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Econ 200 Problem Set 6 Solution

# Econ 200 Problem Set 6 Solution - Answer Key Problem Set 6...

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Answer Key, Problem Set 6 1. Start with stage 2. Firm 1 takes k as given and solves max q 1 (50 2 q 1 2 q 2 2 + k 4 ) . The fi rst-order condition gives q 1 = 48 2 q 2 + k 4 4 . Similarly, fi rm 2’s fi rst-order condition gives: q 2 = 48 2 q 1 4 . Together, these two equations imply q 1 = 8+ k 12 and q 2 = 8 k 24 . This gives fi rm 1 pro fi t of μ 8 + k 12 50 2(16 + k 24 ) 2 + k 4 ¸ k 2 18 = μ 8 + k 12 ¶ μ 16 + k 6 k 2 18 . Maximizing this with respect to k gives fi rm 1’s optimal investment in stage 1. From the fi rst order condition we have: k ( 1 9 1 36 ) = ( 8 6 + 16 12 ) which has a solution of k = 32 . This is a top-dog strategy: the competition in strategic substitues leads fi rm 1 to be “tough” by “overinvesting” in capital. You can verify that investment makes fi rm 1 tough because fi rm 2’s pro fi t is decreasing in fi rm 1’s investment, and that the two fi rms’ quantity choices are strategic substitutes — fi rm 2’s quantity decreases in fi rm 1’s quantity and vice versa.

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Econ 200 Problem Set 6 Solution - Answer Key Problem Set 6...

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