Lecture 9 · Barnett ch. 3 · about 40 min

Elasticity of Demand

Why percentage changes matter more than absolute ones in pricing decisions, the calculus definition of point elasticity, and the exact link between elasticity and whether a price change raises or lowers revenue.

By the end you can

Explain why elasticity uses percentage changes rather than raw slopes
Compute point elasticity of demand from a demand function using its derivative
Classify demand as elastic, inelastic or unit elastic at a given price
Predict the effect of a price change on total revenue from the elasticity

Why not just use the slope?

The slope of a demand curve, $f'(p)$ , mixes units of quantity and units of currency, so it depends on whether quantity is measured in single items or in thousands, and whether price is in kuruş or lira. Elasticity fixes this by comparing percentage changes, a pure number that does not depend on units at all.

Point elasticity of demand

Let $x = f(p)$ be a demand function: quantity demanded as a function of price. Point elasticity measures the percentage response of quantity to a percentage change in price, evaluated at a specific price, using the derivative.

Point elasticity of demandKey formulaFormula sheet →

E(p) = -\,\frac{p\,f'(p)}{f(p)}

$p$: price
$f(p)$: quantity demanded at price p
$f'(p)$: derivative of demand with respect to price

Use it when you have a demand function x = f(p) and need the elasticity at a specific price. The minus sign makes E positive, since f'(p) is normally negative (demand curves slope down).

Elastic, inelastic and unit elastic

Classifying demandRulesFormula sheet →

E(p) \gt 1: \text{ elastic} \qquad E(p) = 1: \text{ unit elastic} \qquad E(p) \lt 1: \text{ inelastic}

$E(p)$: point elasticity of demand at price p

Use it when you need to classify demand at a given price, as a step toward predicting the revenue effect of a price change.

At $p=50$ (from the last check), $E = 0.333 \lt 1$ : demand is inelastic there. At $p = 150$ : $f(150) = 250$ , $f'(150)=-5$ , so $E(150) = \frac{150 \times 5}{250} = 3 \gt 1$ : elastic.

Elasticity and total revenue

This is the calculation that makes elasticity indispensable for pricing: it tells you whether raising price raises or lowers revenue, without needing to try it.

Revenue and elasticityKey resultFormula sheet →

R(p) = p\,f(p) \quad\Longrightarrow\quad R'(p) = f(p)\big[1 - E(p)\big]

$R(p)$: total revenue as a function of price

Use it when you need to predict whether a price increase raises or lowers revenue at the current price.

Since $f(p) \gt 0$ , the sign of $R'(p)$ matches the sign of $1 - E(p)$ :

Inelastic ( $E \lt 1$ ): $R'(p) \gt 0$ . Raising price raises revenue: the percentage drop in quantity is smaller than the percentage rise in price.
Elastic ( $E \gt 1$ ): $R'(p) \lt 0$ . Raising price lowers revenue: quantity falls proportionally more than price rises.
Unit elastic ( $E = 1$ ): $R'(p) = 0$ . Revenue is at a local maximum or minimum with respect to price.

Try it: watch revenue peak at unit elasticity

Every linear demand curve is elastic near the top (high price, low quantity) and inelastic near the bottom (low price, high quantity), with exactly one unit-elastic point in between — precisely where total revenue peaks. This is not a coincidence: it is the formula $R'(p) = f(p)[1-E(p)]$ made visual.

Exam practice

Summary and review

Elasticity uses percentage changes so comparisons do not depend on units.
Point elasticity: $E(p) = -pf'(p)/f(p)$ , for demand $x = f(p)$ .
$E \gt 1$ elastic; $E = 1$ unit elastic; $E \lt 1$ inelastic.
$R'(p) = f(p)[1 - E(p)]$ : revenue rises with price when inelastic, falls with price when elastic, and is stationary at unit elasticity.
A linear demand curve is elastic at high prices, inelastic at low prices, with exactly one unit-elastic point, which is also revenue’s maximum.