# Cross price elasticity of demand

## Cross-price elasticity formula

The cross-price elasticity of demand measures the responsiveness of a good to a change in the price of an alternate good. The cross-price elasticity $E_{X,Y}$ is defined

$\quad E_{X,Y} = (\frac{\frac{Q^{X}_1 - Q^{X}_0}{Q^{X}_0}}{\frac{P^{Y}_1 - P^{Y}_0}{P^{Y}_0}})$

where

$Q^{X}_1$ is the quantity of good X after the price of good Y changes

$Q^{X}_0$ is the quantity of good X before the price of good Y changes

$P^{Y}_1$ is the price of good Y after the price changes.

$P^{Y}_0$ is the price of good Y before the price changes.

## Complementary good

If the cross-price elasticity is negative, it implies that the quantity consumed of a good X decreases after there is a price decrease of Y. This is referred to as a complement good because consumers purchase these goods together. Examples of complement goods are:

• Computers and operating systems
• Fries and ketchup

## Substitute good

If the cross-price elasticity is positive, it implies that the quantity consumed of good X increases after a price increase of good Y. This is referred to as a substitute good because consumers substitute towards purchasing Y instead of good X. Examples of substitute goods are:

• Coke and Pepsi
• Bus tickets and train tickets

### Example 1

Suppose that the cross elasticity of demand for Good X and Good Y is positive 2. What does this tell you about the relationship between these two goods? Given this cross elasticity of demand, if the price of Y increases by 15 percent, what will happen to the demand for Good Y?

Given this knowledge above, we can conclude that good X from the example above is a substitute good for good Y because the elasticity is positive. To calculate what will happen to the demand of good X we plug the value 2 into our formula for elasticity on the left-hand side and 0.15 as the denominator in the right-hand side, giving

$2 = \frac{(\frac{Q^{X}_1 - Q^{X}_0}{Q^{X}_0})}{0.15}$

Multiplying both sides by 0.15 gives

$0.15*2 = (\frac{Q^{X}_1 - Q^{X}_0}{Q^{X}_0})$

which allows us to conclude a 15% increase in the price of good Y will cause the quantity of good X to increase by 30% or 0.3.