Price elasticity of demand
Learning goals
After this page, you can:
- Compute arc elasticity from two price–quantity observations.
- Label demand elastic, inelastic or unit elastic, and say what each means about customers.
- Name the four things that make demand more elastic, and compute income and cross-price elasticity.
The idea
A gym raises its fee from $40 to $48 and loses 10% of its members. The owner thinks it was a disaster. The numbers say the members barely reacted. This page shows you how to read them.
Price elasticity of demand answers one question: if I change my price by 1 percent, by what percent does the quantity I sell change? Quantity goes on top. Price goes on the bottom. Because quantity falls when price rises, the number is negative. Managers usually drop the sign and talk about its size: “elasticity of 0.8” means a 1% price rise loses 0.8% of sales.
The dividing line is 1.
- |ε| > 1 — elastic. Customers are price-sensitive. A 1% price rise loses more than 1% of sales. Think leisure airline seats, one brand of bottled water, restaurant meals.
- |ε| < 1 — inelastic. Customers barely react. Gasoline this month, insulin, your phone plan, the only coffee cart on campus.
- |ε| = 1 — unit elastic. The two percentages match.
Why percent, not units? Suppose a $1 price rise costs you 100 customers. Is that a lot? For a $2 product selling 300 units, it is a disaster. For a $200 product selling 100,000 units, it is nothing. Percentages make the number comparable across products and companies.
Computing it from two points. You rarely know the whole curve. You usually have two observations: a price before and after a change, and the quantities sold. Use the midpoint (arc) formula in the box below. It gives the same answer in both directions. For the gym: %ΔQ = −100 ÷ 950 = −10.5%. %ΔP = 8 ÷ 44 = +18.2%. ε = −10.5 ÷ 18.2 = −0.58. Inelastic: members grumble, but most stay.
If you know the demand line Q = a + b·P, the point elasticity at any price is ε = b × P ÷ Q. Same idea, no second observation needed. The chart below uses this.
What makes demand more elastic. Four things. They are also a checklist for your own product.
- Close substitutes. One gas station in town: inelastic. One of six on the same corner: very elastic.
- Share of the budget. A 10% rise in the price of salt changes nothing. A 10% rise in rent changes behavior.
- Time. Demand is more elastic in the long run. Drivers cannot do much about expensive gas this week. Over three years they buy smaller cars and move closer to work.
- Necessity or luxury, as the buyer sees it. Insulin is inelastic for a diabetic. A third streaming subscription is elastic for almost everyone.
The first item is something a firm partly controls. Brands, loyalty programs, switching costs and differentiation all work by removing close substitutes in the buyer’s mind — that is, by making demand less elastic.
Two cousins. The same ratio works with a different cause on the bottom. Income elasticity = %ΔQ ÷ %Δincome. Positive means a normal good; above 1 means a luxury; negative means an inferior good. Cross-price elasticity = %ΔQ of good A ÷ %Δprice of good B. Positive: substitutes. Negative: complements. Zero: unrelated.
\[ \varepsilon = \frac{\%\Delta Q}{\%\Delta P} \qquad \text{arc: } \%\Delta Q = \frac{Q_2 - Q_1}{(Q_1+Q_2)/2}, \quad \%\Delta P = \frac{P_2 - P_1}{(P_1+P_2)/2} \] Point elasticity on a line Q = a + b·P: ε = b·P ÷ Q · Dividing line: |ε| = 1
Try it
Move the price along one straight demand line. Try these:
- Start at $4.50. Is demand elastic or inelastic here?
- Move down to $1.00. Now what?
- Find the price where |ε| is exactly 1. Where on the line is it?
The same line is elastic at the top and inelastic at the bottom. “This product is inelastic” is only ever true at a price.
Worked example
Problem. A regional pizza chain raises its large-pizza price from $14 to $16. Weekly sales fall from 2,000 to 1,800 pizzas. Elastic or inelastic?
Step 1. Percent change in quantity (midpoint). −200 ÷ 1,900 = −10.5%.
Step 2. Percent change in price (midpoint). 2 ÷ 15 = 13.3%.
Step 3. Divide. |ε| = 10.5 ÷ 13.3 = 0.79. Inelastic.
Follow-up. A second chain, in a town with six pizzerias, makes the same price change. Its sales fall from 2,000 to 1,500. %ΔQ = −500 ÷ 1,750 = −28.6%, so |ε| = 28.6 ÷ 13.3 = 2.1. Elastic. Why the difference? The second chain’s customers have close substitutes nearby.
Check with Excel. =ABS(((Q2-Q1)/AVERAGE(Q1,Q2))/((P2-P1)/AVERAGE(P1,P2)))
Check yourself
A gym raises its fee from $40 to $48 and membership falls from 1,000 to 900. Using the midpoint formula, the price elasticity is about:
ε = %ΔQ ÷ %ΔP, both at midpoints.
Which product most likely has the MOST elastic demand?
Close substitutes are the strongest driver of elasticity.
Demand for gasoline is more elastic over five years than over one month because:
Long-run demand is more elastic because adjustment takes time.
When household income rises 10%, purchases of store-brand pasta fall 4%. Store-brand pasta is:
Income elasticity = %ΔQ ÷ %Δincome; negative = inferior.
The price of Uber rides rises 10% and Lyft’s ridership rises 6%. The cross-price elasticity of Lyft demand with respect to Uber’s price is:
Positive cross-price elasticity → substitutes; negative → complements.
Video
Video coming soon.