# Proof for homogeneity of elasticities

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).

• Did you mean the Engel aggregation? – Herr K. Nov 21 '18 at 3:10
• Whoops wrote the condtion wrong. Edited – EconJohn Nov 21 '18 at 3:56

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 $$$$\frac{\partial x^*}{\partial p_x}p_x+\frac{\partial x^*}{\partial p_y}p_y+\frac{\partial x^*}{\partial I}I=0\cdot x(p_x,p_y,I)=0$$$$ Dividing both sides by $$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}