Relationships with Constant Elasticity

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! That is, 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 $c$.

For example, the demand for electricity in the summertime, which is greater when it’s hot out and people want to run air conditioners, might be represented by an expression like $Q(P,T) = 100P^{-2}T^3$ This would imply that the price elasticity of demand is $-2$, while the elasticity of demand with respect to the temperature is $+3$.

Example: A demand curve with constant elasticity

Supply and demand curves plot quantity ($Q$) as a function of the market price ($P$). Therefore, the formula for the price elasticity of supply or demand is $\epsilon_{Q,P} = {\% \Delta Q \over \% \Delta P} = {dQ \over dP} \times {P \over Q}$ A demand or supply curve with constant elasticity will be of the form $Q(P) = k P^{\epsilon}$ where $k$ is some constant, and $\epsilon$ is the elasticity.

For example, the following diagram shows a demand curve with a constant elasticity of $-2$: specifically, $Q(P) = 3600P^{-2}$. Drag the price up and down to see how the quantity demanded changes, and to see how the elasticity calculations change (and don’t!) as you move along the demand curve.

If you’d like to play around with some more general graphs showing constant elasticity relationships, try out these additional demand and supply curve diagrams.

Relationship of constant elasticity and logs

A lot of what economists do is to try to estimate the relationships between variables: for example, they might want to estimate the relationship between quantity of electricity demanded, its price, and the temperature, in order to know how much hotter temperatures will increase demand, or how much consumers would reduce energy usage if they were charged more for electricity.

A common tool for estimating these kinds of relatioships is linear regression, in which data is plugged into a linear model of the form $y = a + bx_1 + cx_2 + \cdots$ The result of the regression is estimates of the values of $a$, $b$, and $c$.

If economists think the relationship between the variables looks something like $Q(P,T) = aP^bT^c$ then they can transform that equation by taking the natural log of both sides to get $\ln Q = \ln a + b \ln P + c \ln T$ and then use linear regression on the logs of the quantity, price, and temperature data to estimate $a$, $b$, and $c$. In other words: if there exists a multiplicative relationship between variables, then there is a linear relationship between the logs of those variables:

Because of this, economists often use logged variables in their linear regressions; and the results of those regressions are estimates of the elasticity of the endogenous variable with respect to each exogenous variable.