Throughout microeconomic literature i see the following restiriction placed on the nature of elasticities in demand system estimation.
This being for some arbitarary good $x$ we require price elasticities and income elasticities to be:
$$\varepsilon(x,p_x)+\varepsilon(x,p_y)+\varepsilon(x,I)=0$$
What is the proof for this? (I cant seem to find it).
Note that the Marshallian demand function $x^*(p_x,p_y,I)$ is homogeneous degree zero in $(p_x,p_y,I)$ (see here for a proof). According to Euler's theorem for homogeneous function, it follows that \begin{equation} \frac{\partial x^*}{\partial p_x}p_x+\frac{\partial x^*}{\partial p_y}p_y+\frac{\partial x^*}{\partial I}I=0\cdot x(p_x,p_y,I)=0 \end{equation} Dividing both sides by $x^*$, you have \begin{align} \frac{\partial x^*}{\partial p_x}\frac{p_x}{x^*}+\frac{\partial x^*}{\partial p_y}\frac{p_y}{x^*}+\frac{\partial x^*}{\partial I}\frac{I}{x^*}&=0\\ \epsilon_{xx}+\epsilon_{xy}+\epsilon_{xI}&=0 \end{align}
