Economics 335, Spring 1999 Problem Set #7 Name: 1. A monopolist has two sets of customers, group 1 and group 2. The inverse demand for group 1 may be described by P 1 = 200? Q 1, where P 1 is the price that they pay and Q 1 is the amount that they consume. For group 2, the inverse demand is P 2 = 100? 2Q 2. The monopolist faces a constant marginal cost of 40. a. Suppose that the monopolist is unable to distinguish between these two groups of consumers, and must therefore charge them all the same price P = P 1 = P 2. Show that the monopolist s total demand is given by Q = Q 1 + Q 2 = 0ifP 200 200? P if 100 š P š 200 250? 1.5P if 0 š P š 100 b. We can rewrite this demand function to obtain 0ifQ 250 P = 166 2? 2 Q if 100 š Q š 250 3 3 200? Q if Q š 100 Using this, derive the monopolist s revenue and marginal revenue as functions of Q.
c. The following is a graph of the firm s demand curve and marginal revenue curve. On the graph, draw the marginal cost curve and indicate the price and quantity at which the monopolist maximizes profits. Note that the marginal revenue curve is discontinuous. When the quantity increases beyond 100, the second group of consumers enters the market, so marginal revenue jumps up from 0 to 33 1 3. p 240 220 200 180 160 140 120 100 80 MR 60 40 20 MR D O y m y d. Using algebra, prove that the monopolist maximizes his profits when he charges the price P = 120. 2
e. At the price P = 120, what is the monopolist s profit? What is the consumer surplus of each group? Add these to get the total social surplus created. f. Now suppose that the monopolist can distinguish between the two groups and consumers, and can charge them different prices. Draw a graph containing the demand curves and the marginal revenue curves for each of these two markets, as well as the monopolist s marginal cost curve. On your graph, illustrate the profit maximizing prices in each market. 3
g. Using algebra, derive the monopolist s profit maximizing prices P 1 and P 2. (To do this, simply derive the marginal revenue curve for each market, and set marginal revenue equal to marginal cost. h. At these prices P 1 and P 2, what is the monopolist s total profit from the two markets? What is the consumer surplus in each market? Add the monopolist s profit and the two consumer surpluses to get the total social surplus. i. How does your result compare to the result when the monopolist could not price discriminate? t Who gains? t Who loses? t How do they compare in terms of efficiency? (To compare efficiency, you need to look at the total surplus.) 4
2. The manager of a local movie theater suspects that demand for movies depends on when the movie is shown. Early moviegoers, who go to films before 5 p.m., are not willing to pay as much as those who go to movies in the evening. The manager does some market research and discovers that the demand curves for daytime (D), and evening (E) moviegoers are given by Q D = 6? P D Q E = 8? P E The marginal cost of showing a movie is constant and equal to $4 per customer. This includes the costs of ticketing and of cleaning the seats. a. Suppose that the manager charges separate prices for afternoon and evening tickets. i. What is the profit maximizing price policy? ii. What is the price elasticity of demand in each market at these profit maximizing prices? P D = P E = 5
iii. Recall that Therefore, we expect that t t MC = MR = P 1 + 1 P P D 1 + P 1 D = P E 1 + P 1 E Ô P D = 1 + 1 PE P E 1 + 1 Verify that P 1 D P E = 1+ P E holds in this case. 1 1+ P D Given the elasticities you have just calculated, how would you interpret the fact that evening prices are higher and daytime prices are lower? PD iv. How much profit does the theater earn from each type of consumer? v. What is the consumer surplus of each type of consumer? 6
vi. Add the theater s profit to the consumer surpluses to get the total social surplus. b. Now suppose that the manager decides to adopt a two-part pricing scheme. During the day, customers must pay a cover charge for entering the theater lobby, and then a small additional ticket charge for every movie seen. He adopts a similar scheme for the evening shows. i. What is the optimal charge per movie seen? The answer is the same whether a consumer comes during the day or in the evening. (This is a logical question that requires no calculations.) ii. What is the optimal cover charge for consumers who come during the day? iii. What is the optimal cover charge for consumers who come in the evening? iv. How much profit does the theater earn from each type of consumer? 7
