Elasticity Math
Dr. Staffan Canback, Tellusant
To my pleasant surprise, this is my most popular substance-oriented post ever. There is hope.
I sometimes get the question “why do you use logarithms when calculating elasticities?” Rather than answering each time, I decided to answer once and point people to this post. Ask many, answer once.
I assume the reader knows elementary calculus.
\[\mathbf{Elasticity\ \epsilon\ is\ defined\ as\ a\ change\ in\ input\ x\ leads\ to\ a\ change\ in\ output\ y}\] \[\textcolor{#c00000}{\epsilon=\frac{\Delta y}{y}/\frac{\Delta x}{x}}\] \[\mathbf{Take\ \mathrm{\Delta\} \ to\ the\ limit\ 0\ and\ you\ get}\] \[\textcolor{#c00000}{\epsilon=\ \frac{x}{y}\frac{dy}{dx}}\] \[\mathbf{Reconfigure\ the\ equation}\] \[\textcolor{#c00000}{\frac{dy}{y}=\epsilon\frac{dx}{x}}\] \[\mathbf{Integrate,\ remembering\ that\ the\ integral\ of\ \frac{1}{var}=\ln(var)}\] \[\textcolor{#c00000}{\ln{\left(y\right)}=\epsilon\ \cdot\ \mathrm{ln}{\left(x\right)}}\] \[\mathbf{Reordered}\] \[\textcolor{#c00000}{\epsilon =\ln{\left(y\right)}/\ln{\left(x\right)}}\] \[\mathbf{Which\ is\ equal\ to\ the\ first\ equation}\] \[\textcolor{#c00000}{\epsilon =\ \frac{\Delta y}{y}/\frac{\Delta x}{x}}\] \[\mathbf{\mathrm{Q.E.D.}}\]The reason the logarithm is taken is that it is equivalent to working with percentage changes, and it makes for a nice linear regression.¹
The equations shown explain this.² Think, for example, of y=demand and x=price. Then the elasticity tells us how much demand falls if price increases.³
What I have shown is in any relevant text book. Thus, no magic here, just a refresher.
Tellusant, Inc. sometimes works with this linear equation, but in most cases we use a nonlinear approach that is more complicated. As Angus Deaton, Nobel Prize winner in Economics pointed out:
“…[the linear regression model]…is not consistent with utility maximization…and will eventually lead to gross over-prediction”
¹ It also is a transformation that make observations normally distributed in many cases, thus reducing heteroscedasticity.
² Some readers may ask: Where is the constant C in the integration. I set it to zero for simplicity.
³ There are cases when demand increases when price increases. Those cases are rare.
AI was not used.