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Unformatted text preview: Homework #4 Answer Key due September 8 in class Econ 11 &Summer Session C 2008 University of California, Los Angeles Questions 1 and 2 are worth 10 points, while question 3 is worth 5, for a total of 25 points. 1. A perfectly competitive industry has a large number of potential entrants. Each rm has an identical cost structure given by C ( q ) = q 2 & 5 q + 100 (a) What are the average cost and marginal cost functions? At what output level is the average cost minimized? How much is the average cost of production at that output level? ANSWER: AC ( q ) = q & 5 + 100 q MC ( q ) = 2 q & 5 AC ( q ) = MC ( q ) q & 5 + 100 q = 2 q & 5 q 2 = 100 q = 10 Alternatively, min q AC ( q ) occurs where dAC ( q ) dq = 1 & 100 q 2 = 0 q = 10 At q = 10 , the average cost of production is AC (10) = 10 & 5 + 100 10 = 15 (b) Suppose the total market demand is given by Q = 2500 & 50 P . Find the long-run competitive price. Find the number of rms in the industry. What is each rms prot? ANSWER: The long-run competitive price generates zero prot; i.e., occurs where AC ( q ) = MC ( q ) or where AC ( q ) is minimized. Since we know each rm is prot-maximizing, we can nd each rms sup- ply curve, and then the total market supply from the summation of 1 all the individual &rmssupply curves. d & ( q ) dq = P & MC ( q ) = 0 P = 2 q & 5 Then each &rms supply curve is q = 1 2 P + 5 2 and the total supply in the industry is Q S = nq where n is the number of &rms. From part ( a ) , we know that the point where AC ( q ) = MC ( q ) , or where each &rm earns zero pro&t, occurs where q = 10 . Thus the long-run price is P = 2 (10) & 5 = 15 Q S = 10 n Q D = 2500 & 50 (15) = 1750 and since the market must clear, we know the number of &rms: Q D = Q S 1750 = 10 n n = 175 As we stated in the beginning, each &rms pro&t must be zero. To check, & (10) = 15 (10) & (10) 2 + 5 (10) & 100 = 0 (c) Now hold that number of &rms &xed in the short run (suppose entry...
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- Summer '08