v. What is the consumer surplus of each type of consumer? (There is very little math to do here. Think logically and the answer will be obvious.) vi. Add the theater s profit to the consumer surpluses to get the total social surplus. c. How do the two pricing schemes compare? t Which is best for the theater? t Which is best for consumers? t Which is best for efficiency? 8
3. Consider the same monopolist as in question 1. t He has two sets of customers, group 1 and group 2. The inverse demand for group 1 (high demand consumers) may be described by P 1 = 200? Q 1 where P 1 is the price that they pay and Q 1 is the amount that they consume. For group 2 (the low demand consumers), the inverse demand is t t t P 2 = 100? 2Q 2 The monopolist faces a constant marginal cost of 40. We saw in problem 1 that the monopolist could earn higher profits by discriminating between the two groups. Suppose, however that the monopolist cannot tell these two groups apart directly and cannot charge them different prices. Instead, he must craft an indirect strategy for getting the high demand consumers to reveal themselves. He decides to offer his product in two packages, one directed at the low-demand consumers, and one directed at the high-demand consumers. Assume that the inverse demand curves given above are each for individual consumers. Thus, if the price is 20, then each group 1 consumer will consume Q 1 = 200? P = 200? 20 = 180 Each consumer in group 2 will consume Q 2 = 100 2? P = 80 2 = 40 There are an equal number of group 1 and group 2 consumers. The monopolist s objective is therefore to maximize ^1 + ^2 where ^1 is his profit from each consumer in group 1 and ^2 is his profit from each consumer in group 2. A mentioned above, the monopolist s strategy is to offer packages. Package 2 will be bought by low-demand consumers (group 2), at a price just low enough that they are indifferent about buying or not buying (so that consumer surplus is zero). Package 1 will be bought by high-demand consumers (group 1), at a price just low enough that group 1 consumers won t want to buy the package intended for group 2 (who pay a smaller price per unit). The idea is that, if your demand is higher, you are forced to pay more per item, since if you want to pay less per item, you will be unable to buy as much as you want. Thus, high demand consumers will reveal themselves and the monopolist can earn high profits. 9
a. Package 2 (for group 2) gives consumers 10 units of the good. What is the most that group 2 consumers will pay for 10 units? (You need to integrate under the demand curve from 0 to 10). The price p 2 of package 2 will be precisely this number you just calculated. b. Suppose that a consumer in group 1 buys package 2. What is that consumer s surplus? (Again, integrate under the demand curve. Then subtract the purchase price.) c. Package 1 gives consumers 80 units of the good. What would be the most a consumer in group 1 would pay for 80 units (rather than buying nothing at all)? 10
d. Now, taking into account that consumers in group 1 could buy package 2 if they wanted to, what is the most that the monopolist can charge for package 1? This will be a type 1 consumer s utility from consuming 80 units, minus his payoff from buying package 2. All you have to do is subtract the two numbers you just calculated in (b) and (c). e. Calculate the monopolist s total profit if there is one consumer of each type. Verify that it is higher than the monopolist s profit when he does not use package pricing. (You derived this profit level in 1.e.) 11
4. Two fellow criminals are caught by police and interrogated in separate rooms. Each has two choices: collaborate with the police to get a good plea bargain, or stay mum and remain loyal to his friend. Thus, each has the strategy set: collaborate, stay mum. If both stay mum, then they will each be convicted of a minor offense, and will each stay in jail for a year. If one refuses to collaborate, but the other does collaborate, then the collaborator will get off with probation, but the other will go to jail for 10 years. If both collaborate, then each will get 4 years. We can express this game in normal form as follows: Criminal 1 a. Each criminal has a dominant strategy. i. Which strategy is dominant? Criminal 2 collaborate stay mum collaborate -4,-4 0,-10 stay mum -10,0-1,-1 ii. Why is it dominant? b. Describe the equilibrium in dominant strategies: i. What is the equilibrium? (What are the strategies chosen?) ii. What are the equilibrium payoffs? c. There is another strategy combination that makes both players better off. i. Which is it? ii. Explain why that combination won t happen? 12