Macroeconomics Exam Review 191

Macroeconomics Exam Review 191 - Solutions for Foundations...

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2. Assume x = ( 𝑥 1 , 𝑥 2 , 𝑥 3 ) core. Then x must satisfy the following system of inequalities 𝑥 1 + 𝑥 2 1 = 𝑤 ( { 1 , 2 } ) 𝑥 1 + 𝑥 3 1 = 𝑤 ( { 1 , 3 } ) 𝑥 2 + 𝑥 3 1 = 𝑤 ( { 2 , 3 } ) which can be summed to yield 2( 𝑥 1 + 𝑥 2 + 𝑥 3 ) 3 or 𝑥 1 + 𝑥 2 + 𝑥 3 3 / 2 which implies that x exceeds the sum available. This contradiction establishes that the core is empty. Alternatively, observe that the three person majority game is a simple game with no veto players. By Exercise 1.69, its core is empty. 3.263 Assume that the game ( 𝑁, 𝑤 ) is not cohesive. Then there exists a partition { 𝑆 1 , 𝑆 2 , . . . , 𝑆 𝐾 } of 𝑁 such that 𝑤 ( 𝑁 ) < 𝐾 𝑘 =1 𝑤 ( 𝑆 𝑘 ) Assume x core. Then 𝑖 𝑆 𝑘 𝑥 𝑖 𝑤 ( 𝑆 𝑘 ) 𝑘 = 1 , 2 , . . . , 𝐾 Since { 𝑆 1 , 𝑆 2 , . . . , 𝑆 𝐾 } is a partition 𝑖 𝑁 𝑥 𝑖 = 𝐾 𝑘 =1 𝑖 𝑆 𝑘 𝑥 𝑖 𝑁 𝑘 =1 𝑤 ( 𝑆 𝑘 ) > 𝑤 ( 𝑁 ) which contradicts the assumption that x core. This establishes that cohesivity is
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