q p(q) ε d (q) = dp(q) dq C(q) = cq + F. = 1 q q + m max

Transcription

1 Problem Set 2: do four in total. most one can be chosen from Suppose At least one must be chosen from 2-6, and at p(q) = e αq where α < 0. i. Show that the elasticity of price with respect to quantity 1, equals αq. ii. Suppose that the firm s total costs are ε d (q) = dp(q) dq q p(q) C(q) = cq + F. Determine the firm s average total cost; this will equal the Ramsey price. iii. Set the price equal to the firm s average total cost as in ii, and substitute it into the first Ramsey condition, p(q) C (q) p(q) = 1 ε d (q). This gives you the quantity traded, q, as a function of ε, c and F. Solve for the Ramsey price p. iv. How do the Ramsey price and quantity vary with ε, c and F? v. Why is the Ramsey quantity increasing in the fixed cost, F? Explain in words. 2. Suppose there are two firms that each choose how many bottles of fizzy sugar water to produce, q 1 or q 2, who have the same cost per bottle, c. The representative consumer solves max (a 12 ) q q + m q subject to w = pq + m, where total quantity q is the sum of individual production q 1 + q 2. i. In a Cournot equilibrium, how many bottles are produced? ii. What is the socially efficient quantity of bottles to produce? iii. Suppose the government could charge a tax or pay a subsidy for the production of fizzy sugar water. What tax achieves the optimal outcome? iv. Suppose the government imposed a quota on each firm. What is the optimal quota scheme? v. Suppose sugar water imposed a health cost xq on consumers, but this was neglected in their decision-making. What is the optimal tax or subsidy now? 3. There is an apiary and a farm. The apiary makes honey and does not need the farm, while the farm needs the bees to pollinate its flowers. Profits for the apiary are π a (b) = b 1 2 b2 while profits for the farm are π f (q, b) = bq q 2. 1 Recall that (d/dx)e ax = ae ax. 1

2 i. What is the socially optimal quantity of bees b and flowers q? ii. What quantities of bees and flowers do the apiary and farm pick on their own? iii. Suppose the government used a subsidy to provide the bee farmer with incentives to produce the socially optimal level of bees. What is the optimal subsidy? v. What are the Lindahl prices for this market? Provide one set of h s that balance the budget. vi. Show that the Coase theorem holds here, with budget balance: you can pick the h s so that it is individually rational for both parties to participate in the subsidy scheme. (Hint: start with the apiary. What is his payoff if the farm were to opt out?) vii. If there were actually many apiaries or many farms, what would be one challenge with implementing the solution from vi? 4. There are two cities in a given region of the country, and an airport should really be built at one of them. The FAA is trying to get them to build the airport on their own, but is having trouble finding a mutually beneficial arrangement. If an airport is builty in City A or City B, each city gets a payoff u, but the cost of building an airport is city A is c A and the cost of building the airport in city B is c B. Assume that c A < c B, 2u > c A, and u < c A. The FAA cannot compel either city to build the airport. i. When is it efficient to build the airport? Where should it be built? ii. What are the Lindahl prices that implement the efficient outcome? iii. Find a pair of h s that are budget-balanced, but not individually rational. iv. Use the Coase theorem to provide a set of individually rational payments that implement the efficient outcome. Can the budget be balanced, so that the FAA doesn t need to provide extra funds? 5. There is a cafe with a single wifi hotspot. There are two people, i = 1, 2, at the cafe who choose how much data to download, d i, which is between zero and one. The more they download, the slower connection is for the other person, so that 1 s payoff is d 1 (1 d 2 ) and 2 s payoff is d 2 (1 d 1 ), so that the more each person downloads, the worse off the other person is. i. Show that if both people choose d i = 1, neither of them has an incentive to change and reduce consumption. ii. What is the socially optimal level of data consumption? iii. Suppose the cafe owner wants to implement efficient hotspot usage. What Lindahl prices implement efficient wifi usage? iv. Now suppose that each person i = 1, 2 could simply stop using the wifi and get a payoff of zero. Show that there is an individually rational way to implement efficient Internet usage. Can the budget be balanced? v. If the cafe owner was charging for wifi usage, what is the most money she can make by charging the two people for access to the hotspot? 6. You like to listen to music while you study while your roommate does not. If music is playing, you get a payoff of u and she gets a payoff of zero. If music is not playing, you get a payoff of zero and she gets a payoff of v. i. If you have the right to decide whether or not music is played and no money can change hands, what happens? If she gets the right to decide and no money changes hands, what happens? When it is efficient for you to have the decision rights and for her to have the decision rights? ii. Suppose she has the decision rights but u > v. Use the Coase theorem to determine payments 2

