# Cross Elasticity of Demand

If you are learning to play ping pong and you buy new rackets to play, how many ping pong balls would you purchase? These two products are often bought together, which is something economists refer to as complementary goods. The exact answer will differ from person to person not only because their cross-price elasticities of demand are different, but also because their preferences are different, and their consumption decisions react to price changes differently. Eager to learn more? Then read on.

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Cross elasticity of demand, also known as cross-price elasticity of demand, is a measure of the responsiveness of the demanded quantity of one good to a change in the price of another good.

Note that the price elasticity of demand measures the responsiveness of the demanded quantity of one good to the changes in the price of that same good. In contrast, cross-price elasticity of demand measures the responsiveness of the demanded quantity of one good to changes in the price of another good, such as a substitute or a complement.

Imagine two closely related goods such as apples and pears. These goods are usually suitable for interchangeable consumption. Most people would not mind substituting one fruit for another. Now, imagine that the price of pears goes up. Since consumers are likely to prefer both fruits equally, they would shift their consumption towards apples as apples are now cheaper. The proportionate response of the quantity demanded of apples to a proportionate change in the price of pears is what the cross elasticity of demand measures.

## Types of cross elasticity of demand

The value of the cross elasticity of demand will depend on whether the two goods are substitutes, complements, or goods with no apparent relationship.

### Cross elasticity of demand and substitutes

Substitute goods are goods that consumers consider to be identical or similar enough for interchangeable consumption.

The demanded quantity of any two substitutes goes in opposite directions. In other words, the less a person consumes one good, the more they would consume this good’s substitute. The cross-price elasticity of demand for these goods will therefore be positive.

Consider the demand for substitutes such as white potatoes and sweet potatoes. One day, the price of white potatoes rises significantly and you decide that you want to substitute your white potato consumption with sweet potato consumption. Your quantity demand for white potatoes will decrease, leading to an increase in the demanded quantity of sweet potatoes.

Figure 1 illustrates the relationship between substitutes, where Good A is pears and Good B is apples. On the left diagram, an increase in the price of Good A from P1 to P2 leads to a reduction in the quantity demanded from Q1 to Q2, and a movement along the demand curve for Good A. This leads to an outward shift in the demand curve from D1 to D2 of a substitute product - Good B, such that at any given price level, P, Good B is demanded more (Q2 compared to Q1). The diagram on the right illustrates this.

Figure 1. Quantity demand effects of a price increase on substitutes, StudySmarter Originals.

### Cross elasticity of demand and complements

Complementary goods are goods that are consumed jointly or in joint demand.

The demanded quantity for any two complementary goods goes in the same direction. In other words, the more a person consumes one good, the more they would consume this good’s complement. The cross-price elasticity of demand for these goods will therefore be negative.

Consider the demand for complements such as mobile phones and sim cards. Technological advances led to mobile phones being widely accessible at a lower price. This led to an increase in the demanded quantity for phones, leading to an increase in the demanded quantity for sim cards.

Figure 2 illustrates the relationship between complements. Good A is mobile phones and Good B is sim cards. On the left diagram, there is a fall in the price of Good A from P1 to P2, which leads to an increase in the demanded quantity from Q1 to Q2, and a movement along the demand curve for Good A. This leads to an outward shift in the demand curve from D1 to D2 of a complement product - Good B, such that at any given price level, P, Good B is demanded more (Q2 compared to Q1). The diagram on the right illustrates this.

Figure 2. Quantity demanded effects of a price fall on complements, StudySmarter Originals

### Cross elasticity of demand and goods with no apparent relationship

Goods that are neither substitutes nor complements are considered unrelated. In other words, there is no relationship between these goods as they are independent of one another. The value of the cross elasticity of demand between such goods is equal to zero.

## The cross elasticity of demand formula

Cross elasticity of demand is calculated as a percentage change in the quantity demanded of Good A divided by a percentage change in the price of Good B.

The formula for the cross-price elasticity of demand (XED) is:

$XED=\frac{%\Delta QuantityDemandedofGoodA}{%\Delta PriceofGoodB}$

You can find a percentage change in a variable by using the following formula:

$%\Delta =\frac{Newvalue-Oldvalue}{Oldvalue}*100%$

If the goods are substitutes, their cross-price elasticity of demand is going to be positive.

This is because a price change of Good A and quantity demanded of Good B move in the same direction:

• If the price of Good A increases, the quantity demanded of Good B increases.

• If the price of Good A decreases, the quantity demanded of Good B decreases.

The price of Crest toothpaste goes up by 5%, leading to a contraction of demand. Consumers switch to Colgate toothpaste, causing an outward shift in the demand curve and an increase in the quantity demanded by 20%. The cross elasticity of demand between Colgate and Crest toothpastes is therefore: 20% / 5% = 4.

