Cross-Price Elasticity: Substitutes & Complements

Introduction

In our last lesson, we examined how demand responds to changes in consumer income, using income elasticity to classify goods. We now turn to the final piece of our demand elasticity puzzle: the relationship between different goods. This lesson introduces cross-price elasticity of demand (XED), a measure that quantifies how the demand for one good is affected by a change in the price of another.

This concept is crucial for understanding market dynamics, from how a coffee shop is affected by a price change at a nearby tea house, to how the market for electric cars responds to fluctuating gasoline prices. By the end of this lesson, you will be able to calculate the cross-price elasticity of demand and, based on its sign, definitively classify a pair of goods as either substitutes or complements.

Defining and Interpreting Cross-Price Elasticity

Cross-price elasticity of demand measures the percentage change in the quantity demanded of one good (Good X) that results from a one percent change in the price of another good (Good Y).

The general formula mirrors the structure of the other elasticity concepts we've studied:

E_{X Y} = \frac{Percentage Change in Quantity Demanded of Good X}{Percentage Change in Price of Good Y} = \frac{%Δ Q _{X}}{%Δ P _{Y}}

Unlike price elasticity of demand, the sign of the cross-price elasticity coefficient is not predetermined; it can be positive, negative, or zero, and this sign is the key to classifying the relationship between the two goods.

To get a clear overview of these classifications, please watch the following video.

Y1 13) Cross Elasticity of Demand (XED)

The video "Y1 13) Cross Elasticity of Demand (XED)" by EconplusDal provides a concise introduction to the concept and its interpretation.

Watch the initial segments, from the beginning up to the explanation of the signs. Focus on: The core definition of XED. Why a positive XED indicates the goods are substitutes. Why a negative XED indicates the goods are complements.

As the video explains, we can summarize the classifications as follows:

Substitutes ( $E_{X Y} > 0$ ): These are goods used in place of one another (e.g., Coke and Pepsi, butter and margarine). If the price of Good Y increases, consumers buy less of it and switch to its substitute, Good X, causing the quantity demanded of Good X to rise. The price of Y and quantity of X move in the same direction, resulting in a positive elasticity.
Complements ( $E_{X Y} < 0$ ): These are goods used together (e.g., smartphones and apps, cars and gasoline). If the price of Good Y increases, consumers buy less of it, and because the goods are used jointly, they also demand less of Good X. The price of Y and quantity of X move in opposite directions, yielding a negative elasticity.
Unrelated Goods ( $E_{X Y} = 0$ ): A change in the price of one good has no effect on the quantity demanded of the other (e.g., toothpaste and shoes).

While not required for the classification, the magnitude of XED indicates the strength of the relationship. A value greater than 1 (in absolute terms) implies a strong relationship (close substitutes or complements), while a value less than 1 suggests a weak relationship.

Calculation of Cross-Price Elasticity

Your CB2 exam will require you to calculate XED from both discrete data points and from a given demand function. Your quantitative background will be particularly useful for the latter, as it involves the application of partial derivatives.

1. Arc (Midpoint) Method for Discrete Data

When given two price-quantity points, we use the arc elasticity formula to ensure the result is consistent regardless of the direction of the change. The formula is:

E_{X Y} = \frac{( Q _{X 2} - Q _{X 1} )}{( Q _{X 1} + Q _{X 2} ) /2} \div \frac{( P _{Y 2} - P _{Y 1} )}{( P _{Y 1} + P _{Y 2} ) /2} = \frac{Δ Q _{X}}{Δ P _{Y}} \times \frac{P _{Y 1} + P _{Y 2}}{Q _{X 1} + Q _{X 2}}

The following video demonstrates the calculation using simple percentage changes, which forms the basis of the arc method.

Y1 13) Cross Elasticity of Demand (XED)

This part of the EconplusDal video shows two practical examples.

Watch the two calculation examples: Printers and printer ink: A case of complements (E_{XY} = -2.5). MacBook Pro and Sony Vaio: A case of substitutes (E_{XY} = +0.2).

2. Point Method for Demand Functions

When you have a demand equation for Good X that includes the price of Good Y as a variable (e.g., $Q_{X} = f (P_{X}, P_{Y}, I)$ ), you can calculate the point cross-price elasticity. This leverages your familiarity with partial derivatives.

The formula is:

E_{X Y} = \frac{\partial Q _{X}}{\partial P _{Y}} \cdot \frac{P _{Y}}{Q _{X}}

Here, $\frac{\partial Q _{X}}{\partial P _{Y}}$ is the partial derivative of the demand function for Good X with respect to the price of Good Y. It measures how $Q_{X}$ changes as $P_{Y}$ changes, holding all other variables (like the own-price $P_{X}$ and income $I$ ) constant. The sign of this derivative directly tells you the relationship: positive for substitutes, negative for complements.

