## Calculating Cross-Price Elasticity of Demand

This worked example asks you to compute two types of demand elasticities and then to draw conclusions from the results. The initial price and quantity of widgets demanded is (P1 = 12, Q1 = 8). The subsequent price and quantity is (P2 = 9, Q2 = 10). This is all the information needed to compute the price elasticity of demand.

The price elasticity of demand is defined as follows:

$\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}$

From the midpoint formula, we know that:

$\displaystyle\text{percent change in quantity}=\frac{Q_2-Q_1}{(Q_2+Q_1)\div{2}}\times{100}=\frac{10-8}{(10+8)\div{2}}\times{100}=\frac{2}{9}\times{100}=22.2$

And:

$\displaystyle\text{percent change in price}=\frac{P_2-P_1}{(P_2+P_1)\div{2}}\times{100}=\frac{9-12}{(9+12)\div{2}}\times{100}=\frac{-3}{10.5}\times{100}=-28.6$

Therefore:

$\displaystyle\text{Price Elasticity of Demand}=\frac{22.2\text{ percent}}{-28.6\text{ percent}}=-0.77$

Since the elasticity is less than 1 (in absolute value), we can say that the price elasticity of demand for widgets is in the inelastic range.

The cross-price elasticity of demand is computed similarly:

$\displaystyle\text{Cross-Price Elasticity of Demand}=\frac{\text{percent change in quantity of sprockets demanded}}{\text{percent change in price of widgets}}$

The initial quantity of sprockets demanded is 9 and the subsequent quantity demanded is 10 (Q1 = 9, Q2 = 10).

Using the midpoint formula, we can calculate the percent change in the quantity of sprockets demanded:

$\displaystyle\text{percent change in quantity}=\frac{Q_2-Q_1}{(Q_2+Q_1)\div{2}}\times{100}=\frac{10-9}{(10+9)\div{2}}\times{100}=\frac{1}{9.5}\times{100}=10.5$

The percent change in the quantity of sprockets demanded is 10.5%.

The percent change in the price of widgets is the same as above, or -28.6%.

Therefore:

$\displaystyle\text{Cross-Price Elasticity of Demand}=\frac{10.5\text{ percent}}{-28.6\text{ percent}}=-0.37$

Because the cross-price elasticity is negative, we can conclude that widgets and sprockets are complementary goods. Intuitively, when the price of widgets goes down, consumers purchase more widgets. Because they’re purchasing more widgets, they purchase more sprockets.