What Is Cross Price Elasticity of Demand? Formula and Examples

Search Fintrev

Search our guides, reviews and glossary.

Cross price elasticity of demand measures how much the quantity demanded of one good changes when the price of a different good changes. It is calculated as the percentage change in the quantity demanded of good X divided by the percentage change in the price of good Y. The sign of the result shows how the two goods are related: a positive value indicates substitutes, a negative value indicates complements, and a value near zero indicates the goods are largely unrelated. The size of the number shows how strong that relationship is.

The measure is also called cross elasticity of demand and is often abbreviated XED or E_xy.

The Cross Price Elasticity Formula

The basic cross elasticity of demand formula is:

E_xy = % change in quantity demanded of X ÷ % change in price of Y

Written with symbols, where Q_x is the quantity of good X, P_y is the price of good Y and Δ means “change in”:

E_xy = (ΔQ_x / Q_x) ÷ (ΔP_y / P_y) = (ΔQ_x / ΔP_y) × (P_y / Q_x)

The last form is useful because it splits the measure into two parts. The first part, ΔQ_x/ΔP_y, is the raw response in units of X per dollar change in Y’s price. The second part, P_y/Q_x, converts that raw response into percentage terms so the result has no units. Because of that conversion, an elasticity of 0.5 means the same thing whether the goods are sold in cups, gallons or subscriptions.

Midpoint (Arc) Formula

Simple percentage changes give different answers depending on which direction you measure. A move from 100 to 120 is a 20% increase, while a move from 120 back to 100 is a 16.7% decrease. To avoid that, economists often use the midpoint method, which divides each change by the average of the starting and ending values:

• % change in Q_x = (Q₂ − Q₁) ÷ [(Q₁ + Q₂) / 2]
• % change in P_y = (P₂ − P₁) ÷ [(P₁ + P₂) / 2]

The resulting arc elasticity is the same whether the price rises or falls between the two points.

Cross Price Elasticity From a Demand Equation

When demand is given as an equation, you can calculate point cross elasticity directly. Take a hypothetical linear demand function for good X:

Q_x = 500 − 20P_x + 10P_y

The coefficient on P_y (here 10) is ΔQ_x/ΔP_y: each $1 rise in Y’s price adds 10 units of demand for X, holding X’s own price constant. To turn this into an elasticity, multiply by P_y/Q_x at a specific point. If P_x = $10 and P_y = $8, then Q_x = 500 − 200 + 80 = 380, and:

E_xy = 10 × (8 / 380) ≈ 0.21

The positive coefficient tells you X and Y are substitutes in this model. In a linear equation the slope term ΔQ_x/ΔP_y is constant, but the elasticity is not, because P_y/Q_x changes at every point along the curve.

How to Calculate Cross Price Elasticity of Demand

1. Record the starting and ending quantity demanded of good X and the starting and ending price of good Y.
2. Calculate the percentage change in the quantity of X, using the midpoint method above.
3. Calculate the percentage change in the price of Y the same way.
4. Divide the result from step 2 by the result from step 3. Keep the sign, because it tells you whether the goods are substitutes or complements.

Interpreting the Result

A change in the price of Y does not move good X along its own demand curve. It shifts the whole demand curve for X. A rise in the price of a substitute shifts X’s demand curve to the right, so buyers want more X at every price of X. A rise in the price of a complement shifts it to the left. Cross price elasticity measures the size and direction of that shift.

Positive Cross Price Elasticity: Substitutes

If cross price elasticity is positive, the goods are substitutes. When coffee prices rise, some coffee drinkers switch to tea, so the quantity of tea demanded rises. A positive change divided by a positive change gives a positive result.

The size of the coefficient shows how close the substitutes are:

• Weak substitutes have a positive but small coefficient. Tea and coffee are often given as an example: they can replace each other, but many people have a firm preference.
• Strong substitutes have a larger coefficient, such as two brands of toothpaste or chicken burritos from two neighboring restaurants that charge the same price. A price rise at one seller sends a larger share of buyers to the other.
• Perfect substitutes, goods that buyers see as identical, have a cross elasticity that in theory approaches infinity. A tiny price difference would shift all demand to the cheaper option.

Negative Cross Price Elasticity: Complements

Cross price elasticity is negative when the goods are complements, meaning they are used together. When the price of coffee rises, people drink less coffee and buy fewer coffee stir sticks. A negative change in quantity divided by a positive change in price gives a negative result.

Other common pairings include printers and ink cartridges, cars and gasoline, tractors and diesel fuel, game consoles and games, and movie tickets and cinema popcorn. A larger negative number indicates a tighter link. A smaller negative number means the goods are often but not always bought together, as with burgers and fries.

Zero: Unrelated Goods

A coefficient at or near zero means the goods are independent: a rise in the price of eggs, for example, is not expected to change how many olives people buy.

