Ans homework 5 EE 311 1. Suppose that Intel has a monopoly in the market for microprocessors in Brazil. During the year 2005, it faces a market demand curve given by P = 9 - Q, where Q is millions of microprocessors sold per year. Suppose you know nothing about Intel’s costs of production. Assuming that Intel acts as a profit-maximizing monopolist, would it ever sell 7 million microprocessors in Brazil in 2005?
If demand is P 9 Q , then MR 9 2Q . If the firm sets Q 7 , then MR 5 . At this point, if the firm lowered its output it would increase total revenue, and with the lower level of output total cost would fall. Thus, decreasing output would increase profit. Therefore, a profit-maximizing monopolist facing this demand curve would never choose Q 7 .
2. A monopolist faces a demand curve P = 210 - 4Q and initially faces a constant marginal cost MC = 10. a) Calculate the profit-maximizing monopoly quantity and compute the monopolist’s total revenue at the optimal price. b) Suppose that the monopolist’s marginal cost increases to MC = 20. Verify that the monopolist’s total revenue goes down. c) Suppose that all firms in a perfectly competitive equilibrium had a constant marginal cost MC = 10. Find the long-run perfectly competitive industry price and quantity. d) Suppose that all firms’ marginal costs increased to MC = 20. Verify that the increase in marginal cost causes total industry revenue to go up.
a)
With demand P 210 4Q , MR 210 8Q . Setting MR MC implies 210 8Q 10
Q 25 With Q 25 , price will be P 210 4Q 110 . At this price and quantity total revenue will be
TR 110(25) 2,750 .
b)
If MC 20 , then setting MR MC implies 210 8Q 20
Q 23.75
At Q 23.75 , price will be P 115 . At this price and quantity total revenue will be
TR 115(23.75) 2,731.25 . Therefore, the increase in marginal cost will result in lower total revenue for the firm.
c) Competitive firms produce until P = MC, so in this case we know the market price would be P = 10 and the market quantity would be: 210 4Q 10
Q 50 d) In this case, the market price will be P MC = 20, implying that the industry quantity is given by 210 4Q 20
Q 47.50 At this quantity, price will be P 20 . When MC 10 , total industry revenue is 10(50) 500 . With MC 20 , total industry revenue is 20(47.50) 950 . Thus, total industry revenue increases in the perfectly competitive market after the increase in marginal cost.
3. Suppose a monopolist has an inverse demand function given by P = 100Q-1/2. What is the monopolist’s optimal markup of price above marginal cost?
Remember that the demand elasticity in a constant elasticity demand function is the exponent on P when the demand function is written in the regular form, i.e. Q f (P). We can manipulate the inverse demand function to get the regular demand function, Q 10,000P 2 . P MC 1 . So the optimal This implies that the demand elasticity is –2. Therefore, P 2 percentage mark-up of price over marginal cost is ½, or 50 percent. 4. Imagine that Gillette has a monopoly in the market for razor blades in Mexico. The market demand curve for blades in Mexico is P = 968 - 20Q, where P is the price of blades in cents and Q is annual demand for blades expressed in millions. Gillette has two plants in which it can produce blades for the Mexican market: one in Los Angeles and one in Mexico City. In its L.A. plant, Gillette can produce any quantity of blades it wants at a marginal cost of 8 cents per blade. Letting Q1 and MC1 denote the output and marginal cost at the L.A. plant, we have MC1(Q1) = 8. The Mexican plant has a marginal cost function given by MC2(Q2) = 1 + 0.5Q2. a) Find Gillette’s profit-maximizing price and quantity of output for the Mexican market overall. How will Gillette allocate production between its Mexican plant and its U.S. plant? b) Suppose Gillette’s L.A. plant had a marginal cost of 10 cents rather than 8 cents per blade. How would your answer to part (a) change?
a) Profit-maximizing firms generally allocate output among plants so as to keep marginal costs equal. But notice that MC2 < MC1 whenever 1 + 0.5Q2 < 8, or Q2 < 14. So for small levels of output, specifically Q < 14, Gillette will only use the first plant. For Q > 14, the cost-minimizing approach will set Q2 = 14 and Q1 = Q – 14. Suppose the monopolist’s profit-maximizing quantity is Q > 14. Then the relevant MC = 8, and with MR 968 40Q we have 968 40Q 8
Q 24 Since we have found that Q > 14, we know this approach is valid. (You should verify that had we supposed the optimal output was Q < 14 and set MR = MC2 = 1 + 0.5Q, we would have found Q > 14. So this approach would be invalid.) The allocation between plants will be Q2 = 14 and Q1 = 10. With a total quantity Q = 24, the firm will charge a price of P = 968 – 20(24) = 488. Therefore the price will be $4.88 per blade.
b) If MC 10 at plant 1, by the logic in part (a) Gillette will only use plant 2 if Q < 18. It will produce all output above Q = 18 in plant 1 at MC = 10. Assuming Q > 18, setting MR MC implies 968 40Q 10
Q 23.95 (So again, this approach is valid. You can verify that setting MR = MC2 would again lead to Q > 18.) The firm will allocate production so that Q2 = 18 and Q1 = 5.95. At Q = 23.95, price will be $4.89.
