Problem 69 - Calculus Single Variable

Question

Suppose that \(X\) and \(Y\) are items (or goods) that can be purchased at prices \(p_{X}\) and \(p_{Y},\) respectively. Suppose that \(x\) represents the number of units of good \(X,\) and \(y\) represents the number of units of good \(Y\) that a consumer might purchase. The first quadrant of the \(x y\) -plane is known as commodity space in economics. If the consumer has a fixed amount \(C\) that he may allot to the purchase of goods \(X\) and \(Y,\) then the locus of all points \((x, y)\) that represent purchasable combinations of these two goods is known as the consumer's budget line. (It is actually a line segment.) Determine a Cartesian equation for the budget line. What are its intercepts? What is its slope? If the consumer's circumstances change so that he has a different amount \(C^{\prime}\) that he can use toward the purchase of goods \(X\) and \(Y,\) then what is the relationship of the new budget line to the old one?

Step-by-Step Solution

Answer

The budget line equation is \( p_X x + p_Y y = C \), with slope \(-\frac{p_X}{p_Y}\). Changing the budget to \(C'\) shifts the line parallel.

Step 1: Writing the Budget Equation

The consumer's budget equation is given by the formula \( p_X \cdot x + p_Y \cdot y = C \), where \( p_X \) and \( p_Y \) are the prices of goods \( X \) and \( Y \), respectively, and \( C \) is the total budget available.

Step 2: Finding Intercepts

To find the \( x \)-intercept, set \( y = 0 \) in the budget equation, resulting in \( x = \frac{C}{p_X} \). For the \( y \)-intercept, set \( x = 0 \), yielding \( y = \frac{C}{p_Y} \).

Step 3: Calculating the Slope

Transform the budget equation \( p_X x + p_Y y = C \) to the slope-intercept form: \( y = -\frac{p_X}{p_Y}x + \frac{C}{p_Y} \). The slope of the budget line is \( -\frac{p_X}{p_Y} \).

Step 4: Effect of Changing the Budget

If the budget changes to \( C' \), the new budget equation becomes \( p_X x + p_Y y = C' \). The intercepts will change to \( \frac{C'}{p_X} \) for the \( x \)-intercept and \( \frac{C'}{p_Y} \) for the \( y \)-intercept, but the slope remains \( -\frac{p_X}{p_Y} \), so the line shifts parallel.

Key Concepts

Consumer Theory

Consumer Theory is a fundamental concept in economics that examines how individuals decide to allocate their limited resources across different goods and services to maximize their satisfaction. Imagine you have a fixed budget, and you want to spend it on various products that you value. How do you decide what to buy and in what quantities? This is where Consumer Theory comes into play.

The idea is to understand the choices consumers make and how those choices can be represented graphically. One of the main tools used here is the **budget constraint**, which shows the combination of goods a consumer can buy without exceeding their budget. This is often visualized as a budget line, which we'll explore further below.

Intercepts

Intercepts help us figure out where the budget line crosses the axes in a graph. This is crucial for understanding the limits of what a consumer can purchase. Let's break it down:

- **x-intercept**: This is found when you set the quantity of other goods to zero, meaning you only spend on good X. By setting the quantity of Y to zero in the budget equation, we find the maximum amount of X you can buy, which is given by \( x = \frac{C}{p_X} \). This tells you how much of product X you can afford if you don't buy any of product Y.

- **y-intercept**: Similarly, by setting the quantity of X to zero, you will find the maximum amount of Y you can afford, given by \( y = \frac{C}{p_Y} \). This reflects the consumer’s purchasing power for Y when opting out from X.

Understanding intercepts help visualize the trade-offs between the two goods given a fixed budget.

Slope

The slope of the budget line is a key element in consumer decision-making. It describes the rate at which consumers are willing to substitute one good for another while staying within their budget. In mathematical terms, the slope of the budget line is given by \( -\frac{p_X}{p_Y} \). This negative slope indicates that as you increase your purchase of one good, you must decrease your purchase of the other to remain within your budget. For every additional unit of good X you want, you have to give up some units of good Y, and vice versa.

The slope thus reflects the opportunity cost and the trade-off between different goods. A steeper slope signifies a higher relative price of good X compared to good Y, meaning you will need to give up more of good Y to acquire good X.

Economics

In economics, comprehending budget lines is essential for analyzing individual choices in various environments. This helps economists predict how changes in prices or income levels affect consumer behavior. As you might have already noticed, when the consumer's budget changes from \( C \) to \( C' \), while the slope of the budget line remains constant, its position shifts.

This parallel shift signifies new intercepts on the axes, suggesting that a change in income allows more (or fewer) purchases of either good while preserving the trade-off rate. Essentially, the budget line portrays constraints, but shifts or tilts as conditions change, influencing consumer decisions.

By understanding this interplay between constraints (budget lines) and consumer preferences, economics strives to predict consumer responses to policy changes, market conditions, and other economic events.