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November 30th, 2016, 02:34 AM   #1
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So I took this off a course book I'm reading on microeconomics





note: $\displaystyle {x_1} = \phi (\bar z + {y^h} - x_2^h)$


I tried using the chain rule:
$\displaystyle {{dU} \over {dz}} = {{dU} \over {d{x_1}}}{{d{x_1}} \over {dz}} + {{dU} \over {d{x_2}}}{{d{x_2}} \over {dz}}$

where I got the same expression EXCEPT for the first term, so I don't understand where the (-)sign come from:
$\displaystyle - U_1^h({x_1},x_2^h)$

Last edited by skipjack; November 30th, 2016 at 06:15 AM.
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November 30th, 2016, 08:35 AM   #2
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update: So instead of applying the normal chain rule, since:



$\displaystyle \eqalign{
& z = U({x_1},x_2^h) \cr
& {x_1} = g({x_2}) \cr
& {{dz} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}}{{d{x_2}} \over {d{x_2}}} = {{\partial f} \over {\partial {x_1}}}{{d{x_1}} \over {d{x_2}}} + {{\partial f} \over {\partial {x_2}}} \cr} $


, since $\displaystyle {x_1}$ is function of $\displaystyle x_2^h$.
This seems to yield the correct expression, or maybe I've missed something ?

Last edited by Ku5htr1m; November 30th, 2016 at 08:41 AM.
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November 30th, 2016, 03:55 PM   #3
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This is from F.A.Cowell - Microeconomics - Principles and Analysis p.452-453
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