Problem 5 - Applied Calculus

Question

The demand for coffee, \(Q\), in pounds sold per week, is a function of the price of coffee, \(c\), in dollars per pound and the price of tea, \(t\), in dollars per pound, so \(Q=f(c, t)\). (a) Do you expect \(f_{c}\) to be positive or negative? What about \(f_{t} ?\) Explain. (b) Interpret each of the following statements in terms of the demand for coffee: \(f(3,2)=780 \quad f_{c}(3,2)=-60 \quad f_{t}(3,2)=20\)

Step-by-Step Solution

Answer

\(f_c\) is negative and \(f_t\) is positive. Coffee demand decreases as its price increases, and increases as tea's price rises.

Step 1: Understanding Partial Derivatives

The partial derivative of a function with respect to a variable measures how the function changes as that variable changes while keeping other variables constant. In this context, \( f_{c} \) represents how the demand for coffee \(Q\) changes with respect to the price of coffee \(c\), while \( f_{t} \) represents how \(Q\) changes with respect to the price of tea \(t\).

Step 2: Analyzing \(f_c\)

Typically, as the price of an item increases, the demand decreases. Hence, we expect \( f_{c} \) to be negative because as the price of coffee \(c\) increases, the demand for coffee \(Q\) would decrease.

Step 3: Analyzing \(f_t\)

Tea and coffee are often considered substitutes for each other. When the price of tea \(t\) increases, people may switch from tea to coffee, increasing the demand for coffee. Thus, \( f_{t} \) is expected to be positive.

Step 4: Interpreting \(f(3,2)=780\)

The function value \( f(3,2)=780 \) indicates that when the price of coffee is \\(3 per pound and the price of tea is \\)2 per pound, the quantity of coffee demanded is 780 pounds per week.

Step 5: Interpreting \(f_c(3,2)=-60\)

The partial derivative \( f_{c}(3,2)=-60 \) suggests that if the price of coffee increases by \\(1 (from \\)3 to \$4, for example), the demand for coffee will decrease by 60 pounds per week, assuming the price of tea remains constant.

Step 6: Interpreting \(f_t(3,2)=20\)

The partial derivative \( f_{t}(3,2)=20 \) implies that if the price of tea increases by \\(1 (from \\)2 to \$3, for example), the demand for coffee would increase by 20 pounds per week, given the price of coffee remains the same.

Key Concepts

Understanding the Demand Function

A demand function expresses how the quantity demanded of a good, like coffee, is influenced by different factors, such as prices of related goods or income levels. In our example, the demand for coffee depends on the price of coffee itself and the price of tea. This can be captured mathematically by the function \(Q = f(c, t)\), where \(Q\) is the quantity of coffee, \(c\) is the price of coffee, and \(t\) is the price of tea.
This framework helps to understand how changes in these prices can lead to changes in the quantity demanded of coffee. Partial derivatives are useful here as they show us the rate of change of demand with respect to each of these variables, assuming the other variable remains fixed.
By considering the demand function, businesses can predict how certain price changes might affect sales, allowing for better strategic planning.

Exploring Price Elasticity

Price elasticity of demand is a measure of how responsive the quantity demanded of a good is to a change in its price. In our scenario, the partial derivative \(f_c\) provides insights into this responsiveness.
When \(f_c\) is negative, it confirms that coffee has a negative price elasticity, meaning that as coffee becomes more expensive, demand generally drops. This aligns with the law of demand, which states that higher prices typically discourage consumption. Conversely, if \(f_c\) were positive, which is unusual, it would imply that demand increases as price rises—a rare phenomenon known as a Veblen good.

• Elastic Demand: Large response to price change, \(f_c\) is a larger negative number.
• Inelastic Demand: Small response to price change, \(f_c\) is closer to zero.

Understanding elasticity helps businesses and economists set prices that maximize revenue and predict consumer behavior.

Impact of Substitute Goods

Substitute goods are goods that can replace each other in consumption. For example, in our context, tea serves as a substitute for coffee. The presence and pricing of these substitutes can significantly impact the demand for a related good.
Analyzing the partial derivative \(f_t = 20\) indicates that when the price of tea rises, the demand for coffee increases. This positive relationship shows that consumers are likely to switch to coffee when tea becomes more expensive. This behavior is typical when two goods satisfy similar needs, making them strong substitutes.

• If \(f_t\) were zero, tea and coffee would not affect each other's consumption.
• If \(f_t\) were negative, it would suggest an unusual scenario where an increase in tea's price decreases coffee's demand.

Recognizing the role of substitute goods helps companies anticipate changes in demand when a related product's price fluctuates, thus tailoring their marketing or pricing strategy accordingly.