Coleca

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Joe I 05:19, 6 January 2006 (UTC)

this and that

Hello Coleca. For some people, wikipedia is a narcissistic venture. In anyone changes their text, for them it like smudging their very image, they react with derogatory language. In fact, such people actually have much to contribute - and they do - but their narcissism prevents other people from being constructive. C'est la vie. Politis 13:15, 31 January 2006 (UTC)

Showing that the sum of conditional demand elasticity is zero

Start with the definition of the cost function:

$C(w,q)=\min _{x}\{wx:q\geq F(x)\}$

and use the envelope theorem to get

${\frac {\partial C}{\partial w_{i}}}=x_{i}.$

So, we can get the summation

$\sum _{j=1}^{n}\epsilon _{ij}=\sum _{j=1}^{n}{\frac {\partial x_{i}}{\partial w_{j}}}{\frac {w_{j}}{x_{i}}}={\frac {1}{x_{i}}}\sum _{j=1}^{n}{\frac {\partial ^{2}C}{\partial w_{i}\partial w_{j}}}w_{j}$

$={\frac {1}{x_{i}}}\sum _{j=1}^{n}{\frac {\partial ^{2}C}{\partial w_{j}\partial w_{i}}}w_{j}={\frac {1}{x_{i}}}\sum _{j=1}^{n}{\frac {\partial x_{j}}{\partial w_{i}}}w_{j}={\frac {1}{x_{i}}}\left({\frac {\partial }{\partial w_{i}}}\sum _{i=1}^{n}x_{j}w_{j}-x_{i}\right)$

$={\frac {1}{x_{i}}}\left({\frac {\partial C}{\partial w_{i}}}-x_{i}\right)={\frac {1}{x_{i}}}\left(x_{i}-x_{i}\right)=0$

The simpler solution is to notice that a scale change in prices does not change the optimal choice of x, so the conditional demand is homogeneous degree zero. Then by Euler's Theorem

$\sum _{j=1}^{n}{\frac {\partial x_{i}}{\partial w_{j}}}w_{j}=0$

so

$\sum _{j=1}^{n}\epsilon _{ij}=\sum _{j=1}^{n}{\frac {\partial x_{i}}{\partial w_{j}}}{\frac {w_{j}}{x_{i}}}={\frac {1}{x_{i}}}\sum _{j=1}^{n}{\frac {\partial x_{i}}{\partial w_{j}}}w_{j}=0$