User talk:Coleca

Welcome
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Hope to see you around the Wiki! If you have any questions whatsoever, feel free to contact me on my talk page :)  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: $$ C(w,q)=\min_{x}\{w x:q\ge F(x)\} $$ 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 $$ \sum_{j=1}^n\epsilon_{ij}=\sum_{j=1}^n\frac{\partial x_i}{\partial w_j}\frac{w_j}{x_i} =\frac{1}{x_i}\sum_{j=1}^n\frac{\partial x_i}{\partial w_j}w_j=0 $$