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prob3(2) - Problem Set 3(Lecture 4 March 23 ECOS3012...

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Problem Set 3 (Lecture 4, March 23) ECOS3012 - Strategic Behavior, Semester 1, 2011 Exercise 1. Do all the end of chapter problems from the chapter titled “Rationalizability and Iterated Dominance”. Solution . See the last two pages of this file. I think the manual I got these solutions from should work for both the new and old edition – if there is a difference, let me know. Exercise 2. ( Price Competition with differentiated Products ). Recall the way we calculated the reaction functions for Cournot Duopoly using calculus. Firm 1 and Firm 2 compete in two related markets. If the two firms set a price of p 1 > 0 and p 2 > 0 respectively, the respective quantity they can sell is given by the equations q 1 = 12 - p 1 + p 2 and q 2 = 12 - p 2 + p 1 provided q 1 0 and q 2 0. Profits are zero otherwise. Firms maximize profits and each firm’s marginal cost of production is zero. Assume firms compete by simultaneously choosing prices. (a) Express the above as a strategic form game, i.e. write down the payoff functions. Given the payoff functions, do you think that the products of the two firms are more like substitutes or complements? Solution . Holding its own price p 1 fixed, the demand for Good 1, namely q 1 , rises as p 2 increases and correspondingly the demand for Good 2 falls. In other words, the consumers switch from the good whose price has increased to the other. So the good are substitutes. U 1 ( p 1 , p 2 ) = (12 - p 1 + p 2 ) p 1 (1) U 2 ( p 1 , p 2 ) = (12 - p 2 + p 1 ) p 2 (2) (b) Given p 2 , what value of p 1 maximizes the payoff of Firm 1 – i.e. what is the best response of Firm 1 to p 2 ? Now write down the reaction functions. Solution . The best response of Player 1 to a strategy p 2 is the value of p 1 that maximizes her payoff (given p 2 ). In this case, the first order condition for a maximum is given by ∂U 1 ∂p 1 = 12 - 2 p 1 + p 2 = 0 . (3) Solving for p 1 , gives the reaction function of Player 1 as B 1 ( p 2 ) = 12 + p 2 2 (4) 1
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Similarly, the reaction function of Player 2 can be seen to be B 2 ( p 1 ) = 12 + p 1 2 (5) (c) Plot the reaction functions on the xy-plane and see what strategies are rationalizable.
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