Each question is self-contained, and assumptions made in one question do not carry over to other questions, unless explicitly specified.

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1 Economics 21: Microeconomics (Spring 2000) Final Exam Professor Andreas Bentz instructions You can obtain a total of 160 points on this exam. Read each question carefully before answering it. Do not use any books or notes. The use of non-programmable calculators is permitted. You have 120 minutes to answer all questions. Please use the space provided to answer questions; if you need more space, use the back of the page. Each question is self-contained, and assumptions made in one question do not carry over to other questions, unless explicitly specified. Please do not leave the exam room within the final 15 minutes of the exam, except in an emergency. At the end of the exam, you must turn in your answers promptly: if you continue writing I will take off points. Good luck. Name: Class time (circle one): 9 (section 1) 10 (section 2) 1. (15 points) Adam and Eve play the following game: questions 1-11 Eve (player 2): eat apple don't eat apple Adam (player 1): eat apple (4, 2) (1, 1) don't eat apple (4, 0) (3, 2) (a) If it (they) exist(s), what is (are) the dominant strategy equilibrium profile(s)? (If no dominant strategy equilibrium exists, say so.) answer: no dominant strategy equilibrium exists (b) If it (they) exist(s), what is (are) the Nash equilibrium strategy profile(s)? (If no Nash equilibrium exists, say so.) answer: there are two Nash equilibria: (eat, eat) and (don't eat, don't eat) page 1 of 1

2 2. (10 points) After her expulsion from paradise, Eve (along with many others) applies for a job on a big apple cider farm. The farmer knows that some applicants are highly qualified (and should therefore be paid a wage of $10), and that other applicants are badly qualified (and should therefore be paid a wage of $5), but he cannot tell which applicants are the highly and which are the badly qualified ones. The farmer therefore sets an exam with 10 questions. Highly qualified applicants need half an hour's study time to get one question right. Badly qualified applicants need one hour's study time to get one question right. Both types of applicants value their time at $1 per hour. At how many questions should the farmer set the pass/fail level on the exam so that: highly qualified workers study for the exam, get the required number questions right and obtain their wage of $10; and so that badly qualified workers do not find it worthwhile to study for the exam, fail the exam and obtain their wage of $5? answer: for the high productivity types, we want: y 5, which says that y 10 for the low productivity types, we want: y, which says that y 5 so any y between 5 and 10 will do 3. (20 points) Adam has the choice of staying in paradise or leaving paradise. If he stays in paradise, he makes $100 for certain by producing fig-leaves with snap-on elastic bands. If he leaves paradise, he will become an apple-vendor: and with probability 0.5 demand for apples is low (so that he earns $49 as a vendor), and with probability 0.5 demand for apples is high (so that he earns $121 as a vendor). (a) Adam is risk averse. Without knowing anything more about Adam's utility function, can you definitely say whether he will stay in paradise? Explain your answer briefly. answer: He will stay in paradise. The expected value of the lottery (of leaving) is $85, which is less than the $100 he can get for certain. Since he is risk averse he will never take that lottery. A diagram will help you explain this. page 2 of 2

3 (b) Suppose Adam's von Neumann-Morgenstern utility function over wealth (m) is of the following form: u(m) = m. Is Adam risk averse? Why? answer: He is risk averse. His utility function is concave. You can show this by calculating the second derivative (which is -(1/4) m -3/2 ). The second derivative is negative, which shows concavity. (c) Suppose Adam's von Neumann-Morgenstern utility function over wealth (m) is of the following form: u(m) = m. Suppose also that he is expelled from Paradise so that he has to work as an apple vendor, as described above. Clearly, he faces risk (demand for apples is either high or low, each with probability 0.5). He is offered insurance against income loss on the following terms: he can chose how much income to insure. Each dollar that he insures costs him an insurance premium of $0.50. How much consumption will he chose to have when demand is high? When demand is low? Write up, and solve, the Lagrangean for Adam's constrained maximization problem. answer: The (insurance) budget constraint is c L = c H. Where subscript L designates the low demand state, and subscript H designates the high demand state. You get this budget constraint in just the usual way: c L = k + k, and c H = k. You can solve one for k and substitute into the other to obtain the budget constraint above. The Lagrangean is: L = 0.5 c L c H - λ(c L c H ) The first-order conditions are: (i) 0.25/ c L = λ (ii) 0.25/ c H = λ (iii) c L = c H Combining (i) and (ii) gives c L = c H, just as it should, since you are offered insurance on actuarially fair terms. Then substituting that into (iii) gives you the values c L = c H = 85. page 3 of 3