Cross Elasticity Is Not Symmetric

E_xy and E_yx are usually different numbers. A big rise in car prices may sharply reduce gasoline demand over time, while a rise in gasoline prices may affect car purchases differently. The first subscript is the good whose quantity is measured and the second is the good whose price changed.

Worked Example

In this hypothetical example, a café raises the price of a cup of coffee from $4.00 to $5.00. Over the following weeks, with nothing else changed:

• Weekly tea sales rise from 200 cups to 240 cups.
• Weekly stir stick use falls from 1,000 to 850.

Percentage change in coffee’s price (midpoint): $1.00 ÷ $4.50 ≈ 22.2%

Tea: change in quantity = 40 ÷ 220 ≈ 18.2% E_{tea, coffee} = 18.2% ÷ 22.2% ≈ +0.82

The positive sign confirms tea is a substitute for coffee at this café. Because the value is below 1, tea demand rose by a smaller percentage than coffee’s price did.

Stir sticks: change in quantity = −150 ÷ 925 ≈ −16.2% E_{sticks, coffee} = −16.2% ÷ 22.2% ≈ −0.73

The negative sign confirms stir sticks are a complement to coffee.

Using simple percentages instead of midpoints, the tea calculation would give 20% ÷ 25% = 0.80. The difference is small here, but it grows as the price change gets larger, which is why the midpoint method is preferred for discrete changes.

How Cross Price Elasticity Is Used

Pricing decisions. A product with close substitutes has limited room for price rises, because a high positive cross elasticity means customers will switch to rivals. A product with no close substitutes can often be priced higher. Firms test incremental price changes to estimate how much demand they would lose to competitors.

Pricing complementary products together. When two goods are strong complements, a firm that sells both can treat them as one system. Printers are sometimes sold at a low price, even at a loss, on the expectation that buyers will keep purchasing the manufacturer’s ink. Game consoles and games, or razors and blades, follow similar logic.

Managing a product line. A company that sells several similar items can use cross elasticity to check whether a discount on one product mainly attracts new customers or mostly pulls sales away from its own other products, an effect called cannibalization.

Competition policy. Antitrust authorities use cross elasticity to decide which products belong in the same market. If buyers readily switch between two products when one gets more expensive, the products compete with each other, and a merger between their makers may raise concerns. A well-known U.S. Supreme Court case, United States v. E. I. du Pont de Nemours & Co. (1956), turned on whether cellophane competed with other flexible wrapping materials, and the Court’s reasoning relied on the cross elasticity between them.

Limitations of Cross Price Elasticity

It assumes everything else stays constant. The formula isolates the effect of one price on one quantity. In real data, incomes, seasons, advertising, the good’s own price and the prices of other goods all change at the same time. Separating the cross price effect from those influences requires careful statistical work, not just two observations.

It depends on the starting point and the size of the change. Elasticity varies along a demand curve, so a figure measured at one price level may not hold at another. Large price changes can produce different responses from small ones.

It changes over time. Buyers often react more in the long run than in the short run, because switching to a substitute or abandoning a complementary product can take time. A cross elasticity measured over a few weeks may understate the eventual response.

Inflation can distort the inputs. When the general price level is rising, a nominal price increase overstates the real increase. Analysts often adjust prices using an index such as the Consumer Price Index before calculating elasticities over long periods.

The sign mixes two effects. A price change affects buyers through substitution and through purchasing power. When a widely used good gets more expensive, people have less money left for everything else, which can lower demand for other goods even if they are not true complements. Substitutes and complements defined by observed cross elasticity (gross substitutes and gross complements) include both effects.

Labels are not fixed. The same pair of goods can be substitutes for one group of buyers and complements or unrelated for another, and the relationship can shift as tastes or technology change.

Cross Price Elasticity vs. Related Measures

Price elasticity of demand measures how the quantity demanded of X responds to a change in X’s own price, not Y’s. Own price elasticity is normally negative, while cross price elasticity can be positive, negative or zero.

Income elasticity of demand measures how the quantity demanded responds to a change in buyers’ income rather than to any price. It separates normal goods (positive) from inferior goods (negative), a different classification from substitutes and complements.

Cross elasticity of supply is a supply-side measure. It looks at how the quantity of one good that producers supply responds to a change in the price of another good, for instance when farmers shift land from one crop to another after relative prices move.

Marginal rate of substitution describes the rate at which a consumer is willing to trade one good for another while staying equally satisfied. It is a theoretical concept about preferences. Cross price elasticity is a measurable market response to price changes.

FAQs

The Bottom Line

Pricing managers, product line planners and antitrust authorities rely on cross price elasticity to see how goods compete or work together. A high positive value limits room for price rises, while strong complements can be priced as one system. Because the measure varies along the demand curve, shifts over time and mixes substitution with purchasing power effects, any single estimate should be read with its starting point and time frame in mind.