Macroeconomics Exam Review 170

Macroeconomics Exam Review 170 - 2 Assume is convex For any...

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Unformatted text preview: 2. Assume ( ) is convex. For any x / ∈ ( ) there exists w such that w x < inf x ∈ ( ) w x = ( w , ) by the Strong Separation Theorem. Monotonicity ensures that w ≥ and hence x / ∈ ∗ ( ). 3.215 Assume x ∈ ( ) = ∗ ( ). That is w x ≥ ˆ ( w ) for every x Therefore, for any ∈ ℜ + w x ≥ ( w ) for every x which implies that x ∈ ∗ ( ) = ( ). 3.216 A polyhedron = { ∈ : ( ) ≤ , = 1 , 2 ,..., } = ∩ =1 { x ∈ : ( x ) ≤ } is the intersection of a finite number of closed convex sets. 3.217 Each row a = ( 1 , 2 ,... ) of defines a linear functional ( x ) = 1 1 + 2 2 + ⋅⋅⋅ + on ℜ . The set of solutions to x ≤ c...
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