Final%20Exam%20Solutions - Economics 3010 Fall 2008...

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Unformatted text preview: Economics 3010 Fall 2008 Professor Daniel Benjamin Cornell University FINAL EXAM Credit Guide I. True / False / Uncertain Your grade on these questions will depend on the generality, completeness, and persuasiveness of your explanation, not simply on whether the “true” or “false” is correct. The objective here is to provide an answer that convinces; not merely an answer that is “not wrong.” At the very least, make sure that you give clear definitions for the relevant economic terms used in the question. Please use a separate blue book for this section. (5 points) 1) If Ann’s (ordinal) utility function is UA(a, b) = a + b and Bob’s utility function is UB(a, b) = (a + b)3, then Ann and Bob have the same preferences. True. 2 points: An (ordinal) utility function is unique up to a monotonic transformation. 2 points: The function y(x) = x3 is a monotonic transformation, i.e., a strictly increasing function. 1 point: Since Bob’s utility function is a monotonic transformation of Ann’s, they have the same preferences. (5 points) 2) If cars become more energy‐efficient (so that it becomes less expensive to use your car), then individuals’ consumption of gasoline will surely decrease. False. 2 points: [accurate indifference curve / budget constraint diagram of “car services” vs “all other goods,” showing higher energy‐efficiency as a reduction in the price of “car services”] 2 points: [diagram and/or explanation that use of car services could increase] 1 point: If use of car services increases, then individuals’ consumption of gasoline will increase. (5 points) 3) If demand for a good is inelastic, then government policies designed to restrict supply of the good will actually increase revenues accruing to suppliers. True. 2 points: If demand for a good is inelastic, then a 1% increase in price causes a less than 1% decrease in quantity demanded. 3 points: [supply and demand diagram illustrating that revenues will increase when supply falls, and/or explanation that P × Q(P) will increase when Q(P) goes down by a smaller percentage than P increases] (5 points) 4) Clean air is a public good. True. 2 points: A public good is a good that is non‐excludeable and non‐rival. 1 point: A good is excludable if people can be prevented from using it. 1 point: A good is rival if one person’s use of the good diminishes another person’s enjoyment of it. 1 point: People can’t be prevented from using clean air, and one person’s use of it doesn’t diminish another person’s enjoyment of it. (5 points) 5) Total surplus is always higher when a market is perfectly competitive rather than dominated by a monopolist who can perfectly price‐discriminate. False. 2 points: [supply and demand diagram showing that total surplus is maximized when a market is perfectly competitive] 2 points: [demand curve diagram showing that total surplus is also maximized with perfect price discrimination] 1 point: Total surplus is the same in both cases. (5 points) 6) If individuals who already have a lot of cavities are more likely to buy dental insurance, that is an example of adverse selection. True. 2 points: Adverse selection arises when one person knows more about the attributes of a good than another (and, as a result, the uninformed person runs the risk of being sold a good of low quality). 1 point: The insurance company cannot observe the individual’s number of cavities when deciding at what price to offer dental insurance, while the individual knows that his own dental health is relatively low. 2 points: The insurance company therefore runs the risk that the people who buy insurance will be skewed toward those who are “low quality” from the company’s point of view. (5 points) 7) The Net Present Value (NPV) of an investment project (that has immediate costs and generates delayed benefits) will generally fall when the interest rate rises. True. 2 points: NPV = (M0 – P0) + (M1 – P1) / (1 + r) + (M2 – P2) / (1 + r)2 + … + (MT – PT) / (1 + r)T (or equivalent definition) 2 points: Since the investment has immediate costs and delayed benefits, it is the benefits that will be discounted at a rate that is increasing in (1 + r). 1 point: When (1 + r) rises, the benefits will be divided by a larger number, and hence NPV will fall. Intuitively, the opportunity cost of undertaking the investment is higher when r is larger. (5 points) 8) The No‐Arbitrage Condition implies that all risky assets will have the same rate of return. False. 