5. Market demand is P = 64 - (Q/7). A multiplant monopolist operates three plants, with marginal cost functions:
a) Find the monopolist’s profit-maximizing price and output at each plant. b) How would your answer to part (a) change if MC2 (Q2) = 4?
a) Equating the marginal costs at MCT, we have Q = Q1 + Q2 + Q3 = 0.25MCT + 0.5MCT – 1 + MCT – 6, which can be rearranged as MCT = (4/7)Q + 4. Setting MR = MC yields 64 – (2/7)*Q = (4/7)*Q + 4 or Q = 70 and P = 54. At this output level, MCT = 44, implying that Q1 = 11, Q2 = 21, and Q3 = 38.
b) In this case, using plant 3 is inefficient because its marginal cost is always higher than that of plant 2. Hence, the firm will use only plants 1 and 2. Moreover, the firm will not use plant 1 once its marginal cost rises to MC2 = 4, so we can immediately see that it will only produce 4Q1 = 4 or Q1 = 1 unit at plant 1. Its total production can be found by setting MR = MC2, yielding 64 – (2/7)*Q = 4 or Q = 210 and P = 34. So it produces Q1 = 1 unit in plant 1 and Q2 = 209 units in plant 2, while producing no units in plant 3 (i.e. Q3 = 0). 6. Suppose that a monopolist’s market demand is given by P = 100 - 2Q and that marginal cost is given by MC = Q/2. a) Calculate the profit-maximizing monopoly price and quantity. b) Calculate the price and quantity that arise under perfect competition with a supply curve P = Q/2. c) Compare consumer and producer surplus under monopoly versus marginal cost pricing. What is the deadweight loss due to monopoly? d) Suppose market demand is given by P = 180 - 4Q. What is the deadweight loss due to monopoly now? Explain why this deadweight loss differs from that in part (c).
a)
With demand P 100 2Q , MR 100 4Q . Setting MR MC implies 100 4Q .5Q
Q 22.2 (All figures are rounded.) At this quantity, price will be P 55.6 .
b)
A perfectly competitive market produces until P = MC, or 100 2Q .5Q
Q 40 At this quantity, price will be P = 20.
c) Under monopoly, consumer surplus is 0.5(100 – 55.6)(22.2) = 493. Since MC(22.2) = 11.1, producer surplus is 0.5(11.1)(22.2) + (55.6 – 11.1)(22.2) = 1111. Net benefits are 1604. (All figures are rounded.) Under perfect competition, consumer surplus is 0.5(100 – 20)(40) = 1600, and producer surplus is 0.5(20)(40) = 400. Net benefits are 2000. Therefore, the deadweight loss due to monopoly is 396. d) Now setting MR = MC gives 180 8Q 0.5Q
Q 21.2
At this quantity, price is 95.2. Consumer surplus is 0.5(100 – 95.2)(21.1) = 51 and producer surplus is 0.5(10.6)(21.2) + (95.2 – 10.6)(21.2) = 1906. Net benefits are 1957.
Setting P = MC as in perfect competition yields
180 4Q .5Q Q 40 At this quantity, price is 20. Consumer surplus is 0.5(180 – 20)(40) = 3200 and producer surplus is 0.5(20)(40) = 400. Net benefits with perfect competition are 3600. Therefore, the deadweight loss in this case is 1643.
While the competitive solution is identical with both demand curves, the deadweight loss in the first case is far greater. This difference occurs because with the second demand curve demand is less elastic at the perfectly competitive price. If consumers are less willing to change quantity as price increases toward the monopoly level, the firm will be able to extract more surplus from the market.
7. Which of the following are examples of first-degree, second-degree, or third-degree price discrimination? a) The publishers of the Journal of Price Discrimination charge a subscription price of $75 per year to individuals and $300 per year to libraries. b) The U.S. government auctions off leases on tracts of land in the Gulf of Mexico. Oil companies bid for the right to explore each tract of land and to extract oil. c) Ye Olde Country Club charges golfers $12 to play the first 9 holes of golf on a given day, $9 to play an additional 9 holes, and $6 to play 9 more holes. d) The telephone company charges you $0.10 per minute to make a long-distance call from Monday through Saturday and $0.05 per minute on Sunday. e) You can buy one computer disk for $10, a pack of 3 for $27, or a pack of 10 for $75. f) When you fly from New York to Chicago, the airline charges you $250 if you buy your ticket 14 days in advance, but $350 if you buy the ticket on the day of travel.
