The midpoint method is how economics courses calculate elasticity between two points on a curve. Instead of measuring each percentage change from the starting value, you measure it from the average of the starting and ending values. That single change means you get the same elasticity whether the price goes up or comes down, something the ordinary percentage-change method can't promise.
Below you'll find the formula for price elasticity of demand, price elasticity of supply and income elasticity, worked examples for each, a step-by-step calculation table, the total revenue test, the mistakes that cost points on problem sets and a practice problem with the answer. Every example uses made-up numbers chosen for teaching, so you can check each step on your own calculator.
The midpoint method is a way to calculate percentage changes, and therefore elasticities, using the average of two values as the base. Many textbooks call the result arc elasticity, because it measures responsiveness across an arc of the curve between two points rather than at one exact point.
Every 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 method only changes how you calculate those percentages: you use change ÷ average instead of change ÷ starting value.
The midpoint formula economics courses teach is not the geometry midpoint formula, even though it borrows the same averaging step. In geometry, (x1 + x2) ÷ 2 gives you a coordinate. In economics, that average becomes the denominator of a percentage change, and the final answer is one number: the elasticity.
This is the midpoint method economics professors usually mean when a problem gives you two prices and two quantities, and it's the version in the free OpenStax Principles of Economics textbook, Section 5.1. If you want a refresher on why buyers respond to prices in the first place, our summary of Mankiw's 10 principles of economics covers the idea that people respond to incentives.
The midpoint method gives the same answer in both directions because the base of each percentage, the average, doesn't depend on which point you call the start. With simple percentage change, the base switches when you reverse direction, and so does your answer.
Take a campus coffee cart. At \$4 a cup it sells 120 cups a day. At \$5 it sells 90 cups a day.
Same two points, same demand curve, yet you get two different numbers and two different labels. That's the direction problem.
Average quantity is (120 + 90) ÷ 2 = 105, and average price is (\$4 + \$5) ÷ 2 = \$4.50. Quantity changes by 30 ÷ 105 = 28.57%, and price changes by 1 ÷ 4.50 = 22.22%. Elasticity = 28.57 ÷ 22.22 = 1.29, whichever direction you go.
| Method | \$4 to \$5 | \$5 to \$4 |
|---|---|---|
| Starting value as the base | 1.00 (unit elastic) | 1.67 (elastic) |
| Midpoint method (average as the base) | 1.29 (elastic) | 1.29 (elastic) |
The midpoint answer lands between the two simple answers. It isn't the exact elasticity at \$4 or at \$5. It's the average responsiveness across the stretch of the curve between them, which is what a two-point question is really asking for. OpenStax puts it simply: the formula uses the same base, average quantity and average price, for a price increase and a price decrease.
The midpoint formula for price elasticity of demand divides the percentage change in quantity demanded by the percentage change in price, with each percentage measured against the average of its two values.
Percentage change in quantity demanded:
\[ \%\Delta Q = \frac{Q_2 - Q_1}{(Q_1 + Q_2)/2} \times 100 \]
Percentage change in price:
\[ \%\Delta P = \frac{P_2 - P_1}{(P_1 + P_2)/2} \times 100 \]
Price elasticity of demand:
\[ E_d = \frac{\%\Delta Q}{\%\Delta P} \]
In words: subtract the old quantity from the new quantity and divide by the average quantity. Do the same with the two prices. Then divide the quantity result by the price result. Here \(Q_1\) and \(P_1\) are the first quantity and price, and \(Q_2\) and \(P_2\) are the second pair.
Both averages divide by 2, so the 2s cancel, and the midpoint method formula shrinks to a shortcut that's quicker on a calculator:
\[ E_d = \frac{(Q_2 - Q_1)/(Q_2 + Q_1)}{(P_2 - P_1)/(P_2 + P_1)} \]
In words: the change in quantity over the sum of the two quantities, divided by the change in price over the sum of the two prices. Check it with the coffee cart: (−30 ÷ 210) ÷ (1 ÷ 9) = −0.1429 ÷ 0.1111 = −1.29. Same answer, fewer steps.
Report the absolute value for price elasticity of demand unless your instructor says otherwise. Because price and quantity demanded move in opposite directions, the raw result is always negative, and both OpenStax and Lumen Learning's midpoint formula lesson report it as a positive number. So −1.29 becomes 1.29. If your online homework asks for a signed answer, follow its instructions.