The following resources formalize this and show it in practice.

Applications of Partial Derivatives II

This document from Shivaji College provides a concise mathematical definition of cross-price elasticity.

Read the first part of the document, focusing on the sections defining and interpreting cross-price elasticity. Pay attention to how it defines the mathematical formula using partial derivatives and how it explains the relationship between the sign and the classification of goods as substitutes, complements, or unrelated.

Now, let's watch a video that walks through a full calculation using this method.

find Price elasticity cross price elasticity & income elasticity from demand function #elasticity

This video from ECON MATHS demonstrates how to find cross-price elasticity from a multivariate demand function.

Please watch the segment from this timestamp. It clearly shows: How to take the partial derivative of the demand function with respect to the price of the other good. How to substitute the given values for prices and quantities into the formula. How to interpret the resulting negative coefficient as indicating complementary goods.

Application: Practice Problems

Let's apply these methods to problems similar to what you might see on your exam. We will use the provided worksheet of microeconomics problems.

A worksheet with practice problems on elasticity.

Problem 1: Hamburgers and Soft Drinks (Discrete Data)

Question: The price of hamburgers increases from $4.50 to $5.50, causing the quantity demanded of soft drinks to fall from 100 to 85. Calculate elasticity – are hamburgers and soft drinks complements or substitutes?

Solution:
Here, soft drinks are Good X and hamburgers are Good Y.

$Q_{X 1} = 100$ , $Q_{X 2} = 85$
$P_{Y 1} = 4.50$ , $P_{Y 2} = 5.50$

Using the arc formula:

E_{X Y} = \frac{( 85 - 100 )}{( 100 + 85 ) /2} \div \frac{( 5.50 - 4.50 )}{( 4.50 + 5.50 ) /2} = \frac{- 15}{92.5} \div \frac{1}{5}

E_{X Y} = - 0.1622 \div 0.2 = - 0.811

Interpretation: Since $E_{X Y}$ is negative, hamburgers and soft drinks are complements. The price of hamburgers went up, and the demand for soft drinks went down, which is the expected relationship for complementary goods.

Problem 2: Sandals and Sneakers (Demand Function)

Question: From the worksheet (Problem 7): The demand function for sandals is $Q_{s an d a l s} = 40 - 2 P_{s an d a l s} + 3 P_{s n e ak er s} + 0.1 M$ . Suppose that $P_{s an d a l s} = 15$ and $M = 1500$ . What is the cross-price elasticity of sandals when the price of sneakers is $50? (Note: I've adapted the question slightly to ask for point elasticity, a common exam task).

Solution:
Here, sandals are Good X and sneakers are Good Y. We will use the point elasticity formula: $E_{X Y} = \frac{\partial Q _{X}}{\partial P _{Y}} \cdot \frac{P _{Y}}{Q _{X}}$ .

Find the partial derivative:
The coefficient on the $P_{s n e ak er s}$ term gives us the partial derivative.
$\frac{\partial Q _{s an d a l s}}{\partial P _{s n e ak er s}} = + 3$
The positive sign immediately tells us that sandals and sneakers are substitutes.
Find the initial quantity of sandals ( $Q_{X}$ ):
Substitute the given values ( $P_{s an d a l s} = 15$ , $P_{s n e ak er s} = 50$ , $M = 1500$ ) into the demand function.
$Q_{s an d a l s} = 40 - 2 (15) + 3 (50) + 0.1 (1500)$ $Q_{s an d a l s} = 40 - 30 + 150 + 150 = 310$
Calculate the cross-price elasticity:
Now plug the values into the point elasticity formula.
$E_{X Y} = 3 \cdot \frac{50}{310} \approx + 0.484$

Interpretation: The cross-price elasticity is +0.484. Since the value is positive, sandals and sneakers are substitutes. This makes intuitive sense, as a consumer might choose between wearing sandals or sneakers.

Conclusion

In this lesson, we have rounded out our understanding of demand elasticity by focusing on the cross-price dimension. By calculating how the quantity demanded of one good responds to a price change in another, we can rigorously determine the economic relationship between them.

Key Takeaways:

Cross-Price Elasticity of Demand (XED) measures the relationship between two goods.
The sign of the XED coefficient is the key to classification:
- $E_{X Y} > 0$ implies the goods are substitutes.
- $E_{X Y} < 0$ implies the goods are complements.
Calculation can be done using the arc (midpoint) formula for discrete data points or the point elasticity formula ( $E_{X Y} = \frac{\partial Q _{X}}{\partial P _{Y}} \cdot \frac{P _{Y}}{Q _{X}}$ ) when a demand function is known.

With this lesson, you are now proficient in analyzing demand from three critical angles: its sensitivity to own-price changes, income changes, and changes in the prices of related goods. In our next lesson, we will shift our focus from the demand side to the supply side and explore the price elasticity of supply.

Recommended resources