2009 Final Exam Solutions

2009 Final Exam Solutions - Economics 3010 Fall 2009...

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Unformatted text preview: Economics 3010 Fall 2009 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) Exceptions to the Law of Demand are rare. True. 1 point: The Law of Demand states that when price increases, quantity demanded decreases. 3 points: An exception to the Law of Demand requires that the good is inferior, has a large budget share, and has few close substitutes. (Such a good is called a Giffen good. These conditions are implied by the Slutsky equation.) 1 point: Theoretically speaking, since most goods are normal goods‐‐‐and virtually all goods with a large budget share are normal goods‐‐‐exceptions to the Law of Demand should be rare. Practically speaking, exceptions to the Law of Demand are in fact rarely observed. (Either one of these statements is sufficient for credit.) (5 points) 2) Subsidizing a monopolist’s production can increase social surplus. True. 2 points: The monopolist’s profit‐maximizing price is a markup over marginal cost: p = MC / (1 + 1/εd) > MC. That is, the monopolist chooses to produce at a point on the demand curve with lower output than the competitive level (where p = MC). (A helpful diagram can substitute for the equations.) 1 point: The competitive equilibrium maximizes social surplus, whereas there is deadweight loss when the market structure is monopoly. 1 point: Subsidizing the monopolist’s production reduces the monopolist’s marginal cost of production at any level of output, causing the monopolist to reduce its price and increase its output. 1 point: Since output moves closer to the competitive level, social surplus increases. (Note that the amount of the subsidy is a transfer from the government to the firm and does not affect social surplus.) (5 points) 3) The First Fundamental Theorem of Welfare Economics states that even if there is imperfect competition, the general equilibrium allocation of resources will be equal as well as Pareto efficient. False. 2 points: The First Fundamental Theorem of Welfare Economics states that if all markets are perfectly competitive, then the general equilibrium will be Pareto efficient. 2 points: However, there is no guarantee that the equilibrium allocation of resources will be equal. 1 point: Imperfect competition is a violation of the assumptions of perfect competition, so the theorem does not apply in that case. (5 points) 4) As long as getting admitted to an MBA program requires high ability, individuals who have MBAs may earn higher wages in equilibrium, even if they learned nothing in business school. True. 2 points: Employers face an adverse selection problem when deciding whom to hire; it is difficult to determine which job applicants are the high‐ability type (whom employers need to pay more to retain) and which are the low‐ability type (whom employers would prefer to pay less). 2 points: If getting admitted to an MBA program requires high ability, it can act as a (costly) signal of high ability to employers. 1 point: Since employers can tell that business school graduates have high ability, employers will pay them higher wages. (This is called a “separating equilibrium” because attending the MBA program allows the high‐ability types to separate themselves from the low‐ability types, thereby signaling their type to employers.) (5 points) 5) Public goods, goods which are rival and excludable, tend to be overprovided by private markets. False. 1 point: A good is rival if one person’s use of the good diminishes another person’s enjoyment of it. 1 point: A good is excludable if people can be prevented from using it. 1 point: Public goods are goods which are nonrival and nonexcludable. 2 points: Public goods tend to be underprovided by private markets (because individuals do not receive the full value of producing the public good). (5 points) 6) An appropriate tax on pollution‐emitting firms can eliminate the deadweight loss from pollution, while at the same time raising revenue for the government and creating incentives for firms to invent cleaner production technologies. (Please use an appropriate diagram to support your argument.) True. 3 points: [appropriate diagram showing how a Pigouvian tax eliminates deadweight loss, while raising revenue; good economic intuition can substitute for the diagram] 2 points: A tax on pollution creates incentives for firms to invent cleaner production technologies because, if they can do so, they can pay less in taxes for any given amount of output. (5 points) 7) The same number of people will end up buying an extended warranty on a cell phone regardless of whether the default is to include the extended warranty in the purchase (in which case people who do not want it have to opt out) or the default is not to include it (in which case people who want it have to opt in). False. 