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Macroeconomics Exam Review 168

Macroeconomics Exam Review 168 - c 2001 Michael Carter All...

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On the other hand, by Exercise 3.16, 𝑓 attains its maximum at an extreme point of 𝑆 . That is, there exists x 1 ˆ 𝑆 such that 𝑓 ( x 1 ) 𝑓 ( x ) for every x 𝑆 In particular 𝑓 ( x 1 ) 𝑓 ( x 0 ) since x 0 ˆ 𝑆 𝑆 . This contradicts (3.64) since x 1 ˆ 𝑆 . Thus our assumption that 𝑆 ˆ 𝑆 yields a contradiction. We conclude that 𝑆 = ˆ 𝑆 3.210 1. (a) 𝑃 is compact and convex, since it is the product of compact, convex sets (Proposition 1.2, Exercise 1.165). (b) Since x 𝑛 𝑖 =1 conv 𝑆 𝑖 , there exist x 𝑖 conv 𝑆 𝑖 such that x = 𝑛 𝑖 =1 x 𝑖 . ( x 1 , x 2 , . . . , x 𝑛 ) 𝑃 ( x ) so that 𝑃 ( x ) = . (c) By the Krein-Millman theorem (or Exercise 3.207), 𝑃 ( x ) has an extreme point z = ( z 1 , z 2 , . . . , z 𝑛 ) such that z 𝑖 conv 𝑆 𝑖 for every 𝑖 𝑛 𝑖 =1 z 𝑖 = x . since z 𝑃 ( x ). 2. (a) Exercise 1.176 (b) Since 𝑙 > 𝑚 = dim 𝑋 , the vectors y 1 , y 2 , . . . , y 𝑙 are linearly dependent (Exercise 1.143). Consequently, there exists numbers 𝛼 1 , 𝛼 2 , . . . , 𝛼 𝑙 , not all zero, such that
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