If the goods are complements, their cross-price elasticity of demand is going to be negative.

This is because a price change of Good A and quantity demanded of Good B move in the opposite direction:

• If the price of Good A increases, the quantity demanded of Good B decreases.

• If the price of Good A decreases, the quantity demanded of Good B increases.

The price of smartphones increases by 10% leading to a contraction of demand. This causes the demand curve for a complementary good (smartphone apps) to shift inwards, leading to a reduction in the demanded quantity by 20%. Cross elasticity of demand between smartphone apps and smartphones is: -20% / 10% = -2.

### Values of the cross-price elasticity of demand

The table below summarises all the values that the cross-price elasticity of demand (XED) can take.

 Values of XED Quantity demanded (QD) response to a change in the price (P) of another good Description Type of goods XED < -1 P ↑ QD ↓ P ↓ QD ↑ Price of Good A and quantity demanded of Good B move in the opposite direction. Strongcomplements. -1 < XED < 0 P ↑ QD ↓P ↓ QD ↑ Price of Good A and quantity demanded of Good B move in the opposite direction. Weak complements. XED = 0 - Price of Good A and quantity demanded of Good B have no apparent relationship. Unrelated goods. 0 < XED < 1 P ↑ QD ↑P ↓ QD ↓ Price of Good A and quantity demanded of Good B move in the same direction Weaksubstitutes. XED > 1 P ↑ QD ↑P ↓ QD ↓ Price of Good A and quantity demanded of Good B move in the same direction Strongsubstitutes.

## What factors affect the cross elasticity of demand?

There are three factors that affect the value of the cross elasticity of demand.

1. If the goods are substitutes, the value of the cross elasticity of demand is positive.

2. If the goods are complements, the value of the cross elasticity of demand is negative.

3. If the goods are not related, the value of the cross elasticity of demand is equal to zero

Consumer preferences and their budget constraints affect the demand for any good. Whether the preferences or the budget play a decisive role in which product the consumer chooses, will determine whether the goods have a strong or weak relationship.

### Strong and weak substitutes

The magnitude of the cross elasticity of demand for substitute goods is determined based on whether the two goods are strong or weak substitutes.

Consider substitutes such as Assam tea and Ceylon tea. These two types of black tea are strong substitutes as there is only a minor difference between them.

In this case, it is likely that consumers would not mind which type of tea they consume. Therefore, a change in the price of a substitute will outweigh consumer preferences. Thus, if the price of Assam tea increases, consumers are likely to increase the demanded quantity of Ceylon tea. This will result in a higher value of the cross elasticity of demand as the change in the price of one good is likely to significantly affect the quantity demanded of another good.

The value of the cross elasticity of demand for strong substitutes will generally be large and positive.

The price (P) of Assam tea goes up from £2.20 to £2.50 leading to an increase in the quantity demanded (QD) of Ceylon tea from 10 to 18.

Let’s calculate the cross elasticity of demand (XED) between the two goods:

1. Change in the QD of Ceylon tea = (18-10) / 10 = 80%

2. Change in the P of Assam tea = (2.50-2.20) /2.20 = 13.64%

3. XED = 80% / 13.64% = 5.87

Note: XED> 0 as the two goods are substitutes.

Now consider substitutes such as green loose leaf tea and matcha tea powder. There are apparent differences in flavour, texture, and smell between the two goods making them weak substitutes.

A cup of matcha tea is definitely different from a cup of green tea. Source: Unsplash.

In this case, it is likely that the differences between the two products will result in consumer preferences for a particular type of tea outweighing any changes in the price of a substitute product. Thus, if the price of green loose leaf tea decreases, consumers are still likely to stay with their preferred good: matcha. This will result in a lower value of the cross elasticity of demand as the change in the price of one good is unlikely to significantly affect the quantity demanded of another good.

The value of the cross elasticity of demand for weak substitutes will generally be low and positive.

The price (P) of green loose leaf tea goes down from £4.50 to £4 leading to a fall in the quantity demanded (QD) of matcha tea powder from 15 to 14.

Let’s calculate the cross elasticity of demand (XED) between the two goods:

1. Change in the QD of matcha = (14-15) / 15 = -6.66%

2. Change in the P of green loose leaf tea = (4-4.50) /4.50 = -11.11%

3. XED = -6.66% / -11.11% = 0.59

Note: XED> 0 as the two goods are substitutes

### Strong and weak complements

The magnitude of the cross elasticity of demand for complementary goods is determined based on whether the two goods are strong or weak complements.

Consider complements such as fish and chips. These two goods are strong complements as they are very often consumed together as a single dish.

