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Macroeconomics Exam Review 69

Macroeconomics Exam Review 69 - implies 1 ∨ 2 1 2 ≥...

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Unformatted text preview: implies ( 1 ∨ 2 ) + ( 1 ∧ 2 ) ≥ ( 1 ) + ( 2 ) for all ≥ 0. By Exercise 1.186, the set of all supermodular functions is a convex cone in ( ). 2.51 Since is supermodular and is nonnegative definite, ( 1 ∨ 2 ) ( 1 ∨ 2 ) ≥ ( ( 1 ) + ( 2 ) − ( 1 ∧ 2 ) ) ( 1 ∨ 2 ) = ( 2 ) ( 1 ∨ 2 ) + ( ( 1 ) − ( 1 ∧ 2 ) ) ( 1 ∨ 2 ) for any 1 , 2 ∈ . Since and are increasing, this implies ( 1 ∨ 2 ) ( 1 ∨ 2 ) ≥ ( 2 ) ( 1 ∨ 2 ) + ( ( 1 ) − ( 1 ∧ 2 ) ) ( 1 ) (2.33) Similarly, since is nonnegative definite, supermodular, and and increasing ( 2 ) ( 1 ∨ 2 ) ≥ ( 2 ) ( (...
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