Problem 15 - Applied Calculus

Question

People commuting to a city can choose to go either by bus or by train. The number of people who choose either method depends in part upon the price of each. Let \(f\left(P_{1}, P_{2}\right)\) be the number of people who take the bus when \(P_{1}\) is the price of a bus ride and \(P_{2}\) is the price of a train ride. What can you say about the signs of \(\partial f / \partial P_{1}\) and \(\partial f / \partial P_{2}\) ? Explain your answers.

Step-by-Step Solution

Answer

\(\partial f / \partial P_{1} < 0\) and \(\partial f / \partial P_{2} > 0\).

Step 1: Understanding the Partial Derivative

Partial derivative with respect to a variable measures how the function changes as that variable changes, while keeping others constant.

Step 2: Interpret \\( \partial f / \partial P_{1} \\\)

The partial derivative \(\partial f/\partial P_{1}\) represents the change in the number of people taking the bus as the bus fare \(P_1\) changes, while the train fare \(P_2\) stays constant. If the bus fare increases, the number of people choosing the bus is likely to decrease, suggesting that \(\partial f/\partial P_{1} < 0\).

Step 3: Interpret \\( \partial f / \partial P_{2} \\\)

The partial derivative \(\partial f/\partial P_{2}\) measures the change in the number of bus commuters as the train fare \(P_2\) changes, while the bus fare stays constant. If the train fare increases, more people might prefer the bus, implying that \(\partial f/\partial P_{2} > 0\).

Step 4: Conclusion from Partial Derivatives

Based on the above interpretations, increasing the bus fare is expected to decrease the number of bus commuters, while increasing the train fare is expected to increase the number of bus commuters.

Key Concepts

Understanding Price Elasticity

Price elasticity is a crucial concept for understanding how changes in price affect consumer behavior. When dealing with commuting choices, like choosing between a bus or train based on fare changes, price elasticity helps us predict how sensitive commuters are to these changes.

The concept of price elasticity refers to how much the quantity demanded of a good (in our case, bus or train rides) responds to a change in its price. If commuters are very responsive to price changes, we say the price elasticity is high.

• When the bus fare rises and fewer people choose buses, the demand for bus rides is said to be elastic.
• If a rise in bus fare doesn't change the number of commuters significantly, the demand is inelastic.

The negative sign for \( \partial f/\partial P_1 \) in our exercise indicates that bus fare increases lead to a decrease in bus commuters, showing an elastic demand. Conversely, the positive sign for \( \partial f/\partial P_2 \) suggests that when train fares increase, more people switch to buses, indicating cross-elasticity where the price of one good affects the demand for another.

Commuting Choices and Decisions

Commuting choices are influenced by various factors, with price being a major determinant. When people decide how to commute, they consider not only the cost but also comfort, convenience, and time.

In an urban environment where both buses and trains serve as efficient options, commuters weigh their choices based on:

• Cost Efficiency: Lower fare might encourage more people to use that mode of transport.
• Time Saving: Commuters may prefer a slightly more expensive option if it saves time.
• Convenience: Factors such as ease of access to stations, cleanliness, and seating availability play a role.

Changes in fares directly affect these choices. As demonstrated by the partial derivatives \( \partial f/\partial P_1 \) and \( \partial f/\partial P_2 \), cost changes in one mode can shift commuter preference towards the other.

The Role of Mathematical Modeling

Mathematical modeling plays a pivotal role in analyzing and predicting the impact of price changes on commuter choices. By using functions like \( f(P_1, P_2) \), we can systematically explore how variables such as bus or train fares influence commuter behavior.

A well-constructed mathematical model can help in:

• Predicting Outcomes: Quantify how fare changes affect bus versus train usage.
• Policy Making: Provide insights for transportation authorities to make informed decisions on fare adjustments.
• Resource Allocation: Enable better planning for necessary infrastructure or service enhancements based on expected commuter numbers.

The partial derivatives \( \partial f/\partial P_1 \) and \( \partial f/\partial P_2 \) serve as powerful tools within these models, giving a clear sense of direction on how commuting numbers might shift with fare changes.