C.5 Relationships with Constant Elasticity

We’ve just established the general formula for elasticity using calculus is \(\epsilon = \frac{dy}{dx} \times \frac{x}{y}\) There’s a special case of this in which elasticity is constant: specifically, when the relationship between the exogenous and endogenous variables is multiplicative. For example, consider the relationship \(y = ax^b\) In this case \(\frac{dy}{dx} = abx^{b-1}\) so the elastictiy is \(\begin{aligned} \epsilon &= abx^{b-1} \times \frac{x}{ax^b}\\ &= \frac{abx^b}{ax^b}\\ &= b \end{aligned}\) This extends to more than one independent variable! (Note: This is perhaps the most important result for this topic! Read it carefully!) If you see some function of the form \(y = x_1^ax_2^bx_3^c\)the elasticity of $Y$ with respect to $X_1$ is going to be $a$, with respect to $X_2$ is going to be $b$, and with respect to $X_3$ is going to be $c$.

Example 1: One Exogenous Variable

For example, consider a demand function given by \(Q = 4P^{-\frac{1}{2}}\) The price elasticity of demand for this function is \(\begin{aligned} \epsilon_{Q,P} &= -2P^{-\frac{3}{2}} \times \frac{P}{4P^{-\frac{1}{2}}}\\ &= -\frac{2P^{-\frac{1}{2}}}{4P^{-\frac{1}{2}}}\\ &= -\frac{1}{2} \end{aligned}\)

Example 2: Multiple Exogenous Variables

Suppose a firm has the production function $q = f(L,K) = L^\frac{2}{3}K^\frac{1}{3}$. We can define a firm’s output elasticity of labor $\epsilon_{q,L}$ as the percentage increase in output due to a 1% increase in its labor force. Even though this function has multiple arguments, we can use the partial derivative of $q$ with respect to $L$ to calculate the output elasticity of labor, holding capital constant: \(\begin{aligned} \epsilon_{q,L} &= \frac{\partial q}{\partial L} \times \frac{L}{q}\\ &= \frac{2}{3}L^{-\frac{1}{3}}K^\frac{1}{3} \times \frac{L}{L^\frac{2}{3}K^\frac{1}{3}}\\ &= \tfrac{2}{3} \end{aligned}\)