Macroeconomics Exam Review 202

Macroeconomics Exam Review 202 - Solutions for Foundations...

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4.9 To show that 𝑟 is nonlinear, consider 𝑟 ((1 , 2 , 3 , 4 , 5) + (66 , 55 , 75 , 81 , 63)) = 𝑟 (67 , 57 , 78 , 85 , 68) = (85 , 78 , 68 , 67 , 58) = (5 , 4 , 3 , 2 , 1) + (81 , 75 , 67 , 63 , 55) To show that 𝑟 is differentiable, consider a particular point, say (66 , 55 , 75 , 81 , 63). Consider the permutation 𝑔 : 𝑛 → ℜ 𝑛 defined by 𝑔 ( 𝑥 1 , 𝑥 2 , . . . , 𝑥 5 ) = ( 𝑥 4 , 𝑥 3 , 𝑥 1 , 𝑥 5 , 𝑥 2 ) 𝑔 is linear and 𝑔 (66 , 55 , 75 , 81 , 63) = (81 , 75 , 67 , 63 , 55) = 𝑟 (66 , 55 , 75 , 81 , 63) Furthermore, 𝑔 ( x ) = 𝑟 ( x ) for all x close to (66 , 55 , 75 , 81 , 63). Hence, 𝑔 ( x ) approxi- mates 𝑟 ( x ) in a neighborhood of (66 , 55 , 75 , 81 , 63) and so 𝑟 is differentiable at (66 , 55 , 75 , 81 , 63). The choice of (66 , 55 , 75 , 81 , 63) was arbitrary, and the argument applies at every x such that x 𝑖 = x 𝑗 . In summary, each application of 𝑟 involves a permutation, although the particular permutation depends upon the argument, x . However, for any given x 0 with x 0 𝑖 = x 0 𝑗 , the same permutation applies to all x in the neighborhood of x 0 , so that the permutation (which is a linear function) is the derivative of 𝑟 at x 0 . 4.10 Using (4.3), we have for any
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