Cross-Price Elasticity of Demand

The cross-price elasticity of demand measures how responsive consumer demand for one good is to changes in the price of a second good.
A good's own price is an important determinant of demand, but the prices of other goods matter as well. The cross-price elasticity of demand, which is the percentage change in the quantity demanded for one good divided by the percentage change in the price of another, is used to measure the sensitivity of the demand for one good to the price of another. Cross-price elasticity of demand (Eab\text{E}_{ab}) is calculated by dividing the percentage change in the quantity demanded of good A (QDa\text{Q}_{\text{D}a}) by the percentage change in the price of good B (Pb\text{P}_{b}).
Eab=%ΔQDa%ΔPb\text{E}_{ab}=\frac{\%\Delta \text{Q}_{\text{D}a}}{\%\Delta \text{P}_{b}}
The cross-price elasticity of demand may be positive or negative (the absolute value is not used in this elasticity calculation), depending on whether the goods in question are substitutes or complements. When examining the cross-price elasticity between oranges and tangerines, if the price of oranges increases, one would expect buyers to buy fewer oranges and more tangerines. Thus, there would be an increase in the quantity of tangerines demanded. The denominator and numerator would both be positive, so the ratio would also be positive. Therefore, the cross-price elasticity of demand is greater than zero when goods A and B are substitutes. For example, if a 10% rise in the price of oranges resulted in a 15% rise in the demand for tangerines, the two goods would be considered substitutes. This can be expressed mathematically:
Eab(Cross Elasticity)=Percentage Change in QDaPercentage Change in Pb=15%10%=1.5>0\begin{aligned}\text{E}_{ab}\;(\text{Cross Elasticity})&=\frac{\text{Percentage Change in Q}_{\text{D}a}}{\text{Percentage Change in P}_{b}}\\\\&=\frac{15\%}{10\%}\\\\&=1.5 \gt 0\end{aligned}
The opposite case holds true for goods that are complements. If the price of an MP3 player falls, the demand for music downloads rises. This means that the denominator (%ΔPb\% \Delta \text{P}_{b}) is negative, while the numerator (%ΔQDa\%\Delta \text{Q}_{\text{D}a}) is positive. The ratio would then be negative, so the cross-price elasticity of demand will be less than zero (negative) for two goods that are complements.

For example, if the quantity demanded of ketchup goes down 30% in response to an increase in 15% of the price of hot dogs, the cross-price elasticity of demand is –2. A negative cross-price elasticity of demand means these goods are complementary (when hot dogs become more expensive, people respond by cutting their consumption of ketchup as well).