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Problem Set #2 Micro II – Spring Term 2010 SUGGESTED SOLUTIONS 1. (Third Degree Price Discrimination) Suppose that for some good, the marginal cost of production is $5. Demand comes from two sources, senior citizens and the rest of the population. Senior citizen demand is given by the inverse demand function Pseniors = 15 – Q/500. Demand by the rest of the population is given by the inverse demand function Prest = 20 – Q/2000. a. If a monopolist facing these two demands must set a single price for the good, what price would it optimally set? What would be its profit? For setting single price, total demand curve has to be derived. Q = Qs + QR Qs = (15 – Ps )500 Qs = 7500 – 500Ps QR = 40000 – 2000PR Since Q = QS + QR and Ps = PR Q = 7500 – 500P + 40000 – 2000P Q = 47500 – 2500P P as a function of Q P = 47500/2500 – Q/2500 = 19 – Q/2500 MR = 19 – 2Q/2500 = MC = 5 Q = 17500 and putting this in P = 19 – 17500/2500 = $12 Profits = (P – MC) * Q = (12 – 5)*17500 = $1,22,500 b. Suppose this monopolist could set two prices, one for seniors and another for regulars. What prices maximize profit and what is that level of profit? The monopolist can set different prices by engaging in third-degree price discrimination. MRS = 15 – Qs/250 and MRR = 20 – QR/1000 and MC = 5 1. Total output should be divided between groups so that MR for each group is equal 2. Total output is chosen so that MR for each group of consumers is equal to the MC of production MR for each group will be equated to MC 15 – Qs /250 = 5 or QS = 2500 20 – QR/1000 = 5 or QR = 15000 PS = 15 – 2500/500 = $20 PR = 20 – 15000/2000 = $12.5 Profits = 20 * 2500 + 12.5 * 15000 – 17500 * 5 = $1,50,000 NOTE: DO CHECK IF THE TWO CONDITIONS ARE BEING FOLLOWED 2. (Third Degree Price Discrimination) Suppose a firm sells to students and others at a single price of $10 per unit. At this price, it sells 10,000 units in total; 2,000 to students and 8,000 to the others (non-students). At the price of $10, demand by students has elasticity -3, while demand by the others has elasticity -1.5. a. Suppose the firm decided to raise the price it charges non-students by $0.10. At the same time, it will lower the price facing students. If it wants to lower the price charged to students so that it sells (approximately) the same 10,000 units (so that the decrease in demand by nonstudents is balanced by an increase in demand by students), by how much should it decrease its price to students? What will be the new price charged to students? Hint: Use the formula for elasticity of demand. ED = ∆Q/∆P * P/Q; P1 = Students and P2 = Non-students 3 = ∆Q/∆P * 10/2000 ∆Q/∆P = 600 Q = a – 600P a = 2000 + 6000 = 8000 Q1 = 8000 – 600P1 1.5 = ∆Q/∆P * 10/8000 ∆Q/∆P = 1200 Q = a – 1200 P a = 8000 + 12000 = 20000 Q2 = 20000 – 1200P2 If P2 = 10.1 Q2 = 20000 – 1200 * 10.1 = 7880 10000 – 7880 = 2120 = Q1 P1 = 8000/600 – 2120/600 = $9.8 b. What will be the approximate impact on the firm’s profits if it simultaneously raises its price to non-students to $10.10 and lowers the price charged to students as calculated in part a? Hint: Use MR = P(1+1/ed) The output should be divided such that MR1 = MR2 MR1 = 10.1 ( 1 – 1/1.5) = 10.1 * 1/3 = 3.367 MR2 = 9.8 ( 1 – 1/3) = 9.8 * 2/3 = 6.53 Since the current output distribution does not equate MR1 to MR2, it can be concluded that profits will be lower. Prices are not optimal c. Note that we say nothing in this problem about MC. Why don’t you need to know this to answer part b? Equating MR1 and MR2 gives the following relationship that must hold for prices The higher price will be charged to consumer with the lower demand elasticity P1 ( 1 1 E 2 ) P2 ( 1 1 E1 ) 3. You are selling two goods to a market consisting of three customers with reservation prices as follows: Consumer A B C Good 1 10 40 70 Good 2 70 40 10 a. Graph the reservations prices. Are demands negatively or positively correlated? Are they perfectly correlated? b. Suppose the MC1 = MC2 =20. Compute the optimal prices and profits for: i) selling the goods individually, ii) pure bundling, iii) mixed bundling. i) 70 and 70 140 – 40 = 100 ii) 80*3 – 20*6 = 120 iii) 69.95*2 + 80 – 20*4 = 139.9, Therefore go for mixed bundling c. In part b, which was the most profitable strategy? Why? Since the MC are significant, mixed bundling should be a more profitable technique than others d. Suppose MC1 = MC2 = 0. Would you change your pricing strategy (as compared to your answer in part b)? Why or why not? i) P1 = P2 = 40 4 * 40 = $160 ii) Same as in a(ii) except that MC =0 so profits are $240 iii) 69.95*2 + 80 = $219.9, Therefore go for Pure Bundling. This was predicted because MC =0 4. Sal’s satellite company broadcasts TV to subscribers in Los Angeles and New York. The demand functions for each of these two groups are QNY = 60 – 0.25PNY QLA = 100 – 0.50PLA where Q is in thousands of subscriptions per year and P is the subscription price per year. The cost of providing Q units of service is given by C = 1,000 + 40Q where Q = QNY + QLA. a. What are the profit-maximizing prices and quantities for the New York and Los Angeles markets? We know that a monopolist with two markets should pick quantities in each market so that the marginal revenues in both markets are equal to one another and equal to marginal cost. Marginal cost is $40 (the slope of the total cost curve). To determine marginal revenues in each market, we first solve for price as a function of quantity: PNY = 240 - 4QNY and PLA = 200 - 2QLA. Since the marginal revenue curve has twice the slope of the demand curve, the marginal revenue curves for the respective markets are: MRNY = 240 - 8QNY and MRLA = 200 - 4QLA. Set each marginal revenue equal to marginal cost, and determine the profitmaximizing quantity in each submarket: 40 = 240 - 8QNY, or QNY = 25 and 40 = 200 - 4QLA, or QLA = 40. Determine the price in each submarket by substituting the profit-maximizing quantity into the respective demand equation: PNY = 240 - (4)(25) = $140 and PLA = 200 - (2)(40) = $120. 