Price Elasticity of Demand vs Cross-Price Elasticity of Demand
Price Elasticity of Demand and Cross-Price Elasticity of Demand are two Elasticity concepts in AP Economics that students often mix up. Price elasticity of demand measures how responsive quantity demanded is to a change in the good's price. Cross-price elasticity of demand measures how responsive the quantity demanded of one good is to a change in the price of another good. Here is how they compare side by side.
It is the percentage change in quantity demanded divided by the percentage change in price. Demand is elastic when the absolute value is greater than 1 and inelastic when it is less than 1. Goods with many substitutes, that take a large share of income, or judged over a longer time horizon tend to be more elastic.
It is calculated as the percentage change in quantity demanded of Good A divided by the percentage change in price of Good B. If the ratio is positive, the goods are considered substitutes. If the ratio is negative, the goods are considered complements.
Price vs Cross-Price Elasticity: Whose Price Sits in the Denominator
| Price Elasticity of Demand | Cross-Price Elasticity of Demand | |
|---|---|---|
| How many markets the formula touches | One, since the good's own price sits over its own quantity | Two, since one good's price sits over a different good's quantity |
| Meaning of the sign | Negative for any ordinary demand curve, so the absolute value is reported | Positive means substitutes, negative means complements |
| What a reading of zero means | Perfectly inelastic demand, a vertical curve, and buyers carrying the whole burden of a per-unit tax | Two unrelated goods, so one good's demand curve does not move when the other's price changes |
| Effect on the numerator good's curve | A movement along the curve | A shift of the curve to a new position |
| Symmetric between the two goods | No second good exists, so the question never arises | No: tea with respect to coffee differs in size from coffee with respect to tea |
| What the magnitude is used for | Judging revenue effects and how a tax burden splits | Judging how close two products are as rivals |
| Reading it off a stimulus table | One price column and the quantity column beside it | One price column and a quantity column from a different market entirely |
One price change generates both numbers, and only the numerator changes
A shop raises the price of a large coffee from $2.00 to $2.50. Its own coffee sales fall from 90 cups a day to 66, and its tea sales rise from 40 cups to 52. Both elasticities share a denominator: the midpoint change in coffee's price is 0.50 over 2.25, about 22.2 percent. Price elasticity uses coffee's own quantity, 24 over 78, about 30.8 percent downward, giving about 1.39 in absolute value, so coffee demand is elastic. Cross-price elasticity uses tea's quantity, 12 over 46, about 26.1 percent, giving positive 1.17, so tea and coffee are substitutes and reasonably close ones. Writing the two calculations one above the other makes the whole distinction visible: identical denominators, different numerators. Note also that the 24 lost coffee sales and the 12 gained tea sales do not have to match, since some buyers left the shop entirely. Students who mix these up almost always do it by pulling the wrong quantity column out of a stimulus table, not by misremembering a formula, which is why labeling each column with the good it belongs to before touching a calculator is worth the ten seconds it costs.
Cross-price elasticity is not symmetric, so the order of the two goods matters
Swap the roles in the coffee example and the number changes. Tea's price is not what moved, so the figure of 1.17 cannot be reused in the other direction. If tea's price rose 20 percent and coffee sales rose 5 percent, the cross-price elasticity of coffee with respect to tea is 0.25, far below the 1.17 computed the other way. The asymmetry is real rather than an artifact of rounding. The shop sells far more coffee than tea, so a given number of switchers is a large percentage of tea sales and a small percentage of coffee sales. A niche product is highly sensitive to the price of the mass-market product it competes with, while the mass-market product barely registers the traffic. Exam wording is precise about direction for this reason. The phrase cross-price elasticity of demand for tea with respect to the price of coffee names the numerator good first and the denominator good second, and reversing them changes the answer. Price elasticity has no such trap, because there is only one good and only one possible ordering.
Zero is a legitimate reading for both formulas and means something different each time
A cross-price elasticity near zero means two goods are unrelated. Change the price of umbrellas and the quantity of pencils demanded does not move, so the pencil demand curve stays exactly where it was. A price elasticity of zero means perfectly inelastic demand, a vertical curve where buyers purchase the same quantity at every price. The first is a statement about two markets not touching. The second is a statement about one market's buyers having no alternative at all. Exam items deploy them in different places. Perfectly inelastic demand shows up in tax incidence questions, where the answer is that buyers bear the entire burden of a per-unit tax. Zero cross-price elasticity shows up in classification questions, where the three answer choices are substitutes, complements, and unrelated goods, and the sign alone decides which one is correct. Reading a zero as no relationship when the question was about tax burden is a fast way to lose an easy point.
Frequently asked questions
How does cross-price elasticity tell substitutes from complements?
Cross-price elasticity separates them by sign alone. A positive value means substitutes: the price of one good rises, buyers move toward the other, and its quantity demanded rises. A negative value means complements: the price of one good rises, buyers use less of it, and they demand less of the paired good as well. Beef and chicken typically give a positive figure, printers and ink cartridges a negative one. Magnitude then tells you how strong the link is, so a pair at positive 2.4 are far closer substitutes than a pair at positive 0.2.
Why is the cross-price elasticity of A with respect to B different from B with respect to A?
Cross-price elasticity divides one percentage change by another, and a percentage depends on the base it is measured against. When a large market and a small market trade customers, one flow of buyers is a big percentage of the small market and a small percentage of the large one. Ten buyers moving into a tea market averaging 50 cups is 20 percent of it, while those same ten leaving a coffee market averaging 250 cups is 4 percent. Divide each by the same percentage price change and the two elasticities differ by a factor of five, even though a single flow of customers produced both.
Does a change in a good's own price ever shift its own demand curve?
A change in a good's own price never shifts that good's demand curve. It moves buyers along the curve, which is precisely what price elasticity measures. The shift happens on the other good's diagram. When coffee's price rises, coffee's quantity demanded slides up its unchanged curve while the demand curve for tea shifts rightward, and cross-price elasticity puts a number on that shift. Sketching the two diagrams side by side in your notes prevents the most common wording error on this topic.
Live Elasticity graph. Drag the curves, or open the full version.
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