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Macroeconomics Exam Review 217

# Macroeconomics Exam Review 217 - Solutions for Foundations...

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3. Since 𝑆 is open, 𝑇 = 𝑓 1 ( 𝑆 ) is open. Therefore, 𝑇 = 𝑓 ( 𝑆 ) is a neighborhood of 𝑓 ( x 0 ). Therefore, 𝑓 is locally onto. 4.62 Assume to the contrary that there exists x 0 = x 1 𝑆 with 𝑓 ( x 0 ) = 𝑓 ( x 1 ). Let x = x 1 x 0 . Define 𝑔 : [0 , 1] 𝑆 by 𝑔 ( 𝑡 ) = (1 𝑡 ) x 0 + 𝑡 x 1 = x 0 + 𝑡 x . Then 𝑔 (0) = x 0 𝑔 (1) = x 1 𝑔 ( 𝑡 ) = x Define ( 𝑡 ) = x 𝑇 ( 𝑓 ( 𝑔 ( 𝑡 ) ) 𝑓 ( x 0 ) ) Then (0) = 0 = (1) By the mean value theorem (Mean value theorem), there exists 0 < 𝛼 < 1 such that 𝑔 ( 𝛼 ) 𝑆 and ( 𝛼 ) = x 𝑇 𝐷𝑓 [ 𝑔 ( 𝛼 )] x = x 𝑇 𝐽 𝑓 ( 𝑔 ( 𝛼 )) x = 0 which contradicts the definiteness of 𝐽 𝑓 . 4.63 Substituting the linear functions in (4.35) and (4.35), the IS-LM model can be expressed as (1 𝐶 𝑦 ) 𝑦 𝐼 𝑟 𝑟 = 𝐶 0 + 𝐼 0 + 𝐺 𝐶 𝑦 𝑇 𝐿 𝑦 𝑦 + 𝐿 𝑟 𝑟 = 𝑀/𝑃 which can be rewritten in matrix form as ( 1 𝐶 𝑦 𝐼 𝑟 𝐿 𝑦 𝐿 𝑟 ) ( 𝑦 𝑟 ) = ( 𝑍 𝐶 𝑦 𝑇 𝑀/𝑃 ) where 𝑍 = 𝐶 0 + 𝐼 0 + 𝐺 . Provided the system is nonsingular, that is 𝐷 = 1 𝐶 𝑦 𝐼 𝑟 𝐿 𝑦 𝐿 𝑟 = 0 the system can be solved using Cramer’s rule (Exercise 3.103) to yield
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