2.5 points: People’s behavior exhibits inertia; they tend to stick with defaults or the status quo. 2.5 points: Hence if the default is to include the extended warranty, then people will be more likely to buy it. (5 points) 8) It cannot be profit‐maximizing for resort hotels in Egypt to charge higher prices during the low‐occupancy season‐‐‐the winter, when most visitors are wealthy Europeans‐‐‐than during the high‐occupancy season‐‐‐the summer, when locals flock to the resorts in large numbers. False. 1 point: This pricing scheme may be an example of price discrimination. (In particular, second‐degree price discrimination.) 2 points: Since Europeans have higher willingness to pay than locals, it may be profit‐maximizing to charge them a higher price. 2 points: Since the resorts know that most visitors in the winter are Europeans, charging higher rates in the winter may be profit‐maximizing (even though the winter is the low‐occupancy season). II. Brief Problem #1 (10 points) Ronald is a corporate executive with a lot of smarts but few moral scruples. The present discounted value of his current wealth is $81 million, but he has the opportunity to steal $19 million from his company (by funneling money into a fake expense account). He knows that there is only a 20% chance he will get caught, in which case the legal system will require him to pay a fine of $F in addition to returning the full amount he stole. Suppose Ronald is an expected‐utility maximizer with Bernoulli utility function u(w) = w1/2. Graph his Bernoulli utility function. Is he risk‐averse, risk‐ neutral, or risk‐loving? How large does F have to be in order to deter him from embezzling the money? If Ronald is caught, why is it more efficient to punish him with a fine instead of a prison sentence? 2 points: [accurate graph of u(w) = w1/2] 2 points: Ronald is risk‐averse because his Bernoulli utility function is concave. 4 points: Ronald will be deterred if his expected utility from stealing, (.80) (100 mill)1/2 + (0.20) (81 mill – F) 1/2, is less than his expected utility from not stealing, (81 mill)1/2. The F that makes him indifferent is $56 mill; then his expected utility is 9000, regardless of whether he steals or not. So he will be deterred from embezzling the money as long as F > $56 mill. 2 points: If Ronald is caught, it is more efficient to punish him with a fine instead of a prison sentence primarily because with a fine, the loss to Ronald is a gain for the taxpayers. In contrast, a prison sentence entails a loss to Ronald (in terms of unpleasantness of the experience) that is not received as a gain by anyone else. III. Brief Problem #2 (10 points) The Sesame household is composed of two individuals, Ernie and Bert. They engage in home production of two goods, food f (measured in quality units) and tidiness t. Each has 60 minutes per day to devote to home production of food, tidiness, or some combination of both. Their production possibilities frontiers are given by the following table: Bert Time spent Ernie on food prep Food output Tidiness output Food output Tidiness output 0 min 0 6 0 24 10 min 1 5 2 20 20 min 2 4 4 16 30 min 3 3 6 12 40 min 4 2 8 8 50 min 5 1 10 4 60 min 6 0 12 0 For example, this table shows that if Ernie spends 10 minutes on food preparation (and hence 50 minutes on tidying up the house), he would produce 1 unit of food quality and 5 units of tidiness. The total amount of the household’s food quality and tidiness is the amount produced by Ernie plus the amount produced by Bert. Suppose each has utility function u(f, t) = min{f, t}. Who has the comparative advantage in food production? In tidiness production? If each of them were a separate household, how much time would Ernie spend on food production, and how much time would Bert spend? Now that they are together as a single household, what are their optimal time allocations? Explain intuitively how it is possible for them to produce more of both goods when they specialize than if they each produced the goods independently. 2 points: Bert has the comparative advantage in tidiness production; whereas Ernie gives up 1 food‐ quality unit for each unit of tidiness he produces, Bert only gives up ½ food‐quality unit per unit of tidiness. 2 points: Ernie has the comparative advantage in food production; his opportunity cost of producing 1 unit of food quality is 1 unit of tidiness, compared to 2 units of tidiness for Bert. 2 points: If they were separate households, Ernie would spend 30 minutes on food production and 30 minutes on tidiness production, giving him utility u(3, 3) = min{3, 3} = 3. Bert would spend 40 minutes on food production and 20 minutes on tidiness production, giving him utility u(8, 8) = min{8, 8} =8. 2 points: Now that they are together as a single household, it is optimal for Ernie to spend 60 minutes on food production and Bert to spend 30 minutes on food production and 30 minutes on tidiness production. That way, they get a total of 12 units of both food and tidiness. 