Macroeconomics Exam Review 73

# Macroeconomics Exam Review 73 - Solutions for Foundations...

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2.63 As in the proof of the theorem, let 𝜽 1 , 𝜽 2 belong to Θ with 𝜽 2 𝜽 1 . Choose any optimal solutions x 1 𝜑 ( 𝜽 1 ) and x 2 𝜑 ( 𝜽 2 ). We claim that x 2 𝑋 x 1 . Assume otherwise, that is assume x 2 𝑋 x 1 . This implies (Exercise 1.44) that x 1 x 2 = x 1 . Since x 1 x 1 x 2 , we must have x 1 x 1 x 2 . Strictly increasing differences implies 𝑓 ( x 1 , 𝜽 2 ) 𝑓 ( x 1 , 𝜽 1 ) > 𝑓 ( x 1 x 2 , 𝜽 2 ) 𝑓 ( x 1 x 2 , 𝜽 1 ) which can be rearranged to give 𝑓 ( x 1 , 𝜽 2 ) 𝑓 ( x 1 x 2 , 𝜽 2 ) > 𝑓 ( x 1 , 𝜽 1 ) 𝑓 ( x 1 x 2 , 𝜽 1 ) (2.41) Supermodularity implies 𝑓 ( x 1 x 2 , 𝜽 2 ) + 𝑓 ( x 1 x 2 , 𝜽 2 ) 𝑓 ( x 1 , 𝜽 2 ) + 𝑓 ( x 2 , 𝜽 2 ) which can be rearranged to give 𝑓 ( x 1 x 2 , 𝜽 2 ) 𝑓 ( x 2 , 𝜽 2 ) 𝑓 ( x 1 , 𝜽 2 ) 𝑓 ( x 1 x 2 , 𝜽 2 ) Combining this inequality with (2.41) gives 𝑓 ( x 1 x 2 , 𝜽 2 ) 𝑓 ( x 2 , 𝜽 2 ) > 𝑓 ( x 1 , 𝜽 1 ) 𝑓 ( x 1 x 2 , 𝜽 1 ) (2.42) However x 1 and x 2
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