Macroeconomics Exam Review 111

# Macroeconomics Exam Review 111 - Solutions for Foundations...

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Since 𝑔 is homogeneous 𝑔 ( x 1 + x 2 ) = 𝑔 ( x 1 ) + 𝑔 ( x 2 ) which shows that 𝑔 is additive and hence linear. Conversely if 𝑓 ( x ) = 𝑔 ( x ) + y with 𝑔 linear 𝑓 ( 𝛼 x 1 + (1 𝛼 ) x 2 ) = 𝛼𝑔 ( x 1 ) + (1 𝛼 ) 𝑔 ( x 2 ) + 𝑦 = 𝛼𝑔 ( x 1 ) + 𝑦 + (1 𝛼 ) 𝑔 ( x 2 ) + 𝑦 = 𝛼𝑓 ( x 1 ) + (1 𝛼 ) 𝑓 ( x 2 ) 3.40 Let 𝑆 be an aﬃne subset of 𝑋 and let y 1 , y 2 belong to 𝑓 ( 𝑆 ). Choose any x 1 𝑓 1 ( y 1 ) and x 2 𝑓 1 ( y 2 ). Then for any 𝛼 ∈ ℜ 𝛼 x 1 + (1 𝛼 ) x 2 𝑆 Since 𝑓 is aﬃne 𝛼 y 1 + (1 𝛼 ) y 2 = 𝛼𝑓 ( x 1 ) + (1 𝛼 ) 𝑓 ( x 2 ) = 𝑓 ( 𝛼 x 1 + (1 𝛼 ) x 2 ) 𝑓 ( 𝑆 ) 𝑓 ( 𝑆 ) is an aﬃne set. Let 𝑇 be an aﬃne subset of 𝑌 and let x 1 , x 2 belong to 𝑓 1 ( 𝑇 ). Let y 1 = 𝑓 ( x 1 ) and y 2 = 𝑓 ( x 2 ). Then y 1 , y 2 𝑇 . For every 𝛼 ∈ ℜ 𝛼 y 1 + (1 𝛼 ) y 2 = 𝛼𝑓 ( x 1 ) + (1 𝛼 ) 𝑓 ( x 2 ) 𝑇 Since 𝑓 is aﬃne, this implies that 𝑓 ( 𝛼 x 1 + (1 𝛼 ) x 2 ) = 𝛼𝑓 ( x 1 ) + (1 𝛼 ) 𝑓 ( x 2 ) 𝑇 Therefore 𝛼 x 1 + (1 𝛼 ) x 2 𝑓 1 ( 𝑇 ) We conclude that 𝑓 1 ( 𝑇 ) is an aﬃne set.
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