Article — Cross-Price Elasticity Calculator
Cross-price elasticity calculator: substitutes vs complements
Cross-price elasticity of demand is the percent change in quantity demanded of one good divided by the percent change in price of another. A positive value means the two goods are substitutes, a negative value means they are complements, and a value near zero means they are unrelated. The calculator on this page applies the standard point and midpoint formulas and labels the result so a non-economist can read it at a glance.
Economists abbreviate the measure as CPE or EAB, where A is the good whose quantity is changing and B is the good whose price is changing. The order of the two goods matters — the elasticity of beef with respect to chicken is not the same number as the elasticity of chicken with respect to beef.
What cross-price elasticity measures
Demand for most products depends on more than its own price. A shopper who sees butter jump in price may switch to margarine; a driver who sees gasoline jump in price may also delay buying a new SUV. Cross-price elasticity puts a number on those linked responses so firms, regulators, and forecasters can compare them across markets.
The figure is unit-free. Whether quantities are measured in cans or tonnes and prices in dollars or euros, the ratio of two percent changes leaves a pure number. That is why textbooks treat cross-price elasticity as one of the four standard demand elasticities alongside own-price, income, and advertising elasticity.
The U.S. Department of Justice and the Federal Trade Commission use cross-price elasticities in the SSNIP test — “small but significant non-transitory increase in price” — when defining markets in merger reviews. If a 5% hypothetical price rise on one product would push buyers to another, the two goods belong in the same market.
The cross-price elasticity formula
The point formula divides the percent change in quantity demanded of A by the percent change in price of B:
Point E = %ΔQ_A / %ΔP_BMidpoint E = [ΔQ / avg Q] / [ΔP / avg P]Substitutes E > 0Complements E < 0The midpoint version uses the average of the start and end values instead of the starting values. That small change has a useful property — the elasticity is the same whether you read the price move as old-to-new or new-to-old. The point formula is asymmetric in direction, which is why introductory courses often start there but applied work prefers the midpoint approach.
Reading the sign: substitutes vs complements
The sign of cross-price elasticity carries more economic information than the size. A positive number means the two goods substitute for each other in consumer choice. A negative number means they are complements that move together in shopping baskets.
A handful of measured cross-price elasticities have become standard reference points in textbooks and antitrust filings. Coca-Cola and Pepsi sit near +0.7, meaning a 10% rise in Pepsi's price pushes Coke demand up about 7%. Beef and chicken come in near +0.45. Cars and gasoline are complementary at roughly −0.2 to −0.3 in transport-economics studies.
- Coca-Cola ↔ Pepsi = +0.60 to +0.80 (substitutes, sweetened-beverage research)
- Beef ↔ chicken = +0.40 to +0.50 (USDA Economic Research Service)
- Coffee ↔ tea = +0.25 to +0.35 (consumer-panel studies)
- Cars ↔ gasoline = −0.15 to −0.30 (transport-economics literature)
- Bread ↔ butter = −0.20 (complementary household staples)
- Game consoles ↔ games = strongly negative (classic razor-and-blade pairing)
Midpoint method for cross-price elasticity
When the price move is larger than a few percent, point elasticity gives different answers depending on direction. A price rise from $10 to $12 is a 20% increase if you divide by $10, but a $12 to $10 move is a 16.7% decrease if you divide by $12. The midpoint formula avoids that by dividing each change by the average of the two values.
Mathematically:
$$E_arc = [(Q2 - Q1) / ((Q1 + Q2)/2)] / [(P2 - P1) / ((P1 + P2)/2)]$$
The result is direction-independent and slightly more conservative for large moves. Empirical work in food, energy, and transport economics almost always reports midpoint elasticities for that reason.
If you have only two data points and the price change is bigger than 5%, prefer the midpoint method. The point formula is fine for short-window calculus problems and for routine pricing sensitivities where the move is small.
Worked cross-price elasticity examples
Example 1 — beverages, point method. Pepsi raises prices 10%. Coke sales rise 6%. The cross-price elasticity of Coke with respect to Pepsi is +0.6 (= 6% ÷ 10%). Coke and Pepsi are substitutes; the strength is moderate.
Example 2 — fuel and vehicles, midpoint. Gasoline prices climb from $3.00 to $3.60 (average $3.30). Monthly SUV unit sales fall from 100 to 92 (average 96). The percent change in quantity is −8.33% and in price is +18.18%. The cross-price elasticity is −0.46 — SUVs and gasoline are complements.
Example 3 — coffee and tea, point method. Tea's price falls 8% during a promotion. Coffee shop sales drop 2.4%. The elasticity is +0.30 — a weak substitute relationship typical of caffeine sources that satisfy slightly different occasions.
Business uses of cross-price elasticity
Pricing teams use cross-price elasticity to forecast how a competitor's discount will affect their own sales. A retailer that knows the elasticity of its store brand to a national brand can predict how a sale on the national label will dent private-label volume that week.
Product managers also use it for portfolio decisions. If two SKUs in the same lineup are highly substitutable to each other, marketing budgets that lift one mostly cannibalise the other. Highly complementary SKUs justify bundle promotions and joint marketing.
The same study can report a different sign depending on which good is A and which is B. Always state the direction explicitly: “cross-price elasticity of beef with respect to chicken” means a chicken-price change drives a beef-quantity response. Reversing the labels can flip the size if the markets are asymmetric.
Common cross-price elasticity mistakes
Three errors recur in classroom problems and on retailer dashboards. The first is dropping the sign and reporting absolute values. The sign is the whole story; an absolute value strips out the substitute-versus-complement distinction.
The second is mixing units. Cross-price elasticity needs percent changes on both sides. A common slip is to divide a dollar change by a percent change, which produces a meaningless number that depends on price levels.
The third is reading the cross-price elasticity of two goods at one point in time as if it held at every other price level. Cross-price elasticities can shift over time with consumer tastes, income levels, and the availability of new substitutes. Estimates from a 2010 study may no longer hold after a streaming service has entered the market or a new bundle has launched.
The Bureau of Labor Statistics publishes Consumer Expenditure Survey microdata that researchers use to estimate cross-price elasticities for hundreds of consumer-goods pairs. The same survey underpins the Consumer Price Index basket and is one of the most-cited sources for U.S. demand estimation.