Econ 100A (UC Berkeley) Fall 2011 Prof. Santesteban
Final Exam (150 points) Please write your name, ID number, and GSI at the top right of this page. Please write your last name on every other page of this exam. You may use a calculator. No other reference material is allowed, i.e., no cheat sheets. Please show your work. Otherwise, no partial credit will be given. Good luck! Part I: Long Problems. Each question if worth 25 points. Please show all work and explain each step. No credit will be given for just answers or for mathematical scribble that is hard to follow. Problem 1 A firm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the firm’s average cost, meaning that this firm alone (although still a price taker) can earn real economic profits in the long run. a) If the market price is $20 per widget and the firm’s marginal cost is MC=0.4q, where q is the daily widget production for the firm, how many widgets will the firm produce? (10 points)
MC = .4q
p = $20
Set p = MC
20 = .4q
q = 50. 1
b) Suppose a government study has found that the firm’s new process is polluting the air and estimates the social marginal cost of widget production by this firm to be SMC=0.5q. If the market price is still $20, what is the socially optimal level of production for the firm? What should be the rate of a government imposed per‐unit tax to bring about this optimal level of production? (20 points)
SMC = .5q Set p = SMC 20 = .5q
q = 40.
At the optimal production level of q = 40, the marginal cost of production is MC = .4q = .4(40) = 16, so the tax t = 20 −16 = $4. c) Graph your results. (5 points)
Problem 2 Suppose there are two types of used cars: plums and lemons. A plum is worth $3000 to a buyer and $1900 to a seller. A lemon, on the other hand, is worth $1000 to a buyer and $500 to a seller. The fraction of used cars that are plums is ¼ and the fraction that are lemons is ¾. Assume that when buyers and sellers bargain, the agreed upon sales price is always the maximum that buyers are willing to pay. a) What would be the prices for lemons and plums if there were perfect information about used car quality? (4 points) Price Plums = $3000 Price Lemons = $1000 b) What would be the price of a used car if neither buyer nor seller knew whether a particular car was a lemon or a plum? (4 points)
Average WTP = 3000*0.25 + 1000*0.75 = 1500
Average WTS = 1900*0.25 + 500*0.75 = 850
Price of a used car = $1500
c) Assume that buyers cannot tell whether a used car is a plum or a lemon. Sellers know which type of car they own. What will be the market price for used cars? Explain. (4 points) Since the WTP is lower than the reservation price of a plum owner, no plums will be offered. As such, only lemons will sell for $1000. Price of a used car = $1000 d) Now assume that there are as many plums as lemons. Continue to assume that buyers cannot tell if a car is a plum or a lemon. What will be the market price for used cars? Explain. (4 points)
Average WTP = $2000
Now plum owners are willing to sell.
Price of a used car = $2000
e) Continue to assume that the distribution of lemons and plums is 50‐50. The Akerlof Institute offers a new service. For a price P, it will inspect any used car to determine whether it is a plum or a lemon. The inspection is 100% accurate. What is the maximum price P* that owners of plums would be willing to pay to have their cars inspected? [Hint: Bear in mind that if one plum owner is willing to pay P*, all plum owners will be willing to pay P*] (5 points)
Owners of plums will want to pay for the service as long as: 3000 – P* – 1900 >= 2000 ‐ 1900 Hence, P*=1000. At that price, owners of plums are indifferent between buying the service or not. At a price of $1000 (or any P* > 0), owners of lemons would not want to pay for the service. Hence, the maximum price plum owners would be willing to pay is P*=1000.
f) The Akerlof Institute decides to charge P*. How much will plums sell for? And lemons? (4 points) At P*=1000, there’s perfect signaling and plums sell for $3000 and lemons for $1000.
