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Chapter 12: Price Discrimination 1. A price-discriminating monopolist faces the following inverse demand functions: In Market One it is P1 = 20-Q1 where P1 is the price charged in Market 1 and Q1 is the quantity demanded in Market one. In Market Two it is P2 = 15-1.5Q2 where P2 is the price charged in Market 2 and Q2 is the quantity demanded in Market Two. Marginal cost is constant at $5. Find the profit-maximizing quantity and price charged in each market. Calculate profit in each market and joint profit. What would this firm’s price, quantity and profit be if it were constrained to charge the same price to all consumers? Show this outcome on a graph. 2. A price-discriminating monopolist faces the following inverse demand functions: In Market One it is P1 = 80-Q1 and in Market Two it is P2 = 60-Q2 . Marginal cost is constant at $10. Consumers in market two can resell the good to consumers in market one at a cost of $4 per unit. Find the profit-maximizing quantity and price charged in each market subject to the resale constraint. 3. Consider the following inverse demand function: P = 10 – Q. Marginal cost is constant at $2. a. Find the TR and MR functions under perfect-price discrimination. What is profit-maximizing quantity? What is profit? b. The perfect-price discrimination outcome can be attained by an all-or-nothing packaging scheme. Explain how. 4. Consider the following inverse demand function: P =400-2Q. Marginal and average cost are constant at $200. a. Find the monopoly quantity, price and profit. Draw a graph illustrating this outcome. b. If this firm employs a two-part tariff scheme, what will the user charge for this good be? Explain. Your explanation should include a calculation of the profit-maximizing "entrance fee" this charges. What is monopoly profit under this pricing scheme? c. Calculate consumer surplus, producer surplus, and any deadweight loss for (a) and (b). 5. a. What conditions make price discrimination possible? b. A monopoly can sell its good in the US, where the elasticity of demand is -2, and in South Korea, where the elasticity of demand is -4. Its marginal cost is $10. At what price does the monopoly sell its good in each country if resales are not possible? ANSWERS 1. The firm maximizes profit by producing where MR =MC in EACH market. First, find the MR function for each market: TR1=20Q1-Q12 which gives us MR1 = 20-2Q1. For market two: TR2=15Q2-1.5Q22 which gives us MR2 = 15-3Q2. Second, set MR equal to MC and solve for quantities. Market One: 20-2Q1 = 5 which gives us Q1 = 7.5, and Market Two: 15-3Q2 = 5, which gives us Q2= 3.33. Third, plugging these quantities into the demand function gives us prices. P1 = 20-7.5 = $12.5 and P2 = 15-1.5(3.33)= $10. 1= (12.5-5)(7.5) = $56.25. 2=(10-5)(3.33) = $16.67. joint = $56.25+$16.67=$72.92. First, we need to find the market demand curve by horizontally adding the individual demand curves. To do this we must solve for quantity for each buyer: Q1=20-P and Q2 = 10-2/3P. Next, Q = Q1+ Q2 =(20-P)+(10-2/3P)=30-(5/3)P. After doing this, we re-solve for P so that P = 18-(3/5)Q. This allows us to write the market MR equation as MR = 18-(6/5)Q. (NOTE: Given the constraint that the price must be the same for both groups of buyers, you CANNOT get the market MR curve be horizontally summing the individual MR curves—doing this implicitly assumes that the MR in each market will be the same at the profit-maximizing solution. But this would not be true if the price is the same in each market. That is why you must first derive the market demand curve and then get the MR curve from that.) Second, set MR = MC and solve for Q. 18-(6/5)Q=5, and so Q 10.833. Third, Plug Q = 10.833 into market demand curve to get price. P =18-(3/5)(10.833) $11.50. Fourth, find profit. Profit = (11.50-5)(10.833) = $70.42. 2. 1st. Set up the Lagrangian. L= P1Q1 + P2Q2 – MC1Q1-MC2Q2 + [P1-P2-t] where t = cost of reselling good. L = (80-Q1)(Q1) + = (60-Q2)(Q2) - 10Q1 – 10Q2 + [(80-Q1) – (60-Q2)-4] = 70Q1 - Q12 + = 50Q2-Q22 + [16-Q1 +Q2] nd 2 . Find the first order conditions. i. L/Q1 = 70 – 2Q1 - = 0. = 70 – 2Q1 . ii. L/Q2 = 50 –2Q2 + = 0. = -50 + 2Q2 iii. L/ = 16-Q1 + Q2 = 0 Q1 = 16 + Q2. 3rd. i and ii imply: 70 – 2Q1 = -50+ 2Q2 or Q1 = 60 - Q2. Using this along with iii, gives us 16 + Q2 = 60 - Q2. 2Q2 = 44 or Q2 = 22. 4th. Now that we have Q2, we can find Q1, P1 and P2. Q1 = 16 + Q2 = 38. P1 = 80-Q1 = 80- 38 = $42. P2 = 60-Q2 = 60-22= $38. 3.a. Under perfect price discrimination, total revenue equals the entire under the demand curve up to the quantity sold, so TR = ∫10-Q dQ = 10Q –0.5Q2 . MR = dTR/dQ = 10 – Q. Setting MR = MC 10 – Q = 2, or Q =8. Profit is TR – TC = 10(8) –0.5(8)2 – (2)(8) = $32. b. Require the consumer to purchase the good in a package of 8 units or not at all. The all-or-nothing price equals average revenue at a quantity of 8 units AR = 10 – 0.5Q = 10- 0.5(8) = $6. So the price of a package of 8 units is $6*8= $48. Profit = $48 - $16 = $32. 4a. MR = 400-4Q. a. 400-4Q=200 which gives us Q = 50. For price, P=400-2(50)=$300, and for profit we have (300-200)(50)=$5,000. b. The firm should set the user charge (P) equal to MC (which is 200) and charge an entrance fee equal to the consumer surplus that would exist at P=200. Profit = entrance fee+(P-AC)(Q)= 0.5(400-200)(100) + (200-200)(100)= $10,000. c. Single-P monopoly Two-Part Tariff Monopoly Change Consumer Surplus 0.5(400-300)(50)=$2,500 0 -2,500 Producer Surplus (300-200)(50)= $5,000 $10,000 +5,000 Social Total $7,500 $10,000 +2,50 0 The social total is increased under a two-part tariff. In fact, there is no deadweight loss under a two-part tariff (for identical consumers). 5. a. First, the firm must have some degree of market power, b. Resale of the good must be difficult, and c. The firm must be able to identify (or the consumers have to implicitly identify themselves) the different groups of buyers that it wants to price discriminate against. b. PUS = MC/[1+ (1/eUS)] = $10/[0.5] = $20 and PSK = $10/[0.80] = $13.33 Question 2 :Same price to all consumers. $20 P=20-Q (for P>$15) $15 P=18-(3/5)Q (for P<$15) $12 P=$11.50 Dmkt. $10 MR=18-(6/5)Q $5 MC MRmkt 5 10.833 Q Note: Because there is a kink in the market demand curve, there will be a discontinuity in the market MR curve. $ per unit Question 3. Perfect Price Discrimination 10 Pall-or-nothing = $6 $2 MC D = MR 8 AR Q