Macroeconomics Exam Review 68

Macroeconomics Exam Review 68 - Solutions for Foundations...

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2.43 For every 𝜃 2 𝜃 1 Θ, if 𝑥 1 𝐺 ( 𝜃 1 ) and 𝑥 2 𝐺 ( 𝜃 2 ), then 𝑥 1 𝑥 2 𝑥 1 and therefore 𝑥 1 𝑥 2 𝐺 ( 𝜃 1 ). If 𝑥 1 𝑥 2 , then 𝑥 1 𝑥 2 = 𝑥 1 𝑔 ( 𝜃 1 ) 𝑔 ( 𝜃 2 ) and therefore 𝑥 1 𝑥 2 𝐺 ( 𝜃 2 ). On the other hand, if 𝑥 1 𝑥 2 , then 𝑥 1 𝑥 2 = 𝑥 2 𝐺 ( 𝜃 2 ). 2.44 Assume 𝜑 is increasing, and let 𝑥 1 , 𝑥 2 𝑋 with 𝑥 2 𝑥 1 . Let 𝑦 1 𝜑 ( 𝑥 1 ). Choose any 𝑦 𝜑 ( 𝑥 2 ). Since 𝜑 is increasing, 𝜑 ( 𝑥 2 ) 𝑆 𝜑 ( 𝑥 1 ) and therefore 𝑦 2 = 𝑦 1 𝑦 𝜑 ( 𝑥 2 ). 𝑦 2 𝑦 1 as required. Similarly, for every 𝑦 2 𝜑 ( 𝑥 2 ), there exists some 𝑦 𝜑 ( 𝑥 2 ) such that 𝑦 1 = 𝑦 𝑦 2 𝜑 ( 𝑥 1 ) with 𝑦 2 𝑦 1 . 2.45 Since 𝜑 ( 𝑥 ) is a sublattice, sup 𝜑 ( 𝑥 ) 𝜑 ( 𝑥 ) for every 𝑥 . Therefore, the function 𝑓 ( 𝑥 ) = sup 𝜑 ( 𝑥 ) is a selection. Similarly 𝑔 ( 𝑥
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