0
$\begingroup$

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.

$\endgroup$

2 Answers 2

2
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.


Possibly you meant to post this question:
Help with Income Elasticity Exercise in Becker's Economic Theory

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.