Problem 3 Consider two firms, A and B, which simultaneously set prices in each period t = 0, 1, 2, . . . Firm A has marginal cost of CA = 12, while firm B has marginal cost of CB = 20. Market demand in each period is given by Q (P) = 140 – P/2. (a) What are the Bertrand equilibrium prices, output levels, and profits in the single‐period static stage game? (6 points)
(b) What is firm A’s monopoly price, output, and profits in the single‐period static stage game? Same for firm B. (6 points)
(c) Suppose the two firms start to collude in the following way: In t = 0, both charge firm B’s monopoly price, and firm A gets a 70 percent market share, while firm B gets 30 percent. In each subsequent period, they keep price and market shares unchanged unless one firm deviates. If one firm deviates, then both firms charge the Bertrand equilibrium price forever. Calculate firm A’s and firm B’s critical discount factor. (7 points)
(d) Suppose that the actual discount factor of both firms is 0.6. Is the collusive agreement in (c) sustainable? (6 points)
Problem 4 There are two book stores, A and B, in a small town in Switzerland. In Switzerland, books can only be sold at the prices set by the publishers, so there is no price competition at the retail level. Rather, the book stores compete by differentiating themselves in their hours of operation. If store i sells qi books during si hours, then i’s profit is given by 5qi – 100si. In total, 1000 books are sold in Bern per day. The number of books sold by each book store depends on its hours of operation. If store A is open for sa hours and store B for sb hours, then A sells qa = 1000*sa/(sa+sb) and B sells qb = 1000*sb/(sa+sb). a) Suppose the city regulates the book stores and requires them to be open for exactly 8 hours per day. Compute the profits of firms A and B. (8 points)
b) The mayor of the town suggests to deregulate and to allow book stores to either be open for 8 or for 12 hours a day. The city council hires you as a game theorist to predict what the outcome of the deregulation is going to be. What is your prediction if stores have to choose independently whether to be open for 8 or for 12 hours? Is the deregulated outcome efficient (from the point of view of the book stores)? Why or why not? [Hint: Write out the normal form of this game and solve for any Nash equilibria.] (8 points)
c) There is a powerful trade union of book store employees in the town that wants to know whether or not it should oppose deregulation. They hired a game theorist that tells them that they should push for deregulation and that, after deregulation, both book stores will be happy to increase wages such that one hour of operation costs $140 rather than $100 to each book store. Furthermore, in this case, no employee would have to be afraid that the stores will be open for more than 8 hours per day. This sounded crazy to the union and they fired the game theorist. Was the union justified in firing the game theorist? Why or why not? (9 points)
Part II ‐ True/False/Uncertain. Please choose 10 out of 12. Each question if worth 5 points. Please explain your answers using math, graphs, or economic logic. All credit is based on the quality of the explanation. Question 1 DeBeers (an unregulated monopolist) determines that, at current prices and quantities, the elasticity of aggregate demand for diamonds is ‐0.8. DeBeers now knows that it can increase profits by lowering the price.
Question 2 The following extensive‐form game has exactly one Nash Equilibrium.
Question 3 Suppose that a man commutes to work every day either by walking or taking BART. It costs $1 per ride, and the man uses BART four times per month. Now, suppose that BART offers a monthly pass covering unlimited rides for $20. Holding all other factors constant (including the man’s utility function), the man will not find it optimal to purchase the monthly pass. [Hint: Assume that our commuter has a downward sloping demand for use of BART as a function of price. Consider what the marginal cost to the commuter is with pass and without.]
Question 4 If two simultaneous move Bertrand price competitors have different constant marginal costs, then any price between their marginal costs could be a Nash equilibrium price.
ANS: T So long as firm 1 prices just below firm 2 and below firm 2’s MC, firm 2 is best responding by pricing just above firm 1. But firm 1 is best‐responding only if it prices above its own MC.
Question 5 In a duopoly, if you can choose to either be a simultaneous move Cournot competitor or a Stackelberg leader, you will always choose to be a Stackelberg leader. ANS: T You gain a first mover advantage by being a Stackelberg leader. And you can always earn Cournot profits by playing the Cournot quantity as the leader in the sequential game.
Question 6 Suppose a player in a sequential game has 5 potential decision nodes, with 2 possible actions at each node. Then she has 25 possible pure strategies. ANS: F He has
Question 7 Cooperation is difficult to achieve in a Prisoners’ Dilemma because each player thinks the other player might not cooperate. ANS: F Cooperation is difficult to achieve because not cooperating is a dominant strategy for each player ‐‐ i.e. regardless of whether or not the other player cooperates, it is in each player’s interest not to cooperate.
Question 8 Depending on the shape of the marginal cost curve, a monopolist might produce an output level on the elastic or the inelastic part of demand. ANS: F A monopolist always produces on the price inelastic part of demand.
Question 9 In the presence of positive production externalities, a monopolist might produce the efficient output level. ANS: F In the presence of positive externalities, the efficient quantity is larger than the quantity produced if only private MC is set to MR. The quantity where MC=MR is the monopolist quantity. Introducing a positive externality then moves the efficient quantity further out without changing the quantity the monopolist would produce ‐‐ implying the monopolist will deviate even further from efficiency.
Question 10 In the absence of the negative externality from each individual’s contribution to road congestion, roads would not be congested (aside from congestion caused by accidents). ANS: F Just as the optimal level of pollution is not zero ‐‐ the optimal amount of road congestion is not zero.
Question 11 A change in the price of one good cannot leave utility unchanged unless the price change is accompanied by a change in income. ANS: F There are several types of counterexamples to this. For instance, if a consumer is at a corner solution and the price of the good that is not consumed increases, the consumer’s utility is unchanged. Or, if a consumer consumes at an interior solution, an increase in the price of one good could be accompanied by a decrease in the price of the other good such that the consumer’s utility remains unchanged.
Question 12 If labor and capital are perfect complements in production, and capital is fixed in the short run, then short run supply curves are vertical. ANS: T If capital cannot be varied, there is no way to increase production when output price rises.