The idea
The price elasticity of demand is the percentage change in units sold for a 1% change in price. It is almost always negative. At −2, a 1% price rise loses about 2% of units; at −0.5, only about 0.5%.
Revenue is price × units, so elasticity tells you which way revenue moves. When demand is inelastic (between 0 and −1), units fall by a smaller percentage than price rises, and a price rise lifts revenue. When it is elastic (below −1), a rise cuts revenue.
To estimate elasticity from sales at different prices, fit ln(units) on ln(price): the slope is the elasticity, the same at every price. With the cost of one unit, a constant elasticity also gives the profit-maximising price directly.
The formula
b = % change in units% change in price ≈ ln(Q2 ÷ Q1)ln(P2 ÷ P1), P* = cost × b1 + b
- b
- the elasticity: the slope of ln(units) on ln(price)
- Q, P
- units sold and price, at two prices (or many, in a regression)
- cost
- the extra cost of making and selling one more unit
- P*
- the profit-maximising price, when b is below −1 and stays the same at every price
A worked example
A bakery tests its sourdough at two prices: it sells 1,000 loaves a month at $4 and 640 at $5. Each loaf costs $2 to make.
- Elasticity from logs: ln(640 ÷ 1,000) ÷ ln(5 ÷ 4) = −0.446 ÷ 0.223 = −2.0. Demand is elastic.
- So revenue falls with the price rise: $4 × 1,000 = $4,000, against $5 × 640 = $3,200.
- Best price: $2 × (−2.0) ÷ (1 − 2.0) = $4.00.
- Check the profit: at $4, ($4 − $2) × 1,000 = $2,000; at $5, ($5 − $2) × 640 = $1,920.
Keep the price at $4. At an elasticity of −2 the best price is twice the unit cost; the less elastic the demand, the bigger the best markup.
Try it
Price, units and profit
- Revenue if you raise the price 10%
- −9.1%
- Profit at today’s price
- $1,920
- Profit at the best price
- $2,000
- Units at the best price
- 1,000
Cutting the price toward $4.00 would earn $80 more a period: the extra units outweigh the lower margin.
The common mistake
Using simple percentage changes
From $4 to $5 the price rises 25% and units fall 36%, which gives −36 ÷ 25 = −1.44. Measured the other way, from $5 down to $4, units rise 56% while the price falls 20%: −2.81. Simple percentages give a different answer depending on the direction; logs give −2.0 both ways.
Remember
- Elasticity is the percentage change in units for a 1% change in price. Estimate it as the slope of ln(units) on ln(price).
- Inelastic demand (between 0 and −1): a price rise lifts revenue. Elastic demand (below −1): it lowers revenue.
- With a constant elasticity b below −1, the profit-maximising price is cost × b ÷ (1 + b). Near −1 that price runs away, so test it before trusting it.
Check yourself
Three questions from the Learn room’s bank. Your answers update your mastery of this skill, the same as in Practise.
Use it on real numbers
Open Price elasticity, in Marketing › Pricing and try the idea on your own numbers.