Economics 21: Microeconomics (Summer 2002) 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, or the use of only the non-programming and non-graphing functions of a programmable calculator, 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. I must be able to see your work for you to get (full) credit. There are 10 questions on this exam. These 10 questions are independent of each other. 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:. Section (circle one): section 1 (9L) section 2 (10) It s your job to bring US Scare, a recently bankrupt airline, back into business. questions 1-10 1. Suppose US Scare produces its output, y (airline seats), with two inputs: pilots (p) and airplanes (a). Each pilot costs a wage of $100,000, and each airplane costs a rental rate of $10,000,000. US Scare s production technology for the long run looks like this: y = a 1/2 p 1/2. a. (5 points) Write up the Lagrangean for US Scare s long-run cost minimization problem. Do not solve the Lagrangean. If you were to solve the Lagrangean, your answer for US Scare s long-run cost function would be: c(y) = 2,000,000y. answer: L = 100,000p + 10,000,000a - λ( a 1/2 p 1/2 - y) b. (5 points) Suppose that in the short run, US Scare has 100 airplanes, and that number is fixed in the short run. What is US Scare s short-run cost function? answer: The short-run production function is 10 p 1/2. Solving for p, we see that in order to produce y units out output, US Scare needs to use (y/10) 2 pilots. Therefore, the short-run cost function is c(y) = 1,000,000,000 + 100,000(y/10) 2, or c(y) = 1,000,000,000 + 1,000y 2. page 1 of 9
2. (10 points) US Scare serves two distinct types of customers, students and non-students. The demand for airline tickets by students (subscript S ) is y S = 1,000-2p S (where p S is the price charged in the student market, and y S is the quantity sold to students). The demand for airline tickets by non-students (subscript N ) is y N = 1,000 - p N (where p N is the price charged in the non-student market, and y N is the quantity sold to non-students). US Scare knows who is a student and who isn t. US Scare has fixed cost of $1,000 and constant marginal cost of $100 per ticket. If US Scare can act as a third-degree price discriminating monopolist, how many tickets will US Scare sell optimally to students, and which price will it charge? How many tickets to nonstudents and at what price? answer: First, we need the inverse demand functions. For the student market, that s p S = 500 - (1/2)y S, and for the non-student market it is p N = 1,000 - y N. The monopolist s profit is π = (500 - (1/2)y S ) y S + (1,000 - y N ) y N - 100 (y N + y S ) - 1,000. Take the derivative with respect to y S (and set equal to zero) to get 500 - y S = 100, or y S = 400. From this follows that p S = 300. Similarly, for the non-student market: take the derivative with respect to y N (and set equal to zero) to get 1,000-2y N = 100, or y N = 450. From this follows that p N = 550. page 2 of 9
3. (10 points) In fact, US Scare is competing with another airline, Untied, and both are the only two competitors. Both airlines compete in a Cournot fashion (they play a simultaneous-move Cournot game). Each airline has constant marginal cost of $100 per ticket and fixed cost of $1,000. The total market demand for airline tickets is y = 4,000-4p (where y is the quantity sold and p is the price). What is the quantity sold by each of the two competitors in a Nash equilibrium in this Cournot game? answer: First, we need the inverse market demand function: p = 1,000 - (1/4)y. Next, calculate US Scare s best response. US Scare s profit is: (1,000 - (1/4)(y US Scare + y Untied ))y US Scare - 100y US Scare - 1,000 Take the derivative with respect to y US Scare and set equal to zero to obtain: y US Scare = 2(900 - (1/4)y Untied ) And the symmetric best response function can be obtained for Untied. Putting the two together, you obtain: y US Scare = 1,200 and because of symmetry, y Untied = 1,200. page 3 of 9
