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Unformatted text preview: Economics 104B Solution for Problem Set #4 Spring 2011 1. In Bertrand competition firms decide independently what prices to set to maxi- mize their individual profits. Suppose that there are two firms in a market selling the same product. Let D ( p ) = a- bp be the market demand and let c 1 and c 2 be firm 1 and firm 2’s constant marginal costs, respectively. (Assume a,b > 0, and < c 1 < c 2 < a/b .) Let p m be the optimal price for firm 1 when firm 1 is the only seller in the market. Assume p m < c 2 . Does a Bertrand equilibrium exist? If your answer is yes, find a Bertrand equilibrium price. If your answer is no, explain why. Answer: Consider price pair ( p * 1 ,p * 2 ) = ( p m ,c 2 ). Since p m < c 2 , firm 1 is doing the best by charging its monopoly price pm , given that firm 2 is charging price p * 2 = c 2 . On the other hand, given that firm 1 is charging price p * 1 = p m , the fact that p m < c 2 implies that the maximum profit that firm 2 can make is 0. Since price p * 2 = c 2 yields profit 0 to firm 2 given firm 1’s price p * 1 = p m , firm 2 is also doing the best by charging price p * 2 given that firm firm 1 is charging price p * 1 . Thus, ( p * 1 ,p * 2 ) = ( p m ,c 2 ) is a Bertrand equilibrium. 2. A monopolistic firm, call it Firm 1, has decided to introduce a new video game. In deciding what type of manufacturing plant to build, it has the choice of two technologies. Technology A is publicly available which results in total cost func- tion C A ( q ) = 10 + 8 q. Technology B is a proprietary technology developed in firm 1’s research lab. It generates cost function C B ( q ) = 60 + 2 q. Firm 1 must decide which technology to adopt. Market demand for the new game is p ( Q ) = 20- Q , where Q is the total industry output. a. Suppose firm is certain that it can maintain its monopoly position in the market without any threat of entry. Which technology would you advise firm 1 to adopt? What would be firm 1’s profit given this choice?...
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This note was uploaded on 12/26/2011 for the course ECON 104B taught by Professor Qin during the Fall '09 term at UCSB.
- Fall '09