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Course: PBAF 516, Fall 2009
School: Washington
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School Evans of Public Affairs, Mark Long PBAF 516: Microeconomic Policy Analysis Problem Set 5 -- Answers Market Failures Monopolies 1. Suppose that a monopolist faces the following demand curve: Qd = 100 2P. This monopolists cost and marginal cost curves are the following: Cost = 75+10Q+2Q2, MC = 10 + 4Q. a. Rewrite the demand curve as P = Qd 100 = 2P P = 50 0.5Qd b. Graph the demand curve. P MC 50 10...

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School Evans of Public Affairs, Mark Long PBAF 516: Microeconomic Policy Analysis Problem Set 5 -- Answers Market Failures Monopolies 1. Suppose that a monopolist faces the following demand curve: Qd = 100 2P. This monopolists cost and marginal cost curves are the following: Cost = 75+10Q+2Q2, MC = 10 + 4Q. a. Rewrite the demand curve as P = Qd 100 = 2P P = 50 0.5Qd b. Graph the demand curve. P MC 50 10 MR D 100 Q 50 c. Derive an equation for the MR curve. MR = 50 Qd d. Graph the MR curve. (See above) e. Graph the MC curve. (See above) f. Compute the profit maximizing Q. Set MR = MC, and solve for Q: 50 Q = 10 + 4Q 40 = 5Q Q=8 g. Compute the profit maximizing P. Use the demand curve and find the P where Q=8: P = 50 0.5*8 = 46 h. Compute AC at the profit maximizing Q. Is the firm earning positive profit? AC = Cost / Q = (75+10Q+2Q2 )/ Q = 75/Q + 10 + 2Q = 75/8 + 10 + 2*8 = 35.38. Since this is less than P, the firm is earning a positive profit. i. Compute the firms profit maximizing revenue, cost, and profit. Revenue = P * Q = 46 * 8 = 368 Cost = 75+10Q+2Q2 = 75 + 10*8 + 2*8*8 = 283 Profit = Revenue Cost = 368 283 =85. Evans School of Public Affairs, Mark Long PBAF 516: Microeconomic Policy Analysis Profit also can be found using the following formula: Profit = (P-AC)*Q = (46 35.38)*8 = 85. j. Compute the firms MC at the profit maximizing Q. MC = 10 + 4Q = 10 + 4*8 = 42. k. Compute the firms percentage mark-up: (P MC)/P (P MC)/P = (46 42) / 46 = 4 / 46 = 8.7% l. Increase Q by one above the profit maximizing Q, compute the new price, and compute the demand elasticity. P = 50 -0.5*(8+1) = 45.5. = %Qd / %P = [(9-8)/8] / [(46-45.5)/46] = [1/8] / [0.5/46] = 0.125 / 0.011 = 11.5 m. Show that (P MC)/P = 1/. 1/ = 1/11.5 = 8.7% n. What is the efficient quantity? The efficient quantity is where MC = MB (i.e. MC = D). 10+4Q = 50 0.5Q 4.5Q = 40 Q = 8.89 o. What is the deadweight loss created by the DWL monopolist. is defined by the triangle horizontally between Qm and Q* and vertically between MC and MB. At Qm, MB=46 and MC=42. DWL = * (MB - MC) * (Efficient Quantity Monopolist Quantity) = * (46 42) * (8.89 8) = * (4) * (0.89) = 17.78 p. If the government required the monopolist to charge consumers a price of 45.56, would this increase or decrease DWL (hint: think carefully about this one). At a price of 45.56, consumers would demand: 45.56 = 50 0.5Qd Qd =8.88 (i.e., consumers would demand approximately the efficient amount. If this were the end of the story, we would say that deadweight loss is eliminated by the government policy. However, we have to consider the firm's profit at this price: Revenue = P * Q = 45.56 * 8.88 = 404.57 Cost = 75+10Q+2Q2 = 75 + 10*8.88 + 2*8.88*8.88 = 321.51 Profit = Revenue Cost = 404.57 321.51 = 83.06. Evans School of Public Affairs, Mark Long PBAF 516: Microeconomic Policy Analysis In this case, the firm still earns profit, so it would remain in business. It is possible to construct an example where the firm would choose to shut-down at the regulated price in this case, deadweight loss would be increased by the government policy. Browning and Zupan Problems: 11.11 Refer to the figure below: Under unregulated monopoly, output is Qm and price is Pm. With a price ceiling of Pgov, the demand curve the monopolist faces becomes PgovED, and the MR curve becomes PgovEGMR. The new profit-maximizing rate of output is Qgov and the price is Pgov, so output increases and price falls. Output need not always increase with a price ceiling, however. If the price ceiling is set below X the sellers output will fall. P A MC Pm Pgov X G D Qm Qgov 11.13 (P MC)/P = 1/ (P 10)/P = 1/5 P - 10 = 0.2P 0.8P = 10 P = 10/0.8 = \$12.50 MR E 11.21 No. If ATC is at a minimum at \$50, then MC also equals \$50 at that output. If MR=\$60, then MR>MC and the firm should increase its output.
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