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Macroeconomics Exam Review 174

Macroeconomics Exam Review 174 - Solutions for Foundations...

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3.226 Assume y is an eﬃcient production plan in 𝑌 and let 𝑆 = 𝑌 𝑦 . 𝑆 is convex. We claim that 𝑆 ∩ℜ 𝑛 ++ = . Otherwise, if there exists some z 𝑆 ∩ℜ 𝑛 ++ , let y = y + z z 𝑆 implies y 𝑌 while z ∈ ℜ 𝑛 ++ implies y > y contradicting the eﬃciency of y . Therefore, 𝑆 is a convex set which contains no interior points of the nonnegative orthant 𝑛 + . By Exercise 3.225, there exists a price system p such that p 𝑇 x 0 for every x 𝑆 Since 𝑆 = 𝑌 𝑦 , this implies p ( y y ) 0 for every y 𝑌 or py py for every y 𝑌 𝑦 maximizes the producer’s profit at prices p . 3.227 Consider the set 𝑆 = { x ∈ ℜ 𝑛 : x 𝑆 } . 𝑆 int 𝑛 = = 𝑆 int 𝑛 + = From the previous exercise, there exists a hyperplane with nonnegative normal p 0 such that p 𝑇 x 0 for every x 𝑆 Since p 0 , this implies p 𝑇 x 0 for every x 𝑆 3.228 1. Suppose x ( x ). Then, there exists an allocation (
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