Cross-Price Elasticity of Demand | Equation, Derivation and Rearrangements

Q: What are common mistakes with the Cross-Price Elasticity of Demand formula?

Confusing cross-price elasticity with own-price elasticity of demand. Assuming a result of zero implies no relationship when it could imply unrelated goods.

Q: What is a real-world example of the Cross-Price Elasticity of Demand formula?

In an economic or financial decision involving Cross-Price Elasticity of Demand, Cross-Price Elasticity of Demand is used to calculate the XED value from Percentage Change in Quantity of X and Percentage Change in Price of Y. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Q: What are some study tips for the Cross-Price Elasticity of Demand formula?

Remember that the sign (+ or -) is just as important as the magnitude. Ensure you are using percentage changes, not absolute changes, to maintain consistency. Consider that XED can vary along the demand curve, so be aware of the specific price range being analyzed.

Core idea

Overview

A positive XED indicates that two goods are substitutes, as an increase in the price of one leads to an increase in the demand for the other. Conversely, a negative XED identifies complementary goods, where a rise in the price of one reduces the demand for its complement. This metric is essential for firms assessing market competition and portfolio strategy.

When to use: Apply this when determining the competitive relationship between two goods or analyzing the impact of pricing strategy on a related product line.

Why it matters: It allows businesses and policymakers to understand how price shocks in one sector—such as gasoline—propagate to demand in related sectors, like electric vehicles or public transit.

Symbols

Variables

XED = Cross-Price Elasticity of Demand, % $Δ$ $Q_{x}$ = Percentage Change in Quantity of X, % $Δ$ $P_{y}$ = Percentage Change in Price of Y

XED

Cross-Price Elasticity of Demand

Absolute value of demand elasticity

Percentage Change in Quantity of X

%

Percentage Change in Price of Y

%

Walkthrough

Derivation

Derivation of Cross-Price Elasticity of Demand

This derivation defines the cross-price elasticity of demand as the ratio of the relative change in the quantity demanded of one good to the relative change in the price of a related good.

Ceteris paribus: all other factors affecting demand remain constant.
The goods X and Y are related, meaning a change in the price of Y influences the quantity demanded of X.

1

Define Absolute Change

Elasticity is defined as the ratio of the proportional change in quantity to the proportional change in price, expressed as the change in a variable divided by its original value.

X E D = \frac{\frac{Δ Q _{x}}{Q _{x}}}{\frac{Δ P _{y}}{P _{y}}}

Note: Ensure you are using the percentage change form for the denominator and numerator.

2

Convert to Percentage Form

Since multiplying both the numerator and denominator by 100 transforms the decimals into percentages, we arrive at the standard formula for cross-price elasticity.

X E D = \frac{%Δ Q _{x}}{%Δ P _{y}}

Note: Remember that if XED > 0, the goods are substitutes; if XED < 0, they are complements.

Result

X E D = \frac{%Δ Q _{x}}{%Δ P _{y}}

Source: Mankiw, N. G. (2020). Principles of Economics (9th ed.). Cengage Learning.

Free formulas

Rearrangements

Solve for $%Δ Q_{x}$

Make % $Δ$ $Q_{x}$ the subject

%Δ Q_{x} = X E D \cdot %Δ P_{y}

Isolate the percentage change in quantity by multiplying both sides by the percentage change in the price of Y.

Difficulty: 2/5

Solve for $%Δ P_{y}$

Make % $Δ$ $P_{y}$ the subject

%Δ P_{y} = \frac{%Δ Q _{x}}{X E D}

Isolate the percentage change in price by rearranging the elasticity ratio.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of this as a 'lever' connecting two different markets. If you push down on the price of good Y, the quantity of good X moves either in the same direction (like a gear pulling a chain) or the opposite direction (like a see-saw), depending on how tightly they are linked in the consumer's lifestyle.

XED

Cross-Price Elasticity of Demand

The 'sensitivity ratio'; it tells you how much the consumer's interest in good X shifts when the price tag on good Y changes.

Percentage change in quantity demanded of Good X

The result or the reaction: how much more or less of good X people are buying.

Percentage change in price of Good Y

The stimulus: the trigger event happening in the market for a different good.

Signs and relationships

Positive (+): Indicates 'substitutes'; if the price of coffee (Y) rises, people buy more tea (X) instead.
Negative (-): Indicates 'complements'; if the price of printers (Y) rises, people buy fewer ink cartridges (X) because they bought fewer printers.
Zero (0): Indicates 'independent goods'; the price change of one has absolutely no impact on the demand for the other.

One free problem

Practice Problem

If the price of Good Y increases by 10% and the quantity demanded of Good X increases by 5%, what is the cross-price elasticity?

Percentage Change in Quantity of X5 %

Percentage Change in Price of Y10 %

Solve for: XED

Hint: Divide the percentage change in quantity of X by the percentage change in price of Y.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In an economic or financial decision involving Cross-Price Elasticity of Demand, Cross-Price Elasticity of Demand is used to calculate the XED value from Percentage Change in Quantity of X and Percentage Change in Price of Y. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Study smarter

Tips

Remember that the sign (+ or -) is just as important as the magnitude.
Ensure you are using percentage changes, not absolute changes, to maintain consistency.
Consider that XED can vary along the demand curve, so be aware of the specific price range being analyzed.

Avoid these traps

Common Mistakes

Confusing cross-price elasticity with own-price elasticity of demand.
Assuming a result of zero implies no relationship when it could imply unrelated goods.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

This derivation defines the cross-price elasticity of demand as the ratio of the relative change in the quantity demanded of one good to the relative change in the price of a related good.

Apply this when determining the competitive relationship between two goods or analyzing the impact of pricing strategy on a related product line.

It allows businesses and policymakers to understand how price shocks in one sector—such as gasoline—propagate to demand in related sectors, like electric vehicles or public transit.

Confusing cross-price elasticity with own-price elasticity of demand. Assuming a result of zero implies no relationship when it could imply unrelated goods.

In an economic or financial decision involving Cross-Price Elasticity of Demand, Cross-Price Elasticity of Demand is used to calculate the XED value from Percentage Change in Quantity of X and Percentage Change in Price of Y. The result matters because it helps compare incentives, policy effects, market outcomes, or financial decisions in context.

Remember that the sign (+ or -) is just as important as the magnitude. Ensure you are using percentage changes, not absolute changes, to maintain consistency. Consider that XED can vary along the demand curve, so be aware of the specific price range being analyzed.

References

Sources

Mankiw, N. G. (2020). Principles of Economics (9th ed.). Cengage Learning.
Pindyck, R. S., & Rubinfeld, D. L. (2017). Microeconomics (9th ed.). Pearson.
Varian, H. R. (2014). Intermediate Microeconomics: A Modern Approach (9th ed.). W. W. Norton & Company.