How cross-price and income elasticity work
Ordinary price elasticity asks how a good's own price moves its sales. Its two siblings ask about everything else on the demand curve's "held constant" list. Cross-price elasticity asks: when butter gets more expensive, what happens to margarine? When printers get cheaper, what happens to ink cartridges? When gas prices spike, what happens to transit ridership? The math is the same ratio of percentage changes — but here you keep the sign, because the sign is the answer. Positive means the goods compete for the same job (substitutes); negative means they are used together (complements). Income elasticity swaps in income as the driver and sorts goods into inferior, necessity, and luxury.
The two formulas
Exy = %ΔQA ÷ %ΔPB
Ei = %ΔQ ÷ %ΔIncome
midpoint form: %ΔX = (X₂ − X₁) ÷ [(X₁ + X₂) ÷ 2] × 100
where QA is the quantity of the good you are watching and PB the price of the other good. Dividing by the average of the two values (midpoint method) makes the answer identical whichever direction the change runs. Read the sign: Exy > 0 substitutes, Exy < 0 complements; Ei < 0 inferior, 0 to 1 normal necessity, Ei > 1 luxury.
Worked example
Take the classic pairing: butter's price rises from $2.00 to $2.50, and margarine sales climb from 100 to 120 units as shoppers switch. Here is the midpoint calculation, step by step:
| Step | Amount |
|---|---|
| Butter (good B) price changemidpoint method: change ÷ average of the two prices = 22.22% | $2.00 → $2.50 |
| Margarine (good A) quantity changemidpoint method: change ÷ average of the two quantities = 18.18% | 100 → 120 units |
| ÷ Exy = %ΔQ of A ÷ %ΔP of B18.18% ÷ 22.22% | +0.8182 |
| = Exy of +0.8182 — the goods are substitutespositive sign: butter got pricier and buyers switched to margarine | +0.8182 |
Computed with this calculator's default settings — open the tool above and you'll see the same numbers, then swap in your own data or flip to income mode.
From Engel's tables to market definition
Income elasticity has one of the oldest empirical pedigrees in economics. In 1857 the Prussian statistician Ernst Engel tabulated working families' budgets and found that the share of income spent on food falls as income rises — food is a necessity with income elasticity below 1, a regularity still known as Engel's law. The same arithmetic drives modern decisions: a grocery chain stocking more private-label brands is betting those goods are inferior (demand rises when incomes fall), while an airline adding premium cabins is betting on luxury-sized elasticities. Cross-price elasticity earns its keep in antitrust, where a high Exy between two products is evidence they compete in the same market — and in pricing, where a printer maker who knows ink is a strong complement can sell the printer near cost.
This tool is the sibling of ourprice elasticity of demand calculator— same math, different question. To see what elastic demand is worth to buyers in dollar terms, try theconsumer surplus calculator, or work the supply side of the market with themarginal cost calculator.
Frequently asked questions
What is cross-price elasticity of demand?
Cross-price elasticity (Exy) measures how the quantity demanded of one good responds when the price of a different good changes: the percentage change in quantity of good A divided by the percentage change in the price of good B. Unlike ordinary price elasticity, the sign is the point of the exercise. A positive Exy means the goods are substitutes, a negative Exy means they are complements, and a value near zero means the two markets barely touch. Regulators use it to define markets in antitrust cases, and firms use it to predict how a rival’s price move will hit their own sales.
How do I tell substitutes from complements?
Read the sign. Substitutes satisfy the same want, so when good B gets pricier, buyers switch to good A and its quantity rises — butter and margarine, or one streaming service and another. That co-movement makes Exy positive. Complements are consumed together, so a pricier good B makes the whole bundle more expensive and demand for good A falls — printers and ink cartridges, or gasoline and big SUVs. That opposite movement makes Exy negative. This calculator labels magnitudes at or below 0.1 as "unrelated" — a presentation convention for numbers too small to read anything into, not an official cutoff.
What is income elasticity of demand?
Income elasticity (Ei) asks the same question with income as the driver: the percentage change in quantity demanded divided by the percentage change in consumer income, holding prices fixed. It sorts goods by how demand travels with prosperity. A positive Ei marks a normal good (people buy more as they earn more); a negative Ei marks an inferior good (people buy less as they earn more). The size matters too: an Ei above 1 means spending on the good grows faster than income itself, which is the formal definition of a luxury.
What are inferior and luxury goods?
An inferior good is one people buy less of as income rises because they can now afford something they prefer — instant noodles, secondhand furniture, or bus trips for someone who buys a car. Its income elasticity is negative. A luxury has income elasticity above 1: restaurant meals, foreign vacations, and premium brands typically absorb a growing share of the budget as income climbs. Between the two sit normal necessities, with Ei between 0 and 1 — food overall, utilities, basic clothing — which grow with income but more slowly. The same good can move categories across income levels and countries.
Why use the midpoint (arc) method?
The simple point method divides each change by its starting value, so the same two data points give a different elasticity depending on which one you call "before". The midpoint method divides by the average of the two values instead, which makes the answer identical in both directions — the reason AP Economics and most intro textbooks require it for elasticity between two observed points. This calculator offers both: use midpoint for coursework and two-point data, and point when you genuinely mean "starting from here, a small change".
Sources
- OpenStax, Principles of Economics 3e — open textbook (see the chapters on elasticity)
- Marshall, Principles of Economics (1890) — Book III, on elasticity of wants (Library of Economics and Liberty)
The official figures this page quotes are drawn from the primary sources above — check them (or a qualified professional) before relying on a result.
Disclaimer: This calculator is foreducation and illustration only. An elasticity computed from two observed points assumes nothing else changed between them — real-world estimates control for the good's own price, other prices, income, and seasonality all at once. The substitutes / complements and inferior / luxury labels are textbook readings of a two-point calculation, not a demand study, and nothing here is pricing, financial, or tax advice.