4. US Scare and Untied are planning to merge, forming a monopoly (AA - Amalgamated Airlines). In order to assess the impact of the new monopoly airline, the Federal Trade Commission needs to know which price the new monopolist (Amalgamated) would charge. The market demand for airline tickets is y = 4,000-4p. Amalgamated s marginal cost would be constant at $100 per ticket, and it would have a fixed cost of $1,000. a. (10 points) If Amalgamated behaved as a single-price (ordinary) monopolist, which quantity would this monopolist produce, and which price would it charge? answer: Again, we need the inverse demand function, p = 1,000 - (1/4)y. The monpolist s profit is π = (1,000 - (1/4)y)y - 100y - 1,000. Maximize with respect to y to get 1,000 - (1/2)y - 100 = 0, or y = 1,800. The price is therefore p = 550. b. (10 points) If Amalgamated behaved as a perfectly price-discriminating monopolist, which quantity would it sell? answer: If Amalgamated behaved as a perfectly price-discriminating monopolist it would charge a different price to each customer, and different prices for different units, but would sell output up to where marginal cost (100) is equal to the price that the last consumer is just willing to pay, that is 100 = 1,000 - (1/4)y, or y = 3,600. c. (10 points) Suppose Amalgamated were to charge a price of $550 per unit of output. What is the elasticity of market demand at that price? answer: The output at a price of $550 is 1,800 units. The derivative of demand with respect to price (dy/dp) is -4 in this case. The price elasticity of demand is (dy/dp)(p/y), or -4(550/1,800) = -1.2. page 4 of 9
5. Suppose the proposed merger between US Scare and Untied is not allowed by the FTC. US Scare needs to buy insurance against aircraft accidents. Suppose the probability of an accident is 0.1. If there is no accident, US Scare will have profits of $100,000. If there is an accident, US Scare will have profits of only $50,000. Suppose US Scare s vnm utility function over profits (π) is u(π) = ln(π), where ln denotes the natural logarithm. a. (5 points) Show, by calculating the appropriate derivative, whether US Scare is riskneutral, risk-averse, or risk-loving. answer: The second derivative of ln(π) is -1/π 2, which is negative, so US Scare is riskaverse. b. (20 points) US Scare is offered actuarially fair insurance (insurance at a rate γ = 0.1). Calculate how US Scare will insure: that is, calculate how much money US Scare will choose to have in the good and in the bad state. Do this by writing up, and solving, the appropriate Lagrangean. answer: L = 0.1 ln(c b ) + 0.9 ln(c g ) - λ(0.1 c b + 0.9 c g - 0.1 50,000-0.9 100,000) FOC: (i) 0.1(1/c b ) = λ 0.1 (ii) 0.9(1/c g ) = λ 0.9 (iii) 0.1 c b + 0.9 c g = 0.1 50,000 + 0.9 100,000 from (i) and (ii) follows c b = c g, and putting this into (iii) you get c g = 5,000 + 90,000, or c g = c b = 95,000. page 5 of 9
6. (10 points) US Scare needs to hire new pilots. There are type 1 (productive) and type 2 (less productive) pilots - and US Scare wants to hire type 1 pilots into Captain positions, and type 2 pilots into First Officer positions. A Captain position pays a salary of $100,000. A First Officer position pays $80,000. However, US Scare does not know whether any given applicant is a type 1 pilot or a type 2 pilot. US Scare designs a test to help it distinguish between type 1 and type 2 pilots. In order to get one question right, a type 1 pilot needs to spend $100 on training materials and tuition. In order to get one question right, a type 2 pilot needs to spend $200 on training materials and tuition. How many questions should US Scare require pilot applicants to get right, so that it can distinguish between type 1 and type 2 pilots (i.e. so that there exists a separating equilibrium)? answer: For the type 1 pilots you want: 100,000-100q > 80,000, or q < 200. For type 2 pilots you want: 80,000 > 100,000-200q, or q > 100. So any number of questions between 100 and 200 will produce a separating equilibrium. 7. (10 points) US Scare thinks about different ways of purchasing a new plane. It has three options. Option (a): Option (b): Option (c): US Scare pays $50 million today, and then there are no more payments. US Scare pays $20 million today, $20 million next year, and $20 million the year after that (i.e. in two years time), and then there are no more payments. US Scare pays nothing this year, but next year, and every year thereafter (until perpetuity) pays $6 million per year. The interest rate is 10%. Calculate the present value of each of the payment plans and decide which option US Scare should take. answer: The present values are: Option (a): $50,000,000 Option (b): $20,000,000 + $20,000,000/(1.1) + $20,000,000/(1.1) 2 = $54,710,743.80 Option (c): $6,000,000/0.1 = $60,000,000 Therefore, US Scare should take option (a). page 6 of 9
