Macroeconomics Exam Review 178

Macroeconomics Exam Review 178 - Solutions for Foundations...

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3.235 Assume system I has a solution, that is there exists ˆ x ∈ℜ ? + such that ? ˆ x = 0 , c ˆ x > 0 , ˆ x 0 Then x = ˆ x / c ˆ x satisFes the system ? x = 0 , cx =1 , x 0 (3.70) which is equivalent to x ? ± = 0 , xc =1 , x 0 (3.71) Suppose y ∈ℜ ² satisFes ? y c Multiplying by x 0 gives x ? ± y xc Substituting (3.71), this implies the contradiction 0 1 We conclude that system II cannot have a solution if I has a solution. Now, assume system I has no solution. System I is equivalent to (3.70) which in turn is equivalent to the system ( ? c ) x = ( 0 1 ) or ± x = b (3.72) where ± = ( ? c ) is ( ² +1) × ³ and b = ( 0 1 ) ∈ℜ ² +1 . If (3.72) has no solution, there exists (by the ±arkas alternative) some
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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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