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Macroeconomics Exam Review 158

# Macroeconomics Exam Review 158 - Solutions for Foundations...

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3.176 Let 𝑓 be the production function. If 𝑓 is homothetic, there exists (Exercise 3.175) a linearly homogeneous function and strictly increasing function 𝑔 such that 𝑓 = 𝑔 . 𝑐 ( w , 𝑦 ) = min x { w 𝑇 x : 𝑓 ( x ) 𝑦 } = min x { w 𝑇 x : 𝑔 ( ( x )) 𝑦 } = min x { w 𝑇 x : ( x ) 𝑔 1 ( 𝑦 ) } = 𝑔 1 ( 𝑦 ) 𝑐 ( w , 1) by Exercise 3.166. 3.177 Let 𝑓 : 𝑆 → ℜ be positive, strictly increasing, homothetic and quasiconcave. By Exercise 3.175, there exists a linearly homogeneous function : 𝑆 → ℜ and strictly increasing function 𝑔 𝐹 ( 𝑅 ) such that 𝑓 = 𝑔 . = 𝑔 1 𝑓 is positive, quasiconcave (Exercise 3.148) and homogeneous of degree one. By Proposition 3.12, is concave and therefore 𝑓 = 𝑔 is concavifiable. 3.178 Since 𝐻 𝑓 ( 𝑐 ) is a supporting hyperplane to 𝑆 at x 0 , then 𝑓 ( x 0 ) = 𝑐 and either 𝑓 ( x ) 𝑐 = 𝑓 ( x 0 ) for every x 𝑆 or 𝑓 ( x ) 𝑐 = 𝑓 ( x 0 ) for every x 𝑆 3.179 Suppose to the contrary that y = ( ℎ, 𝑞 ) int 𝐴
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