Cross Elasticity (XED) Equation, Derivation & Rearrangements

Q: What are common mistakes with the Cross Elasticity (XED) formula?

Swapping the goods in the formula.

Q: What is a real-world example of the Cross Elasticity (XED) formula?

Tea price rises 10%, coffee demand rises 5%. XED = 0.5 (Substitutes).

Q: What are some study tips for the Cross Elasticity (XED) formula?

A positive result indicates the goods are substitutes. A negative result indicates the goods are complements. A result near zero suggests the products are unrelated. Focus on the sign first to identify the relationship, then the magnitude for the strength.

Core idea

Overview

Cross Elasticity of Demand (XED) measures the responsiveness of the quantity demanded for one good to a change in the price of another good. It is a fundamental metric used to classify the relationship between two products as substitutes, complements, or independent of one another.

When to use: Apply this formula when analyzing how a competitor's pricing strategy impacts your sales volume or how the cost of a primary good affects its accessories. It assumes a ceteris paribus environment where consumer income and the price of the product itself remain constant.

Why it matters: This metric helps firms identify their closest competitors and plan production based on shifts in related markets. For regulators, it is a critical tool for defining market boundaries and determining if a merger might lead to a monopoly by reducing consumer alternatives.

Symbols

Variables

XED = XED, \%\Delta QD_A = %Δ Q Good A, \%\Delta P_B = %Δ P Good B

X E D

XED

V a r iab l e

%Δ Q D_{A}

%Δ Q Good A

%Δ P_{B}

%Δ P Good B

Walkthrough

Derivation

Derivation: Cross Price Elasticity (XED)

XED measures the interdependence of goods.

1

Ratio formula:

How the demand for good A reacts to price changes in good B.

X E D = \frac{%Δ Q D _{A}}{%Δ P _{B}}

Result

X E D = \frac{%Δ Q D _{A}}{%Δ P _{B}}

Source: A-Level Economics — Elasticities

Free formulas

Rearrangements

Solve for $X E D$

Make XED the subject

X E D = \frac{%Δ Q D _{A}}{%Δ P _{B}}

XED is already the subject of the formula.

Difficulty: 1/5

Solve for $%Δ Q D_{A}$

Make %Delta QDA the subject

%Δ Q D_{A} = X E D \times %Δ P_{B}

Start from the Cross Elasticity of Demand (XED) formula. To make the percentage change in quantity demanded for good A (`\% $Δ$ $Q D_{A}$ `) the subject, multiply both sides by the percentage change in price for good B (`\% $Δ$ $P_{B}$ `).

Difficulty: 2/5

Solve for $%Δ P_{B}$

Make %Delta PB the subject

%Δ P_{B} = \frac{%Δ Q D _{A}}{X E D}

Start from the Cross Elasticity of Demand (XED) formula. To make %Delta PB the subject, first clear the denominator by multiplying, then divide by XED.

Difficulty: 2/5

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

Visual intuition

Graph

The graph is a straight line passing through the origin, representing a proportional relationship where the slope is determined by the fixed percentage change in the price of good B. For an economics student, this linear shape means that a larger percentage change in the quantity demanded of good A corresponds to a higher cross elasticity value, indicating a more significant responsiveness between the two goods. The most important feature is that the linear relationship means doubling the percentage change in quant

Graph type: linear

Why it behaves this way

Intuition

Imagine two distinct demand curves, where a shift in the supply or demand of one good causes a measurable shift in the demand curve of the other, indicating their market relationship.

XED

Operational measure of the sensitivity of demand for one good to a price change in another.

Quantifies if products are substitutes (positive XED), complements (negative XED), or unrelated (zero XED).

The percentage change in the quantity of good A demanded by consumers.

Represents the proportional shift in consumer purchases of good A.

The percentage change in the market price of good B.

Represents the proportional change in the cost of good B to consumers.

Signs and relationships

XED > 0: A positive value indicates that goods A and B are substitutes. As the price of good B rises, consumers demand more of good A.
XED < 0: A negative value indicates that goods A and B are complements. As the price of good B rises, consumers demand less of good A.
XED = 0: A zero value indicates that goods A and B are unrelated. A price change in B does not significantly affect the demand for A.

Free study cues

Insight

Canonical usage

Cross Elasticity of Demand (XED) is a dimensionless ratio, representing the responsiveness of the percentage change in quantity demanded of one good to the percentage change in the price of another good.

Common confusion

A common confusion is attempting to assign units to XED or other elasticity measures. Since it is a ratio of percentage changes, the units effectively cancel out, resulting in a pure, dimensionless number.

Dimension note

Cross Elasticity of Demand is inherently dimensionless because it is calculated as the ratio of two percentage changes. Each percentage change is itself a dimensionless ratio (change in quantity/original quantity or

Unit systems

dimensionless · Represents the percentage change in the quantity demanded of good A. This is a dimensionless ratio, often expressed as a percentage.

dimensionless · Represents the percentage change in the price of good B. This is a dimensionless ratio, often expressed as a percentage.

One free problem

Practice Problem

A local coffee shop observes that when the price of tea (Good B) increases by 10%, the quantity demanded for their coffee (Good A) increases by 5%. Calculate the Cross Elasticity of Demand.

%Δ P Good B10 %

%Δ Q Good A5 %

Solve for: $X E D$

Hint: Divide the percentage change in demand for coffee by the percentage change in the price of tea.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Tea price rises 10%, coffee demand rises 5%. XED = 0.5 (Substitutes).

Study smarter

Tips

A positive result indicates the goods are substitutes.
A negative result indicates the goods are complements.
A result near zero suggests the products are unrelated.
Focus on the sign first to identify the relationship, then the magnitude for the strength.

Avoid these traps

Common Mistakes

Swapping the goods in the formula.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

XED measures the interdependence of goods.

Apply this formula when analyzing how a competitor's pricing strategy impacts your sales volume or how the cost of a primary good affects its accessories. It assumes a ceteris paribus environment where consumer income and the price of the product itself remain constant.

This metric helps firms identify their closest competitors and plan production based on shifts in related markets. For regulators, it is a critical tool for defining market boundaries and determining if a merger might lead to a monopoly by reducing consumer alternatives.

Swapping the goods in the formula.

Tea price rises 10%, coffee demand rises 5%. XED = 0.5 (Substitutes).

A positive result indicates the goods are substitutes. A negative result indicates the goods are complements. A result near zero suggests the products are unrelated. Focus on the sign first to identify the relationship, then the magnitude for the strength.

References

Sources

Mankiw, N. Gregory. Principles of Economics.
Sloman, John, et al. Economics.
Wikipedia: Cross elasticity of demand
Mankiw, N. Gregory. Principles of Economics. 9th ed. Cengage Learning, 2021.
McConnell, Campbell R., Stanley L. Brue, and Sean M. Flynn. Economics: Principles, Problems, and Policies. 23rd ed.
A-Level Economics — Elasticities

Cross Elasticity (XED)

Overview

Variables

Derivation

Ratio formula:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Income Elasticity of Demand (YED)

Price Elasticity of Demand (PED)

Frequently Asked Questions

Sources