Macroeconomics Exam Review 203

Macroeconomics Exam Review 203 - Solutions for Foundations...

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4.12 DeFne the function ( ? )= ± ( (8 , 8) + ? (1 , 1) ) =(8+ ? ) 1 / 3 (8 + ? ) 2 / 3 =8+ ? The directional derivative of ± in the direction (1 , 1) is ² (1 , 1) ± (8 , 8) = lim ? 0 ( ? ) (0) ? =1 Generalization of this example reveals that the directional derivative of ± along any ray through the origin equals 1, that is ² x 0 ± [ x 0 ] = 1 for every x 0 . Economically, this means that increasing inputs in the same proportions leads to a proportionate increase in output, which is the property of constant returns to scale. We will study this property of homogeneity is some depth in Section 4.6. 4.13 Let p = ± ( x 0 ). Each component of p represents the action of the derivative on an element of the standard basis { e 1 , e 2 ,..., e ± } (see proof of Theorem 3.4) ³ ² = ²± [ x 0 ]( e ² ) Since e ² =1, [ x 0 ]( e ² ) is the directional derivative at x 0 in the direction e ² (Exer- cise 4.10) ³ ² = [ x 0 ]( e ² ² e ? ( x 0 ) But this is simply the ´ partial derivative of ± (Exercise 4.11) ³ ² = [ x 0 ]( e ²
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