The cross-price elasticity of demand shows the responsiveness of the quantity demanded of one good to changes in the price of a related good.

The cross-price elasticity of demand measure the responsiveness of change in quantity demanded of one good to change in the price of other goods.

Mathematically, it is calculated as:

$CED=\frac{\%∆Q_x}{\%∆P_y}$

Where $\%∆Q_x$ is the percentage change in quantity demanded of goods x.

$\%∆P_y$ is the percentage change in the price of good Y.

Unlike the price elasticity of demand, the sign of the coefficient of cross-price elasticity is very significant.

The coefficient of cross-price elasticity is used to determine whether the two goods compared are complements or substitutes.

When the cross-price elasticity of demand is positive; the increase in the price for one good (say good A) would lead to an increase in the quantity demanded of another (say good B) and a decrease in the price of good A would lead to a decrease in quantity demanded of good B

This is a characteristic of a **substitute good.** One common example of substitute goods is Pepsi and coca-cola.

On the other hand, when an increase in the price of one good(say good A) causes a decrease in the quantity demanded of another(say good B) or when a decrease in the price of good A leads to an increase in the quantity good B demanded, then cross-price elasticity of demand is negative.

**Complimentary goods **usually have a negative cross-elasticity of demand. Notebooks and pens are examples of complementary goods.

In some rare cases, the coefficient of cross-price elasticity of demand is zero. This means the two goods involved are unrelated or independent of each other.

For example, if a $10\%$ rise in the price of a laptop causes a zero change in the quantity of Pepsi demanded, then, Pepsi and the laptop are unrelated.

Enough of the lecturing, let's do some calculations ðŸ˜Š

__Example 1__

__Example 1__

Determine the cross-price elasticity of demand if a 10% rise in the quantity demanded by Pepsi is caused by a 20% rise in the price of coca-cola.

**Solution:**

Recalled that:

$CED=\frac{\%∆Q_p}{\%∆P_c}$$

$CED=\frac{10}{20}$

$CED=0.5$

The cross-price elasticity of demand is positive, hence, Pepsi and coca-cola are substitutes.

__Example 2__

__Example 2__

A rise in the price of the notebook from €9 to €11 causes a decrease in the number of pens demanded from 10000 to 8000. Calculate the cross-price elasticity of demand.

**Solution:**

Using the midpoint method

$\%∆Q_p=\frac{8000-10000}{\frac{8000+10000}{2}}\times 100$

$\%∆Q_p=\frac{-2000}{9000}\times 100$

$\%∆Q_p=-22.2$

Solving for the percentage change in the price of the notebook

$\%∆P_n=\frac{11-9}{\frac{11+9}{2}}\times100$

$\%∆P_n=\frac{2}{10}\times100$

$\%∆P_n=20$.

$CED=\frac{-22.2}{20}$

$CED=-1.11$

The cross-price elasticity coefficient is negative, hence we can say the two goods are complimentary.

__Example 3__

__Example 3__

Determine the cross-price elasticity of demand If a 10% rise in the price of the laptop doesn't cause a change in the quantity of pen demanded.

**Solution:**

$CED=\frac{\%∆Q_p}{\%∆P_l}$

$CED=\frac{0}{10}$

$CED=0$

The coefficient of CED is zero. Hence, we can say the two goods are unrelated.

**Related post**

- How to solve the average cost, total cost, and marginal cost
- How to solve price elasticity of demand
- Components of economic cost

**Relevance of cross-price elasticity of demand**

All companies, whether small or large are concerned about the effect rival pricing strategy will have on the demand for their product.What effect will a rival price increase have on the demand for our product?

What happens to the demand for competitors' goods if we decrease our goods price?

This and more would be answered through cross-price elasticity of demand?

To illustrate, a company that sells pens would also be interested in the price of complementary goods like a notebook. This is because a rise in the price of notebooks would decrease the quantity of pens' demanded.