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Unformatted text preview: Some Homework Answers Problem: Osborne, Problem 60.1 There are two firms with constant unit costs. The inverse demand func- tion is P = A- q 1- q 2 where q 1 and q 2 are the outputs of firms 1 and 2. We consider a variant of Cournot’s duopoly game in which firm 1 chooses its output to maximize its market share subject to not making a loss, while firm 2 seeks to maximize its profits. Answer: Let us suppose that firm 1 has unit costs c 1 and firm 2 has unit costs c 2 . (One might interpret the problem as asking only about the case where c 1 = c 2 , but let’s see if we can handle this more general problem.) What is the best response function of firm 1? Given firm 2’s output, q 2 , in order to maximize his market share, he just wants to maximize his output (subject to not making a loss). Firm 1 will not make a loss, so long as the price is at least as great as his unit cost. Since the price is P = A- q 1- q 2 , the largest quantity that firm 1 can choose without losing money is q 1 such that A- q 1- q 2 = c 1 . Rearranging terms, we find that firm 1’s best response function is q 1 = b 1 ( q 2 ) = A- c 1- q 2 . (1) What about firm 2’s reaction function? Firm 2 seeks to maximize its profits, which are Pq 2- c 2 q 2 = ( A- q 1- q 2 ) q 2- c 2 q 2 . A necessary condition for q 2 > 0 to maximize 2’s profits is that the derivative of its profits with...
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This note was uploaded on 12/25/2011 for the course ECON 171 taught by Professor Charness,g during the Fall '08 term at UCSB.
- Fall '08