.2 Suppose we have perfect competition in both the labor market and in the output market. Show that this leads to the equilibrium wage being equal to the marginal revenue product of labor, i.e. that W = MRP. Exercise 10.2.3 a) How will the equilibrium criterion in Exercise 10.2.2 change if the firm is a monopolist in the output market? b) Will the firm, in that case, demand more, less, or an equal amount of labor? Exercise 10.2.4 a) Suppose the output market is, again, a perfectly competitive market and that there are many workers. However, let the firm be a monopsonist in the labor market. As compared to Exercise 10.2.2, will the firm demand more, less, or an equal amount of labor? In your answer, you can assume that the supply of labor increases with higher wages. b) Will the equilibrium wage be affected? In that case, how