Macroeconomics Exam Review 206

Macroeconomics Exam Review 206 - The function is the...

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Unformatted text preview: The function is the composition of with and , ( x , y ) = ( ( x ) , ( y )) By the chain rule, the derivative of is [ x , y ] = [ 1 , z 2 ] ( [ x ] , [ x ] ) = ( z 1 , [ y ]) + ( [ x ] , z 2 ) = [ x ] [ y ]) + ( y ) [ x ] where z 1 = ( x ) and z 2 = ( y ). 4.25 For = 1, ( ) = is linear and therefore (Exercise 4.6) [ ] = 1 ( [ ]( ) = ). For = 2, let ( ) = so that ( ) = 2 = ( ) ( ). Using the product rule [ x ] = ( ) ( ) + ( ) ( ) = 2 Now assume it is true for 1 and let ( ) = 1 , so that ( x ) = ( ). By the product rule [ x ] = [ ] + ( )1 By assumption [...
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