To calculate elasticity using the midpoint method, label your two points, find the two changes, divide each change by its average and then divide the quantity percentage by the price percentage. This table runs through every step with the coffee cart numbers.
| Step | What You Do | Coffee Cart Example |
|---|---|---|
| 1 | Label the points (P1, Q1) and (P2, Q2) | P1 = \$4, Q1 = 120; P2 = \$5, Q2 = 90 |
| 2 | Change in quantity: Q2 − Q1 | 90 − 120 = −30 |
| 3 | Average quantity: (Q1 + Q2) ÷ 2 | (120 + 90) ÷ 2 = 105 |
| 4 | Percentage change in quantity | −30 ÷ 105 × 100 = −28.57% |
| 5 | Change in price: P2 − P1 | \$5 − \$4 = \$1 |
| 6 | Average price: (P1 + P2) ÷ 2 | (\$4 + \$5) ÷ 2 = \$4.50 |
| 7 | Percentage change in price | 1 ÷ 4.50 × 100 = 22.22% |
| 8 | Divide step 4 by step 7 | −28.57 ÷ 22.22 = −1.29 |
| 9 | Take the absolute value and classify | 1.29 is greater than 1, so demand is elastic |
Two habits save marks. First, carry at least four decimal places until the last step. Rounding 22.22% to 22% too early gives 1.30 instead of 1.29, and an auto-graded system may mark that wrong. Second, keep both percentages on the same scale: either multiply both by 100 or neither. The 100s cancel in the ratio, but dividing 28.57 by 0.2222 gives nonsense.
If you have a long demand schedule, put Q1 in cell A2, Q2 in B2, P1 in C2 and P2 in D2, then use =((B2-A2)/((A2+B2)/2))/((D2-C2)/((C2+D2)/2)). Wrap it in ABS() if you want the absolute value. Our Excel homework help tutors can show you how to fill it down a whole table.
Suppose a town's gas price rises from \$3.20 to \$3.60 a gallon, and weekly sales fall from 50,000 to 48,000 gallons.
Because 0.35 is less than 1, demand is inelastic over this range. A 1% price increase cuts the quantity demanded by only about 0.35%. That fits a product with few substitutes in the short run, and OpenStax notes that the demand for energy is somewhat inelastic in the short run but much more elastic in the long run.
Compare the absolute value of your answer with 1. Above 1, quantity responds by a bigger percentage than price, so demand is elastic. Below 1, it responds by a smaller percentage, so demand is inelastic. Exactly 1 is unit elastic.
| Absolute Value | Label | What It Means | Effect of a Price Increase on Total Revenue |
|---|---|---|---|
| 0 | Perfectly inelastic | Quantity doesn't change at all | Rises |
| Between 0 and 1 | Inelastic | Quantity changes by a smaller percentage than price | Rises |
| Exactly 1 | Unit elastic | Quantity and price change by the same percentage | Stays the same |
| Greater than 1 | Elastic | Quantity changes by a larger percentage than price | Falls |
| Infinite | Perfectly elastic | Any price increase wipes out quantity demanded | Falls to zero |
The two extreme rows are the polar cases described in OpenStax Section 5.2 on perfect elasticity and perfect inelasticity. You'll rarely calculate them from data, but they show up on graphing questions.
The total revenue test lets you check a price elasticity of demand answer with simple multiplication. Total revenue is price times quantity. If price and total revenue move in the same direction, demand is inelastic. If they move in opposite directions, demand is elastic. If revenue doesn't change, demand is unit elastic. OpenStax Section 5.3 on elasticity and pricing sets out the same rule.
With the midpoint method this check always agrees with your elasticity, because a midpoint elasticity below 1 means revenue moves with price, and one above 1 means it moves against price. The simple method can disagree: it labeled the coffee cart unit elastic going from \$4 to \$5, even though revenue fell. The test only applies to price elasticity of demand, not to supply or income elasticity.
A straight-line demand curve has the same slope everywhere, but its elasticity changes as you move along it. Curtis and Irvine's open textbook chapter on price responsiveness explains that elasticity is high at high prices, low at low prices and equal to 1 at the midpoint of a linear demand curve. OpenStax shows this with one demand schedule: the midpoint elasticity is 0.45 between \$60 and \$70 but 1.47 between \$120 and \$130. Never judge elasticity by how steep a graph looks, since the axis scale changes the picture.
The price elasticity of supply midpoint formula is the same calculation with quantity supplied in place of quantity demanded:
\[ E_s = \frac{(Q_{s2} - Q_{s1}) \,/\, \left[(Q_{s1} + Q_{s2})/2\right]}{(P_2 - P_1) \,/\, \left[(P_1 + P_2)/2\right]} \]
In words: the percentage change in quantity supplied, using the average quantity as the base, divided by the percentage change in price, using the average price as the base.