8. US Scare is thinking about making its passengers pay separately for in-flight food and for increased legroom. For that reason it wants to know what its passenger s demand looks like. A typical US Scare airline passenger has the following utility function over in-flight food (f) and legroom (l): u(f, l) = ln(f) + l, where ln denotes the natural logarithm. Legroom costs a passenger p L per unit and food costs p F per unit. A passenger has wealth m. a. (5 points) What is the marginal rate of substitution (in-flight food is on the horizontal axis and legroom on the vertical axis)? answer: MRS = -(1/f)/1 = -(1/f) b. (5 points) Write up the Lagrangean for a passenger s utility maximization problem, subject to their budget constraint. Do not solve this Lagrangean. answer: L = ln(f) + l - λ(p L l + p F f - m) c. (5 points) This passenger s demand for legroom that you could derive from your answer to part (a) is the following: l(p L, p F, m) = (m/p L ) - 1. Is legroom a normal or an inferior good? Show by calculating the appropriate derivative. answer: take the derivative with respect to income to obtain 1/p L > 0, so it is a normal good. d. (20 points) This passenger s demand for in-flight food that you could derive from your answer to part (a) is the following: f(p F, p L, m) = p L /p F. Suppose US Scare considers lowering the price of in-flight food. Describe in words the intuition behind the income, substitution, and total effects from a price fall in general (for any good, normal or inferior). For the specific example of in-flight food for the consumer with the utility function given above, is there anything more specific you can say about the size of either the income or the substitution effect? answer: If the price of a good falls, the substitution effect says to consume more of the good since it is now relatively cheaper (that is, its relative price, relative to all other goods, has fallen). The income effects says that since you are now richer (your income buys you more, that is the purchasing power of your income has increased), you should consume more of all normal goods, and less of all inferior goods. Therefore, for a normal good income and substitution effects go in the same direction (more consumption), and for an inferior good, the two effects go in the opposite direction (we cannot say whether the overall result will be more consumption or less consumption [the latter case being the one of a Giffen good]). In the specific case of in-flight food, in-flight food does not depend on income (the utility function is quasilinear). So therefore there will be no income effect. The overall effect is made up purely of the substitution effect, so that the overall effect says to consume more. (more space to continue on next page) page 7 of 9
(this page is intentionally blank for you to continue your answer) page 8 of 9
9. (10 points) A potential competitor, America West (the only airline at which the autopilot is referred to as the designated driver ) is thinking of entering the market. Here is the game between US Scare and America West (America West is player 1, US Scare is player 2): (node A): America West (player 1) enter stay out (node B): US Scare (player 2) (0, 10) keep price high lower price (4, 4) (node C): America West (player 1) stay in exit (-2, -2) (-1, 5) What is the subgame perfect equilibrium in this game? Fill in the two blanks below: This is America West s equilibrium strategy: stay out at A, exit at C This is US Scare s equilibrium strategy: lower price at B 10. (10 points) US Scare knows that without you as a consultant, they will succeed with a probability of 99% (and therefore make profits of $100,000,000 [in words, $100 million]), and they will fail with a probability of 1% (and therefore make profits of $0 [zero]). With you as a consultant they will succeed for sure (i.e. make $100,000,000), but they have to pay your salary. How much is the most salary that you can ask for (the highest salary US Scare is just willing to pay)? US Scare s vnm utility function over money, m, is u(m) = m. answer: US Scare s expected utility from not hiring you is 0.01 0 + 0.99 100,000,000 = 9,900. The certainty equivalent therefore is 9,900 2 = 98,010,000. So having 98,010,000 is equally as good as being exposed to the lottery, so your salary can be at most: 100,000,000 - salary = 98,010,000, or salary = 1,990,000. page 9 of 9