1 point: Arbitrage means buying some of one asset and selling another to realize a positive return at no risk. 1 point: The No‐Arbitrage Condition states that arbitrage is not possible. 2 points: The No‐Arbitrage Condition implies that all riskless assets will have the same rate of return (or, that all risk‐adjusted assets will have the same rate of return). 1 point: Risky assets will have a higher rate of return than riskless assets. II. Brief Problem #1 (10 points) Imagine that Native Alaskans have been hit hard by the rising costs of heating oil. A state agency is considering two alternative solutions: (i) Subsidize the Native Alaskans’ purchases of heating oil. (ii) Take the money that would have been spent on the subsidy program and give it to the Native Alaskans. Assuming administrative costs of each option is zero, which of the two plans would the aid recipient prefer? Which of the two plans leads to higher use of heating oil by the aid recipient? In answering these questions, you can assume that the aid recipient’s optimum is interior (i.e., not a corner solution). Draw a budget constraint / indifference curve diagram to illustrate your answers. 2 points: [accurate budget constraint / indifference curve diagram of option (i), showing the subsidy as a decrease in the price of heating oil] 2 points: [accurate budget constraint / indifference curve diagram of option (ii), showing the lump sum transfer as an increase in income sufficient to make the new optimum under option (i) affordable] 3 points: The aid recipient would prefer the lump sum transfer because he can get onto a higher indifference curve. The economic intuition is that he must be at least as well off under the lump sum transfer because (by construction) he can still afford what he would choose under the subsidy. But other options are now available that make him better off. 3 points: The aid recipient would consume more heating oil under option (i). Relative to (i), (ii) is a pure substitution effect (there is no income effect because, by construction, the aid recipient can still afford what he would choose under (i)). Since (i) causes a lower relative price of heating oil, the aid recipient will consume more of it under (i). III. Brief Problem #2 (10 points) Bob Vila sells building materials in a competitive industry. He receives $100 for each unit of material sold. The table below presents how output changes as additional units of labor are hired. Bob hires workers from a competitive labor market. Units of Labor Labor Output 1 5 2 9 3 12 4 14 5 15 How many employees should Bob hire if the market‐determined wage per work period is $150? How (if at all) would your answer change if the going wage increased to $250? How (if at all) would your answer change if Bob also has a weekly fixed cost of $400 in addition to his variable cost of labor? What would the opportunity cost of Bob’s own time have to be in order for it to be optimal for him to permanently close his business? Explain the economic intuition for your answers. 2 points: Bob should hire an additional worker if that worker’s marginal revenue product (MRP) exceeds that worker’s marginal cost. 2 points: The MRP of the 3rd worker is $300 (which equals (12 – 9) × $100); the MRP of the 4th worker is $200 (which equals (14 – 12) × $100); and the MRP of the 5th worker is $100 (which equals (15 – 14) × $100). 1 point: If the market‐determined wage per work period is $150, Bob should hire 4 workers. 1 point: If the market‐determined wage per work period is $250, Bob should hire 3 workers. 2 points: If Bob also has a weekly fixed cost of $400, this would not affect the MRP of labor, so it would not affect the optimal number of workers. But it might mean that Bob should go out of business if his profit were negative even at the optimal number of workers. 