2 points: When they specialize, they can each focus on producing the good for which they have a comparative advantage. Since Ernie has a lower opportunity cost of producing a unit of food quality, they can jointly sacrifice less tidiness by having Ernie specialize in food production. Similarly, since Bert has a lower opportunity cost of producing a unit of tidiness, they can jointly sacrifice less food quality by having Bert specialize in tidiness. 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. There is a small Pacific island in which rice is the staple food. Rice is produced by 100 farmers with identical production functions. As a result of the available technology and prevailing input prices, each farmer’s short run cost function for planting and harvesting rice is c(y) = 64 + 4y + y2, where y is units of rice, and cost is denominated in the local currency. (5 points) (a) Judging from the cost function, can you infer whether there is a (at least one) fixed factor of production? Show that a farmer’s short run average cost (AC), average variable cost (AVC), and marginal cost (MC) functions are given by: AC(y) = 64/y + 4 + y, AVC(y) = 4 + y, MC(y) = 4 + 2y. 1 point: Yes, there is at least one fixed factor of production. 1 point: The existence of a fixed factor can be inferred from the “64” fixed cost term, which does not depend on output. 1 point: AC(y) = c(y)/y = 64/y + 4 + y. 1 point: AVC(y) is calculated from only the variable cost part: (4y + y2)/y = 4 + y. 1 point: MC(y) = ∂c(y)/∂y = 4 + 2y. (5 points) (b) Graph the firm’s AC, AVC, and MC curves. Judging from the AC curve, is there a region where the underlying production function is increasing returns to scale? Constant returns to scale? Decreasing returns to scale? Explain. With reference to your graph, explain why you will not need to worry about the shut‐down condition when deriving a farmer’s supply function. 1 point: [accurate graph of each of the curves] 2 points: The AC curve is initially decreasing, then increasing. This implies that the underlying production function is IRS at low output, then DRS at higher output. 2 points: Ordinarily, a farmer would set output y* to equate MC(y*) = p, except when the shut‐down condition applies. The shut‐down condition is that the farmer will produce y* = 0 when AVC(y*) > p. Here, the MC curve and the AVC curve intersect at y = 0, so the shut‐down condition would only apply in situations where the farmer would want to set zero output anyway based on the MC(y*) = p condition alone. (2 points) (c) Show that each farmer’s short run supply function is if p ≥ 4 y(p) = { (p – 4)/2 { 0 if p < 4. 1 point: Since we can ignore the shut‐down condition, we can solve MC(y*) = 4 + 2y* = p for y* to get y* = (p – 4)/2. 1 point: Sincethe farmer cannot produce negative output, there is a corner solution of y* = 0 when p < 4. (1 point) (d) Show that the short‐run market supply function is S(p) = { 50p – 200 if p ≥ 4 { 0 if p < 4. 1 point: Since there are 100 farmers (and there is no entry or exit in the short run), S(p) = 100 y(p), which gives the solution. (2 points) (e) Suppose the demand function for rice is given by: D(p) = { 1000 – 10p if p ≤ 100 { 0 if p > 100. Show that the competitive equilibrium price is p* = 20, and the equilibrium market quantity is Y* = 800. 1 point: Solving S(p) = 50p – 200 = 1000 – 10p = D(p) for p gives p* = 20. (Since the solution is between 4 and 100, the boundary conditions do not bind.) 1 point: Y* = S(p*) = 50(20) – 200 = 800. (Alternatively, Y* = D(p*) =1000 – 10(20) = 800.) (3 points) (f) Show that, at the short run equilibrium, each farmer produces 8 units of rice. What is each farmer’s profit? Is the short run equilibrium also the long run equilibrium? 1 point: Each farmer produces y(20) = (20 – 4)/2 = 8 units of rice. (Alternatively, you can get credit by noting that since the farmers are identical, each must produce the same amount of output. Since total output is 800, each farmer’s output is 800/100 = 8.) 1 point: At the short run equilibrium, profit = p*y* – c(p*) = (20)(8) – [64 + 4(8) + (8)2] = 0. 1 point: Since zero profit is the condition for long run equilibrium, the short run equilibrium is also the long run equilibrium. (4 points) (g) Define what it means for demand to be inelastic. Show that, at the equilibrium price, the price elasticity of demand is ‐1/4. Why would you expect the demand for rice to be inelastic in this economy? 1 point: Demand is inelastic if a 1% increase in price causes quantity demanded to fall by less than 1%. 2 points: Here, the elasticity of demand is εd = (∂D(p*)/∂p) (p* / D(p*)) = (‐10) (20 / 800) = ‐1/4. Hence demand is indeed inelastic at the equilibrium. 1 point: Since rice is the staple food, there are probably few close substitutes. As a result, I would expect demand to be inelastic. (5 points) (h) Now, suppose that just after the rice harvest, but before the rice is brought to market, a tsunami strikes the island and destroys 1/8 of each farmer’s rice crop. Assuming that it is prohibitively expensive to sell the rice elsewhere, and it cannot be stored to be sold later, explain why the supply curve will now be perfectly inelastic at a quantity of 700 units when the rice is brought to market. Draw a supply and demand diagram illustrating the new equilibrium. Show that new (short run) equilibrium price and quantity are p* = 30 and Y* = 700. 