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Macroeconomics Exam Review 28

Macroeconomics Exam Review 28 - 1.162 Let ℭ be a...

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Unformatted text preview: 1.162 Let ℭ be a collection of convex sets and let x , y belong to ∩ ∈ ℭ . for every ∈ ℭ , x , y ∈ and therefore x + (1 − ) y ∈ for all 0 ≤ ≤ 1 (since is convex). Therefore x + (1 − ) y ∈ ∩ ∈ ℭ . 1.163 Fix some output . Assume that x 1 , x 2 ∈ ( ). This implies that both ( , − x 1 ) and ( , − x 2 ) belong to the production possibility set . If is convex ( , − x 1 ) + (1 − )( , − x 2 ) = ( + (1 − ) , x 1 + (1 − ) x 2 ) = ( , x 1 + (1 − ) x 2 ) ∈ for every ∈ [0 , 1]. This implies that x 1 + (1 − ) x 2 ∈ ( ). Since the choice of was arbitrary, this implies that ( ) is convex for every . 1.164 Assume 1 and 2 are convex sets. Let = 1 + 2 . Suppose x , y belong to ....
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