2 points: If Bob hires 3 workers, his total revenue is $1200 (12 units of labor output × $100), and his labor costs are $750 (3 units of labor input × $250). Hence revenue minus accounting costs is $1200 ‐ $750 ‐ $400 = $1150. If the opportunity cost of Bob’s time is at least $50/week, then Bob should close his business. IV. Multi‐Part Problem #1 Note: This question is designed so that if you skip part of the question, you still have enough information (stated explicitly earlier in the question) to answer later parts of the question. Please answer as many parts as you can. Consider the market for news in a small town. Suppose there are fixed costs, C = 20, associated with running a newspaper (e.g., hiring reporters, establishing distribution routes, contracting with printing companies), but the marginal cost of printing additional copies of the newspaper is zero. However, the number of copies of the newspaper to be printed must be decided in advance because setting up the infrastructure takes time. The town residents care about getting news, but not whom they get it from; regardless of the number of newspaper‐producing firms, all newspapers are perfect substitutes for the consumers. The (inverse) demand function for the total number of newspapers purchased from all firms, y, is given by: p(y) = 24 – 2y. (5 points) (a) Assuming the market were perfectly competitive, draw a supply/demand diagram. Show that the competitive equilibrium price and quantity are p*comp = 0 and y*comp = 12. What would a competitive newspaper‐producing firm’s profit be? Explain why perfect competition cannot persist in this market. 1 point: [accurate supply / demand diagram, showing the supply curve as perfectly elastic at a price of zero] 2 points: The equilibrium price and quantity are determined by p(y*comp) = 24 – 2 y*comp = 0, which implies p*comp = 0 and y*comp = 12. 1 point: A competitive newspaper‐producing firm’s profit would be p*comp y*comp – C = ‐20. 1 point: Perfect competition cannot persist in this market because the firms would earn losses and exit (at least in the long run). (7 points) (b) Suppose that a single firm, the Leader Ledger, emerges as a monopolist in this market. Show that the Ledger’s marginal revenue curve is given by MR(y) = 24 – 4y. Show that monopoly price and quantity are p*monop = 12 and y*monop = 6. On a diagram, illustrate the demand curve, the monopolist’s marginal revenue curve, and the monopolist’s profit. 1 point: For the monopolist, TR(y) = y × p(y) = 24y – 2y2. 1 point: MR(y) = ∂TR(y)/∂y = 24 – 4y. 1 point: The monopoly output solves MR(y*monop) = 24 – 4 y*monop = 0 = MC(y*monop). Hence y*monop = 24/4 = 6. 1 point: The monopoly price solves p*monop = p(y*monop) =24 – 2 y*monop =24 – 2(6) = 12. 3 points: [1 point each for the demand curve, the monopolist’s marginal revenue curve, and the monopolist’s profit] (3 points) (c) Now suppose a new firm, the Follower Journal, makes plans to enter the market. As the incumbent, the Ledger can set its own output level, y1, before the Journal sets its output level, y2. Show that the Journal’s marginal revenue function is given by MR2(y1, y2) = 24 – 2y1 – 4y2. Show that the Journal’s optimal output as a function of the Ledger’s output is y2*(y1) = (12 – y1) / 2. 1 point: The Follower Journal’s total revenue is TR2(y1, y2) = y2 × p(y1 + y2) = 24 y2 – 2 y2(y1 + y2) = 24 y2 – 2y1y2 – 2y22. 1 point: The Follower Journal’s marginal revenue is MR2(y1, y2) = ∂TR2(y1, y2)/∂y2 = 24 – 2y1 – 4y2. 1 point: The Follower Journal’s output solves MR2(y1, y2*) = 24 – 2y1 – 4y2* = 0 = MC(y2*). Hence y2*(y1) = (24 – 2y1)/4 = (12 – y1) / 2. (5 points) (d) Show that the Ledger’s marginal revenue function (taking into account the Journal’s response optimal response from part c, y2*(y1)) is MR1(y1) = 12 – 2y1. Show that the Stackelberg equilibrium price and quantities are p*stac = 6, y1*stac = 6, and y2*stac = 3. Explain why the Ledger will survive in this market, while the Journal will go out of business. 1 point: The Leader Ledger’s total revenue is TR1(y1) = y1 × p(y1 + y2*(y1)) = 24 y1 – 2 y2*(y1) (y1 + y2*(y1)) = 24 y1 – 2 y1 y2*(y1) – 2y12 = 24 y1 – 2 y1 ((12 – y1) / 2) – 2y12 = 24 y1 – 12 y1 + y12 – 2y12 = 12 y1 – y12. 1 point: The Leader Ledger’s marginal revenue is MR1(y1) = dTR1(y1)/dy1 = 12 – 2y1. 2 points: The Leader Ledger’s output solves MR1(y1*stac) = 12 – 2 y1*stac = 0 = MC(y1*stac). Hence y1*stac = 6. Moreover, y2*stac = y2*(6) = (12 – 6) / 2 = 3. Finally, the equilibrium price is determined by p(y1*stac + y2*stac) = p(6 + 3) = 24 – 2(9) = 6. 