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Macroeconomics Exam Review 193

Macroeconomics Exam Review 193 - Solutions for Foundations...

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3.268 Let ( 𝑁, 𝑤 1 ) and ( 𝑁, 𝑤 2 ) be balanced games. By the Bondareva-Shapley theorem, they have nonempty cores. Let x 1 core( 𝑁, 𝑤 1 ) and x 2 core( 𝑁, 𝑤 2 ). That is, 𝑔 𝑆 ( x 1 ) 𝑤 1 ( 𝑆 ) for every 𝑆 𝑁 𝑔 𝑆 ( x 2 ) 𝑤 2 ( 𝑆 ) for every 𝑆 𝑁 Adding, we have 𝑔 𝑆 ( x 1 ) + 𝑔 𝑆 ( x 2 ) = 𝑔 𝑆 ( x 1 + x 2 ) 𝑤 1 ( 𝑆 ) + 𝑤 2 ( 𝑆 ) for every 𝑆 𝑁 which implies that x 1 + x 2 belongs to core( 𝑁, 𝑤 1 + 𝑤 2 ). Therefore ( 𝑁, 𝑤 1 + 𝑤 2 ) is balanced. Similarly, if x core( 𝑁, 𝑤 ), then 𝛼 x belongs to core( 𝑁, 𝛼𝑤 ) for every 𝛼 ∈ ℜ + . That is ( 𝑁, 𝛼𝑤 ) is balanced for every 𝛼 ∈ ℜ + . 3.269 1. Assume otherwise. That is assume there exists some y 𝐴 𝐵 . Taking the first 𝑛 components, this implies that e 𝑁 = 𝑆 𝑁 𝜆 𝑠 e 𝑆 for some ( 𝜆 𝑆 0 : 𝑆 𝑁 ). Let = { 𝑆 𝑁 𝜆 𝑆 > 0 } be the set of coalitions with strictly positive weights. Then is a balanced family of coalitions with weights 𝜆 𝑆 (Exercise 3.266). However, looking at the last coordinate, y 𝐴 𝐵 implies 𝑆 ∈ℬ 𝜆 𝑠 𝑤 ( 𝑆 ) = 𝑤 ( 𝑁 ) + 𝜖 > 𝑤 ( 𝑁 ) which contradicts the assumption that the game is balanced. We conclude that
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