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I get that they are both gauging responsiveness, but I don't particularly understand why they are the same formula.

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    $\begingroup$ Price elasticity is just a comparison of changes in quantity to changes in price. The supply and demand functions are both functions from these two variables, but their elasticities will end up being different since the changes are happening along different curves. You could always have a subscript to avoid ambiguity: $e_d = \frac{\frac{dQ_d}{Q}}{\frac{dP}{P}}$ $\endgroup$ – Kitsune Cavalry Sep 14 '17 at 2:36
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The elasticity is a very general concept. Consider an arbitrary function

$$y=f(x_1,x_2, \cdots , x_n) $$

The elasticity of $y$ with respect to $x_i$ is

$$ e_{y,x_i} = \frac{\partial y}{\partial x_i}\frac{x_i}{y} $$

It measures the percentage change of $y$ when variable $x_i$ changes by 1% - a measure of responsiveness, as you say. It can be applied to anything, including the demand and the supply, where you get the common concepts like own-price, cross-price, income elasticity, etc. The sign of the elasticity is given by the sign of the derivative, which depends on the function $f(\cdot)$.

Elasticities applied to functions other than standard demand and supply abound. For example, individual factors of production, relative factors of production, investment, taxation, preferences, city size, et cetera.

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