Econ 200 Problem Set 5 Solution

Econ 200 Problem Set 5 Solution - Answer Key: Problem Set 5...

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Answer Key: Problem Set 5 March 31, 2005 Question 1 a) From the lemma taught on Feb. 24th, we know that with f xed prices and n f rms, in any Nash Equilibrium, each peripheral f rm locates at the same position as some other f rm and no f rm’s market share is smaller than any half-market of any other f rm. Therefore in our example of two f rms, both will locate at the middle of the unit interval: x 1 = x 2 = 1 2 . For proof, f rst note that there is no pro f table deviation for any of them (by deviating to any location other than 1 / 2 when their opponent locates at 1 / 2 ,the f rmmakeslesspro f t). Second there can not be any other Nash equilibrium. This can easily be shown by contradiction. Suppose there is some Nash equilibrium in which x i < 1 2 . Then x j <x i can not be a best response because it would be more pro f table for f rm j to move to x j x i . Now if x j = x i then f rm j can make more pro f tbymov ingto x i + ε ,fora small positive number ε. If x j >x i , f rm j can increase its pro f tbymov ing to x j ε, for any ε<x j x i .So x i < 1 2 can not be part of an equilibrium. The same argument applies for x i > 1 2 . Hence x i = x j = 1 2 is the only Nash equilibrium. b) Let us solve the game backward. Without loss of generality, assume f rm 1 moves f rst and x 1 1 2
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This note was uploaded on 07/18/2008 for the course ECON 200 taught by Professor Philiphaile during the Spring '08 term at Yale.

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Econ 200 Problem Set 5 Solution - Answer Key: Problem Set 5...

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