In this case, it is likely that consumers would strongly prefer to eat fish together with chips, therefore a change in the price of a complement will outweigh consumer preferences. So if the price of fish decreases they are likely to increase their quantity demanded of chips as well. This will result in a higher value of the cross elasticity of demand as the change in the price of one good is likely to affect the quantity demanded of another good significantly.

The value of the cross elasticity of demand for strong complements will generally be high and negative

The price (P) of fish goes down from £9.20 to £8.80 leading to an increase in the quantity demanded (QD) of chips from 20 to 30.

Let’s calculate the cross elasticity of demand (XED) between the two goods:

1. Change in the QD of chips = (30-20) / 30 = 33.33%

2. Change in the P of fish = (8.80-9.20) /9.20 = -4.35%

3. XED = 33.33% / -4.35% = -7.66

Note: XED <0 as the two goods are complements.

Consider another example of complements such as pasta and basil pesto sauce. These two goods are weak complements as consumers can eat pasta with a variety of different sauces apart from the basil pesto sauce.

In this case, it is likely that the differences between the two products will result in consumer preferences for a particular type of product outweighing any changes in the price of a complementary product. Thus, if the price of pasta increases, consumers are likely to still buy the basil pesto sauce they prefer. This will result in a lower value of the cross elasticity of demand as the change in the price of one good is unlikely to significantly affect the quantity demanded of another good.

The value of the cross elasticity of demand for weak complements will generally be low and negative.

The price (P) of pasta goes up from £1.30 to £1.50 leading to a fall in the quantity demanded (QD) of basil pesto sauce from 20 to 19.

Let’s calculate the cross elasticity of demand (XED) between the two goods:

1. Change in the QD of basil pesto sauce = (19-20) / 19 = -5.26%

2. Change in the P of pasta = (1.50-1.30) / 1.30 = 15.38%

3. XED = -5.26% / 15.38% = -0.34

Note: XED <0 as the two goods are complements.

The cross-price elasticity of demand is a useful measure that allows us seeing how the consumer quantity demanded of one product will respond to price changes of another product. Businesses use cross elasticity of demand to determine how strong their product brand image is. The businesses increase the price of their products by a small margin, and if the demand shifts significantly towards competitors’ products, it is a sign that they need to work on brand loyalty. The firms will then invest more in customer loyalty programmes, packaging, and advertising to increase brand loyalty.

## Cross Elasticity of Demand - Key takeaways

• Cross elasticity of demand, also known as the cross-price elasticity of demand, is a measure of the responsiveness of the quantity demanded of one good to a change in the price of another good.
• Substitute goods are goods that consumers consider to be identical or similar enough for interchangeable consumption.
• Complementary goods are goods that are consumed jointly or in joint demand.
• Cross-price elasticity of demand is calculated as a percentage change in the quantity demanded of Good A divided by a percentage change in the price of Good B.
• If the goods are substitutes, their cross-price elasticity of demand is going to be positive. This is because a change in the price of Good A and the quantity demanded of Good B move in the same direction.
• If the goods are complements, their cross-price elasticity of demand is going to be negative. This is because a change in the price of Good A and the quantity demanded of Good B move in the opposite direction.

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What is cross elasticity of demand?

Cross elasticity of demand is a measure of the responsiveness of the demanded quantity of one good to a change in the price of another good.

How to calculate cross elasticity of demand?

Cross elasticity of demand is calculated as a percentage change in the quantity demanded of Good A divided by a percentage change in the price of Good B.

What is an example of cross elasticity of demand?

Consider the demand for substitutes such as pears and apples. If the price of pears rises significantly, consumers may choose to substitute pear consumption with apple consumption. The pears’ demanded quantity will decrease, leading to an increase in the apples’ demanded quantity. Cross elasticity of demand will measure how strongly the demanded quantity of apples will react to a change in the price of pears.

What factors affect cross elasticity of demand?

The value of the cross elasticity of demand is affected by three factors:

1. If the goods are substitutes the value of the cross elasticity of demand is positive.

2. If the goods are complements the value of the cross elasticity of demand is negative.

3. Goods which are not related have the value of the cross elasticity of demand equal to zero.

What is the difference between price elasticity of demand and cross elasticity of demand?

The price elasticity of demand measures the responsiveness of the quantity demanded of one good to the changes in the price of that same good. In contrast, cross elasticity of demand measures the responsiveness of the quantity demanded of one good to changes in the price of another good, such as a substitute or a complement.

## Test your knowledge with multiple choice flashcards

If the goods are substitutes their cross-price elasticity of demand can be…

If the goods are complements their cross-price elasticity of demand can be...

A change in the price of a substitute Good A and quantity demanded of Good B move …

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