1 point: The supply curve will now be perfectly inelastic at a quantity of 700 units (the amount remaining after the tsunami) because the units have already been produced and cannot be sold elsewhere or later. Hence the farmers will be willing to sell that rice at any positive price. 1 point: [accurate supply and demand diagram, depicting perfectly inelastic supply] 2 points: The new equilibrium price and quantity satisfy S(p) = 700 = 1000 – 10p = D(p), giving p* = 30 and Y* = 700. (4 points) (i) What is each farmer’s profit in this new, post‐tsunami short run equilibrium? (Remember that all of the costs of planting and harvesting 8 units of rice have already been sunk. Assume that these are all of the costs, so only the farmer’s revenue is affected by the tsunami.) Explain why, even though the farmers as a whole benefit from natural disasters, any individual risk‐ averse farmer would still have positive willingness to pay for crop‐yield insurance (which pays out when natural disasters reduce crop yields). Applying the logic of this problem in another context, explain why labor unions may increase union members’ incomes by restricting the number of hours they are allowed to work for their employer. 1 point: New profit = new revenue – old cost = (7)(30) – 160 = 50. 2 points: Even though the farmers as a whole benefit from the reduction in market supply, each individual farmer would be better off if his own supply had not been reduced. Therefore, a risk‐averse farmer would be willing to pay some amount of money for crop‐yield insurance, which can reduce the variability of his income (and hence reduce the variability of his consumption). 1 point: Labor unions may restrict the number of hours that union members are allowed to work in order to reduce the supply of labor facing the employer, raising the equilibrium wage. If the demand for labor is inelastic, the union members’ incomes will increase. (Note: The demand for labor is likely to be inelastic in this context because contracts between the labor union and the employer often require the employer to hire only labor union members; hence there are few close substitutes for a given worker.) 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. A motorist has noticed a strange noise coming from her engine whenever she drives her car. She is deciding whether to go bring the car to a mechanic to have the engine checked out. If she does not go to the mechanic, then her monetary payoff (i.e., measured in dollars) is ‐$50, and the mechanic’s monetary payoff is $0. If she leaves her car with the mechanic, then the mechanic can either work hard, in which case he will find and fix the problem, or he can shirk, in which case he will not find the problem, and he will tell the motorist that nothing is wrong but charge her for his time anyway. If the mechanic works hard, then the motorist’s monetary payoff is $50, and the mechanic’s monetary payoff is $50. If the mechanic shirks, then the motorist’s monetary payoff is ‐$150, and the mechanic’s monetary payoff is $100. Both players have Bernoulli utility function u(m) = m, where m is the player’s monetary payoff. Each player maximizes his or her own expected utility. (6 points) (a) Draw the extensive‐form representation for this game, with the payoffs denoted in terms of Bernoulli utilities. Use backwards induction to find the unique subgame‐perfect Nash equilibrium, and explain your reasoning. 2 points: DRAW EXTENSIVE FORM 2 points: In the subgame where the motorist goes to the mechanic, the mechanic’s optimal action is to shirk. Given that the mechanic will shirk, it is optimal for the motorist not to go to the mechanic. 2 points: The unique subgame‐perfect Nash equilibrium is (don’t go to mechanic, shirk). (6 points) (b) Draw the normal‐form representation for this game, with the payoffs denoted in terms of Bernoulli utilities. Find the unique pure‐strategy Nash equilibrium, and explain your reasoning. 2 points: DRAW NORMAL FORM 2 points: The unique pure‐strategy Nash equilibrium is (don’t go to mechanic, shirk). 2 points: If the mechanic’s strategy is to shirk, then it is indeed optimal for the motorist not to go. If the motorist’s strategy is not to go, then the mechanic is indifferent between his two actions, hence (don’t go to mechanic, shirk) is a Nash equilibrium. Note that there cannot be a pure‐strategy equilibrium in which the mechanic works hard; in that case, the motorist’s optimal action is to go to the mechanic, but then the mechanic prefers to shirk. (4 points) (c) Show that there is a continuum of mixed‐strategy Nash equilibria, where the motorist’s strategy is to not go to the mechanic, and the mechanic’s strategy is to shirk with probability ½ ≤ p < 1. We need to show that each player’s strategy is optimal, given the other player’s strategy. 