Macroeconomics Exam Review 35

# Macroeconomics Exam Review 35 - Solutions for Foundations...

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1.182 Let 𝑛 be the dimension of a convex set 𝑆 in a linear space 𝑋 . Then 𝑛 = dim aff 𝑆 and there exists a set { x 1 , x 2 , . . . , x 𝑛 +1 } of aﬃnely independent points in 𝑆 . Define 𝑆 = conv { x 1 , x 2 , . . . , x 𝑛 +1 } Then 𝑆 is an 𝑛 -dimensional simplex contained in 𝑆 . 1.183 Let w = ( 𝑤 ( { 1 } ) , 𝑤 ( { 2 } ) , . . . , 𝑤 ( { 𝑛 } )) denote the vector of individual worths and let 𝑠 denote the surplus to be distributed, that is 𝑠 = 𝑤 ( 𝑁 ) 𝑖 𝑁 𝑤 ( { 𝑖 } ) 𝑠 > 0 if the game is essential. For each player 𝑖 = 1 , 2 , . . . , 𝑛 , let y 𝑖 = w + 𝑠 e 𝑖 be the outcome in which player 𝑖 receives the entire surplus. ( e 𝑖 is the 𝑖 th unit vector.) Note that 𝑦 𝑖 𝑗 = { 𝑤 ( { 𝑖 } ) + 𝑠 𝑗 = 𝑖 𝑤 ( { 𝑖 } ) 𝑗 = 𝑖 Each y 𝑖 is an imputation since 𝑦 𝑖 𝑗 𝑤 ( { 𝑗 } ) and 𝑗 𝑁 𝑦 𝑖 𝑗 = 𝑗 𝑁 𝑤 ( { 𝑗 } ) + 𝑠 = 𝑤 ( 𝑁 ) Therefore { y 1 , y 2 , . . . , y 𝑛 } ⊆ 𝐼 . Since 𝐼 is convex (why ?), 𝑆 = conv { y 1 , y 2 , . . . , y 𝑛 } ⊆ 𝐼 . Further, for every 𝑖, 𝑗 𝑁 the vectors y 𝑖 y 𝑗 = 𝑠 ( e 𝑖 e 𝑗 ) are linearly independent. Therefore
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