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33 - 5 α-c 2 To see why draw out the best response curves...

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ECON 3600 Problem Set 3 1. Draw a 3x3 strategic form game with no pure strategy Nash Equilibrium. Explain why there is no pure strategy NE in your game. Solution : There are many possible games. A classic one is rock-paper-scissors. r s p r 0 , 0 1 , - 1 - 1 , 1 s - 1 , 1 0 , 0 1 , - 1 p 1 , - 1 - 1 , 1 0 , 0 2. Exercise 58.1 in Osborne’s An Introduction to Game Theory Solution : Firm 1’s profits are: q 1 ( α - c 1 - q 1 - q 2 ) To find Firm 1’s best response find the FOC: α - c 1 - q 2 - 2 q 1 = 0 which simplifies to: q 1 * = α - c 1 - q 2 2 Similarly, q 2 * = α - c 2 - q 1 2 Solving the two equations with two unknowns, we get: q 1 * = α - 2 c 1 + c 2 3 and q 2 * = α - 2 c 2 + c 1 3 If c 1 > c 2 , then q 1 * < q 2 * . As c 2 falls, q 1 * falls and q 2 * rises. Total output rises. Price falls. 1
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Tricky part: However, this is only true for c 1 0 . 5 ( α + c 2 ) . If c 1 > 0 . 5 ( α + c 2 ) , then Firm 1 produces
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Unformatted text preview: . 5 ( α-c 2 ) . To see why, draw out the best response curves for this situation. You will notice that Firm 1’s BR curve is completely below Firm 2’s BR curve, and thus the only place the BR curves intersect is where q 1 = 0. 3. Exercise 59.1 Solution : Firm 1’s profits are: q 1 ( α-2 q 1-q 2 ) To find Firm 1’s best response find the FOC: α-q 2-4 q 1 = which simplifies to: q 1 * = α-q 2 4 Similarly, q 2 * = α-q 1 4 Solving the two equations with two unknowns, we get: q 1 * = q 2 * = α 5 2...
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