Price and quantity supplied normally move in the same direction, so the answer is usually positive and there's no sign to drop. You classify it with the same cutoffs: above 1 is elastic, below 1 is inelastic and exactly 1 is unit elastic.
When the price of a flat of strawberries rises from \$18 to \$22, a farm increases the flats it brings to market from 800 to 1,200.
Supply is elastic over this range, since each 1% rise in price brings about a 2% rise in quantity supplied. For a second check, OpenStax works an apartment example in which rent rising from \$650 to \$700 a month raises units supplied from 10,000 to 13,000, for a supply elasticity of about 3.5.
Supply also tends to get more elastic with time, because firms find it easier to expand production over several years than over a few months. That's the idea behind the long-run supply curve.
The income elasticity midpoint formula measures how quantity demanded responds to a change in income, with average income as the base:
\[ E_I = \frac{(Q_2 - Q_1) \,/\, \left[(Q_1 + Q_2)/2\right]}{(I_2 - I_1) \,/\, \left[(I_1 + I_2)/2\right]} \]
In words: the percentage change in quantity demanded divided by the percentage change in income, where \(I_1\) and \(I_2\) are the old and new incomes and both percentages use averages as the base.
Here the sign carries the meaning, so keep it. OpenStax Section 5.4 on elasticity in areas other than price calls a good with positive income elasticity a normal good and one with negative income elasticity an inferior good. Curtis and Irvine's chapter on income elasticity adds cutoffs for size:
A household's yearly income rises from \$50,000 to \$70,000. The change is \$20,000 and the average is \$60,000, so income rises by 33.33% on the midpoint basis.
If you dropped the minus sign on the noodles out of habit from demand problems, you'd call an inferior good normal and lose the mark.
Cross-price elasticity swaps income for the price of a different good. You divide the midpoint percentage change in the quantity of good A by the midpoint percentage change in the price of good B. OpenStax notes that substitutes have positive cross-price elasticities and complements have negative ones, so keep the sign here as well.
Most lost points on elasticity questions come from the same few slips, and each one is easy to catch once you know to look for it.
If your answers keep missing the key and you can't see why, a second pair of eyes on your setup usually finds it in minutes. Our economics homework help and microeconomics tutors work through elasticity problems with you, and you can book a tutor to review your own work and explain each step.
Try this midpoint method problem on paper before you read the solution.
Problem: A university raises the price of a semester parking permit from \$150 to \$250. Permits sold fall from 2,200 to 1,800.
Simple percentage change uses the starting value as the base, so the same two points give a different elasticity depending on whether price rises or falls. In the coffee cart example, that means 1.00 one way and 1.67 the other. The midpoint method divides each change by the average of the two values, so the base is identical in both directions and you get one answer, 1.29, either way.
Yes, in intro economics the two names describe the same calculation. Arc elasticity measures responsiveness over a stretch, or arc, of a curve between two points, and the midpoint method is the standard way to compute it: each percentage change uses the average of the two values as its base. Some textbooks use one name and some use the other, so treat them as interchangeable unless your instructor says otherwise.
For price elasticity of demand, usually yes. Price and quantity demanded move in opposite directions, so the raw answer is negative, and textbooks such as OpenStax report the absolute value. For income and cross-price elasticity, never drop the sign, because it tells you whether a good is normal or inferior, or whether two goods are substitutes or complements. Price elasticity of supply is normally positive already, so there is nothing to drop.
Point elasticity measures responsiveness at one spot on a curve, usually with calculus or the slope of a linear demand curve. The midpoint method measures average responsiveness between two points. The closer together the two points are, the closer the midpoint result gets to the point elasticity there. Intro courses usually hand you two price and quantity pairs, which is why the midpoint method is the one you will use most.
No. The geometry midpoint formula finds the coordinates halfway between two points: (x1 + x2) ÷ 2 and (y1 + y2) ÷ 2. The economics version uses those same averages, but only as the bases of two percentage changes. Its output is a single number, the elasticity, not a point on a graph. If an economics question asks for a midpoint elasticity and you give a coordinate pair, you have used the wrong formula.
Yes. Any elasticity that divides one percentage change by another can use midpoint averages: price elasticity of demand, price elasticity of supply, income elasticity, cross-price elasticity and the wage elasticity of labor supply covered in OpenStax. The steps never change. Find each change, divide it by the average of its two values, then divide the percentage change of the responding variable by the percentage change of the variable that caused it.
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