C.4 Calculating Elasticity Using Calculus

We can rewrite our general formula \(\epsilon_{Y,X} = \frac{\frac{\Delta y}{y}}{\frac{\Delta x}{x}}\) as \(\epsilon_{Y,X} = \frac{\Delta y}{\Delta x} \times \frac{x}{y}\) If we take the limit as the change in the exogenous variable $\Delta x$ gets smaller, we write this in the standard calculus way: \(\lim_{\Delta X \rightarrow 0} \epsilon_{Y,X} = \frac{dy}{dx} \times \frac{x}{y}\) If we know the formula for the relationship between the exogenous variable $X$ and the endogenous variable $Y$ – that is, if $y = f(x)$ – then we can write the instantaneous elasticity as a function of $x$ as \(\epsilon(x) = f^\prime (x) \times \frac{x}{f(x)}\) Note that this is all written as a function of the exogenous variable: for example, if we were to think of the price elasticity of demand, we would ask what the elasticity is at a certain price $P$ (the exogenous variable), not at a certain quantity $Q$ (the endogenous variable).

Example

Suppose that the relationship between $X$ and $Y$ may be given by the function $y = f(x) = 10 + 2x + \frac{1}{4}x^2$. Then the elasticity of $Y$ with respect to $X$ is \(\epsilon_{Y,X} = \frac{dy}{dx} \times \frac{x}{y} = (2 + \tfrac{1}{2}x) \times \frac{x}{10 + 2x + \frac{1}{4}x^2} = \frac{2x + \frac{1}{2}x^2}{10 + 2x + \frac{1}{4}x^2}\) This isn’t constant; it varies as $x$ changes. However, there is another special case worth examining: the case of constant elasticity.