1 point: The Leader Ledger’s profit is TR1(y1*stac) – C = 12(6) – 62 – 20 = 16, which is positive, while the Follower Journal’s profit is TR2(y1*stac, y2*stac) – C = 24(3) – 2(6)(3) – 2(32) – 20 = ‐2, which is negative. Hence the Ledger will survive in thismarket, while the Journal will go out of business. (7 points) (e) Instead of going out of business, the Follower Journal is acquired by National News Corporation (NNC). NNC has a strict corporate policy requiring affiliated newspapers to act as aggressive competitors in any market. Hence, the Ledger and the Journal now simultaneously choose their production capacities. Explain why the two firms’ marginal revenue functions (taking into account the other’s response) are MR1(y1, y2) = 24 – 2y2 – 4y1 and MR2(y1, y2) = 24 – 2y1 – 4y2. Show that the Cournot equilibrium price and quantities are p*cour = 8, y1*cour = 4, and y2*cour = 4. What will each firm’s profit be? 1 point: We can find Firm 2’s marginal revenue exactly like we did in part c, since Firm 2 takes firm 1’s output as given. Hence MR2(y1, y2) = ∂TR2(y1, y2)/∂y2 = 24 – 2y1 – 4y2. 1 point: Firm 1’s marginal revenue is symmetric: MR1(y1, y2) = 24 – 2y2 – 4y1. 3 points: The Cournot equilibrium quantities are found by simultaneously setting both firms’s marginal revenue equal to marginal cost (which is zero): 24 – 2y1*cour – 4y2*cour = 0 and 24 – 2y2*cour – 4y1*cour = 0. Solving these gives y1*cour = 4, and y2*cour = 4. 1 point: The equilibrium price is p*cour = p(y1*cour + y2*cour) = 24 – 2(8) = 8. 1 point: Both firms earn profit p*cour y1*cour – C = (8)(4) – 20 = 12. (3 points) (f) Suddenly, the Internet becomes widely available to consumers. Now that there is no need to distribute print copies of the news, the only decision the firms must make is how much to charge consumers for an on‐line newspaper subscription. Explain why the Bertrand equilibrium prices are p1*bert = p2*bert = 0. Explain why one of the firms will go out of business. 1 point: p1*bert = p2*bert = 0 is an equilibrium. If either firm deviates to a positive price, he will lose his market share. If either deviates to a negative price, he will capture the entire market, but lose money on each transaction. 1 point: Moreover, this is the only equilibrium. Suppose to the contrary that there were an equilibrium where one firm offered a positive price, say p1*bert =ρ > 0. Then the other firm’s best response is a price infinitesimally smaller than ρ, say p2*bert =ρ – є (where є is a tiny, positive number). But this cannot be an equilibrium because both firms are paying the fixed cost C, while Firm 2 is capturing all the demand, so Firm 1 could profitably deviate by offering price p1*bert =ρ – 2є > 0. 1 point: At the equilibrium, both firms are earning profit ‐C = ‐20 < 0, so one of the firms will go out of business (leaving the other as a profitable monopolist). V. Multi‐Part Problem #2 Note: This question is designed so that if you skip part of the question, you still have enough information (stated explicitly earlier in the question) to answer later parts of the question. Please answer as many parts as you can. Two firms have agreed to collude. Now each must decide whether to keep prices high (as agreed) or to cheat. If both firms keep prices high, then each will earn profit of $2 billion. If both cheat, then each will earn profit of $0 billion. If one keeps prices high and the other cheats, then the cheater will earn $3 billion, while the other firm earns ‐$3 billion. (5 points) (a) Draw the normal form for this game (assuming that each firm seeks only to maximize profit). Is there a dominant strategy? What is the Nash equilibrium? 1 point: The normal form is: Keep Cheat Keep (2, 2) (‐3, 3) Cheat (3, ‐3) (0, 0) Firm 1 Firm 2 1 point: A dominant strategy is a strategy that is a best‐response to every strategy by the other player. 1 point: Each firm has a dominant strategy to cheat. 1 point: A Nash equilibrium is a strategy profile such that each player’s strategy is a best‐response to the other player’s strategy. 1 point: The Nash equilibrium is (Cheat, Cheat). (4 points) (b) Suppose that the two firms can write a legally‐enforced contract. What does the Coase theorem predict will occur? 2 points: The Coase theorem says that if the parties can write a contract and transactions costs are not too high, the parties will private negotiate a socially efficient (i.e., potentially Pareto efficient) outcome. 2 points: The Coase theorem predicts that the socially efficient outcome will occur: (Keep, Keep). (5 points) (c) Unfortunately for the firms, a contract promising (illegal) anti‐competitive behavior (as in part b) will not be enforceable in court. However, suppose that the firms’s cheating decision is repeated. In particular, they will face the same decision about whether to cheat every year from now on, forever, and the interest rate is 10%. Explain why the following two observations are true: • Maintaining collusion forever has a present value of $20 billion. • Consider this sequence of events: in the first year, firm 1 cheats, while firm 2 keeps prices high; thereafter, both firms always cheat. The present value of this sequence of events is $3 billion for firm 1 and ‐$3 billion for firm 2. 