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Macroeconomics Exam Review 85

Macroeconomics Exam Review 85 - Solutions for Foundations...

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5. This implies that ˜ 𝑥 ˜ 𝑧 . Therefore ˜ 𝑥 = ˜ 𝑧 𝜑 𝑥 ) ˜ 𝑥 is a fixed point of 𝜑 . 6. Since 𝐸 𝑀 , ˜ 𝑥 = inf 𝑀 is the least fixed point of 𝜑 . 2.116 1. Let 𝑆 𝐸 and 𝑠 = sup 𝑆 . For every 𝑥 𝑆 , 𝑥 𝜑 ( 𝑥 ). Since 𝜑 is increasing, there exists some 𝑧 𝑥 𝜑 ( 𝑠 ) such that 𝑧 𝑥 𝑥 . 2. Let 𝑧 = sup 𝑧 𝑥 . Then (a) Since 𝑧 𝑥 𝑥 for every 𝑥 𝑆 , 𝑧 = sup 𝑧 𝑥 sup 𝑥 = 𝑠 (b) 𝑧 𝜑 ( 𝑠 ) since 𝜑 ( 𝑠 ) is a complete sublattice. 3. Define 𝑆 = { 𝑥 𝑋 : 𝑥 𝑠 for every 𝑠 𝑆 } 𝑆 is the set of all upper bounds of 𝑆 in 𝑋 . Then 𝑆 is a complete lattice, since 𝑆 = ( 𝑠 ) 4. Let 𝜇 : 𝑆 𝑆 be the correspondence 𝜇 ( 𝑥 ) = 𝜑 ( 𝑥 ) 𝜓 ( 𝑥 ) where 𝜓 : 𝑆 𝑆 is the constant correspondence defined by 𝜓 ( 𝑥 ) = 𝑆 for every 𝑥 𝑆 . Then (a) Since 𝜑
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