Economics 311 Midterm Exam

Transcription

1 Time allowed: 2 hours Economics 311 Midterm Eam Do FOUR questions (all questions have equal weight). John Kennan, December 19, 201 IMPORTANT: Eplain your answers carefully. A good diagram is often more effective than a lot of words (but you must eplain what the diagram means). You get no credit for unsupported assertions or guesses. Write as if you are trying to convince an intelligent pern who does not already know the answers. If your answers would not convince such a pern, it will be assumed that you do not fully understand the answers. 1. The Phoeni Moons, a professional football team, has a stadium which seats 30,000 people. All seats are identical. The optimal ticket price is $5, yet this results in an average attendance of only 20,000. (a) Eplain how it can be profitable to leave 10,000 seats empty. In order to sell more seats it is necessary to reduce the price charged for all of the seats. If MR = MC with me seats empty, then selling more seats would mean that cost increases more than revenue (e.g. if MC = 0, selling more seats would mean that revenue falls). (b) Net week the Moons play the Tucn Turkeys, who have offered to buy an unlimited number of tickets at $ each to be reld only in Tucn. How many tickets should be ld to Tucn to maimize profits? (i) 30,000, (ii) more than 10,000, (iii) 10,000, (iv) less than 10,000, (v) none. Eplain your answer. (ii) more than 10,000, assuming that marginal cost is less than $ (if demand is highly elastic, marginal revenue could be above $ with p = 5; in that case nothing changes; but it is more plausible to assume that the marginal cost of selling seats is negligible). There is now an opportunity cost of selling seats in Phoeni, the profit maimizing choice is where MR =. (c) Given your answer to part (b) above, what price should the Moons charge their own fans, to maimize profit? (i) $5, (ii) more than $5, (iii) between $ and $5, (iv) $, (v) less than $. Eplain your answer. (ii) more than $5, assuming that marginal cost is less than $. the quantity ld is reduced, a higher price can be charged. Since 1

2 2. Design a contract to maimize the epected profits received by a risk-neutral principal who will hire a risk-averse agent. The agent s utility function is u (c, e) = log (c) e where e is effort (high or low), and c is consumption, which is equal to the wage payment specified in the contract. The principal can observe gross revenue, but cannot observe the agent s effort. The agent has an outside option that is a sure thing with utility level 1 2. The low effort level is zero, and the high effort level is 1 2. Gross revenue depends on the agent s effort level. If effort is high, revenue R is 20 with probability 1 5, and 25 with probability 5. If effort is low, R is 20 with probability 3 5, and 25 with probability 2 5. The contract specifies wages for each realization of output. Epected profit given high effort is π = p i (R i w i ) Choose {w i } to maimize this, subject to an incentive constraint and a participation constraint p i log (w i ) e p 0 i log (w i ) p i log (w i ) e u 0 High effort means R is 20 with probability 1 5, and R is 25 with probability 5, epected revenue is 2. Low effort means R is 20 wp 3/5 and R is 25 wp 2/5, epected revenue is 22. After paying for the agent s effort, the epected net payoff from high effort is 23½, this is what the principal wants to implement, unless it costs too much to provide the incentive. The individual rationality (aka participation) constraint is: where w is the wage in logs. 1 5 log (W 20) + 5 log (W 25) = 0 w 20 = w 25 The incentive compatibility constraint is then: 3 5 w w 25 = w w 25 = 5 2 2

3 This gives w 25 = 1 w 20 = 1 ( ) profit is ep ( 1) 5 ep 1 = If the principal chooses to implement ( a) constract such that the agent supplies low effort, then profit is 22 ep 1 2 = So the high-effort constract is more profitable. 3. A consumer has an income of $2, 000 per month, which is used to rent an apartment, and to buy food. Apartments can be rented at a monthly rate of $1 per square foot. The consumer s utility function is u (f, ) = log (f) 1000 where f is food, measured in pounds, and is apartment size, measured in square feet. (a) If the price of food is $ per pound, what is the optimal consumption plan? Marginal utility of each good is infinite when the quantity is zero, corner lutions can t be optimal. Equating marginal utility per dollar gives Using the budget constraint I = 1 fp f p f f = = p and since I = 2, 000 and p = 1 this implies 0 = (1000) (2000) = ( ) ( 1000) So = 1000 and then f = 1000 = 250. (b) If the price of food rises to $8 per pound, does the consumer rent a smaller apartment? No. The lution for above was obtained without using any information about the price of food. 3

4 . There are 3 people who live net to a lake. They all care about the quality of the water in the lake, but me care more than others. They have quasilinear preferences represented by utility functions u 1 (y, ) = y + log () u 2 (y, ) = y + 3 log () u 3 (y, ) = y + 5 log () where is the water quality and y is a (private) consumption good. Each pern has an endowment of 2 units of the y good. Each pern can make the water cleaner, at a cost: c (q) = 1 2 q2 where q represents the quantity of the private good used to clean the water, meaning that consumption of the private good is y = q2. The water quality is determined by the equation = q 1 + q 2 + q 3 where q i is the choice made by pern i, taking the others choices as given. (a) Find a Nash equilibrium. Write the utility functions as u i (y, ) = y + a i log () Taking q 2 and q 3 as given, pern 1 chooses q to maimize The first-order condition is So q 1 = a 1, and similarly Adding these equations gives 2 = 9, and = 3, and u 1 (y, ) = q2 + a 1 log (q + q 2 + q 3 ) a 1 q = q + q 2 + q 3 = a 1 q i = a i = q 1 = 1 3 q 2 = 1 q 3 = 5 3

5 (b) Is the equilibrium outcome efficient? No (because there is a nonpecuniary eternality). Cleaner water benefits all of the consumers, but each consumer only takes the individual benefit into account when deciding how much of the y good is allocated to water-cleaning. For eample, setting q 1 = 3 would yield =, and if pern 5 then gives 1 unit of the y good to pern 1, then pern 3 is better off (because the marginal utility of the clean water is 1 for this pern when = 5, and above 1 when is lower). This change al makers pern 2 better off. Efficiency requires that the sum of the marginal utilities is equal to the marginal cost, and the marginal cost must be the same for each pern: = A monopolist faces the demand curve 3 = 9 P = Q where Q is the annual quantity ld, and P is the price, in dollars. Labor is the only input, and the labor supply curve is perfectly elastic at a wage of $2 per hour. The production function is where L is hours worked. Q = 10L 5000 (a) Find the profit-maimizing price and quantity. Marginal Revenue is Cost is with and marginal cost is MR = 60 5 Q C = wl Q 2 = 10L 5000 C = w ( Q ) MC = wq 5 Since w = 2, equating MC and MR gives 60 5 Q = 2Q 5 the optimal quantity is Q = 50, and the price is p = 0. 5

6 (b) Suppose a price ceiling of $35 is imposed. What is the new profit maimizing plan, and how much profit is made? Now marginal revenue is $35 up to the point where the demand price falls below the ceiling; this point lves the equation 35 = Q Q = At this point marginal cost is 25, it is optimal to produce up to this point. Selling more would mean that marginal revenue falls below marginal cost, the profit-maimizing point is at Q =