Problem 25 - Applied Calculus

Question

If \(q\) is the quantity of chicken demanded as a function of the price \(p\) of beef, the cross-price elasticity of demand for chicken with respect to the price of beef is defined as \(E_{\text {cross }}=|p / q \cdot d q / d p|\). What does \(E_{\text {cross }}\) tell you about the sensitivity of the quantity of chicken bought to changes in the price of beef?

Step-by-Step Solution

Answer

The cross-price elasticity \(E_{\text{cross}}\) indicates the sensitivity of chicken demand to changes in beef price: \(E_{\text{cross}} > 1\) suggests strong substitution, \(E_{\text{cross}} < 1\) implies weak substitution.

Step 1: Understand the Concept

The cross-price elasticity of demand measures how the quantity demanded of one good (chicken) responds to a change in the price of another good (beef). A positive cross-price elasticity indicates that the goods are substitutes, while a negative value suggests they are complements.

Step 2: Identify Components of the Formula

The formula for cross-price elasticity is given as \(E_{\text{cross}} = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right| \). Here, \(p\) is the price of beef, \(q\) is the quantity of chicken demanded, and \(\frac{dq}{dp}\) is the derivative of the quantity of chicken with respect to the price of beef.

Step 3: Interpret \(\frac{dq}{dp}\)

The term \(\frac{dq}{dp}\) represents how the quantity of chicken demanded changes as the price of beef changes. If \(\frac{dq}{dp} > 0\), an increase in the price of beef leads to an increase in the demand for chicken; if \(\frac{dq}{dp} < 0\), an increase in the price of beef leads to a decrease in the demand for chicken.

Step 4: Analyze the Formula

Using the formula \(E_{\text{cross}} = \left| \frac{p}{q} \cdot \frac{dq}{dp} \right| \), the magnitude of \(E_{\text{cross}}\) indicates the sensitivity of the demand for chicken to changes in the price of beef. A larger value of \(E_{\text{cross}}\) suggests greater sensitivity, meaning the demand for chicken is significantly affected by the price changes in beef.

Step 5: Conclude Based on \(E_{\text{cross}}\)

If \(E_{\text{cross}} > 1\), it implies strong sensitivity, indicating that chicken is a good substitute for beef. Conversely, a value of \(E_{\text{cross}} < 1\) suggests less sensitivity, meaning chicken is not a strong substitute for beef. Thus, \(E_{\text{cross}}\) helps to decide if the two commodities are substitutes or complements.

Key Concepts

Substitute Goods

Substitute goods are two products that consumers can use in place of each other. When the price of one good rises, consumers might switch to the substitute product, increasing its demand. For example, if the price of beef goes up, more people may opt to buy chicken.

This relationship between the two goods is measured by the cross-price elasticity of demand. If this elasticity is positive, it indicates that the two goods are substitutes. An increase in the price of beef leads to an increase in the quantity demanded of chicken.

• A larger positive cross-price elasticity means the two goods are strong substitutes.
• The consumer reaction is noticeable if they prefer switching due to a price increase.

Understanding the concept of substitute goods is crucial for businesses to strategize pricing and remain competitive in markets with multiple alternatives.

Complementary Goods

Complementary goods are products that are often used together. The demand for one good can increase when the demand for its complement rises. For instance, if burgers (a complement to beef) become more expensive, the demand for beef might decrease.

In terms of cross-price elasticity, complementary goods have a negative value. When the price of one increases, the demand for the other decreases due to their linked nature in consumption.

• A significant negative cross-price elasticity suggests that the two goods are often used in conjunction.
• Businesses may bundle complementary goods to encourage consumer purchases.

Recognizing complementary relations helps businesses and policymakers understand how interconnected product markets influence consumer choices.

Elasticity of Demand

Elasticity of demand quantifies how responsive the quantity demanded is to changes in economic variables, such as price. It's not only focused on cross-price elasticity but also on how demand changes with the good's own price, the consumer's income, and more.

The elasticity of demand is expressed as a ratio and can be:

• Elastic: Demand changes significantly with price changes (elasticity greater than 1).
• Inelastic: Demand changes little with price changes (elasticity less than 1).

For cross-price elasticity, it helps determine whether goods are substitutes or complements based on whether elasticity is positive or negative.
By understanding the elasticity of demand, companies can make informed decisions on pricing strategies, maximizing revenue, and forecasting market trends.