4 4. (15 points) Adam quits his job as an apple vendor and instead uses the skills he has acquired in paradise (making clothes consisting of fig leaves and elastic bands) to produce skimpy green t-shirts for Dartmouth students. He uses labor (l) and cotton (k) to produce t-shirts. Labor costs $8 and cotton costs $2. Adam's production function for t-shirts (y) is as follows: y = lk. (a) In the short run, Adam can only use the amount of labor he employs (cotton is fixed at the level k* = 4). What is Adam's short-run total cost function? Explain briefly. answer: c S (y) = $8 (y/4) + $2*4, or: c S (y) = 2y + 8. (b) In the long run, both the amount of labor and the amount of cotton Adam uses can be varied freely. What is Adam's long-run (total) cost function? answer: Write down the Lagrangean for cost-minimization: L = 8l + 2k - λ(lk - y) The first-order conditions are: (i) 8 = λk (ii) 2 = λl (iii) lk = y Combining (i) and (ii) gives you 4 = k/l. Multiplying both sides of that by l 2, you get 4l 2 = lk. From (iii) you know that lk = y, so you know that 4l 2 = y, or l = 1/2 y. Using this, you can now solve for k, and you obtain k = 2 y. Calculating the cost function from this: c(y) = 8l + 2k, which is: c(y) = 4 y + 4 y, or c(y) = 8 y. page 4 of 4

5 5. (10 points) Suppose Adam's (short-run) cost function for t-shirts (y) looks somewhat like this: c(y) = 2y + 8. His t-shirts are bought by undergraduates and postgraduates, and each of these two separate markets have their own demand curve for t-shirts. Postgraduates' (inverse) demand curve for t-shirts is p(y pg ) = 20 - y pg. Undergraduates' (inverse) demand curve for t-shirts is p(y ug ) = y ug. Suppose further that Adam is a monopolist in the market for green t-shirts and that he can (third-degree) price discriminate, that is, he can treat the undergraduate and the postgraduate markets differently. Which prices would he charge to undergraduates, and which to postgraduates? answer: profit is [20 - y pg ] y pg + [ y ug ] y ug - 2(y pg + y ug ) - 8 Maximizing with respect to y pg gives: 20-2y pg -2 = 0, or y pg = 9. Putting this into the (inverse) demand function, you get the price of p pg = 11. Maximizing with respect to y ug gives: 10 - y ug -2 = 0, or y ug = 8. Putting this into the (inverse) demand function, you get the price of p pg = (15 points) Eve sees that Adam is making profits, and enters the market for green t-shirts as a competitor to Adam. Now there are two firms in this industry: Adam Inc., and Eve Inc., and these two firms compete in quantities (Cournot competition). Suppose for this question that the (inverse) total market demand curve for t-shirts (y) is: p(y) = 14 - y. Adam and Eve both have the same cost function; Adam's cost function is c(y A ) = 2y A + 8, and Eve's cost function is c(y E ) = 2y E + 8. What are the quantities produced by Adam and Eve in equilibrium? answer: Eve's profit is [14 - (y E + y A )] y E - 2y E - 8. Differentiating with respect to y E to find her best response function, we get: 14-2y E - y A - 2 = 0, or y E = (12 - y A )/2. Adam's profit is [14 - (y E + y A )] y A - 2y A - 8. Differentiating with respect to y A to find his best response function, we get: 14-2y A - y E - 2 = 0, or y A = (12 - y E )/2. Using y A to substitute into y E, we get y E = (12 - (12 - y E )/2)/2 or y E = 4. Substituting this result into Adam's best response function, we get (no surprise here - this has to be the same because Adam and Eve run identical companies) y A = 4. page 5 of 5

