Elasticity Math

Dr. Staffan Canback, Tellusant

This is my most popular substance-oriented post ever.

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.

$Elasticity math$

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.

[2025-06-25]