Question

According to data from the U.S. Department of Energy, sales of the fuel- efficient Toyota Prius hybrid fell from 158,574 vehicles sold in 2008 to 139,682 in \(2009 .\) Over the same period, according to data from the U.S. Energy Information Administration, the average price of regular gasoline fell from $$\$ 3.27$$ to $$\$ 2.35$$ per gallon. Using the midpoint method, calculate the cross-price elasticity of demand between Toyota Prii (the official plural of "Prius" is "Prii") and regular gasoline. According to your estimate of the cross-price elasticity, are the two goods complements or substitutes? Does your answer make sense?

Short answer

Cross-price elasticity is negative, indicating that Toyota Prii and gasoline are complements. This makes sense as a decrease in the price of gasoline reduces the demand for fuel-efficient vehicles like the Toyota Prius.

Step-by-step solution

Derive Quantity Percentage Change

Using the midpoint method, calculate the percentage change in quantity demanded for the Toyota Prii. The formula for the percentage change using the midpoint method is: \[ \text{Percentage Change in Quantity} = \frac{Q_2 - Q_1}{(Q_2 + Q_1)/2} \times 100 \% \] Where \(Q_1\) is the initial quantity and \(Q_2\) is the final quantity. Substituting the given quantities: \[ \text{Percentage Change in Quantity} = \frac{139,682 - 158,574}{(139,682 + 158,574)/2} \times 100 \% \]

Derive Price Percentage Change

Calculate the percentage change in the price of gasoline. The formula for the percentage change using the midpoint method is similar to the quantity one: \[ \text{Percentage Change in Price} = \frac{P_2 - P_1}{(P_2 + P_1)/2} \times 100 \% \] Where \(P_1\) is the initial price (\$3.27) and \(P_2\) is the final price (\$2.35). Substituting the given prices: \[ \text{Percentage Change in Price} = \frac{2.35 - 3.27}{(2.35 + 3.27)/2} \times 100 \% \]

Calculate the Cross-Price Elasticity of Demand

The cross-price elasticity of demand (CPED) is given by the ratio of the percentage change in quantity demanded to the percentage change in price. \[ CPED = \frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}} \] Substituting the values calculated in the previous steps to determine the CPED.

Determine the Relationship Between Goods

The sign of the CPED will indicate the type of goods. If CPED is negative, the goods are complements; if positive, they are substitutes. After calculating the CPED, observe its sign to determine the relationship between Prii and gasoline.

Analyze the Results and Conclude

Interpret the correlation according to the calculated elasticity and make sense of it in the context of the relationship between hybrid cars and the price of gasoline.

Key concepts

Midpoint Method

The midpoint method is a technique used to calculate the percentage change in quantity or price without placing undue emphasis on either the starting or ending values. This approach aims to provide a more balanced measurement by using the average of the initial and final values as the base for percentage calculations. For students, mastering the midpoint method is essential for accurately determining changes and elasticities in economics.

To apply the midpoint method, you use the formula: \[\begin{equation}\text{Percentage Change} = \frac{New Value - Old Value}{(New Value + Old Value) / 2} \times 100\text{%}\right.\right.\right}\right.\end{equation}\]This formula reduces bias towards one side of the change and provides a symmetric measure of variation. When working with percentage changes in quantity or price, using the midpoint method ensures that the elasticity measure is consistent, no matter whether prices are rising or falling or whether quantities are increasing or decreasing.

Complements and Substitutes

Understanding the relationship between goods is pivotal in economics, as it shapes consumers' behavior. Complements and substitutes are two categories that describe how the demand for one product is affected by the change in the price of another.

Complements are goods that are often used together, like tennis rackets and tennis balls. If the price of one complement goes down, consumers might buy more of both, leading to a negatively signed cross-price elasticity of demand. On the other hand, substitutes are goods that can replace each other, such as butter and margarine. An increase in the price of one substitute might lead to an increase in the demand for the other, reflected in a positive cross-price elasticity.

Determining whether goods are complements or substitutes is critical in making sense of market dynamics and predicting consumer reactions to price changes. For students, recognizing these relationships can clarify complex economic interactions and enhance their understanding of market behavior.

Price Elasticity

Price elasticity measures how much the quantity demanded of a good responds to a change in its price. This concept is not just about whether demand increases or decreases, but by how much. It’s a powerful tool for understanding the sensitivity of consumers to price changes and hence, the potential impact on sales and revenue.

The general formula for elasticity is \[\begin{equation}\text{Elasticity} = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}\right.\right.\right}\right.\end{equation}\]Specifically, cross-price elasticity of demand (CPED) measures the responsiveness of the quantity demanded for one good when the price of another good changes. If CPED is greater than 0, the goods are substitutes; if it is less than 0, they are complements. This distinction helps in forecasting the effects of pricing strategies and understanding competitive dynamics.

For students, gauging price elasticity is essential for making informed predictions about how changes in pricing might influence market demand and competitive positioning.