Macroeconomics Exam Review 141

# Macroeconomics Exam Review 141 - Solutions for Foundations...

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? 1. Then ± ( ²³ 1 +(1 ² ) ³ 2 )=( ²³ 1 ² ) ³ 2 ) ? =( ²³ 1 ² ) ³ 2 )( ²³ 1 ² ) ³ 2 ) ? 1 ( ²³ 1 ² ) ³ 2 )( ²³ ? 1 1 ² ) ³ ? 1 2 )( s i n c e ³ ? 1 is convex) = ² 2 ³ ? 1 + ² (1 ² ) ³ ? 1 1 ³ 2 + ² (1 ² ) ³ 1 ³ ? 1 2 ² ) 2 ³ ? 2 = ²³ ? 1 ² ) ³ ? 2 ²³ ? 1 (1 ² ) ³ ? 2 + ² 2 ³ ? 1 + ² (1 ² ) ³ ? 1 1 ³ 2 + ² (1 ² ) ³ 1 ³ ? 1 2 ² ) 2 ³ ? 2 = ²³ ? 1 ² ) ³ ? 2 ² (1 ² ) ( ³ ? 1 ³ 1 ³ ? 1 2 ³ ? 1 1 ³ 2 + ³ ? 2 ) = ²³ ? 1 ² ) ³ ? 2 ² (1 ² ) ( ³ ? 1 1 ( ³ 1 ³ 2 ) ³ ? 1 2 ( ³ 1 ³ 2 ) ) = ²³ ? 1 ² ) ³ ? 2 ² (1 ² ) ( ( ³ 1 ³ 2 )( ³ ? 1 1 ³ ? 1 2 ) ) Since ³ ± is monotonic (Example 2.53) ³ ? 1 1 ³ ? 1 2 0 ⇐⇒ ³ 1 ³ 2 0 and therefore ( ³ 1 ³ 2 )( ³ ? 1 1 ³ ? 1 2 ) 0 We conclude that ± ( ²³ 1 ² ) ³ 2 ) ²³ ? 1 ² ) ³ ? 2 = ²± ( ³ 1 )+(1 ² )( ³ 2 ) ± is convex for all ? =1 , 2 ,... . 3.121 For given x 1 , x 2 ´ , de±ne µ :[0 , 1] ´ by µ ( )=(1 ) x 1 + x 2 Then µ (0) = x 1 , µ (1) = x 2 and = µ ± . Assume ± is convex. For any 1 2 [0 , 1], let µ ( 1 )= ¯ x 1 and µ ( 2 x 2 For any ² [0 , 1] µ ± ²¶ 1 ² ) 2 ² = ² ¯ x 1 ² x 2 ± ²¶ 1 ² ) 2 ² = ± ± ² ¯
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## This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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