Elasticity Mathematics: Point vs Arc Elasticity, Cross-Price Elasticity and the Lerner Index
Elastic and inelastic get thrown around loosely, but elasticity is a precise number with a precise formula — and which formula you use changes the answer. Get the mathematics right and elasticity stops being a vocabulary word and becomes a tool that prices products, predicts tax burdens and measures monopoly power.
A supermarket raises the price of bread 10% and sells 2% less. It raises the price of a luxury cheese 10% and sells 25% less. Same price change, wildly different responses — and the difference is worth millions in revenue decisions. Elasticity puts a number on that responsiveness.
Most students meet elasticity as percentage change in quantity divided by percentage change in price and think they are done. They are not, because there are at least three ways to compute that ratio, they give different answers, and knowing which to use when is what separates a confident answer from a muddled one.
The core definition
PED = percentage change in quantity demanded over percentage change in price
Because demand curves slope down, PED is almost always negative. Economists often quote the absolute value, but the sign carries meaning.
| Absolute PED | Category | Interpretation |
|---|---|---|
| > 1 | Elastic | Quantity responds more than proportionally |
| = 1 | Unit elastic | Quantity responds proportionally |
| < 1 | Inelastic | Quantity responds less than proportionally |
| = 0 | Perfectly inelastic | Quantity does not respond at all |
| infinity | Perfectly elastic | Any price rise collapses quantity to zero |
The trouble starts when you compute the percentage changes, because percentage change depends on what you divide by. That is what the arc formula fixes.
Arc elasticity: the midpoint method
Suppose price rises from 10 to 12 pounds and quantity falls from 100 to 80. The percentage change in price is plus 20% measured from 10, but minus 16.7% measured from 12. Same move, two different percentages. Elasticity between two points should not depend on which you call the start.
The arc elasticity or midpoint method divides changes by the average of the two values:
arc PED = [change in Q over average Q] divided by [change in P over average P]
Using the midpoint of each variable as the base makes the result identical whether price rises or falls.
Worked example. Price 10 to 12, quantity 100 to 80. Change in Q = minus 20, average Q = 90, so percentage change = minus 22.2%. Change in P = plus 2, average P = 11, so percentage change = plus 18.2%. Arc PED = minus 22.2% over 18.2% = minus 1.22. Elastic. Computing the reverse direction gives exactly the same minus 1.22 — the whole point of the arc method.
When to use arc elasticity. Whenever you have two separate points and want a single elasticity for the range between them — typically when a question gives before and after data. The midpoint base makes the answer direction-independent.
Point elasticity: elasticity at a single point
Elasticity varies continuously along a demand curve — it is different at every point. To find it at one specific point, use point elasticity, and this is where calculus enters. It uses the derivative dQ/dP — the slope at that point — in place of finite changes:
point PED = (dQ/dP) times (P over Q)
dQ/dP is the derivative of quantity with respect to price; P over Q is price over quantity at the point of interest.
Worked example. Demand is Q = 200 minus 5P. Find elasticity at P = 20. Here dQ/dP = minus 5. At P = 20, Q = 200 minus 100 = 100. So PED = (minus 5)(20 over 100) = minus 1. Exactly unit elastic. Move to a different price and the elasticity changes, even though the slope stays minus 5 — because the P over Q ratio changes.
Common error — thinking a straight-line demand curve has constant elasticity. It does not. A linear demand curve has constant slope but continuously changing elasticity. Near the top (high price, low quantity) it is elastic; near the bottom it is inelastic; and it passes through unit elasticity exactly at the midpoint. Confusing constant slope with constant elasticity is one of the most common and costly errors in the topic.
Elasticity varies along a linear demand curve
Take Q = 200 minus 5P at several prices:
| Price | Quantity | PED = minus 5(P/Q) | Region |
|---|---|---|---|
| 36 | 20 | minus 9.0 | Highly elastic |
| 30 | 50 | minus 3.0 | Elastic |
| 20 | 100 | minus 1.0 | Unit elastic |
| 10 | 150 | minus 0.33 | Inelastic |
| 4 | 180 | minus 0.11 | Highly inelastic |
Same curve, same slope, elasticity ranging from minus 9 to minus 0.11. The midpoint is always where unit elasticity sits and, not coincidentally, where total revenue peaks.
Elasticity connects to revenue, taxation and market power in ways that reward careful study. The Economics Made Simple bundle works through every elasticity type with revenue diagrams and past-paper questions, while the Case Studies collection shows real estimated elasticities for goods from petrol to cinema tickets.
Elasticity and total revenue
The most useful thing elasticity does is predict revenue. Total revenue is price times quantity, and the two move in opposite directions when price changes. Which wins depends on elasticity.
The revenue rule. If demand is elastic (absolute PED above 1), a price cut raises revenue and a price rise lowers it. If inelastic (absolute PED below 1), a price rise raises revenue. Revenue is maximised at unit elasticity, where the effects offset. This is why cinemas discount (elastic) but water companies can charge more (inelastic).