a) Third degree – the firm is charging a different price to different market segments, individuals and libraries. b) First degree – each consumer is paying near their maximum willingness to pay.
c) Second degree – the firm is offering quantity discounts. As the number of holes played goes up, the average expenditure per hole falls. d) Third degree – the firm is charging different prices for different segments. Business customers (M-F) are being charged a higher price than those using the phone on Sunday, e.g., family calls. e) Second degree – the firm is offering a quantity discount. f) Third degree – the airline is charging different prices to different segments. Those who can purchase in advance pay one price while those who must purchase with short notice pay a different price. 8. Suppose a profit-maximizing monopolist producing Q units of output faces the demand curve P = 20 - Q. Its total cost when producing Q units of output is TC = 24 + Q2. The fixed cost is sunk, and the marginal cost curve is MC = 2Q. a) If price discrimination is impossible, how large will the profit be? How large will the producer surplus be? b) Suppose the firm can engage in perfect first-degree price discrimination. How large will the profit be? How large is the producer surplus? c) How much extra surplus does the producer capture when it can engage in first-degree price discrimination instead of charging a uniform price?
a)
If price discrimination is impossible the firm will set MR MC . 20 2Q 2Q
Q5 At this quantity, price will be P 15 , total revenue will be TR 75 , total cost will be TC 49 , and profit will be 26 . Producer surplus is total revenue less non-sunk cost, or, in this case, total revenue less variable cost. Thus producer surplus is 75 5 50 . 2
b) With perfect first-degree price discrimination the firm sets P MC to determine the level of output. 20 Q 2Q
Q 6.67 The price charged each consumer, however, will vary. The price charged will be the consumer’s maximum willingness to pay and will correspond with the demand curve. Total revenue will be the area underneath the demand curve out to Q = 6.67 units, or 0.5(20 – 13.33)(6.67) + 13.33(6.67) = 111.16. Since the firm is producing a total of 6.67 units, total cost will be TC 68.49 . Profit is then
42.67 , while producer surplus is revenue less variable cost, or 111.16 6.672 66.67 .
c) By being able to employ perfect first-degree price discrimination the firm increases profit and producer surplus by 16.67. 9. Fore! is a seller of golf balls that wants to increase its revenues by offering a quantity discount. For simplicity, assume that the firm sells to only one customer and that the demand for Fore!’s golf balls is P = 100 - Q. Its marginal cost is MC = 10. Suppose that Fore! sells the first block of Q1 golf balls at a price of P1 per unit. a) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 20 and P1 = 80. b) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 30 and P1 = 70. c) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 40 and P1 = 60. d) Of the three options in parts (a) through (c), which block tariff maximizes Fore!’s total profits?
a) We can represent the marginal willingness to pay for each unit beyond Q1 = 20 as P = 100 – (20 + Q2) = 80 – Q2. The associated marginal revenue is then MR = 80 – 2Q2, so the profit maximizing second block is MR = MC: 80 – 2Q2 = 10. Thus Q2 = 35 and P2 = 80 – 35 = 45. So the firm sells the first 20 units at a price of $80 apiece, while the firm sells any quantity above 20 at $45 apiece. The firm’s total profit will be (80 – 10)*20 + (45 – 10)*35 = $2625. b) The marginal willingness to pay for each unit beyond Q1 = 30 is P = 70 – Q2. So MR = 70 – 2Q2 and we have MR = MC: 70 – 2Q2 = 10. Thus Q2 = 30 and P2 = 40. The firm’s total profit will be (70 – 10)*30 + (40 – 10)*30 = $2700. c) The marginal willingness to pay for each unit beyond Q1 = 40 is P = 60 – Q2. So MR = 60 – 2Q2 and we have MR = MC: 60 – 2Q2 = 10. Thus Q2 = 25 and P2 = 35. The firm’s total profit will be (60 – 10)*40 + (35 – 10)*25 = $2625. d) The option in part (b) yields the highest profits, of $2700. 10. Suppose that Acme Pharmaceutical Company discovers a drug that cures the common cold. Acme has plants in both the United States and Europe and can manufacture the drug on either continent at a marginal cost of 10. Assume there are no fixed costs. In Europe, the demand for the drug is QE = 70 - PE, where QE is the quantity demanded when the price in Europe is PE. In the United States, the demand for the drug is QU = 110 - PU, where QU is the quantity demanded when the price in the United States is PU. a) If the firm can engage in third-degree price discrimination, what price should it set on each continent to maximize its profit? b) Assume now that it is illegal for the firm to price discriminate, so that it can charge only a single price P on both continents. What price will it charge, and what profits will it earn? c) Will the total consumer and producer surplus in the world be higher with price discrimination or without price discrimination? Will the firm sell the drug on both continents?
