Question

The following numbers measure the trade-off between grades and income: $$ \begin{array}{lcccc} \begin{array}{l} \text { Total } \\ \text { Hours } \end{array} & \begin{array}{c} \text { Hours } \\ \text { Studying } \end{array} & \text { GPA } & \begin{array}{c} \text { Hours } \\ \text { Working } \end{array} & \text { Income } \\ \hline 60 & 30 & 2.0 & 30 & \$ 150 \\ 60 & 10 & 1.0 & 50 & \$ 250 \\ 60 & 0 & 0.0 & 60 & \$ 300 \\ \hline \end{array} $$ a. Calculate the opportunity cost of an increase in the number of hours spent studying in order to earn a 3.0 grade point average (GPA) rather than a 2.0 GPA. (Assume linear relationships.) b. Is the opportunity cost the same for a move from a 0.0 GPA to a 1.0 GPA as it is for a move from a 1.0 GPA to a 2.0 GPA? c. What is the opportunity cost of an increase in income from \(100 to \)150?

Short answer

Answer: The opportunity cost of increasing income from $100 to $150 is a decrease in GPA from 3.0 to 2.0.

Step-by-step solution

Find the relationship between studying hours, working hours, GPA, and income

Using the given data, we can see that the total hours is constant (60 hours). Also notice that the data points have a linear relationship. Let's assume the following equations for the relationship: - GPA = m1 * (Total Hours Studying) + b1 - Income = m2 * (Total Hours Working) + b2 We can see from the data that b1 = b2 = 0. Now, let's find m1 and m2. For GPA: 2.0 = m1 * 30, 1.0 = m1 * 10 => m1 = 2/30 = 1/10 For Income: 150 = m2 * 30, 250 = m2 * 50 => m2 = 150/30 = 5 Now we have the equations: - GPA = 1/10 * (Total Hours Studying) - Income = 5 * (Total Hours Working)

Calculate the required studying hours and working hours for 3.0 GPA

To calculate the required studying hours for a 3.0 GPA, we can use the equation we found in the previous step: 3.0 = 1/10 * (Total Hours Studying) => Total Hours Studying = 30 Since the total hours is 60: Total Hours Working = 60 - 30 = 30

Calculate the opportunity cost

Opportunity cost can be defined as the cost of next best alternative given up. In this case, it is the income earned from working that must be given up to achieve a 3.0 GPA. Income at 2.0 GPA = 5 * 30 = $150 Income at 3.0 GPA = 5 * 30 = $150 The opportunity cost of increasing the GPA from 2.0 to 3.0 is $0, since there is no reduction in income. b. Is the opportunity cost the same for a move from a 0.0 GPA to a 1.0 GPA as it is for a move from a 1.0 GPA to a 2.0 GPA?

Compare different opportunity costs

Opportunity cost from 0.0 GPA to 1.0 GPA: Income at 0.0 GPA = 5 * 60 = $300 Income at 1.0 GPA = 5 * 50 = $250 Opportunity cost = \(300 - \)250 = $50 Opportunity cost from 1.0 GPA to 2.0 GPA: Income at 1.0 GPA = 5 * 50 = $250 Income at 2.0 GPA = 5 * 30 = $150 Opportunity cost = \(250 - \)150 = $100 The opportunity cost for a move from a 0.0 GPA to a 1.0 GPA is \(50, while the opportunity cost for a move from a 1.0 GPA to a 2.0 GPA is \)100. Therefore, the opportunity costs are not the same. c. What is the opportunity cost of an increase in income from \(100 to \)150?

Calculate the opportunity cost of increasing income

We can use the inverse relationship between income and studying hours to find the change in GPA that corresponds to the increase in income. Income at \(100 = 5 * 40 = \)100 Income at \(150 = 5 * 30 = \)150 Now let's find the corresponding GPAs: GPA at 40 working hours (20 studying hours) = 1/10 * 20 = 2.0 GPA at 30 working hours (30 studying hours) = 1/10 * 30 = 3.0 The opportunity cost of increasing income from \(100 to \)150 is a decrease in GPA from 3.0 to 2.0.

Key concepts

Trade-Off Analysis

Trade-off analysis is a crucial concept in economics that involves comparing the costs and benefits of a decision. It's the process of understanding what you must forego when choosing one alternative over another.

Let's consider the scenario of a student balancing work and study hours. With a finite number of hours in a day (60 hours, in our exercise), each additional hour spent studying is one less hour earning income, and vice versa. This is a classic example of a trade-off. For a student considering whether to allocate more time to study or work, trade-off analysis helps to understand the opportunity cost, which is the next best alternative forgone—in this case, either grades or income.

Understanding Opportunity Cost

Opportunity cost isn't always about money—it could be grades, leisure time, or any other benefit given up. In the given exercise, reducing study time to work more hours means earning more money at the expense of a lower GPA. Conversely, studying more may increase the GPA but decrease the income. To analyze trade-offs, it's essential to calculate these opportunity costs, as the student did for different GPA levels, to make informed decisions balancing academic achievements with financial needs.

Linear Relationships in Economics

Linear relationships are fundamental in economics as they provide a simple yet powerful way to understand the direct correlation between variables. The exercise illustrates this with a straightforward relationship between study hours, work hours, GPA, and income. Here, linear equations represent how a change in one variable directly affects another.

The assumption of linearity allows for the creation of a predictive model; in this case, for every unit increase in study hours, there is a consistent change in the GPA, and for every unit increase in work hours, there's a consistent change in income.

Applying Linearity in Calculations

By calculating the slopes of these relationships (given by the coefficients in the formulas), we see the rate of change: how much GPA is expected to increase per hour of study, and how much income is expected to increase per hour of work. This makes it possible to predict outcomes and understand the trade-offs involved in allocation time for studying and working.

Cost-Benefit Analysis in Education

Cost-benefit analysis in education helps students make decisions that align with their personal goals, taking into account both the quantitative and qualitative aspects. It's about weighing the tangible benefits, such as increased income, against intangible ones, like knowledge or a higher GPA, and vice versa.

In the educational setting of our exercise, a student might consider the financial aid they'd receive for a higher GPA, or the job prospects and potential salary increases that come with higher grades. Equally, they may consider the immediate financial benefit of working more hours.

Integrating Qualitative Benefits

Moreover, when students understand the cost-benefit implications, they can strategize their time to maximize their outcomes—whether that's pursuing additional tutoring for challenging subjects or allocating more hours to a part-time job. Education involves a range of non-monetary benefits that can be harder to quantify, such as personal development and skill acquisition, but they should also be integrated into a cost-benefit analysis.