1. Total output should be divided between groups so that MR for each group is equal TRUE: QNY = 25, MRNY = 40 and QLA = 40, MRLA = 40, MRLA = MRNY 2. Total output is chosen so that MR for each group of consumers is equal to the MC of production: TRUE: MRLA = MCT and MRNY = MCT b. As a consequence of a new satellite that the Pentagon recently deployed, people in Los Angeles receive Sal’s New York broadcasts, and people in New York receive Sal’s Los Angeles broadcasts. As a result, anyone in New York or Los Angeles can receive Sal’s broadcasts by subscribing in either city. Thus Sal can charge only a single price. What price should he charge, and what quantities will he sell in New York and Los Angeles? Given this new satellite, Sal can no longer separate the two markets, so he now needs to consider the total demand function, which is the horizontal summation of the LA and NY demand functions. Above a price of 200 (the vertical intercept of the demand function for Los Angeles viewers), the total demand is just the New York demand function, whereas below a price of 200, we add the two demands: QT = 60 – 0.25P + 100 – 0.50P, or QT = 160 – 0.75P. Rewriting the demand function results in P 160 1 Q. 0.75 0.75 2 Now total revenue = PQ = (213.3 – 1.3Q)Q, or 213.3Q – 1.3Q , and therefore, MR = 213.3 – 2.6Q. Setting marginal revenue equal to marginal cost to determine the profitmaximizing quantity: 213.3 – 2.6Q = 40, or Q = 65. Substitute the profit-maximizing quantity into the demand equation to determine price: 65 = 160 – 0.75P, or P = $126.67. Although a price of $126.67 is charged in both markets, different quantities are purchased in each market. QNY 60 0.25126.67 28.3 and QLA 100 0.50126.67 36.7. Together, 65 units are purchased at a price of $126.67 each. c. In which of the above situations, (a) or (b), is Sal better off? In terms of consumer surplus, which situation do people in New York prefer and which do people in Los Angeles prefer? Why? Sal is better off in the situation with the highest profit. Under the market condition in 4a, profit is equal to: = QNYPNY + QLAPLA - (1,000 + 40(QNY + QLA)), or = (25)($140) + (40)($120) - (1,000 + 40(25 + 40)) = $4,700. Under the market conditions in 4b, profit is equal to: = QTP - (1,000 + 40QT), or = (126.67)(65) - (1,000 + (40)(65)) = $4633.33. Therefore, Sal is better off when the two markets are separated. Consumer surplus is the area under the demand curve above price. Under the market conditions in 4a, consumer surpluses in New York and Los Angeles are: CSNY = (0.5)(240 - 140)(25) = $1250 and CSLA = (0.5)(200 - 120)(40) = $1600. Under the market conditions in 4b the respective consumer surpluses are: CSNY = (0.5)(240 – 126.67)(28.3) = $1603.67 and CSLA = (0.5)(200 – 126.67)(36.7) = $1345.67. The New Yorkers prefer 4b because the equilibrium price is $126.67 instead of $140, thus giving them a higher consumer surplus. The customers in Los Angeles prefer 4a because the equilibrium price is $120 instead of $126.67. 5. A cable TV company offers, in addition to its basic service, two products: a Sports Channel (Product 1) and a Movie Channel (Product 2). Subscribers to the basic service can subscribe to these additional services individually at the monthly prices P1 and P2, respectively, or they can buy the two as a bundle for the price PB, where PB < P1 + P2. (They can also forego the additional services and simply buy the basic service.) The company’s marginal cost for these additional services is zero. Through market research, the cable company has estimated the reservation prices for these two services for a representative group of consumers in the company’s service area. These reservation prices are plotted (as x’s) in Figure 11.16, as are the prices P1, P2, and PB that the cable company is currently charging. The graph is divided into regions, I, II, III, and IV. Figure 11.16 a. Which products, if any, will be purchased by the consumers in region I? In region II? In region III? In region IV? Explain briefly. Product 1 = sports channel. Product 2 = movie channel. Region Purchase Reservation Prices I nothing r1 < P1, r2 < P2, r1 + r2 < PB II sports channel r1 > P1, r2 < PB - P1 III movie channel r2 > P2, r1 < PB - P2 IV both channels r1 > PB - P2, r2 > PB - P1, r1 + r2 > PB To see why consumers in regions II and III do not buy the bundle, reason as follows: For region II, r1 > P1, so the consumer will buy product 1. If she bought the bundle, she would pay an additional PB - P1. Since her reservation price for product 2 is less than PB - P1, she will choose only to buy product 1. Similar reasoning applies to region III. Consumers in region I purchase nothing because the sum of their reservation values are less than the bundling price and each reservation value is lower than the respective price. In region IV the sum of the reservation values for the consumers are higher than the bundle price, so these consumers would rather purchase the bundle than nothing. To see why the consumers in this region cannot do better than purchase either of the products separately, reason as follows: since r1 > PB - P2 the consumer is better off purchasing both products than just product 2, likewise since r2 > PB - P1, the consumer is better off purchasing both products rather than just product 1.