<p>Let <span class="math-container">$x_i(p_1, \ldots, p_i, \ldots, p_n, m)$</span> be the demand function of good <span class="math-container">$i$</span> in terms of all prices and income <span class="math-container">$m$</span>. Then we know that this function is homogeneous of degree 0 in prices and income. So by Euler's theorem, we have that:
<span class="math-container">$$
\sum_j \frac{\partial x_i}{\partial p_j} p_j + \frac{\partial x_i}{\partial m} m = 0.
$$</span>
Dividing both sides by <span class="math-container">$x_i$</span> gives:
<span class="math-container">$$
\begin{align}
&amp;\sum_j \frac{\partial x_i}{p_j}\frac{p_j}{x_i} + \frac{\partial x_i}{\partial m} \frac{m}{x_i} = 0,\\
\leftrightarrow &amp; \varepsilon^i_i = - \sum_{j \ne i} \varepsilon^j_i - \varepsilon^i_m
\end{align}
$$</span>
Where <span class="math-container">$\varepsilon^i_j$</span> is the <span class="math-container">$p_j$</span>-elasticity of good <span class="math-container">$i$</span>.</p>
<p>So the own price elasticity is the negative of the sum of all the cross price elasticities minus the income elasticity.</p>

CC BY-SA 4.0
2023-07-25T09:39:34.123Z
2023-07-25T09:39:34.123Z
tdm
<blockquote>
<p>I encountered the claim</p>
</blockquote>
<p>Hard to give a decent response, since no source is given, hence no context is available for the claim!</p>
<blockquote>
<p>equal to the sum of all cross-price elasticities</p>
</blockquote>
<p>This is not generally true, e.g.; demands derived from Cobb-Douglas type preferences will provide counterexamples to these claims, as a good's own-price-elasticity will be negative, while their cross-price-elasticity will be zero.</p>
<blockquote>
<p>Also on an intuitive level.</p>
</blockquote>
<p>Hard to provide non-mathematical intuition for why the statement is true when the statement is, in fact, false.</p>
<hr />
<p>Possibly you meant to post this question: <br>
<a href="https://economics.stackexchange.com/q/2983/1601">Help with Income Elasticity Exercise in Becker's Economic Theory</a></p>

CC BY-SA 4.0
2023-06-12T13:39:38.780Z
2023-06-12T13:39:38.780Z
Giskard
<p>I encountered the claim stated in the title, without it being further substantiated. I'm looking for some elaboration on it, assuming it's true. Also on an intuitive level.</p>

<elasticity><cross-price-elasticity>
Is the price elasticity of a product equal to the sum of all cross-price elasticities of demand for that product? Why?
Nasan
# Is the price elasticity of a product equal to the sum of all cross-price elasticities of demand for that product? Why?

(Asked by: username_0 on 2023-06-12)

I encountered the claim stated in the title, without it being further substantiated. I'm looking for some elaboration on it, assuming it's true. Also on an intuitive level.

**Tags:** elasticity, cross-price-elasticity


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## 2 answers

### Answer by username_1 on 2023-07-25. Score: 2 | Accepted answer

Let $x\_i(p\_1, \ldots, p\_i, \ldots, p\_n, m)$ be the demand function of good $i$ in terms of all prices and income $m$. Then we know that this function is homogeneous of degree 0 in prices and income. So by Euler's theorem, we have that:
$$
\sum\_j \frac{\partial x\_i}{\partial p\_j} p\_j + \frac{\partial x\_i}{\partial m} m = 0.
$$
Dividing both sides by $x\_i$ gives:
$$
\begin{align}
&\sum\_j \frac{\partial x\_i}{p\_j}\frac{p\_j}{x\_i} + \frac{\partial x\_i}{\partial m} \frac{m}{x\_i} = 0,\\
\leftrightarrow & \varepsilon^i\_i = - \sum\_{j \ne i} \varepsilon^j\_i - \varepsilon^i\_m
\end{align}
$$
Where $\varepsilon^i\_j$ is the $p\_j$-elasticity of good $i$.

So the own price elasticity is the negative of the sum of all the cross price elasticities minus the income elasticity.


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### Answer by username_2 on 2023-06-12. Score: 0

> I encountered the claim

Hard to give a decent response, since no source is given, hence no context is available for the claim!

> equal to the sum of all cross-price elasticities

This is not generally true, e.g.; demands derived from Cobb-Douglas type preferences will provide counterexamples to these claims, as a good's own-price-elasticity will be negative, while their cross-price-elasticity will be zero.

> Also on an intuitive level.

Hard to provide non-mathematical intuition for why the statement is true when the statement is, in fact, false.

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Possibly you meant to post this question:   
[Help with Income Elasticity Exercise in Becker's Economic Theory](https://economics.stackexchange.com/q/2983/1601)


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