Cross-Price Elasticity of Demand
Cross-price elasticity of demand measures the responsiveness of the quantity demanded of one good as a result of a change in the price of another good. In other words, it reflects the fact that the quantity demanded of a good is affected by changes in the prices of other goods. More specifically, it is the percentage change in the quantity demanded of one good in response to a percentage change in the price of another good. It is calculated as the percentage change in the quantity demanded of the first good divided by the percentage change in the price of the second good.
Cross-Price Elasticity of Demand and the Relationship Between Goods
Whether the cross-price elasticity of demand is positive or negative depends on whether the goods are substitutes or complements.
Substitute Goods
Two goods are substitutes when they are consumed in place of each other. Consequently, when two goods are substitutes, the cross-price elasticity of demand is positive. This is because an increase in the price of one good results in an increase in the quantity demanded of the other good. Thus, the percentage change in the quantity demanded of good 1 and the percentage change in the price of good 2 move in the same direction.
Complementary Goods
On the other hand, when two goods are complements, meaning they are typically consumed together, the cross-price elasticity of demand is negative. This is because an increase in the price of one good leads to a decrease in the quantity demanded of the other good. As a result, the percentage change in the quantity demanded of good 1 and the percentage change in the price of good 2 move in opposite directions.
How to Calculate Cross-Price Elasticity of Demand?
The cross-price elasticity of demand is defined as:
\[ E_{d, cross} = \frac{\text{Percentage change in the quantity demanded of good 1}}{\text{Percentage change in the price of good 2}} \]
The change in the quantity demanded (\(\Delta Q_1\)) and in the price (\(\Delta P_2\)) is calculated as the difference between the final value and the initial value. That is:
\[ \Delta Q_1 = Q_1^{\text{final}} - Q_1^{\text{initial}} \]
\[ \Delta P_2 = P_2^{\text{final}} - P_2^{\text{initial}} \]
This can be expressed in terms of absolute changes as follows:
\[ E_{d, cross} = \frac{\frac{\Delta Q_1}{Q_1^{\text{initial}}}}{\frac{\Delta P_2}{P_2^{\text{initial}}}} \]
By multiplying both sides of the original equation by \(\frac{P_2^{\text{initial}}}{\Delta P_2}\), the result is:
\[ \left( E_{d, cross} \cdot \frac{\Delta P_2}{P_2^{\text{initial}}} \right) = \left( \frac{\Delta Q_1}{Q_1^{\text{initial}}} \cdot \frac{P_2^{\text{initial}}}{\Delta P_2} \right) \]
Now, we can see that the multiplication of \(\frac{\Delta P_2}{P_2^{\text{initial}}}\) with \(\frac{P_2^{\text{initial}}}{\Delta P_2}\) simplifies and cancels, resulting in:
\[ E_{d, cross} = \frac{\Delta Q_1}{\Delta P_2} \cdot \frac{P_2^{\text{initial}}}{Q_1^{\text{initial}}} \]
Where:
- \(\Delta Q_1\) = change in the quantity demanded of good 1 (final value minus initial value)
- \(\Delta P_2\) = change in the price of good 2 (final value minus initial value)
- \(P_2^{\text{initial}}\) = initial price of good 2
- \(Q_1^{\text{initial}}\) = initial quantity demanded of good 1
This formula is useful for calculating cross-price elasticity of demand using data on changes in quantities and prices.
Example of Calculating Cross-Price Elasticity of Demand
Let us consider a specific example to calculate the cross-price elasticity of demand.
Suppose that:
- Initial quantity demanded of good 1 (\(Q_1^{\text{initial}}\)): 100 units
- Final quantity demanded of good 1 (\(Q_1^{\text{final}}\)): 70 units
- Change in the quantity demanded of good 1 (\(\Delta Q_1\)): \(Q_1^{\text{final}} - Q_1^{\text{initial}} = 70 - 100 = -30\) units
- Initial price of good 2 (\(P_2^{\text{initial}}\)): 25 monetary units
- Final price of good 2 (\(P_2^{\text{final}}\)): 20 monetary units
- Change in the price of good 2 (\(\Delta P_2\)): \(P_2^{\text{final}} - P_2^{\text{initial}} = 20 - 25 = -5\) monetary units
Substituting these values into the formula:
\[ E_{d, cross} = \frac{\Delta Q_1}{\Delta P_2} \cdot \frac{P_2^{\text{initial}}}{Q_1^{\text{initial}}} \]
Substituting the values:
\[ E_{d, cross} = \frac{-30}{-5} \cdot \frac{25}{100} \]
We calculate each part:
- \(\frac{-30}{-5} = 6\)
- \(\frac{25}{100} = 0.25\)
Now, multiply both results:
\[ E_{d, cross} = 6 \cdot 0.25 = 1.5 \]
Therefore, the cross-price elasticity of demand is 1.5, indicating that a 1% increase in the price of good 2 results in a 1.5% increase in the quantity demanded of good 1.
This indicates that goods 1 and 2 are substitutes, as an increase in the price of one leads to an increase in the demand for the other. When the cross elasticity is positive (as in this case, \(E_{d, cross} = 1.5\)), it means the goods are related such that a price increase in one results in higher demand for the other.