Price Elasticity of Demand

By the end of this article, you should be able to: define price elasticity and compute it, say when a price rise raises revenue and when it raises profit, explain why one elasticity number can mislead, and estimate elasticity when you can't randomise the price.

What elasticity measures

Price elasticity of demand is how strongly the amount people buy responds to the price. It's the percentage change in quantity for each 1% change in price:

$E = \frac{\%\ \text{change in quantity}}{\%\ \text{change in price}}$

It's negative for almost every product, because a higher price means fewer buyers. People often quote just its size, $|E|$ .

Uber's economists measured it for UberX in 2015, using almost 50 million app sessions in its four largest US cities. At one price step, the price rose 8.3% and the share of riders who went ahead with the trip fell about 3%:

$E \approx \frac{-3\%}{8.3\%} \approx -0.36$

Across the price steps they studied, most elasticities fell between −0.4 and −0.6. In their words, "demand is quite inelastic."

Why the size of E decides what a price rise does

Revenue is price times quantity. Raise the price 1% and quantity changes by $E$ %, so revenue changes by about $(1 + E)$ %. That gives three cases:

Elasticity	Name	Raise the price 10%	Revenue
$-0.5$	inelastic, $\lvert E \rvert \lt 1$	quantity falls about 5%	rises about 5%
$-1$	unit elastic	quantity falls about 10%	about flat
$-2$	elastic, $\lvert E \rvert \gt 1$	quantity falls about 20%	falls about 10%

Read top to bottom: when buyers barely react, a price rise brings in more money. When they react one-for-one, revenue doesn't move. When they react more than the price changed, the price rise costs you revenue.

At Uber's −0.5, raising fares would raise revenue. So why doesn't a company with inelastic demand just keep raising the price?