Foundationsmicroeconomics
Price Elasticity of Demand: The Midpoint Formula and Determinants
Overview
Price elasticity of demand (Ed) measures how strongly quantity demanded responds to a change in the good's own price. This lesson teaches the standard formula and the midpoint (arc elasticity) method that fixes the direction problem, works through a full numerical example, explains the five determinants (substitutes, necessity vs. luxury, budget share, time horizon, market definition), and shows the total-revenue test that links elasticity to business pricing decisions.
Learning objectives
- Compute Ed with the simple and midpoint formulas.
- Explain why the midpoint method is preferred.
- List and apply the five determinants of price elasticity of demand.
- Use the total-revenue test to predict revenue effects of price changes.
The formula
Ed = (% change in quantity demanded) / (% change in price). Compute each percentage change relative to a base: if price rises from 12 and quantity falls from 100 to 80, % change in Q = -20/100 = -20%, % change in P = 2/10 = 20%, so Ed = -20/20 = -1. Economists usually report the absolute value: |Ed| = 1, unit elastic. The sign reminds us the relationship is inverse; the magnitude tells us the strength.
Ed = (%ΔQd) / (%ΔP). Use absolute value when classifying: >1 elastic, <1 inelastic, =1 unit elastic.
The midpoint (arc elasticity) method
The simple formula has a flaw: going from 12 gives a different elasticity than going from 10, because the base changes. The midpoint method fixes this by using the average as the base: % change = (new - old) / ((new + old)/2). Price $10->12, quantity 100->80: %ΔP = 2/11 = 18.2%, %ΔQ = -20/90 = -22.2%, Ed = -22.2/18.2 = -1.22 (elastic). Same answer in both directions -- which is why textbooks and exams use the midpoint method.
Worked example, step by step
A cinema raises ticket prices from 10; attendance falls from 500 to 400 per week. Step 1: midpoint of price = 2, so %ΔP = 2/9 = 22.2%. Step 2: midpoint of quantity = 450, change = -100, so %ΔQ = -100/450 = -22.2%. Step 3: Ed = -22.2/22.2 = -1.0 -- unit elastic. Interpretation: revenue is unchanged (500x4,000; 400x4,000), exactly what unit elasticity predicts. Practice this three-step routine until it is mechanical.
The five determinants
(1) Availability of substitutes: more substitutes -> more elastic (Coke vs. 'soda'). (2) Necessity vs. luxury: necessities (insulin, salt) are inelastic; luxuries (vacations) are elastic. (3) Share of budget: big-ticket items (cars) are more elastic than trivial ones (pepper). (4) Time horizon: longer horizons -> more elastic, as consumers find alternatives (gasoline: inelastic this month, elastic over years as people buy efficient cars). (5) Definition of the market: narrow ('iPhone') is more elastic than broad ('smartphones'), which is more elastic than 'all phones'.
Substitutes, Necessity, Budget share, Time, Market definition -- the broader and longer-run the lens, the more elastic the demand.
The total-revenue test
Total revenue = price x quantity, and elasticity predicts what a price change does to it. Elastic demand (|Ed|>1): price rise -> revenue falls (quantity loss dominates); price cut -> revenue rises. Inelastic (|Ed|<1): price rise -> revenue rises; price cut -> revenue falls. Unit elastic: revenue unchanged. Intuition: with inelastic demand, buyers barely leave, so the higher price applies to nearly the same quantity. Firms with market power therefore raise prices on inelastic segments -- and cut them where demand is elastic.
Key points
- Ed = %ΔQd / %ΔP; classify by absolute value: >1 elastic, <1 inelastic, =1 unit elastic.
- Midpoint method uses averages as bases, giving the same elasticity in both directions.
- Determinants: substitutes, necessity vs. luxury, budget share, time horizon, market definition.
- Elastic: price rise -> revenue falls. Inelastic: price rise -> revenue rises. Unit elastic: no change.
- Elasticity varies along a linear demand curve -- it is not the same as slope.
Common misconceptions
✓ Truth: Slope is not elasticity. Elasticity = % changes, slope = absolute changes, and elasticity varies along any straight-line demand curve (elastic at high prices, inelastic at low prices). Two curves with different slopes can have identical elasticity at a point.
✓ Truth: Without it, elasticity depends on direction -- $10->12 and $12->10 give different answers for the same demand behavior, which is incoherent for analysis. The midpoint method is the standard precisely because it removes that arbitrariness.
✓ Truth: Usually, but context rules: a 'luxury' with no substitutes can be inelastic for its fans, and a necessity can be elastic if a close substitute exists (a specific brand of a necessary drug vs. generics). Determinants interact; apply them, do not memorize labels.
Practice questions
Q1. Price falls from $20 to $16; quantity demanded rises from 100 to 140. Using the midpoint method, Ed is approximately:
- -0.5 (inelastic).
- -1.5 (elastic).
- -2.0 (elastic).
- -1.0 (unit elastic).
Midpoint P = 18, %ΔP = -4/18 = -22.2%. Midpoint Q = 120, %ΔQ = 40/120 = 33.3%. Ed = 33.3 / -22.2 = -1.5, so |Ed| = 1.5 > 1: elastic. Quantity responded more than proportionally.
Q2. Which good likely has the most elastic demand?
- Insulin for diabetics.
- Table salt.
- A specific brand of luxury sneakers.
- Tap water.
The sneakers are a luxury with many close substitutes in a narrowly defined market -- all pushing toward elastic. Insulin, salt, and tap water are necessities with few substitutes and small budget shares -- classically inelastic.
Q3. A firm with inelastic demand raises its price. Total revenue will:
- Fall, because customers leave.
- Rise, because the price gain outweighs the small quantity loss.
- Stay exactly the same.
- Fall to zero.
Inelastic means |Ed| < 1: the percentage quantity loss is smaller than the percentage price gain, so price x quantity rises. This is the total-revenue test -- it is why firms with pricing power raise prices on inelastic segments.
Click a question to reveal the answer and explanation.
Frequently asked questions
When do I use the absolute value of elasticity?
When classifying demand as elastic/inelastic/unit-elastic, always use the absolute value -- the sign just confirms the inverse price-quantity relationship. Keep the negative sign in the formula itself, then take |Ed| for the classification step.
Why does elasticity change along a straight-line demand curve?
Because the percentage bases change. At high prices, a $1 change is a small % of price but quantity is small, so %ΔQ is large -- elastic. At low prices, the same $1 is a large % of price while %ΔQ is small -- inelastic. The midpoint is unit elastic. Same line, different elasticities.
What are perfectly elastic and perfectly inelastic demand?
Perfectly elastic (|Ed| = infinity) is a horizontal curve: any price above the going rate loses all sales -- the case of a price-taking firm. Perfectly inelastic (Ed = 0) is a vertical curve: quantity is fixed regardless of price, like a life-saving drug with no substitute in the very short run.
How does the total-revenue test work with a price cut?
Mirror image: with elastic demand, a price cut raises revenue (new buyers flood in); with inelastic demand, a price cut lowers revenue (barely any new buyers, and everyone pays less). Firms cut prices to gain share only where demand is elastic.