2 points: Suppose the mechanic’s strategy is to shirk with some probability p. Then the motorist’s expected utility of going to the mechanic is p u(‐$150) + (1 – p) u($50) = p(‐150) + (1 – p) (50). The motorist’s expected utility of not going, u(‐$50) = ‐50, is at least as large as the motorist’s expected utility of going, as long as ½ ≤ p < 1. 2 points: Given that the motorist isn’t going to the mechanic, the mechanic can do at least as well with the strategy “shirk if the motorist comes” than with the strategy “work hard if the motorist comes” because he gets expected utility u($0) = 0 in either case. (3 points) (d) Which outcome of the game maximizes the sum of the players’ monetary payoffs? Explain why that outcome is the socially efficient outcome‐‐‐that is, explain how it is a potential Pareto improvement relative to every other possible outcome of the game. 1 point: The outcome of the game maximizes the sum of the players’ monetary payoffs is ($50, $50), which occurs when the motorist goes to the mechanic, and the mechanic works hard. 2 points: That outcome is a potential Pareto improvement relative to every other possible outcome of the game since, at that outcome, it is possible to redistribute money so that relative to any other outcome, both players are weakly better off and at least one is strictly better off. For example, suppose we take $60 from the motorist and give it to the mechanic so that the outcome is now (‐$10, $110). Then both players are strictly better off relative to the (‐$150, $100) outcome from (go to mechanic, shirk), and both players are strictly better off relative to the (‐$50, $0) outcome from the motorist not going to the mechanic. (3 points) (e) Suppose the players could write an enforceable contract. What outcome does the Coase theorem predict they will agree to? Explain. 1 point: The Coase theorem predicts that bargaining will lead to an efficient outcome: in this case, (go to mechanic, work hard). 2 points: The reason is that the players will write a contract involving a payment between the parties that makes the agreed‐on outcome a Pareto improvement over the other possible outcomes. For example, the contract could stipulate that the players take actions (go to mechanic, work hard), and in addition, the motorist agrees to pay $60 to the mechanic. As per part (e) above, this generates a Pareto efficient outcome, so both players would agree to this contract. (4 points) (f) We have been assuming that when the motorist goes to the mechanic, she knows right away which outcome occurred; hence she can discern whether the mechanic worked hard or shirked. For this part of the question only, suppose it is impossible for the motorist or anyone else to discern whether the mechanic worked hard or shirked. (She will eventually find out if her car is broken, but not until much later, when the outcome cannot necessarily be traced back to the mechanic’s behavior.) Explain why in this case, the players would not agree to a contract that stipulates that the mechanic will work hard. Explain how this situation is an example of moral hazard. 2 points: The motorist would not agree to a contract that stipulates that the mechanic will work hard because the mechanic would have an incentive to cheat on the contract and shirk. The terms of the contract would be unenforceable because no one can discern whether the mechanic worked hard. 2 points: Moral hazard is when a principal’s payoff depends on an agent’s actions, and it is impossible to write a contract on the agent’s actions (perhaps because unobservable). This situation fits the definition exactly, where in this case, the motorist is the principal, and the mechanic is the agent. (g) Now suppose that the mechanic cares about the fairness of the transaction in (4 points) addition to caring about his own payoff. That is, suppose the mechanic (Player 2) still gets Bernoulli utility 0 if no transaction occurs, but if motorist goes to the mechanic, then the mechanic has Bernoulli utility function: u2(m1, m2) = m2 – ½ |m1 – m2|, where m2 is the mechanic’s monetary outcome, and m1 is the motorist’s (Player 1) monetary outcome (and |m1 – m2|, the absolute value of the difference in outcomes, is the unfairness of the transaction). The motorist continues to care only about her own outcome: u1(m1, m2) = m1. Draw the extensive form for this modified game, with the payoffs denoted in terms of the new Bernoulli utilities. Show that the unique subgame‐perfect Nash equilibrium is for the motorist to go to the mechanic, and the mechanic works hard. 2 points: DRAW EXTENSIVE FORM 2 points: If the motorist goes to the mechanic, the mechanic’s expected utility from shirking is u2(‐$150, $100) = 100 – ½ |‐150 – 100| = ‐25, and the mechanic’s expected utility from working hard is u2($50, $50) = 50 – ½ |50 – 50| = 50, so the mechanic would work hard. Given that, the motorist’s expected utility from going to the mechanic is u1($50, $50) = 50, which is higher than u1(‐$50, $0) = ‐50, the motorist’s expected utility from not going. The result follows by backwards induction. ...
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