Problem 26 - 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 demandfor 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

\(E_{\text{cross}}\) indicates how sensitive chicken demand is to beef price changes, showing if they are substitutes.

Step 1: Understanding Cross-Price Elasticity

The cross-price elasticity of demand measures how the quantity demanded of one good (chicken) changes in response to a change in the price of another good (beef). It shows the sensitivity of the demand for one product when the price of a related product changes.

Step 2: Elasticity Formula Interpretation

Given the formula \( E_{\text{cross}} = |\frac{p}{q} \cdot \frac{dq}{dp}| \), \( p \) represents the price of beef, \( q \) is the quantity of chicken demanded, \( \frac{dq}{dp} \) is the derivative of the quantity demanded of chicken with respect to the price of beef. This formula helps determine the percentage change in the quantity of chicken demanded for a percentage change in the price of beef.

Step 3: Analyzing Elasticity Values

The magnitude of \( E_{\text{cross}} \) indicates the strength of the relationship: \( E_{\text{cross}} = 0 \) implies no cross-elasticity, meaning chicken demand is unaffected by beef price. If \( E_{\text{cross}} > 0 \), chicken and beef are substitutes, and an increase in the price of beef leads to a higher demand for chicken. If \( 0 < E_{\text{cross}} < 1 \), the sensitivity is low. If \( E_{\text{cross}} > 1 \), the sensitivity is high.

Step 4: Interpreting Absolute Value

The use of absolute value in \( E_{\text{cross}} \) suggests that we are only interested in the magnitude of the sensitivity and not the direction, ensuring the measure is non-negative regardless of the product relationship.

Key Concepts

Demand Sensitivity

When we talk about demand sensitivity, we're essentially looking at how responsive the quantity demanded of a product is to changes in the price of another product. Specifically, cross-price elasticity helps us see this sensitivity when two goods are related. In our example, we're observing how changes in the price of beef affect the demand for chicken. The cross-price elasticity formula, \( E_{\text{cross}} = \left|\frac{p}{q} \cdot \frac{dq}{dp}\right| \), captures this relationship. Here, \( p \) is beef's price, and \( q \) indicates the quantity of chicken demanded. If the calculated elasticity is greater than 0, chicken demand is indeed sensitive to beef prices, signaling a substitute relationship. This means just a small change in beef's price could noticeably impact chicken sales.

• A high cross-price elasticity (greater than 1) implies high demand sensitivity, suggesting consumers quickly switch to chicken when beef prices rise.
• A low elasticity (less than 1) hints at weaker sensitivity, indicating fewer people switching even if beef prices change.

Substitute Goods

Substitute goods are products that can be used in place of each other. When the price of one goes up, consumers might opt for the other, which is often a cheaper or convenient alternative.
In the context of cross-price elasticity, if the elasticity between two goods, like chicken and beef, is positive, they're substitutes. This implies that an increase in the price of beef, for example, might lead to a rise in the demand for chicken. The degree to which people switch from beef to chicken depends on how substitutable these goods are perceived to be.

• If the elasticity is high, consumers see chicken as a close substitute for beef, easily switching when prices hike.
• If it's low, consumers don't consider chicken as good enough replacement, possibly due to differences in taste, preference, or cuisine purpose.

Price Change Impact

The impact of a price change in one product on the demand for another is crucial for businesses and consumers alike. The cross-price elasticity helps predict these effects. When a price change occurs, the relative impact on the demand for a substitute good can vary greatly based on several factors.

• The overall magnitude of an elasticity greater than zero indicates that a rise in the price of beef will likely lead to an increase in chicken demand. The stronger this value, the more significant the impact.
• A greater than one elasticity means consumers will greatly increase their chicken consumption in response to a beef price increase.
• For values less than one, the change in chicken demand will be smaller, indicating a less pronounced reaction to beef's price variation.

Understanding these dynamics allows businesses to strategize pricing, marketing tactics, and even product placement to better meet consumer behavior and maximize sales.