Cross-Price Elasticity Of Demand

Why Compare Demand With Another Good’s Price?

A change in one product’s price can be accompanied by a change in demand for a different product. To measure the responsiveness of that demand, keep the two goods straight: the quantity change is for the good being studied, while the price change is for the other good. This comparison is called cross-price elasticity of demand.

The measure is a ratio of percentage changes, not a ratio of dollar changes or unit changes. That makes it independent of whether quantity is measured in individual items, packs, or hundreds of units. It also means that both percentage changes must be calculated consistently. When the data give two observations rather than a single starting point, the midpoint method provides a symmetric comparison.

Definition: Cross-price elasticity of demand measures the percentage responsiveness of quantity demanded of good $X$ to a percentage change in the price of a different good $Y$, with other relevant determinants held constant. For a small change, its point form is $$ \varepsilon_{Q_X,P_Y} = \frac{\partial Q_X}{\partial P_Y}\frac{P_Y}{Q_X}. $$ For finite changes measured from a specified starting observation, it is calculated as the percentage change in $Q_X$ divided by the percentage change in $P_Y$. The result is signed; do not take its absolute value unless a question explicitly asks for an absolute magnitude.

Here $Q_X$ is the quantity demanded of good $X$, and $P_Y$ is the price of good $Y$. The notation helps prevent a common mix-up: the price in the denominator is not the price of good $X$. In a calculation using observed changes, both the quantity and price changes should refer to the same before-and-after period.

For a percentage change based on a specified starting value, divide the change by that starting value. For two observations with no designated base period, the midpoint method instead divides each change by the average of its two corresponding values. The method used for quantity and price must be consistent.

The Midpoint Calculation

Let the two observations be $(Q_{X,1},P_{Y,1})$ and $(Q_{X,2},P_{Y,2})$. The midpoint percentage change in quantity uses the average of the two quantities as its denominator. The midpoint percentage change in price uses the average of the two prices. Their ratio is the midpoint, or arc, cross-price elasticity:

\varepsilon_{Q_X,P_Y}^{\text{mid}} = \frac{\dfrac{Q_{X,2}-Q_{X,1}}{(Q_{X,1}+Q_{X,2})/2}} {\dfrac{P_{Y,2}-P_{Y,1}}{(P_{Y,1}+P_{Y,2})/2}}.

This calculation requires positive midpoint quantity and price, and a nonzero price change. If the price does not change, the percentage change in price is zero, so this ratio cannot be used to estimate a response to that price change. The midpoint percentage change is also not defined if its average quantity or average price is zero. In the usual setting with positive quantities and prices, these denominators are positive.

Why the Midpoint Method Is Symmetric

The midpoint method is useful because it gives the same percentage change when the direction of comparison is reversed. To see why, consider two positive values $A$ and $B$. The midpoint percentage change from $A$ to $B$ is $(B-A)/((A+B)/2)$. Reversing the comparison gives $(A-B)/((B+A)/2)$. The denominators are identical, while the numerators have opposite signs. Thus, reversing the comparison changes only the sign of each percentage change. When both quantity and price comparisons are reversed, both signs change, so their ratio is unchanged.

The method is not an assertion that the price change caused the quantity change. It is a way of calculating a consistent percentage-change ratio. For the result to describe the response of demand to the other good’s price, other relevant influences—such as the price of good $X$, income, and preferences—must be held constant or otherwise accounted for. If several influences changed at once, the ratio of observed changes alone does not isolate the effect of $P_Y$.

Worked Example: Two-Observation Calculation for a Beverage

In a hypothetical market, the price of coffee rises from $4.00 to $5.00 per bag. Over the same period, weekly demand for tea rises from 120 to 132 bags. Assume other relevant demand conditions are unchanged. Calculate the midpoint cross-price elasticity of tea demand with respect to coffee’s price.

Method. Use midpoint percentage changes for both tea quantity and coffee price, since the data provide two observations and no base period is specified.

Step 1: Calculate the midpoint percentage change in tea quantity. The quantity change is $132-120=12$ bags. The average quantity is $(120+132)/2=126$ bags. Therefore, $$ \frac{12}{126}=\frac{2}{21}\approx 0.095238=9.5238\%. $$

Step 2: Calculate the midpoint percentage change in coffee price. The price change is $$5.00-$4.00=$1.00$. The average price is $($4.00+$5.00)/2=$4.50$. Therefore, $$ \frac{1.00}{4.50}=\frac{2}{9}\approx 0.222222=22.2222\%. $$

Step 3: Divide the percentage changes. $$ \varepsilon_{Q_{\text{tea}},P_{\text{coffee}}}^{\text{mid}} = \frac{2/21}{2/9} = \frac{9}{21} = \frac{3}{7} \approx 0.4286. $$

Check. Both calculated percentage changes are positive, so their ratio is positive. The answer is a ratio of percentage changes, not $12/1$, which would mix bags and dollars. The method also avoids the common mistake of using different bases for the two percentage changes.