3 that are individually rational, budget-balanced, and implement the efficient outcome. iii. What is one problem with implementing this solution in practice? 7. This question is about non-renewable resources, like oil. Suppose there are two generations of representative consumers, 1 and 2. There is a price-taking, representative firm with marginal cost 1 of extracting the resource. A market meets in period 1 where the resource is traded, and again in period 2. We ll show that giving the resource to a monopolist can improve social welfare. Suppose both generations of consumers t = 1, 2 solve max(2 1 q t 2 q t)q t + m t subject to p t q t + m t = w to decide how much to purchase. The total amount of the non-renewable resource is 1, so that q 1 + q 2 1. i. To solve this, start in period 2. What is the market clearing price and quantity, given whatever is left over from generation 1 s consumption? Characterize the price-taking equilibrium in period 1. When will the market end up consuming all the resource, and when won t it? ii. What is the efficient outcome? iii. If there is a monopolist, supply will automatically be restricted, thereby ensuring that less of the good is consumed in period one. But monopolies restrict supply to make profits, so it s not clear this outcome will be better. How much does a profit-maximizing monopolist sell in periods 1 and 2? iv. What is the optimal tax on the first generation to ensure the efficient outcome occurs? 8. Download Shapiro s The 2010 Horizontal Merger Guidelines: from hedgehog to fox in forty years, Antitrust Law Journal, i. Read section 1. How do merger evaluations change over the course of the twentieth century? ii. Read pages What is the big change in the merger guidelines in 2010? iii. Read pages Explain what diversion ratios are and why they were the focus of anti-trust cases. iv. Read page 66 and Why are diversion ratios no longer the focus of anti-trust cases? Why is HHI not helpful for diagnosing the impact of mergers in some markets? v. Read page What is the hypothetical monopolist test? What are the strengths and weaknesses of defining markets in this way? vi. In class we talked about networked goods, like Facebook or Google or the iphone marketplace. HHI was clearly not useful in understanding whether or not the huge market shares observed in the tech industry were good or bad. Do the ideas Shapiro describes more satisfactorily address the case of networked goods, or not? 9. Download Coase s The Problem of Social Cost, Journal of Law and Economics, i. Read about the doctor and the confectioner (p. 8-10). Explain the problem and the court s decision. What does Coase think of the decision? ii. On page 13, he asks, Who caused the smoke nuisance? and the provides a surprising answer. Read enough of the details of the case to understand what he s arguing. Do you agree or not? iii. In the last full paragraph on page 13, he draws a distinction between judges and economists. Explain what he s saying. iv. On page 15, read from But it has to be remembered... in the first paragraph to the end of the paragraph. Explain what he s saying. 3

