ps11sol_09

# ps11sol_09 - Cornell University Economics 3130 Problem Set...

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Cornell University Economics 3130 Problem Set 11 Solutions 1. In a given market, there are two ﬁrms i = { 1 , 2 } that face the demand curve Q D = 16 - p for p 16 and Q D = 0 for p > 16. As usual, Q D is the aggregate quantity demanded. The ﬁrms have no ﬁxed cost. Each has a marginal cost of production of c i > 0. The ﬁrms compete by simultaneously choosing an amount of (identical) goods x i to produce (fractions are allowed), and the amount each produces must be less than or equal to some high number ¯ x . For parts (a)-(d), assume that c 1 = c 2 = 4. (a) What is the cost function for ﬁrm i ? c(w,y) = 4y (b) What are the possible actions for ﬁrm i ? a i = [0 , ¯ x ] (c) What is ﬁrm i ’s best response function? BR i ( x j ) = 6 - x j 2 (d) Which actions for ﬁrm i are strictly dominated? Can this game be solved through iterated elimination of strictly dominated strategies/actions. If so, solve it. If not, explain why not. Any strategy of x 1 > 6 is dominated. If ﬁrm 2 chooses to produce zero, then the best response is x i (0) = 6. By the same logic, any strategy of of x 2 > 6 is dominated. Knowing that x 1 6, ﬁrm 2 will never choose x 2 < 3. The same is true for ﬁrm 1. Knowing that x 1 [3 , 6], ﬁrm 2 will never choose x 2 > 4 . 5. The same is true for ﬁrm 1. Knowing that x 1 [3 , 4 . 5], ﬁrm 2 will never choose x 2 < 3 . 75. The same is true for q 2 . You should recognize a pattern. After n eliminations, a range of 12 2 n remains. In the limit, the range contains only the single strategy x i = 4. 1

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(e) What are the pure strategy Nash equilibria? What are the proﬁts of each ﬁrm in each equilibrium?
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ps11sol_09 - Cornell University Economics 3130 Problem Set...

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