The midpoint formula for elasticity measures how strongly quantity responds to a price change by dividing each change by the average of the starting and ending values. That one adjustment gives the same price elasticity of demand whether the price rises or falls, which is why most economics courses require the midpoint method on homework and exams. Below you will see the formula, why the simple percentage method gives two different answers, a full worked example, how to read the result and how the same formula works for income, cross-price and supply elasticity.
What Is the Midpoint Formula for Elasticity?
Elasticity compares two percentage changes. Price elasticity of demand, for example, is the percentage change in quantity demanded divided by the percentage change in price. The midpoint formula, also called the arc elasticity formula, calculates each percentage change using the average of the two values instead of the starting value:
Percentage change in quantity = (Q₂ − Q₁) ÷ [(Q₁ + Q₂) ÷ 2]
Percentage change in price = (P₂ − P₁) ÷ [(P₁ + P₂) ÷ 2]
Price elasticity of demand = percentage change in quantity ÷ percentage change in price
Because the averages appear in both the numerator and denominator, you can also write it in one line: E = [(Q₂ − Q₁) ÷ (Q₁ + Q₂)] ÷ [(P₂ − P₁) ÷ (P₁ + P₂)]. The factors of 2 cancel. Both versions give the same answer, so use whichever form your textbook shows.
The result for price elasticity of demand is negative, because price and quantity demanded move in opposite directions. Most courses report the absolute value, so an elasticity of −1.5 is described as 1.5. Check your instructor’s convention, because some exams want the sign kept.
The OpenStax Principles of Economics chapter on price elasticity uses the same midpoint method, so if your course uses that free textbook, the notation here will match.
Why the Simple Percentage Change Formula Gives Two Answers
The usual way to calculate a percentage change divides by the starting value. That works for one direction of change, but elasticity is meant to describe a stretch of the demand curve, not a direction of travel. Here is what goes wrong.
Suppose a campus coffee cart raises the price of a cold brew from $4 to $6, and daily sales fall from 120 cups to 80 cups.
| Method | % change in quantity | % change in price | Elasticity |
|---|---|---|---|
| Simple formula, $4 to $6 | (80 − 120) ÷ 120 = −33.3% | (6 − 4) ÷ 4 = 50.0% | 0.67 |
| Simple formula, $6 to $4 | (120 − 80) ÷ 80 = 50.0% | (4 − 6) ÷ 6 = −33.3% | 1.50 |
| Midpoint method, either way | −40 ÷ 100 = −40.0% | 2 ÷ 5 = 40.0% | 1.00 |
The simple formula says demand is inelastic going up and elastic coming down, for the same two points. That contradiction is the whole reason the midpoint method exists. By dividing by the averages, 100 cups and $5, it measures the percentage change the same way in both directions and gives one answer: an elasticity of exactly 1.
How to Use the Midpoint Method Step by Step
Here is the cold brew example worked the way most instructors want it shown. Write each step, because partial credit usually depends on the setup, not just the final number.
- Label the two points. Point 1: P₁ = $4, Q₁ = 120. Point 2: P₂ = $6, Q₂ = 80. It does not matter which point you call 1, because the midpoint method gives the same result either way.
- Find the change and the average for quantity. Change: 80 − 120 = −40. Average: (120 + 80) ÷ 2 = 100. Percentage change: −40 ÷ 100 = −0.40, or −40%.
- Find the change and the average for price. Change: 6 − 4 = 2. Average: (4 + 6) ÷ 2 = 5. Percentage change: 2 ÷ 5 = 0.40, or 40%.
- Divide. Elasticity = −40% ÷ 40% = −1.0. In absolute value, the price elasticity of demand is 1.0.
- Interpret it. An elasticity of 1 means demand is unit elastic over this range: the percentage change in quantity equals the percentage change in price.
A quick self-check: if you swap the points and repeat the calculation, you should get the same number. If you do not, you have divided by a starting value somewhere instead of an average.