1 point: The present value (PV) of a cash stream (m0, m1, m2, …) is m0 + m1 / (1 + r) + m2 / (1 + r)2 + … 1 point: The PV of the infinite, constant stream (m, m, m, …) is m (1 + r) / r. [NOTE: The question as stated is incorrect. The PV should be $22 billion. Full credit for either saying the PV is m (1 + r) / r, or (as implied by the question), m / r.] 1 point: Maintaining collusion forever gives the constant stream (2, 2, 2, …), and hence has PV = 2 / (0.1) = 20. [NOTE: Full credit for the correct $22 billion.] 1 point: This sequence of events gives firm 1 the cash stream (3, 0, 0, …), and hence has PV = 3. 1 point: This sequence of events gives firm 2 the cash stream (‐3, 0, 0, …), and hence has PV = ‐3. (4 points) (d) In the repeated game described in part c, consider the “punishment strategy”: keep a high price as long as the other firm has always maintained a high price in the past; otherwise cheat. Explain why, if both players follow this punishment strategy, it is optimal for each player to collude in every period. Explain why it follows that it is a Nash equilibrium (of this infinitely‐repeated game) for both firms to follow this punishment strategy. 2 points: If both firms follow this punishment strategy, then consider a firm’s decision about whether to collude or to cheat in a given period. If each firm colludes forever, then the firm’s PV is 22, while if a firm cheats, then the firm’s PV is 3. (These are the calculations from part c.) Hence it is optimal for each firm to collude in every period. 2 points: If both firms follow this punishment strategy, then the equilibrium outcome is that each will collude in every period. It would reduce the PV of profit to deviate from this strategy (by cheating in some period). Hence it is a Nash equilibrium (of this infinitely‐repeated game) for both firms to follow this punishment strategy. (7 points) (e) Now suppose once again that there is no repetition; the firms must make a once‐ and‐for‐all decision about whether to collude. However, each firm elects a new Chief Executive Officer (CEO), and the two CEOs happen to be friends. In particular, each CEO does not only care about his own success but also cares about his friend’s success, so the players’s Bernoulli utility functions are: u1 = π1 + ½ π2 u2 = π2 + ½ π1. With the payoffs being the players’s Bernoulli utilities, explain why the normal form of the game is now: Keep Cheat Keep (3, 3) (‐1.5, 1.5) Cheat (1.5, ‐1.5) (0, 0) Firm 1 Firm 2 Find the three Nash equilibria. 2 points: The payoffs to (Keep, Keep) are (2 + ½(2), 2 + ½(2)) = (3, 3). The payoffs to (Keep, Cheat) are (‐ 3 + ½(3), 3 + ½(‐3)) = (‐1.5, 1.5), and symmetrically for (Cheat, Keep). The payoffs to (Cheat, Cheat) are (0 + ½(0), 0 + ½(0)) = (0, 0). 3 points: The two pure‐strategy Nash equilibria are (Keep, Keep) and (Cheat, Cheat). [The circle method is fine for finding the equilibria.] 2 points: To find the mixed‐strategy Nash equilibrium, suppose Firm 2 plays Keep with probability q. Firm 1’s expected utility from playing Keep is q(3) + (1 – q)(‐1.5) = 4.5q – 1.5. Firm 1’s expected utility from playing Cheat is q(1.5) + (1 – q)(0) = 1.5q. In a mixed‐strategy Nash equilibrium, these two expected utilities must be equal: 4.5q – 1.5 = 1.5q. Hence q = ½. Since the payoffs are symmetric, both firms play Keep with probability ½. (5 points) (f) One of the CEOs is worried about potential miscoordination, so she takes the initiative and sets her price before her friend does. Draw the extensive form for this sequential‐move game (that has the same Bernoulli utility payoffs as in part e). What is the unique subgame perfect equilibrium? 2 points: [accurate diagram] 3 points: Solving the game by backwards induction: If Firm 1 has played Cheat, it is optimal for Firm 2 to play Cheat. If Firm 1 has played Keep, it is optimal for Firm 2 to play Keep. Knowing this, it is optimal for Firm 1 to play Keep. Hence the unique subgame perfect equilibrium is (Keep, Keep). [The circle method of backwards induction is fine for finding the equilibrium.] ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell University (Engineering School).

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