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Macroeconomics Exam Review 8

Macroeconomics Exam Review 8 - Solutions for Foundations of...

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1.89 Assume 𝑆 is bounded, and let 𝑑 = 𝑑 ( 𝑆 ). Choose any 𝑥 𝑆 . For all 𝑦 𝑆 , 𝜌 ( 𝑥, 𝑦 ) 𝑑 < 𝑑 + 1. Therefore, 𝑦 𝐵 ( 𝑥, 𝑑 + 1). 𝑆 is contained in the open ball 𝐵 ( 𝑥, 𝑑 + 1). Conversely, assume 𝑆 is contained in the open ball 𝐵 𝑟 ( 𝑥 ). Then for any 𝑦, 𝑧 𝑆 𝜌 ( 𝑦, 𝑧 ) 𝜌 ( 𝑦, 𝑥 ) + 𝜌 ( 𝑥, 𝑧 ) < 2 𝑟 by the triangle inequality. Therefore 𝑑 ( 𝑆 ) < 2 𝑟 and the set is bounded. 1.90 Let 𝑦 𝑆 𝐵 𝑟 ( 𝑥 0 ). For every 𝑥 𝑆 , 𝜌 ( 𝑥, 𝑦 ) < 𝑟 and therefore 𝜌 ( 𝑥, 𝑥 0 ) 𝜌 ( 𝑥, 𝑦 ) + 𝜌 ( 𝑦, 𝑥 0 ) < 𝑟 + 𝑟 = 2 𝑟 so that 𝑥 𝐵 2 𝑟 ( 𝑥 0 ). 1.91 Let y 0 𝑌 . For any 𝑟 > 0, let y = y 𝑟 be the production plan which is 𝑟 units less in every commodity. Then, for any y 𝐵 𝑟 ( y ) 𝑦 𝑖 𝑦 𝑖 𝜌 ( y , y ) < 𝑟 for every 𝑖 and therefore y < y 0 . Thus 𝐵 𝑟 ( y ) 𝑌 and so y int 𝑌 = .
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