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 ? represents the homothetic preference relation . For any x 1 , x 2 ± ? ( x 1 )= ? ( x 2 )= x 1 x 2 = ² x 1 ² x 2 = ? ( ² x 1 )= ? ( ² x 2 ) for every ²> 0 Conversely, if ? is a homothetic functional, x 1 x 2 = ? ( x 1 )= ? ( x 2 )= ? ( ² x 1 )= ? ( ² x 2 )= ² x 1 ² x 2 for every ²> 0 3.173 Suppose that ³ = ´ where ´ is strictly increasing and is homogeneous of degree µ . Then ˆ ( x )= ( ( x ) ) 1 /? is homogeneous of degree one and ³ ´ ˆ where ˆ ´ ( )= ´ ( ? ) ) is increasing. 3.174 Assume x 1 , x 2 ± with ³ ( x 1 )= ´ ( ( x 1 )) = ´ ( x 2 )) = ³ ( x 2 ) Since ´ is strictly increasing, this implies that ( x 1 )= ( x 2 ) Since is homogeneous ( ² x 1 )= ² ? ( x 1 )= ² ? ( x 2 )= ( ² x 2 ) for some µ . Therefore ³ ( ² x 1 )= ´ ( ( ² x 1 )) = ´ ( ( ² x 2 )) = ³ ( ² x 2 ) 3.175 Let x 0 = 0 be any point in ± , and de±ne
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## This note was uploaded on 01/16/2012 for the course ECO 2024 taught by Professor Dr.dumond during the Fall '10 term at FSU.

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