Problem 16 - Applied Calculus

Question

In the 1940 s the quantity, \(q\), of beer sold each year in Britain was found to depend on \(I\) (the aggregate personal income, adjusted for taxes and inflation), \(p_{1}\) (the average price of beer), and \(p_{2}\) (the average price of all other goods and services). Would you expect \(\partial q / \partial I, \partial q / \partial p_{1}, \partial q / \partial p_{2}\) to be positive or negative? Give reasons for your answers.

Step-by-Step Solution

Answer

\(\frac{\partial q}{\partial I}\) is positive, \(\frac{\partial q}{\partial p_1}\) is negative, \(\frac{\partial q}{\partial p_2}\) is likely positive or uncertain.

Step 1: Understand the Variables

Let's consider what each variable represents:- \(q\): Quantity of beer sold.- \(I\): Aggregate personal income (adjusted for taxes and inflation).- \(p_1\): Average price of beer.- \(p_2\): Average price of all other goods and services.

Step 2: Analyzing \(\frac{\partial q}{\partial I}\)

The partial derivative \(\frac{\partial q}{\partial I}\) measures the change in quantity of beer sold with respect to a change in income. As people's income increases, they are usually able to afford more goods, including beer. Thus, we expect \(\frac{\partial q}{\partial I}\) to be **positive**.

Step 3: Analyzing \(\frac{\partial q}{\partial p_{1}}\)

The partial derivative \(\frac{\partial q}{\partial p_1}\) measures the change in quantity of beer sold with respect to a change in the price of beer. In most cases, as the price of beer increases, the quantity demanded tends to decrease, so we would expect \(\frac{\partial q}{\partial p_1}\) to be **negative**.

Step 4: Analyzing \(\frac{\partial q}{\partial p_{2}}\)

The partial derivative \(\frac{\partial q}{\partial p_2}\) measures the change in quantity of beer sold with respect to a change in the price of other goods and services. If the price of other goods increases, people might spend less on them and more on beer, which could increase beer sales, suggesting \(\frac{\partial q}{\partial p_2}\) could be **positive**. However, if people reduce overall spending, this could mean reduced beer sales, making it difficult to determine a definitive sign without specific data.

Key Concepts

Partial Derivatives

In calculus, partial derivatives are a way to explore how a multivariable function changes as one of its variables changes, while the others are kept constant. This is extremely useful in understanding economic models where many factors interdependently affect each other.
For example, if we consider the function for beer sales, where the quantity of beer sold, \( q \), depends on aggregate income \( I \), the price of beer \( p_1 \), and the price of other goods \( p_2 \), partial derivatives help us isolate the effect of each individual factor:

• \( \frac{\partial q}{\partial I} \) measures how sales change as income changes.
• \( \frac{\partial q}{\partial p_1} \) shows the sales change due to beer prices.
• \( \frac{\partial q}{\partial p_2} \) examines changes due to the price of other goods.

Understanding partial derivatives allows economists to predict consumer behavior under various conditions, making them crucial for effective economic forecasting.

Economics in Calculus

Calculus is a powerful tool in economics, enabling analysts to model and predict changes in different markets. Using derivatives, economists can understand how changes in specific variables like price or income affect broader economic measures such as demand or supply.
In the context of our beer sales example, calculus helps us interpret the sensitivity of beer demand to changing economic conditions:

• An increase in \( I \), indicated by a positive \( \frac{\partial q}{\partial I} \), reflects increased purchasing power and likely results in more beer consumption.
• A negative \( \frac{\partial q}{\partial p_1} \) reveals the inverse relation between price and demand, a principle known as the law of demand.

By applying calculus, economists can construct predictive models that account for complex interactions in economic systems. This is essential for forming policies and strategies that aim to stabilize or stimulate economic activities.

Supply and Demand Analysis

Part of economic analysis involves understanding how supply and demand interact within a market. Calculus, with its ability to model continuous change, is invaluable for examining these interactions
In supply and demand analysis, we often look at how the quantity of a good demanded or supplied shifts in reaction to price changes or income fluctuation:

• A decrease in beer prices should theoretically lead to higher demand, which is shown by \( \frac{\partial q}{\partial p_1} < 0 \).
• Income changes, represented by \( \frac{\partial q}{\partial I} \), suggest that greater income generally increases demand.
• The effect of other goods' prices on beer sales, \( \frac{\partial q}{\partial p_2} \), could vary depending on whether those goods are substitutes or complements to beer.

Understanding these principles helps businesses and policymakers make informed decisions regarding pricing, marketing, and economic regulation. Tools such as partial derivatives allow a concise and quantitative exploration of these dynamics.