Question

Use the following data to work Problems 12 to 15 Natalia has a budget of \(\$ 24\) a month to spend on fruit juice and books. The price of fruit juice is \(\$ 3\) a bottle, and the price of a book is \(\$ 6\) Calculate the equation for Natalia's budget line (with the quantity of fruit juice on the left side).

Short answer

The equation is \[ x = 8 - 2y \]

Step-by-step solution

Understand the Problem

Identify the given information: Natalia's budget is \(24, the price of fruit juice is \)3 per bottle, and the price of a book is $6 each. We need to derive the budget line equation.

Define Variables

Let the quantity of fruit juice be denoted by \(x\) and the quantity of books by \(y\).

Formulate the Budget Equation

Using the general budget constraint formula \(P_x \times x + P_y \times y = \text{Budget}\), where \(P_x\) and \(P_y\) are the prices of fruit juice and books, respectively.

Substitute the Values

Substitute \(P_x = 3\), \(P_y = 6\), and \ \text{Budget} = 24 \ into the formula: \[3x + 6y = 24\]

Solve for Quantity of Fruit Juice

Rearrange the equation to isolate \(x\): \[3x = 24 - 6y\]

Simplify the Equation

Divide both sides of the equation by 3: \[x = 8 - 2y\]

Key concepts

Budget Constraint

In everyday life, we constantly make choices about how to spend our limited funds.
The concept of a budget constraint helps us understand the trade-offs involved in these choices.
A budget constraint represents all possible combinations of goods and services that a consumer can afford with their limited budget.
For Natalia, her budget constraint is illustrated by the equation we derived: \(3x + 6y = 24\).
Here, \(x\) is the quantity of fruit juice, \(y\) is the quantity of books, and the total must equal her budget of 24 dollars.
This equation tells us the maximum amount of fruit juice and books Natalia can buy without exceeding her budget. By rearranging and solving the equation, we find the specific combinations of goods she can afford.
For example, if she buys only fruit juice, she'll get 8 bottles (since \(3(8) = 24\)). If she buys only books, she'll get 4 books (since \(6(4) = 24\)).
If she wants a mix of both, she'll need to balance the quantities so that the total cost does not exceed 24 dollars.

Marginal Cost

Marginal cost is the additional cost incurred from consuming one more unit of a good or service.
In Natalia's case, the marginal cost of one more bottle of fruit juice is 3 dollars, while the marginal cost of one more book is 6 dollars. This concept is crucial when making decisions about how to allocate limited resources effectively.
For example, if Natalia's initial purchase is 4 bottles of fruit juice and 2 books and she considers buying another book, she needs to evaluate if the additional benefit of the book is worth the extra 6 dollars she'll spend.
Understanding marginal cost helps Natalia (and consumers in general) decide the optimal mix of goods within their budget.

Opportunity Cost

Opportunity cost refers to the value of the next best alternative that must be foregone to pursue a certain action.
For Natalia, deciding to buy a book means she foregoes the opportunity to spend that money on fruit juice. In our exercise, each book costs 6 dollars, which is the price of two bottles of fruit juice (since \(6 = 3 \times 2\)). The opportunity cost of buying a book is therefore two bottles of fruit juice.
Similarly, if she buys a bottle of fruit juice, she gives up the opportunity to spend those 3 dollars on something else, such as another part of a book.
Opportunity cost is not just the price but what you sacrifice in order to make a particular decision.

Consumer Choice Theory

Consumer choice theory explores how individuals make decisions to allocate their resources—time, money, and effort—among various goods and services to maximize their satisfaction or utility.
Consumers like Natalia will decide on the combination of goods—fruit juice and books—that provides the greatest utility within their budget constraint.
This theory incorporates the concepts of marginal utility (the added satisfaction from consuming an additional unit of a good) and diminishing marginal utility (the decrease in added satisfaction as more of a good is consumed).
Natalia's decisions are influenced by the prices of goods, her income, and her preferences. For her, the ideal combination of fruit juice and books is one that maximizes her utility given her 24 dollar budget. That’s why solving the budget constraint equation helps understand the feasible options she has.