This links to marginal revenue through MR = P(1 + 1/PED). At unit elasticity, MR = 0 — adding output no longer raises revenue, which is the revenue peak.
Income and cross-price elasticity
Change what is in the denominator and you get a new elasticity. Income elasticity (YED) measures responsiveness to income: percentage change in Q over percentage change in income. YED above 0 means a normal good; below 0 an inferior good. Among normal goods, above 1 is a luxury, between 0 and 1 a necessity.
Cross-price elasticity (XED) measures how demand for one good responds to the price of another. A positive XED means substitutes — when the price of tea rises, demand for coffee rises. A negative XED means complements — when the price of printers rises, demand for ink falls.
| Elasticity | Sign/value | Classification |
|---|---|---|
| YED | > 1 | Luxury good |
| YED | 0 to 1 | Necessity |
| YED | < 0 | Inferior good |
| XED | > 0 | Substitutes |
| XED | < 0 | Complements |
Common error — ignoring the sign on XED and YED. For price elasticity the sign is always negative and often dropped. But for income and cross-price elasticity, the sign is the answer — it tells you whether a good is inferior or normal, substitute or complement. Reporting XED as 0.8 without the positive sign, or taking its absolute value, throws away the entire economic content.
The Lerner Index: elasticity as market power
The Lerner Index captures the markup of price over marginal cost as a fraction of price, and ties it to elasticity:
Lerner Index = (P minus MC) over P = minus 1 over PED
The proportional markup equals the reciprocal of the absolute price elasticity the firm faces.
A profit-maximising firm sets MR = MC, and since MR = P(1 + 1/PED), rearranging gives the markup as minus 1 over PED. The more inelastic the demand, the larger the sustainable markup.
Example. A firm facing PED = minus 2 has a Lerner Index of 0.5 — price is double marginal cost. A firm facing PED = minus 5 (more competition) manages only 0.2. A perfectly competitive firm faces perfectly elastic demand, giving a Lerner Index of 0 — price equals marginal cost, no market power.
Why this matters. The Lerner Index turns market power into a number derived entirely from elasticity. It explains why firms invest so heavily in making demand less elastic — through branding, differentiation and lock-in. Every loyalty scheme is, mathematically, an attempt to lower the absolute PED and raise the sustainable markup.
Case Study — Why Cigarette Taxes Work as Revenue but Fail as Deterrents
Cigarette demand is famously inelastic — estimates put the price elasticity for adult smokers between minus 0.3 and minus 0.5. Because demand is inelastic, a tax that raises prices sharply cuts quantity only modestly. For a government, this is ideal revenue: smokers keep buying, so the tax reliably raises money.
But the same inelasticity undermines the public-health goal. If the aim is to make people quit, an inelastic response is exactly what you do not want — the price rises but consumption barely falls. The good is taxed heavily because demand is inelastic, but that inelasticity limits how much the tax changes behaviour.
There is a revealing exception. Youth smoking is considerably more price-elastic than adult smoking — younger, poorer, less addicted consumers respond more to price. So cigarette taxes are far more effective at preventing young people from starting than at making established smokers quit. The policy works best precisely where demand is most elastic, which is the lesson of the entire topic: the elasticity number tells you what the policy will actually do.
Research Spotlight — Estimating Elasticities From Real Data
The challenge: the formulas assume you know the demand curve. In reality economists must estimate elasticities from messy market data — one of the founding problems of econometrics.
The core difficulty is the identification problem, recognised in the 1920s. When you observe price and quantity moving together, you are seeing the intersection of shifting supply and shifting demand. A naive regression of quantity on price does not recover the demand elasticity — it recovers a meaningless blend of both curves. Philip Wright’s 1928 work on this problem introduced what we now call instrumental variables, isolating movements along the demand curve caused purely by supply shifts.
Modern estimates confirm the theory’s intuitions: necessities like petrol and electricity show inelastic short-run demand (around minus 0.2 to minus 0.4), while luxuries and goods with close substitutes are elastic. Elasticities are consistently larger in the long run — people can switch to fuel-efficient cars given time, but not overnight.
Why this matters for you: the point-elasticity formula requires knowing dQ/dP — the demand slope — which is exactly what is hard to observe. This connects elasticity directly to regression analysis and the identification problem. The mathematics is clean; measuring it from real data is where the genuine difficulty, and much of modern empirical economics, lives.
AP & Cambridge A-Level Exam Technique
1. Match the method to the data. Two points given means arc elasticity. A single point and a demand equation means point elasticity with calculus.
2. Use the midpoint base for arc elasticity. Divide by the average of the two prices and quantities, not one endpoint.
3. For point elasticity, get dQ/dP not dP/dQ. If demand is written P = f(Q), invert or differentiate carefully — a common slip.
4. Keep the sign for YED and XED. The sign is the classification. Negative XED means complements; negative YED means inferior good.
5. Remember linear demand has changing elasticity. Constant slope, changing elasticity, unit elastic at the midpoint. Never call a straight-line demand curve elastic or inelastic as a whole.
