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 \(\$ 6\) What is the relative price of a bottle of fruit juice in terms of a book and what is the opportunity \(\operatorname{cost}\) of a bottle of juice?

Short answer

The relative price is \(\frac{1}{2}\) of a book. The opportunity cost of a bottle of juice is \(\frac{1}{2}\) book.

Step-by-step solution

Identify the Given Data

Natalia has a monthly budget of \(\$24\). The price of fruit juice is \(\$3\) per bottle, and the price of a book is \(\$6\) per book.

Calculate the Relative Price

The relative price of a bottle of fruit juice in terms of a book is the ratio of the price of fruit juice to the price of a book. This can be calculated as\[\text{Relative price} = \frac{\text{Price of fruit juice}}{\text{Price of a book}} = \frac{\$3}{\$6} = \frac{1}{2}\].

Determine the Opportunity Cost

The opportunity cost of a bottle of fruit juice is the quantity of books that could be bought with the same amount of money spent on one bottle of fruit juice. Since each bottle of fruit juice costs \(\$3\) and each book costs \(\$6\), the opportunity cost is:\[\text{Opportunity cost} = \frac{\text{Price of fruit juice}}{\text{Price of a book}} = \frac{\$3}{\$6} = \frac{1}{2}\text{ book}\].

Key concepts

relative price

In economics, the relative price is a way to compare the cost of two goods. It's expressed as the ratio of the price of one good to the price of another good. In our example, we have the price of fruit juice and the price of a book. The relative price tells us how many units of one good can be exchanged for one unit of another good.

In the exercise, fruit juice costs \$3 per bottle and books cost \$6 per book. To find the relative price of fruit juice in terms of books, we divide the price of the fruit juice by the price of a book:

\[\text{Relative price} = \frac{\text{Price of fruit juice}}{\text{Price of a book}} = \frac{\$3}{\$6} = \frac{1}{2}\]

This means that one bottle of fruit juice is equal to half a book. It helps us understand how the prices of these goods compare to each other.

budget constraint

A budget constraint shows the combinations of goods a person can buy given their budget and the prices of the goods. It helps us see the trade-offs between different goods. In this exercise, Natalia's budget constraint can be visualized as follows:

Natalia has \$24 to spend each month on fruit juice and books. Fruit juice costs \$3 per bottle, and books cost \$6 each. To understand her budget constraint, we need to find out how many units of fruit juice and books she can purchase with her budget.

If Natalia spends all her money on fruit juice, she can buy:
\[\frac{\$24}{\$3 \text{ per bottle}} = 8 \text{ bottles of fruit juice}\]

If she spends all her money on books, she can buy:
\[\frac{\$24}{\$6 \text{ per book}} = 4 \text{ books}\]

The budget line represents all possible combinations of books and fruit juice that she can buy without exceeding her \$24 budget. Any point on this line shows a combination of books and fruit juice that equals her total budget.

price ratio

The price ratio is closely related to the concept of relative price. It's the ratio of the price of one good to another and indicates how many units of one good can be exchanged for a unit of another. This ratio is important when considering trade-offs and making decisions about consumption.

In the problem, we can see that:
\[\text{Price ratio} = \frac{\text{Price of fruit juice}}{\text{Price of a book}} = \frac{\$3}{\$6} = \frac{1}{2}\]

From this ratio, we know that one book is worth two bottles of fruit juice. Understanding the price ratio helps consumers make informed decisions about how much of each good to purchase based on their preferences and budget.

opportunity cost calculation

Opportunity cost is a fundamental concept in economics. It represents the value of the next best alternative that one gives up when making a choice. In this exercise, it measures what Natalia sacrifices when she chooses to buy one good over the other.

To calculate the opportunity cost of buying a bottle of fruit juice, we need to see how many books she is giving up. Given that a bottle of fruit juice costs \$3 and a book costs \$6, we use the following formula:

\[\text{Opportunity cost} = \frac{\text{Price of fruit juice}}{\text{Price of a book}} = \frac{\$3}{\$6} = \frac{1}{2} \text{ book}\]

Thus, the opportunity cost of buying one bottle of fruit juice is half a book. This means Natalia would have to give up half a book to buy one bottle of fruit juice. Understanding opportunity cost helps people make better financial decisions by considering what they must forego when choosing one option over another.