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Macroeconomics Exam Review 67

# Macroeconomics Exam Review 67 - Solutions for Foundations...

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(d) Suppose x z 1 x and x z 2 x with z 1 x = z 2 x . Then either z 1 x > z 2 x or z 1 x < z 2 x . Without loss of generality, assume z 2 x > z 1 x . Then monotonicity and transitivity imply x z 2 x z 1 x x which is a contradiction. Therefore z x is unique. Let 𝑧 x denote the scale of z x , that is z x = 𝑧 x 1 . For every x ∈ ℜ 𝑛 + , there is a unique z x x and the function 𝑢 : 𝑛 + → ℜ given by 𝑢 ( x ) = 𝑧 x is well-defined. Moreover x 2 x 1 ⇐⇒ z x 2 z x 1 ⇐⇒ 𝑧 x 2 𝑧 x 1 ⇐⇒ 𝑢 ( x 2 ) 𝑢 ( x 1 ) 𝑢 represents the preference order . 2.39 1. For every 𝑥 1 ∈ ℜ , ( 𝑥 1 , 2) 𝐿 ( 𝑥 1 , 1) in the lexicographic order. If 𝑢 represents 𝐿 , 𝑢 is strictly increasing and therefore 𝑢 ( 𝑥 1 , 2) > 𝑢 ( 𝑥 1 , 1). There exists a rational number 𝑟 ( 𝑥 1 ) such that 𝑢 ( 𝑥 1 , 2) > 𝑟 ( 𝑥 1 ) > 𝑢 ( 𝑥 1 , 1). 2. The preceding construction associates a rational number with every real number 𝑥 1 ∈ ℜ . Hence 𝑟 is a function from
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