Worked Calculations With Different Directions of Change

A decrease in price or quantity must retain its negative sign in the percentage change. Dividing two signed percentage changes determines the sign of the elasticity. The following examples illustrate how to calculate the ratio when the observed changes move in different directions.

Worked Example: Both Quantity and Price Fall

A hypothetical streaming service observes that the price of a related service falls from $20 to $18 per month. At the same time, demand for its own subscription changes from 500 to 460 subscriptions per week. Assume other relevant conditions are unchanged. Find the midpoint cross-price elasticity of demand for the service being studied.

Method. Calculate both midpoint percentage changes, retaining the negative signs, and then divide.

Step 1: Quantity change. The change is $460-500=-40$, and the average quantity is $(500+460)/2=480$. Thus, $$ \frac{-40}{480}=-\frac{1}{12}\approx -0.083333=-8.3333\%. $$

Step 2: Price change. The change is $$18-$20=-$2$, and the average price is $($20+$18)/2=$19$. Thus, $$ \frac{-2}{19}\approx -0.105263=-10.5263\%. $$

Step 3: Calculate the ratio. $$ \varepsilon_{Q_X,P_Y}^{\text{mid}} = \frac{-1/12}{-2/19} = \frac{19}{24} \approx 0.7917. $$

Check. The two percentage changes are both negative, so their ratio is positive. Do not discard either negative sign before division; doing so may hide whether the changes moved in the same or opposite directions. The calculation uses the price of the related service in the denominator, not the price of the service whose demand is being measured.

Worked Example: Quantity Falls While the Other Price Rises

In a hypothetical market, a console’s price rises from $300 to $330, and weekly demand for a particular game falls from 240 to 210 downloads. Assume other relevant demand conditions are unchanged. Calculate the midpoint cross-price elasticity of game demand with respect to the console’s price.

Method. Apply the midpoint calculation to both variables and preserve the sign of each change.

Step 1: Calculate the game-quantity percentage change. The change is $210-240=-30$ downloads, and the average quantity is $(240+210)/2=225$. Therefore, $$ \frac{-30}{225}=-\frac{2}{15}\approx -0.133333=-13.3333\%. $$

Step 2: Calculate the console-price percentage change. The change is $$330-$300=$30$, and the average price is $($300+$330)/2=$315$. Therefore, $$ \frac{30}{315}=\frac{2}{21}\approx 0.095238=9.5238\%. $$

Step 3: Divide quantity responsiveness by price change. $$ \varepsilon_{Q_{\text{game}},P_{\text{console}}}^{\text{mid}} = \frac{-2/15}{2/21} = -\frac{21}{15} = -1.4. $$

Check. A negative quantity percentage change divided by a positive price percentage change gives a negative elasticity. The answer is $-1.4$, not $1.4$; replacing the signed changes with their absolute values would lose information. Each percentage change is calculated using its own midpoint, not the midpoint of the other variable.

Common Mistakes and Checks

The calculation is short, but a small setup error can change its meaning. Before finalizing a result, check which good’s quantity appears in the numerator and which good’s price appears in the denominator. Then verify that both percentage changes use the same convention and that the final sign follows from the signed changes.

Using the wrong price. Cross-price elasticity pairs demand for $X$ with the price of $Y$. Using the price of $X$ instead calculates a different elasticity.
Mixing percentage-change methods. For example, using an initial-value denominator for quantity and a midpoint denominator for price does not produce a consistent midpoint calculation.
Dropping negative signs. The definition uses signed percentage changes. Taking absolute values can turn a negative result into a positive one.
Dividing raw changes. A change in bags divided by a change in dollars is not an elasticity. Calculate the percentage change in each variable first.
Treating an observed ratio as automatically causal. If other demand conditions changed, the ratio alone cannot separate their effects from the effect of the other good’s price.

Key takeaway: Cross-price elasticity is the signed percentage change in demand for one good divided by the percentage change in another good’s price. With two observations and no designated base period, use the midpoint denominator for both variables, keep the signs, and verify that the other good’s price is the one in the denominator.

Check Your Understanding

For each problem, use the stated method and keep signed changes. Assume other relevant demand conditions are unchanged unless the question says otherwise.

The price of a different good rises from $10 to $12, while demand for the good being studied rises from 80 to 88 units. Calculate the midpoint cross-price elasticity. (Answer: $0.5238$)
Demand for good $X$ falls by 6% while the price of good $Y$ rises by 15%. Calculate the cross-price elasticity using the given percentage changes. (Answer: $-0.4000$)
The price of good $Y$ rises from $8 to $10, while demand for good $X$ falls from 150 to 135 units. Calculate the midpoint cross-price elasticity. (Answer: $-0.4737$)
The price of a related product falls from $25 to $20, while demand for the product being studied rises from 96 to 120 units. Calculate the midpoint cross-price elasticity. (Answer: $-1.0000$)
The price of good $Y$ rises from $14.00 to $15.40, while demand for good $X$ falls from 420 to 378 units. Calculate the midpoint cross-price elasticity, showing the midpoint percentage change for each variable. (Answer: $-1.1053$)

What You'll Learn