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Macroeconomics Exam Review 27

# Macroeconomics Exam Review 27 - Solutions for Foundations...

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and 𝛼𝑥 + (1 𝛼 ) 𝑦 < 𝛼𝑏 + (1 𝛼 ) 𝑏 Therefore 𝑎 < 𝛼𝑥 +(1 𝛼 ) 𝑦 < 𝑏 and 𝛼𝑥 +(1 𝛼 ) 𝑦 ( 𝑎, 𝑏 ). ( 𝑎, 𝑏 ) is convex. Substituting for < demonstrates that [ 𝑎, 𝑏 ] is convex. Let 𝑆 be an arbitrary convex set in . Assume that 𝑆 is not an interval. This implies that there exist numbers 𝑥, 𝑦, 𝑧 such that 𝑥 < 𝑦 < 𝑧 and 𝑥, 𝑧 𝑆 while 𝑦 / 𝑆 . Define 𝛼 = 𝑧 𝑦 𝑧 𝑥 so that 1 𝛼 = 𝑦 𝑥 𝑧 𝑥 Note that 0 𝛼 1 and that 𝛼𝑥 + (1 𝛼 ) 𝑧 = 𝑧 𝑦 𝑧 𝑥 𝑥 + 𝑦 𝑥 𝑧 𝑥 𝑧 = 𝑦 / 𝑆 which contradicts the assumption that 𝑆 is convex. We conclude that every convex set in is an interval. Note that 𝑆 may be a hybrid interval such ( 𝑎, 𝑏 ] or [ 𝑎, 𝑏 ) as well as an open ( 𝑎, 𝑏 ) or closed [ 𝑎, 𝑏 ] interval. 1.161 Let ( 𝑁, 𝑤 ) be a TP-coalitional game. If core( 𝑁, 𝑤 ) = then it is trivially convex. Otherwise, assume core( 𝑁, 𝑤 ) is nonempty and let x 1 and x 2 belong to core( 𝑁, 𝑤
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