4 v. Page 16, first full paragraph: why does he think firms and organizations exist? Put another way, why isn t all economic activity conducted through markets? What do you think of his claim? vi. Page 17, first full paragraph: what is the role of government in Coase s world? vii. Page 28, last paragraph, and continuation on the next page: What does Coase think about the way legal and economic systems interact? viii. Last paragraph: What is the change he is advocating? This was written in Do you think his advice has been followed? Provide a particular market, firm, judicial, or legislative example, and discuss whether Coase would criticize or approve of it. 10. Download Nordhaus Climate Clubs: Overcoming Free-riding in international climate policy, American Economic Review, i. Explain what he means when he writes about free-riding, and how the Westphalian system lead to the failure of the Kyoto protocol. (p ) ii. Explain what a Climate Club is. What is the difference between a club good and a public good? iii. In his basic model, where is the externality problem in the math (circle it in the first-order necessary condition)? How would this be solved with Lindahl pricing and the Coase theorem? (p ) iv. How is his discussion of the repeated game on p similar to our model of collusion in class? v. On page 1351, he discusses how taxes (prices) and quotas (quantities) have fared in practice. What is his opinion? His discussion is very brief: further develop what you think he might be alluding to, or provide evidence for his claims. vi. On page 1362 to 1363 he discusses why he thinks the Kyoto protocol failed. What are his main findings? Why did the US fail to ratify? Do you agree with his argument? vii. Read his conclusion. What do you think of all this? Can Climate Clubs solve the free-rider problem? 11. There is a monopolist firm who invests in R&D, r, then produces a quantity of goods q for consumers. Producing q units costs nothing, while R&D costs (1/3)r 3 (imagine a computer program that can be distributed for free over the Internet). Consumers value the product as max rq 1 q 2 q2 + m where m + pq = w. Notice that the more the firm invests in R&D, the more consumers value the good. i. For an arbitrary r 0, solve for the optimal price and profits for the monopolist, ignoring R&D costs. Compute the monopolist s profit in terms of r. Now, pick the profit-maximizing level of R&D. ii. What quantity and R&D levels would the social planner pick 2? Compare these to the levels from part i. iii. If you could only subsidize R&D, what would the socially optimal research subsidy be? Does it achieve the socially optimal outcome? iv. Can you design the Lindahl prices so that the consumers get all the gains from trade and the scheme is budget balanced? 12. In class, we showed a particular kind of collusive agreement could work: if you cooperate with me, I will cooperate with you, but if you stab me in the back, I will stop cooperating forever. 2 Note that the equation x = x 2 has two solutions: x = 0 and x = 1. 4

5 This is very unforgiving. Can we find a simple model of collusive agreements where the punishment is for a fixed length of time, an in Porter (1983)? Suppose the payoff to cooperating is 1, the payoff to cheating when the other player cooperates is d > 1, and the payoff to competing is zero. There are two players who both discount at a rate 1 > δ > 0. One player says to the other, if you cooperate with me, I will cooperate with you, but if you stab me in the back, I will be competitive for T periods and then try cooperating with you again. i. Compute the payoff of cooperating forever. ii. For a punishment length of T periods, compute the payoff of deviating, assuming that after a deviation the players return to cooperating forever after T periods. iii. When is cooperating better than deviating? Try to solve for T. iv. Under what condition can the players always find a T large enough that they can effectively collude? How does this compare to the result in class? 13. How do economists really estimate demand and measure market power? Get the sample data set from the class web page; it has whether or not a bunch of consumers purchased the good and the price. Suppose there is a utility-maximizing representative household, and a profitmaximizing monopolist with marginal cost c charging price p that generated the data. i. A given consumer i gets utility u i = 1 βp + ε i from buying the monopolist s product where ε i is an error term for i, and a payoff of zero if it buys no item. So household i makes the purchase if 1 βp + ε i 0, but does not make a purchase if 1 βp + ε i < 0. The problem is that the data do not include u i, only whether or not the household made the purchase. We need to make another assumption about ε i in order to estimate β. The simplest model that economists use is logit demand: the probability that a given consumer buys the good is e1 βp D(p) = 1 + e 1 βp and the probability a given consumer refuses to buy the good is This in turn implies that and taking logs implies that 1 D(p) = e 1 βp. D(p) 1 D(p) = e1 βp, ( ) D(p) log = 1 βp. 1 D(p) Then we can get an estimate of β by computing the sample market share, s, and setting ˆβ = 1 ( ( )) s 1 log. p 1 s 5

6 Compute ˆβ for the sample dataset. ii. The monopolist maximizes profits, solving This gives a first-order necessary condition Show this equals and that it can be simplified to max D(p)(p c). p D(p) + D (p)(p c) = 0. e 1 βp βe1 βp + (p c) = 0, 1 + e1 βp (1 + e 1 βp ) 2 c = p 1 + e1 βp. β Use your estimate of β from part i, the price in the data, and this equation to recover an estimate of the monopolist s true marginal cost, ĉ. iii. Compute the Lerner index for this monopolist. What proportion of the price is pure profit, and what proportion is cost? What is the elasticity of demand at the price charged by the monopolist? iv. Open the second tab of the data sheet: it includes ε i for every household. Compute the loss in consumers surplus from charging the monopoly price instead of your estimated firm marginal cost. iii. This approach uses a structural model to recover fundamentals like the monopolist s true marginal cost from market data. Briefly describe one problem this approach might face in practice, and how you might address it. 6