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

Question

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.

Answer 1

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.

Answer 2

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