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HW Set 3 Solns

# HW Set 3 Solns - INTERMEDIATE MICROECONOMIC THEORY ECON301...

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INTERMEDIATE MICROECONOMIC THEORY ECON301 Fall 2010 Third Problem Set (Answers) 1. Suppose a monopolist produces its output subject to the following total cost equation: C = 100 + 10Q + 2Q 2 , and the demand curve it faces is p = 90 - 2Q. Determine the price, quantity, and profit for this firm. Answer: Set marginal cost equal to marginal revenue => MC = 10 + 4Q = MR = 90 – 4Q => 8Q = 80 and Q* = 10. P* = 90 – 2Q = 90 – 20 = 70. Profit equals revenue minus cost => PQ – C = 70(10) – (100 + 10(10) + 2(10) 2 ) = 700 – (100 + 100 + 200) = 700 – 400 = 300. 2. Assume that the inverse demand equation a monopoly faces is P = 40 - Q, and marginal cost is constant at \$5 per unit. a. Draw a graph showing the demand curve, marginal revenue curve and marginal cost curve. b. What are the monopolist’s profit-maximizing output and price? c. Calculate the consumer surplus, producer surplus and social welfare at the monopoly output and price. d. What is the value of Lerner’s index at the profit-maximizing price and quantity? Show your work. Answers: The demand curve has a P-intercept of 40 and a Q-intercept of 40. The marginal revenue equation is MR = 40 – 2Q. The MR curve has a MR-intercept of 40 and a Q-intercept of 40. Marginal cost is a horizontal line at MC = 5. The firm maximizes profits by setting MR equal to MR => 40 – 2Q = 5 => Q* = 35/2 = 17.5. The monopolist’s price is P* = 40 – 17.5 = 22.5. Consumer surplus equals (40 – 22.5)(17.5)/2 = 153.13. Producer surplus is (22.5 – 5)(17.5) = 306.25. The Lerner’s index equals (P – MC)/P = (22.5 – 5)/22.5 = 17.5/22.5 = 0.778. Consumer surplus equals (40 – 22.5)(17.5)/2. Producer surplus equals (22.5 – 5)(17.5) = (17.5) 2 . 1

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3. Assume that a firm has a monopoly. Its demand curve is given by the equation P = 60 – Q. It produces its output subject to the following short-run cost equation: C = Q 2 + 20. a. Draw a graph of the monopolist’s demand curve, short-run marginal cost curve and marginal revenue curve and determine the profit-maximizing price and quantity. b. Now assume that there is a \$4 per unit specific tax. Determine the new profit- maximizing price and quantity. c. Determine what effect the \$4 specific tax has had on consumer surplus, producer surplus, tax revenues, and social welfare. Be sure to calculate the dollar amounts. Answers: The marginal revenue equation is MR = 60 – 2Q. Since MR equals MC at the profit-maximizing output level, 60 – 2Q = 2Q => Q* = 60/4 = 15. P* = 60 – Q = 60 – 15 = 45. If there is a \$4 specific tax, SMC = 2Q + 4. In this case, MR = MC => 60 – 2Q = 2Q + 4 => 4Q = 60 – 4 => Q* = 56/4 = 14. Consumers pay a price equal to P* = 60 – Q = 60 – 14 = 46. The monopolist receives a net price of 46 – 4 = 42. The government collects a revenue of 4(14) = 56. Both consumer and producer surplus decrease due to the tax. Without the tax CS = (60 – 45)(15) /2 = 121.5. PS = 15(45) – (15) 2 = 450. With the tax CS = (60 – 46)(14)/2 = 98. PS = 14(46) – (14(14) + 14(4)) = 392.
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