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Macroeconomics Exam Review 157

# Macroeconomics Exam Review 157 - Solutions for Foundations...

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3.172 Continuity is a necessary and suﬃcient condition for the existence of a utility function representing (Remark 2.9). Suppose 𝑢 represents the homothetic preference relation . For any x 1 , x 2 𝑆 𝑢 ( x 1 ) = 𝑢 ( x 2 ) = x 1 x 2 = 𝑡 x 1 𝑡 x 2 = 𝑢 ( 𝑡 x 1 ) = 𝑢 ( 𝑡 x 2 ) for every 𝑡 > 0 Conversely, if 𝑢 is a homothetic functional, x 1 x 2 = 𝑢 ( x 1 ) = 𝑢 ( x 2 ) = 𝑢 ( 𝑡 x 1 ) = 𝑢 ( 𝑡 x 2 ) = 𝑡 x 1 𝑡 x 2 for every 𝑡 > 0 3.173 Suppose that 𝑓 = 𝑔 where 𝑔 is strictly increasing and is homogeneous of degree 𝑘 . Then ˆ ( x ) = ( ( x ) ) 1 /𝑘 is homogeneous of degree one and 𝑓 = ˆ 𝑔 ˆ where ˆ 𝑔 ( 𝑦 ) = 𝑔 ( 𝑦 𝑘 ) ) is increasing. 3.174 Assume x 1 , x 2 𝑆 with 𝑓 ( x 1 ) = 𝑔 ( ( x 1 )) = 𝑔 ( x 2 )) = 𝑓 ( x 2 ) Since 𝑔 is strictly increasing, this implies that ( x 1 ) = ( x 2 ) Since is homogeneous ( 𝑡 x 1 ) = 𝑡 𝑘 ( x 1 ) = 𝑡 𝑘 ( x 2 ) = ( 𝑡 x 2 ) for some 𝑘 . Therefore 𝑓 ( 𝑡 x 1 ) = 𝑔 ( ( 𝑡 x 1 )) = 𝑔 ( ( 𝑡 x 2 )) = 𝑓 ( 𝑡 x 2 ) 3.175 Let x 0 = 0 be any point in 𝑆
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