a) With third-degree price discrimination the firm should set MR MC in each market to determine price and quantity. Thus, in Europe setting MR MC 70 2QE 10
QE 30 At this quantity, price will be PE 40 . Profit in Europe is then
E ( PE 10)QE (40 10)30 900 . Setting MR MC in the US implies 110 2QU 10 QU 50 At this quantity price will be PU 60 . Profit in the US will then be
U ( PU 10)QU (60 10)50 2500 . Total profit will be 3400 .
b) If the firm can only sell the drug at one price, it will set the price to maximize total profit. The total demand the firm will face is Q QE QU . In this case Q 70 P 110 P
Q 180 2 P The inverse demand is then P 90 0.5Q . Since MC 10 , setting MR MC implies
90 Q 10 Q 80 At this quantity price will be P 50 . If the firm sets price at 50, the firm will sell QE 20 and
QU 60 . Profit will be 50(80) 10(80) 3200 .
c) The firm will sell the drug on both continents under either scenario. If the firm can price discriminate, total consumer surplus will be 0.5(70 – 40)30 + 0.5(110 – 60)50 = 1700 and producer surplus (equal to profit) will be 3400. Thus, total surplus will be 5100. If the firm cannot price discriminate, consumer surplus will be 0.5(70 – 50)20 + 0.5(110 – 50)60 = 2000 and producer surplus will be equal to profit of 3200. Thus, total surplus will be 5200.
11. You are the only European firm selling vacation trips to the North Pole. You know only three customers are in the market. You offer two services, round trip airfare and a stay at the Polar Bear Hotel. It costs you 300 euros to host a traveler at the Polar Bear and 300 euros for the airfare. If
you do not bundle the services, a customer might buy your airfare but not stay at the hotel. A customer could also travel to the North Pole in some other way (by private plane), but still stay at the Polar Bear. The customers have the following reservation prices for these services:
a) If you do not bundle the hotel and airfare, what are the optimal prices PA and PH, and what profits do you earn? b) If you only sell the hotel and airfare in a bundle, what is the optimal price of the bundle PB, and what profits do you earn? c) If you follow a strategy of mixed bundling, what are the optimal prices of the separate hotel, the separate airfare, and the bundle (PA, PH, and PB, respectively) and what profits do you earn?
a) Without bundling, the best the firm can do is set the price of airfare at $800 and the price of hotel at $800. In each case the firm attracts a single customer and earns profit of $500 from each for a total profit of $1000. The firm could attract two customers for each service at a price of $500, but it would earn profit of $200 on each customer for a total of $800 profit, less profit than the $800 price. b) With bundling, the best the firm can do is charge a price of $900 for the airfare and hotel. At this price the firm will attract all three customers and earn $300 profit on each for a total profit of $900. The firm could raise its price to $1000, but then it would only attract one customer and total profit would be $400. Notice that with bundling the firm cannot do as well as it could with mixed bundling. This is because while a) the demands are negatively correlated, a key to increasing profit through bundling, b) customer 1 has a willingness-topay for airfare below marginal cost and customer 3 has a willingness-to-pay for hotel below marginal cost. The firm should be able to do better with mixed bundling c) Because customer 1 has a willingness-to-pay for airfare below marginal cost and customer 3 has willingness-to-pay for hotel below marginal cost, the firm can potentially earn greater profits through mixed bundling. In this problem, if the firm charges $800 for airfare only, $800 for hotel only, and $1000 for the bundle, then customer 1 will purchase hotel only, customer 2 will purchase the bundle, and customer 3 will purchase airfare only. This will earn the firm $1400 profit, implying that mixed bundling is the best option in this problem. 12. You operate the only fast-food restaurant in town, selling burgers and fries. There are only two customers, one of whom is on the Atkins diet and the other on the Zone diet, whose willingness to pay for each item is displayed in the following table. For simplicity, assume you have zero fixed and marginal costs for each item.
a) If x = 1 and you do not bundle the two products, what are your profit-maximizing prices PB and PF? Calculate total surplus under this outcome. b) Now assume only that x > 0. Instead, suppose that you hired an economist who tells you that the profit-maximizing bundle price (for a burger and fries) is $8, while if you sold the items individually (and did not offer a bundle) your profit-maximizing price for fries would be greater than $3. Using this information, what is the range of possible values for x?
a) You should sell two burgers for PB = 5, and one order of fries for PF = 3. Total surplus is then PS + CS = (10 + 3) + (3 + 0) = 16. b) In order for the profit-maximizing bundle price to be $8, it must be true that 8 + x < 2*8, i.e. that x < 8. In order for the profit-maximizing price of fries to be greater than $3, it must be true that x > 2*3, or x > 6. Thus, we know that 6 ≤ x < 8.