Interpreting the Result: Elastic, Inelastic and Unit Elastic
Once you have the absolute value of the elasticity, the classification follows directly. These categories, and the extreme cases, are covered in the OpenStax section on polar cases of elasticity.
| Absolute value of elasticity | Classification | What it means |
|---|---|---|
| Greater than 1 | Elastic | Quantity changes by a larger percentage than price. Buyers are sensitive to price. |
| Equal to 1 | Unit elastic | Quantity changes by the same percentage as price. |
| Less than 1 | Inelastic | Quantity changes by a smaller percentage than price. Buyers are not very sensitive to price. |
| 0 | Perfectly inelastic | Quantity does not change at all when price changes. |
| Infinite | Perfectly elastic | Any price increase drops quantity demanded to zero. |
Goods with few substitutes, such as gasoline or prescription medicine, tend to have inelastic demand in the short run. Goods with many close substitutes, or purchases that are a large share of a budget, tend to be more elastic. Time matters too: demand usually becomes more elastic over a longer period, because people find alternatives.
The Total Revenue Test
Elasticity tells a business what will happen to total revenue, which is price times quantity, when it changes its price. The rule is simple, and it is a favorite exam question:
- Elastic demand: price and total revenue move in opposite directions. Raising the price lowers revenue.
- Inelastic demand: price and total revenue move in the same direction. Raising the price raises revenue.
- Unit elastic demand: total revenue does not change when price changes.
The cold brew example confirms it. At $4, revenue is $4 × 120 = $480. At $6, revenue is $6 × 80 = $480. Revenue is unchanged, exactly as unit elastic demand predicts. The OpenStax section on elasticity and pricing shows how firms use this logic when they set prices.
Two More Worked Examples
Practice with an inelastic case and an elastic case makes the pattern clear. Both use the midpoint method.
Example 1: Gasoline (Inelastic Demand)
A family buys 100 gallons of gas a month at $3.00 per gallon. When the price rises to $3.60, they cut back to 95 gallons.
- Percentage change in quantity: (95 − 100) ÷ 97.5 = −5.13%
- Percentage change in price: (3.60 − 3.00) ÷ 3.30 = 18.18%
- Elasticity: −5.13% ÷ 18.18% = −0.28, or 0.28 in absolute value
Demand is inelastic. The total revenue test agrees: spending rises from $300 to $342 when the price goes up, because the family barely reduces how much it buys.
Example 2: Concert Tickets (Elastic Demand)
A venue sells 200 tickets at $50. It cuts the price to $40 and sells 300 tickets.
- Percentage change in quantity: (300 − 200) ÷ 250 = 40.0%
- Percentage change in price: (40 − 50) ÷ 45 = −22.2%
- Elasticity: 40.0% ÷ −22.2% = −1.8, or 1.8 in absolute value
Demand is elastic, so lowering the price raises revenue: from $10,000 to $12,000. Fans have other ways to spend an evening, so they respond strongly to price.
Using the Midpoint Formula for Income, Cross-Price and Supply Elasticity
The same averaging trick works for every elasticity you meet in an introductory course. Only the variables change. The OpenStax section on elasticity in areas other than price covers these in more depth.
| Elasticity | Formula (midpoint percentages) | Example | Result |
|---|---|---|---|
| Income elasticity of demand | % change in quantity ÷ % change in income | Income rises from $40,000 to $50,000; restaurant meals rise from 8 to 12 a month: 40.0% ÷ 22.2% | 1.8: a normal good, and income elastic |
| Cross-price elasticity of demand | % change in quantity of good A ÷ % change in price of good B | Coffee rises from $4 to $5; tea sales rise from 100 to 120: 18.2% ÷ 22.2% | +0.82: substitutes |
| Price elasticity of supply | % change in quantity supplied ÷ % change in price | Price rises from $10 to $12; quantity supplied rises from 500 to 700: 33.3% ÷ 18.2% | 1.83: elastic supply |
Two sign rules matter here, unlike with price elasticity of demand. A positive income elasticity means a normal good and a negative one means an inferior good. A positive cross-price elasticity means substitutes, and a negative one means complements. Keep the sign for these two.
Common Mistakes on Midpoint Elasticity Problems
- Dividing by the starting value. This is the simple percentage formula, and it gives a different answer depending on direction. Always divide by the average.
