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Macroeconomics Exam Review 224

Macroeconomics Exam Review 224 - Solutions for Foundations...

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Differentiating with respect to 𝑥 𝑗 𝑘𝐷 𝑗 𝑓 ( x ) = 𝑛 𝑖 =1 𝐷 𝑖𝑗 𝑓 ( x ) 𝑥 𝑖 + 𝐷 𝑗 𝑓 ( x ) or ( 𝑘 1) 𝐷 𝑗 𝑓 ( x ) = 𝑛 𝑖 =1 𝐷 𝑖𝑗 𝑓 ( x ) 𝑥 𝑖 𝑗 = 1 , 2 , . . . , 𝑛 Multiplying each equation by 𝑥 𝑗 and summing ( 𝑘 1) 𝑛 𝑗 =1 𝐷 𝑗 𝑓 ( x ) 𝑥 𝑗 = 𝑛 𝑗 =1 𝑛 𝑖 =1 𝐷 𝑖𝑗 𝑓 ( x ) 𝑥 𝑖 𝑥 𝑗 = x 𝐻 x By Euler’s theorem, the left hand side is ( 𝑘 1) 𝑘𝑓 ( x ) = x 𝐻 x 4.85 If 𝑓 is homothetic, there exists strictly increasing 𝑔 and linearly homogeneous such that 𝑓 = 𝑔 (Exercise 3.175). Using the Chain Rule and Exercise 4.78 𝐷 𝑥 𝑖 𝑓 ( 𝑡 x ) = 𝑔 ( 𝑓 ( 𝑡 x )) 𝐷 𝑥 𝑖
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