Macroeconomics Exam Review 73

Macroeconomics Exam Review 73 - 2.63 As in the proof of the...

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Unformatted text preview: 2.63 As in the proof of the theorem, let 1 , 2 belong to Θ with 2 ≿ 1 . Choose any optimal solutions x 1 ∈ ( 1 ) and x 2 ∈ ( 2 ). We claim that x 2 ≿ x 1 . Assume otherwise, that is assume x 2 ∕ ≿ 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 ( x 1 , 2 ) − ( x 1 , 1 ) > ( x 1 ∧ x 2 , 2 ) − ( x 1 ∧ x 2 , 1 ) which can be rearranged to give ( x 1 , 2 ) − ( x 1 ∧ x 2 , 2 ) > ( x 1 , 1 ) − ( x 1 ∧ x 2 , 1 ) (2.41) Supermodularity implies ( x 1 ∨ x 2 , 2 ) + ( x 1 ∧ x 2 , 2 ) ≥ ( x 1 , 2 ) + ( x 2 , 2 ) which can be rearranged to give ( x 1 ∨ x 2 , 2 ) − ( x 2 , 2 ) ≥ ( x 1 , 2 ) − ( x 1 ∧ x 2 , 2 ) Combining this inequality with (2.41) givesCombining this inequality with (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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