- Flipping the ratio. Elasticity is the quantity change divided by the price change, not the other way around.
- Mixing up the points. If Q₂ goes with P₂ in the numerator, it must go with P₂ in the denominator too. Mismatched points can flip the sign or the size of the answer.
- Confusing elasticity with slope. Along a straight-line demand curve, the slope is constant but elasticity changes: demand is more elastic at high prices and less elastic at low prices.
- Dropping the sign when it matters. Taking the absolute value is fine for price elasticity of demand but not for income or cross-price elasticity.
- Rounding too early. Keep at least three significant figures until the final step, or an answer close to 1 can land on the wrong side of the elastic line.
Elasticity questions also show up in data form, where you calculate elasticities from a table or a regression. If that part of your course is the challenge, our statistics homework help covers the data analysis, and our math homework help covers the calculus version of elasticity, which uses derivatives at a single point instead of two points.
If you are stuck on a problem set, our tutors can work through your elasticity questions step by step. See our economics homework help page, which includes a worked example that combines a per-unit tax with the midpoint formula, or post your economics assignment to compare bids from tutors.
How Elasticity Shows Up in Tax and Pricing Questions
Once you can calculate elasticity, many exam questions ask you to use it. Two applications come up again and again.
Who pays a tax. When the government puts a per-unit tax on a good, the burden falls more heavily on whichever side of the market is less elastic. If demand is inelastic, as with cigarettes or gasoline in the short run, buyers cannot easily cut back, so most of the tax shows up as a higher price. If demand is elastic, sellers absorb more of the tax because raising the price would drive buyers away.
Setting a price. A business facing inelastic demand can raise revenue by raising its price, while a business facing elastic demand may raise revenue by cutting it, as the concert ticket example showed. This is why airlines charge different fares to business and leisure travelers: the two groups have very different elasticities.
In both cases, the midpoint formula for elasticity is the first step. Calculate the number, classify it, and then reason from the classification to the answer the question is really asking for.
Frequently Asked Questions
Why Do Economists Use the Midpoint Formula for Elasticity?
Because the simple percentage formula gives a different elasticity depending on whether the price rises or falls between the same two points. The midpoint formula divides each change by the average of the two values, so the percentage change is measured the same way in both directions. That gives one consistent elasticity for the stretch of the demand curve between the two points, which is what the measure is meant to describe.
Is the Midpoint Formula the Same as Arc Elasticity?
Yes. Arc elasticity is the elasticity measured over a segment, or arc, of a demand or supply curve between two points, and the midpoint formula is the standard way to calculate it. Point elasticity is different: it measures elasticity at a single point on the curve using calculus, and it is used more in intermediate and upper-level economics courses.
Should Price Elasticity of Demand Be Negative or Positive?
The raw result is negative, because price and quantity demanded move in opposite directions under the law of demand. Most introductory courses report the absolute value, so an elasticity of −1.5 is called 1.5 and described as elastic. Some instructors want the negative sign kept, so follow the convention in your textbook or lecture notes, and say which one you are using on exams.
What Does an Elasticity of 1 Mean?
An elasticity of 1 in absolute value means demand is unit elastic over that range: the percentage change in quantity is exactly the same as the percentage change in price. In that case total revenue does not change when the price changes, because the gain from the higher price is exactly offset by the loss from selling fewer units, as in the $4 to $6 cold brew example.
When Should I Use Point Elasticity Instead of the Midpoint Method?
Use the midpoint method when you have two observed points, such as two prices and two quantities from a table or a word problem. Use point elasticity when you have a demand function and need the elasticity at one specific price, which requires the derivative of quantity with respect to price multiplied by the ratio of price to quantity. If a problem gives two points, your instructor almost always expects the midpoint method.
Can the Midpoint Formula Be Used for Supply?
Yes. Price elasticity of supply uses the same formula with quantity supplied in place of quantity demanded. Because supply curves usually slope upward, the result is positive. A value above 1 means supply is elastic, so producers respond strongly to price, and a value below 1 means supply is inelastic, which is common in the short run when production cannot expand quickly.
