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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 fi xed prices and n fi rms, in any Nash Equilibrium, each peripheral fi rm locates at the same position as some other fi rm and no fi rm’s market share is smaller than any half-market of any other fi rm. Therefore in our example of two fi rms, both will locate at the middle of the unit interval: x 1 = x 2 = 1 2 . For proof, fi rst note that there is no pro fi table deviation for any of them (by deviating to any location other than 1 / 2 when their opponent locates at 1 / 2 , the fi rm makes less pro fi 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 fi table for fi rm j to move to x j x i . Now if x j = x i then fi rm j can make more pro fi t by moving to x i + ε , for a small positive number ε. If x j > x i , fi rm j can increase its pro fi t by moving 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 fi rm 1 moves fi rst and x 1 1 2 .
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