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Unformatted text preview: Economics 405/505 Introduction to Game Theory Rui Zhao Applications of Extensiveform Games 1 Stackelberg Model of Duopoly 1.1 General Model Two firms produce identical products. Firm 1 chooses its output q 1 first. Firm 2, after observing q 1 , chooses its output q 2 . The market price then is determined by an inverse demand function P = f ( Q ) , where Q = q 1 + q 2 is the total output. This is an extensiveform game of perfect information: players: two firms actions: q 1 ≥ ,q 2 ≥ payoffs: firm 1, π 1 = Pq 1 C 1 ( q 1 ); firm 2, π 2 = Pq 2 C 2 ( q 2 ) . Strategies: a strategy of firm 1 is simply a choice q 1 ; a strategy of firm 2 is a plan for choosing q 2 given firm 1’s output q 1 , so it is a function q 2 ( q 1 ). 1.2 Lineal demand and constant unit cost Both firms have constant unit cost c , so total costs are C 1 ( q 1 ) = cq 1 , C 2 ( q 2 ) = cq 2 . Market price is given by P = a Q : if Q ≤ a 0 : if Q > a where a > 0 is some constant. Assume a > c . We solve this game using backward induction. Step 1. Given firm 1’s output q 1 , find firm 2’s best choice q * 2 ( q 1 ) . Firm 2’s best choice q * 2 ( q 1 ) should maximize its profit,so is a solution to the following problem: max q 2 q 2 ( a q 1 q 2 ) cq 2 Take the firstorder derivative with respect to q 2 ( q 1 is constant!) and set it equal to zero: a q 1 2 q * 2 c = 0 q * 2 = a c q 1 2 . Since output is bigger than or equal to zero, we have q * 2 = a c q 1 2 : if q 1 ≤ a c 0 : if q 1 > a c 1 In summary, at a node q 1 ≤ a c , firm 2’s best choice is q * 2 ( q 1 ) = a c q 1 2 , at a node q 1 > a c , firm 2’s best choice is q * 2 ( q 1 ) = 0 ....
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This note was uploaded on 09/14/2011 for the course ECON 505 taught by Professor Zhao during the Spring '11 term at SUNY Albany.
 Spring '11
 Zhao
 Game Theory

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