Problem 6 - Applied Calculus

Question

The quantity \(Q\) (in pounds) of beef that a certain community buys during a week is a function \(Q=f(b, c)\) of the prices of beef, \(b\), and chicken, \(c\), during the week. Do you expect \(\partial Q / \partial b\) to be positive or negative? What about \(\partial Q / \partial c\) ?

Step-by-Step Solution

Answer

\(\partial Q / \partial b\) is negative; \(\partial Q / \partial c\) is positive.

Step 1: Understanding the Function

The function \(Q = f(b, c)\) describes the quantity of beef bought based on the prices of beef \(b\) and chicken \(c\). The goal is to determine how changes in these prices affect the quantity \(Q\).

Step 2: Analyzing Partial Derivative with respect to b

The partial derivative \(\frac{\partial Q}{\partial b}\) symbolizes the rate at which the quantity of beef purchased changes as the price of beef itself \(b\) changes, keeping the other factor (price of chicken \(c\)) constant.

Step 3: Determining the Sign of \(\partial Q / \partial b\)

Typically, if the price of beef \(b\) increases, the quantity of beef purchased \(Q\) decreases, following the law of demand. Thus, we expect \(\frac{\partial Q}{\partial b}\) to be negative, indicating an inverse relationship.

Step 4: Analyzing Partial Derivative with respect to c

The partial derivative \(\frac{\partial Q}{\partial c}\) signifies the rate of change in the quantity of beef bought as the price of chicken \(c\) changes, while the price of beef \(b\) is constant.

Step 5: Determining the Sign of \(\partial Q / \partial c\)

If the price of chicken \(c\) goes up, consumers might buy more beef instead, thus increasing the quantity \(Q\) of beef. Therefore, \(\frac{\partial Q}{\partial c}\) is expected to be positive, showing a direct relationship.

Key Concepts

Price Elasticity

Price elasticity measures how the quantity demanded of a good responds to a change in price. When the price of a good changes, consumers adjust the quantity they purchase. Elasticity can be of two types, elastic and inelastic.

Consider beef in our example, where

• Elastic: A significant change in quantity demanded for a small change in price.
• Inelastic: Quantity demanded changes very little with a large change in price.

The elasticity of demand can be understood by the partial derivatives calculated in the solution.

• Negative partial derivative \(\frac{\partial Q}{\partial b}\): Indicates that with an increase in the price of beef, the quantity demanded decreases (law of demand). Consumers find beef less affordable, reducing consumption.
• Positive partial derivative \(\frac{\partial Q}{\partial c}\): Shows that as the price of chicken increases, more beef is purchased. Consumers substitute beef for the now more expensive chicken.

Understanding price elasticity helps businesses determine pricing strategies and forecast consumer behavior.

Functions of Multiple Variables

Functions of multiple variables involve two or more independent variables affecting a dependent variable. Here, the function \Q = f(b, c)\ consists of beef and chicken prices influencing how much beef is purchased.

Each variable can change the output differently, and this is where partial derivatives become crucial. By taking partial derivatives:

• You isolate the effect of one independent variable (keeping others constant).
• You understand the relationship between each variable and the dependent variable.

For instance, in our exercise:

• The derivative \(\frac{\partial Q}{\partial b}\) focuses on the effect of changing beef prices on beef demand.
• The derivative \(\frac{\partial Q}{\partial c}\) highlights how chicken prices affect beef consumption.

This method allows economists and analysts to simplify complex systems and predict the impact of changes in specific variables.

Law of Demand

The law of demand is an economic principle that states as the price of a good increases, the quantity demanded generally decreases, ceteris paribus (all else being equal). This inverse relationship helps explain consumer purchasing behavior under the assumption of rational preference.

In this context:

• For beef, as mentioned, \(\frac{\partial Q}{\partial b} < 0\) implies that when beef's price rises, the demand falls.
• This occurs because consumers may switch to different goods that are now relatively cheaper, illustrating the substitution effect.
• Conversely, \(\frac{\partial Q}{\partial c} > 0\) highlights the opportunity for substitution, such as opting for more beef if chicken becomes more costly.

The law of demand is foundational in understanding how price changes influence market demand, ensuring retailers and policymakers anticipate and respond to consumer needs efficiently.