<p>Note that the Marshallian demand function <span class="math-container">$x^*(p_x,p_y,I)$</span> is homogeneous degree zero in <span class="math-container">$(p_x,p_y,I)$</span> (see <a href="https://economics.stackexchange.com/q/12308/42">here</a> for a proof). According to Euler's theorem for homogeneous function, it follows that 
<span class="math-container">\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}</span>
Dividing both sides by <span class="math-container">$x^*$</span>, you have 
<span class="math-container">\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^*}&amp;=0\\
\epsilon_{xx}+\epsilon_{xy}+\epsilon_{xI}&amp;=0
\end{align}</span></p>

CC BY-SA 4.0
2018-11-21T16:36:02.413Z
2018-11-21T16:36:02.413Z
Herr K.
<p>Throughout microeconomic literature i see the following restiriction placed on the nature of elasticities in demand system estimation. </p>

<p>This being for some arbitarary good <span class="math-container">$x$</span> we require price elasticities and income elasticities to be:</p>

<p><span class="math-container">$$\varepsilon(x,p_x)+\varepsilon(x,p_y)+\varepsilon(x,I)=0$$</span></p>

<p>What is the proof for this? (I cant seem to find it).</p>

CC BY-SA 4.0
2018-11-21T03:10:09.143Z
Did you mean the [Engel aggregation](https://economics.stackexchange.com/q/18839/42)?
Herr K.
CC BY-SA 4.0
2018-11-21T03:56:37.237Z
Whoops wrote the condtion wrong. Edited
EconJohn
<microeconomics><consumer-theory><elasticity>
Proof for homogeneity of elasticities
EconJohn
# Proof for homogeneity of elasticities

(Asked by: username_0 on 2018-11-21)

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).

**Tags:** microeconomics, consumer-theory, elasticity

> **Comments:**
> **username_1**: Did you mean the [Engel aggregation](https://economics.stackexchange.com/q/18839/42)?  
> **username_0**: Whoops wrote the condtion wrong. Edited  

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## 1 answers

### Answer by username_1 on 2018-11-21. Score: 3 | Accepted answer

Note that the Marshallian demand function $x^\*(p\_x,p\_y,I)$ is homogeneous degree zero in $(p\_x,p\_y,I)$ (see [here](https://economics.stackexchange.com/q/12308/42) 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}


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