Macroeconomics Exam Review 31

Macroeconomics Exam Review 31 - c 2001 Michael Carter All...

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If 𝛽 𝑖 > 0, then 𝛼 𝑖 /𝛽 𝑖 𝛼 𝑗 /𝛽 𝑗 𝑡 and therefore 𝛼 𝑖 𝑡𝛽 𝑖 0 If 𝛽 𝑖 0 then 𝛼 𝑖 𝑡𝛽 𝑖 > 0 for every 𝑡 > 0. Therefore 𝛼 𝑖 𝑡𝛽 𝑖 0 for every 𝑡 and 𝛼 𝑖 𝑡𝛽 𝑖 = 0 for 𝑖 = 𝑗 . Therefore, (1.23) represents x as a convex combination of only 𝑛 1 points. 4. This process can be repeated until x is represented as a convex combination of at most dim 𝑆 + 1 elements. 1.176 Assume x is not an extreme point of 𝑆 . Then there exists distinct x 1 and x 2 in S such that x = 𝛼 x 1 + (1 𝛼 ) x 2 Without loss of generality, assume 𝛼 1 / 2 and let y = x 2 x . Then x + y = x 2 𝑆 . Furthermore x y = x x 2 + x = 2 x x 2 = 2( 𝛼 x 1 + (1 𝛼 ) x 2 ) x 2 = 2 𝛼 x 1 + (1 2 𝛼 ) x 2 𝑆 since 𝛼 1 / 2. 1.177 1. For any x = ( 𝑥 1 , 𝑥 2 ) 𝐶 2 , there exists some 𝛼 1 [0 , 1] such that 𝑥 1 = 𝛼 1 𝑐 + (1 𝛼 1 )( 𝑐 ) = (2 𝛼 1 1) 𝑐 In fact, 𝛼 1 is defined by 𝛼 1 =
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