6. Link elasticity to revenue explicitly. Elastic means cut price to raise revenue; inelastic means raise price. State the direction and reason.
7. Use the Lerner Index for market-power questions. Markup = minus 1 over PED.
Practice Questions
Question 1 — Arc elasticity (5 marks)
When the price rises from 8 to 12 pounds, quantity falls from 500 to 300. (a) Calculate the arc PED. (b) State whether demand is elastic or inelastic.
(a) Change in Q = minus 200, average Q = 400, so minus 50%. Change in P = plus 4, average P = 10, so plus 40%. Arc PED = minus 50% over 40% = minus 1.25. [4]
(b) Absolute value above 1, so elastic. [1]
(b) Absolute value above 1, so elastic. [1]
Question 2 — Point elasticity (6 marks)
Demand is Q = 400 minus 8P. (a) Find point elasticity at P = 25. (b) Find it at P = 40. (c) Comment.
(a) dQ/dP = minus 8. At P = 25, Q = 200. PED = (minus 8)(25/200) = minus 1.0 (unit elastic). [2]
(b) At P = 40, Q = 80. PED = (minus 8)(40/80) = minus 4.0 (elastic). [2]
(c) Despite constant slope, elasticity rises from minus 1.0 to minus 4.0 as price increases, because the P over Q ratio rises — the curve is not uniformly elastic or inelastic. [2]
(b) At P = 40, Q = 80. PED = (minus 8)(40/80) = minus 4.0 (elastic). [2]
(c) Despite constant slope, elasticity rises from minus 1.0 to minus 4.0 as price increases, because the P over Q ratio rises — the curve is not uniformly elastic or inelastic. [2]
Question 3 — Cross-price and income elasticity (6 marks)
(a) When coffee’s price rises 20%, tea demand rises 8%. Calculate and interpret the cross-price elasticity. (b) When incomes rise 10%, bus travel falls 4%. Calculate and interpret the income elasticity.
(a) XED = plus 8% over plus 20% = plus 0.4. Positive means coffee and tea are substitutes; the value 0.4 shows the relationship is weak. [3]
(b) YED = minus 4% over plus 10% = minus 0.4. Negative means bus travel is an inferior good. [3]
(b) YED = minus 4% over plus 10% = minus 0.4. Negative means bus travel is an inferior good. [3]
Question 4 — Lerner Index and evaluation (8 marks)
A firm faces demand elasticity of minus 2.5 and marginal cost of 30 pounds. (a) Use the Lerner Index to find the profit-maximising price. (b) Evaluate the claim that a firm can always increase profit by raising its price.
(a) Lerner Index = minus 1 over (minus 2.5) = 0.4. So (P minus 30) over P = 0.4, giving 0.6P = 30, P = 50 pounds. [3]
(b) Evaluation: [5] The claim is generally false. Whether a price rise raises profit depends on elasticity. Where demand is elastic (as here, minus 2.5), a rise cuts quantity more than proportionally, reducing revenue. The optimum is the single price (50 pounds) where MR = MC; above it, marginal revenue falls below marginal cost. Only where demand is inelastic would a rise raise revenue — but a profit-maximising firm never operates on the inelastic part of its demand curve, because MR would be negative. Market power caps the sustainable markup at minus 1 over PED. Caveat: this is static; dynamic considerations (market share, entry deterrence, brand investment to lower future elasticity) can justify other prices. Judgement: the claim confuses having market power with always benefiting from higher prices. Firms raise profit not by raising price without limit but by making demand less elastic, which raises the optimal price the Lerner Index permits.
Summary
Elasticity is one idea — proportional responsiveness — computed three ways. Arc elasticity uses the midpoint base to measure responsiveness between two points without depending on direction. Point elasticity uses the derivative, (dQ/dP)(P/Q), at a single point, revealing that a linear demand curve has constant slope but continuously changing elasticity, passing through unit elasticity at its midpoint where revenue peaks.
Swapping the denominator gives income elasticity (classifying goods as normal, inferior or luxury) and cross-price elasticity (substitutes positive, complements negative, the sign carrying the whole meaning). And the Lerner Index, markup = minus 1 over PED, turns elasticity into a direct measure of monopoly power — explaining why every brand campaign is really an attempt to make demand less elastic. Clean formulas throughout; the genuine difficulty is measuring these numbers from real market data.
References
- Lerner, A.P. (1934) The concept of monopoly and the measurement of monopoly power, Review of Economic Studies, 1(3), pp. 157-175.
- Marshall, A. (1890) Principles of Economics. London: Macmillan.
- Chaloupka, F.J. and Warner, K.E. (2000) The economics of smoking, Handbook of Health Economics, vol. 1B. Amsterdam: Elsevier.
- Wright, P.G. (1928) The Tariff on Animal and Vegetable Oils. New York: Macmillan.
- Varian, H.R. (2014) Intermediate Microeconomics. 9th edn. New York: W.W. Norton.