6 7. (20 points) After a few years, Eve decides to retire from the t-shirt business and move to Florida. This is what happens on the flight to Florida: a hijacker enters the cockpit wielding a bomb and poses the following ultimatum: "Either you give me 10 million dollars and redirect this flight to Cuba, or I will explode this bomb now, killing everyone on board." In extensive form, the game that the hijacker offers the airline looks like this: (payoffs are given in utility units) airline (player 1) give in don't give in (0, 8) hijacker (player 2) explode don't explode (-1, -1) (2, 2) (a) Find the Nash equilibrium strategy profile(s) in this game answer: (give in, explode) and (don't give in, don't explode) (b) Find the subgame perfect equilibrium strategy profile(s) in this game. answer: (don't give in, don't explode) (c) Which Nash equilibrium (or equilibria) is (are) not subgame perfect? Why is this? Give a brief intuitive explanation. answer: The (give in, explode) equilibrium is unreasonable (and therefore not subgame perfect). The equilibrium here is based on a non-credible threat: if the airline were to not give in, the hijacker would have no incentive to explode the bomb - he would die also (payoff -1). 8. (10 points) Back to Adam. In the meantime Adam has left the t-shirt business and started the internet company adamsapple.com (organic groceries home delivery). Adam doesn't actually work - he employs Mary (who, as you know, has experience with home deliveries). Mary can spend high effort (which costs her 2 utility units) or low effort (which costs her 0 utility units) in working for Adam, but Adam can't observe how hard Mary works. What Adam can observe, is the profit Mary makes for the company. If Mary works hard, she makes $100 profits for the firm with probability 0.5 and $25 profits for the firm with probability 0.5. If she spends low effort, she makes $100 profits for the firm with probability 0.2 and $25 profits for the firm with probability 0.8. Her utility function of wealth (m) and cost-of-effort (e) is: u(m, e) = m e. Mary also always has the option to quit and do something else, which gives her a utility of 4. Write down the individual rationality and the incentive compatibility constraints, that is, the constraints within which Adam can choose an appropriate (wage) incentive scheme for Mary. (IR): answer: 0.5 w w (IC): answer: 0.5 w w w w 25-0 page 6 of 6

7 9. (20 points) By now, Eve is happily settled in Florida, and she has wealth of m to live from. She only really consumes two goods, oranges (which as you know, are not the only fruit but she has had some bad experiences with apples in the past), and sun lotion. Her utility function over oranges (o) and sun lotion (s) is u(o, s) = o + ln(s). Oranges cost p o and sun lotion costs p s. (a) Derive Eve's demand curve for sun lotion. Does this tell you that sun lotion is a Giffen or non-giffen good? Why? answer: Write the Lagrangean for Eve's utility maximization problem: L = o + ln(s) - λ(p o o + p s s - m) The first-order conditions are: (i) 1 = λp o (ii) 1/s = λp s (iii) p o o + p s s = m Combining (i) and (ii) we obtain s = p o /p s, and substituting into (iii) we get o = m/p o - 1. The demand curve for sun lotion is s = p o /p s. As p s increases, s decreases (the derivative with respect to p s is negative), so the good is a non-giffen good. (b) Derive Eve's Engel curve for oranges. Does this tell you that oranges are normal or inferior goods for Eve? Why? answer: As you've shown in part (a), the demand function for oranges is o = m/p o - 1. The Engel curve emphasizes the connection between m and o, and it is upward sloping - which indicates a normal good: the derivative of o with respect to m is positive. 10. (5 points) Eve decides to invest some of her money in bonds. She is considering buying a perpetuity (or "consol") with a coupon of $50. The interest rate is 10%. You are Eve's financial adviser. How much would you advise Eve is the maximum she should be willing to pay for this perpetuity? answer: The present value of a perpetuity is coupon/interest rate. Here it is: $50/0.1, or $500. So the maximum Eve should be willing to pay for this perpetuity is $500. page 7 of 7

8 11. (20 points) "The managers of a firm will always take the action that is in the best interest of all of the firm's shareholders, regardless of what their preferences over timed consumption are." Using the example of investment decisions, comment on this statement. Bear in mind that your answer may differ according to the assumption you make about whether or not shareholders can observe what the manager does. answer (key points): I would expect you to show the Fisher Separation Theorem in some detail: draw the diagram, explain how different investment decisions can be made, and how they result in different endowments for shareholders. Then explain how different shareholders like different endowments if they cannot lend or borrow. But with the existence of a perfect capital market (where lending and borrowing is possible at the same interest rate), all shareholders want the same thing: they want managers to maximize the present value of the company. This is easy when shareholders can observe what managers do, but generally that will not be the case (for instance, because of the free-rider problem associated with monitoring sometimes this is called the "Grossman-Hart problem" I discussed in class). When shareholders cannot monitor, they need to give incentives: they need to reward good performance and punish for bad performance. One way this could be done is by rewarding managers (at least in part) in share options. But this imposes risk on the manager (the agent) and generally that is not optimal if the manager is risk averse. So here is an example of the basic tradeoff in moral hazard models: optimal risk sharing versus incentives. More generally, asymmetric information gives you an inefficiency, and one that cannot be overcome: shareholders can acquire information about managers (by giving them the appropriate incentives they know that managers work hard), but that comes at a cost: there is non-optimal risk sharing. page 8 of 8