Cross-Price Elasticity of Demand

The Cross-Price Elasticity of Demand measures the rate of response of quantity demanded of one good, due to a price change of another good. If two goods are substitutes, we should expect to see consumers purchase more of one good when the price of its substitute increases. Similarly if the two goods are complements, we should see a price rise in one good cause the demand for both goods to fall. Your course may use the more complicated Arc Cross-Price Elasticity of Demand formula. If so you'll need to see the article on Arc Elasticity. The common formula for the Cross-Price Elasticity of Demand (CPEoD) is given by:

CPEoD = (% Change in Quantity Demand for Good X)/(% Change in Price for Good Y)

Calculating the Cross-Price Elasticity of Demand
You're given the question: "With the following data, calculate the cross-price elasticity of demand for good X when the price of good Y changes from $9.00 to $10.00." Using the chart below

We know that the original price of Y is $9 and the new price of Y is $10, so we have Price(OLD)=$9 and Price(NEW)=$10. From the chart we see that the quantity demanded of X when the price of Y is $9 is 150 and when the price is $10 is 190. Since we're going from $9 to $10, we have QDemand(OLD)=150 and QDemand(NEW)=190. You should have these four figures written down:

Price(OLD)=9
Price(NEW)=10
QDemand(OLD)=150
QDemand(NEW)=190

To calculate the cross-price elasticity, we need to calculate the percentage change in quantity demanded and the percentage change in price. We'll calculate these one at a time.

Calculating the Percentage Change in Quantity Demanded of Good X
The formula used to calculate the percentage change in quantity demanded is:
[QDemand(NEW) - QDemand(OLD)] / QDemand(OLD)

By filling in the values we wrote down, we get:

[190 - 150] / 150 = (40/150) = 0.2667

So we note that % Change in Quantity Demanded = 0.2667 (This in decimal terms. In percentage terms this would be 26.67%).

Calculating the Percentage Change in Price of Good Y
The formula used to calculate the percentage change in price is:
[Price(NEW) - Price(OLD)] / Price(OLD)

We fill in the values and get:

[10 - 9] / 9 = (1/9) = 0.1111

We have our percentage changes, so we can complete the final step of calculating the cross-price elasticity of demand.

Final Step of Calculating the Cross-Price Elasticity of Demand
We go back to our formula of:
CPEoD = (% Change in Quantity Demanded of Good X)/(% Change in Price of Good Y)

We can now get this value by using the figures we calculated earlier.

CPEoD = (0.2667)/(0.1111) = 2.4005

We conclude that the cross-price elasticity of demand for X when the price of Y increases from $9 to $10 is 2.4005.

How Do We Interpret the Cross-Price Elasticity of Demand?
The cross-price elasticity of demand is used to see how sensitive the demand for a good is to a price change of another good. A high positive cross-price elasticity tells us that if the price of one good goes up, the demand for the other good goes up as well. A negative tells us just the opposite, that an increase in the price of one good causes a drop in the demand for the other good. A small value (either negative or positive) tells us that there is little relation between the two goods.
Often an assignment or a test will ask you a follow up question such as "Are the two goods complements or substitutes?". To answer that question, you use the following rule of thumb:

•If CPEoD > 0 then the two goods are substitutes

•If CPEoD =0 then the two goods are independent (no relationship between the two goods

•If CPEoD < 0 then the two goods are complements

In the case of our good, we calculated the cross-price elasticity of demand to be 2.4005, so our two goods are substitutes when the price of good Y is between $9 and $10.