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Problem set 1: Preferences and Utility Exercise 1: Suppose the utility function for two goods, X and Y, has the Cobb-Douglas form utility= U(X, Y) = Vx . Y. a. Graph the U = 10 indifference curve associated with this utility function. b. If X = 5, what must ) equal to be on the U = 10 indifference curve? What is the MRS at this point? c. In general, develop an expression for the MRS for this utility function. Show how this can be interpreted as the ratio of the marginal utilities for X and Y. d. Consider a logarithmic transformation of this utility function: U' = log U where log is the logarithmic function to base 10. Show that for this transformation the U' = 1 indifference curve has the same properties as the U' = 10 curve calculated in part (a) and (b). What is the general expression for the MRS of this transformed utility function? Exercise 2: Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether they obey the assumption of diminishing MRS): a. U = 3X + Y. b. U = V X - Y. C. U = VX2 + Yz. d. U = vx2 _ yz e. U = X2/3p1/3 f. U = log* + logY. Exercise 3: Consider the following utility functions: a U(X, Y) = XY. b. U(X, Y ) = x2yz c. U(X, Y) = InX + InY. Show that each of these has a diminishing MRS, but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?equal to price. 1 1. In two-part pricing and block pricing. the key quantity is found by setting marginal cost a. True b. False 12. In the Sweezy model of oligopoly, it is possible for marginal cost to change slightly with a. True no resulting change in the profit maximizing output or price. b. False 13. Which of the following statements is incorrect regarding a Cournot oligopoly? Hint: consider the formula given for the profit maximizing price of a Cournot oligopoly that does not have knowledge of its demand function. a. If there are more firms in the industry the price will be closer to the firm's marginal cost. ~ b. If the demand for the good is more elastic, then the price and the marginal cost will be closer to each other. c. If the firm's marginal cost decreases, this will cause the optimal selling price to decrease. d. None of the above answers this question; that is, all of the above statements are correct. 14. Which of the following is correct if a market is contestable? a. Firms have market power. b. There are substantial barriers to entry by new firms. c. Firms earn zero economic profit. d. There are substantial sunk costs. e. Some firms have access to technology that is unavailable to other firms. 15. Rent seeking a. causes firms to produce the output found under perfect competition b. is the primary cause of externalities . involves money paid to decision makers to influence behavior d. results in less market power for the rent seeker 16. Assume that the internal marginal cost of producing steel is MCinema = 14Q and the external marginal cost is MCexternal = 6Q. The inverse demand function for steel is P = 140 -80. What is the socially efficient level of output? 1 = MC 140 - 84 = 200 b 6 8 d. 10MC = 40 22. The (inverse) market demand function in a homogeneous product Cournot duopoly is as follows: P = 400 - 20(Q1 + Q2). The total cost functions are TC = 300 + 40Q, for firm one and TC = 300 + 16002 for firm two. What is the reaction function for firm one? a. Q1 = 20 + 0.5Q2 b. Q1 = 18- 0.75Q2 c. Q1 = 6-0.502 TR, = P, 4 , = 1400 - 20 ( 01 + 62 ) 16, = 4004- 200,- 201204 d. Q1 -9-0.502 MR = 460- 40 4 - 20 12 = 40 0 12 0- 0.5 42 18= 4070-200, 1-204 NE - 400 - 201 1 - 4012 = 40 40Q, - 360- 201 702 =0-0501 for firm one? 23. Using the information from the previous exercise, what is the profit maximizing output a. 2 9. =5-0.5(9-0.5 RX) b. 6 8 d. 10 0. 75 0% = 95 DA = 6 24. Many restaurants, such as Mcdonald's and Starbucks offer free Wi-fi in their locations. This free service is an example of which pricing strategy? a. third-degree price discrimination b. cross-subsidization c. penetration pricing d. commodity bundling e. predatory pricing 25. Assume that a monopoly's price elasticity of demand is -3. If the firm's marginal cost is $14, what price should the firm charge in order to maximize profit? a. $15 b. $18 c. $21 1 + ( - 37 d. $241. Which of the following statements about consumer surplus are correct? (a) Consumer surplus measures the difference between consumers' willingness to pay and price. (b) Consumer surplus is calculated as the area below the inverse demand curve and above the market price. (c) Both (a) and (b) are correct. 2. It is estimated that the market for aluminum exhibits a price elasticity of demand of e = -0.3. Which of the following statements is correct? (a) If demand for aluminum increases by 1 percent, the market price of alu- minum will decrease by 0.3 percent. (b) If the price of aluminum increases by one percent, the quantity demanded of aluminum will decrease by 0.3 percent. (c) Demand for aluminum is elastic. (d) The market for aluminum is a monopoly.3. Market demand is characterized by the demand curve Q = 12 - 4p. Which of the following is closest to the elasticity of demand at the price p = 2? (a) e= - (demand is inelastic). (b) = -1 (demand is unit elastic). (c)