Problem 24 - Applied Calculus

Question

Each person tries to balance his or her time between leisure and work. The tradeoff is that as you work less your income falls. Therefore each person has indifference curves which connect the number of hours of leisure, \(l\), and income, \(s .\) If, for example, you are indifferent between 0 hours of leisure and an income of $$\$ 1125$$ a week on the one hand, and 10 hours of leisure and an income of $$\$ 750$$ a week on the other hand, then the points \(l=0\), \(s=1125\), and \(l=10, s=750\) both lie on the same indifference curve. Table \(9.11\) gives information on three indifference curves, I, II, and III. $$\begin{array}{l} \text { Weekly income } \quad \text { Weekly leisure hours }\\\ \begin{array}{c|c|c|c|c|c} \hline \text { I } & \text { II } & \text { III } & \text { I } & \text { II } & \text { III } \\ \hline 1125 & 1250 & 1375 & 0 & 20 & 40 \\ \hline 750 & 875 & 1000 & 10 & 30 & 50 \\ \hline 500 & 625 & 750 & 20 & 40 & 60 \\ \hline 375 & 500 & 625 & 30 & 50 & 70 \\ \hline 250 & 375 & 500 & 50 & 70 & 90 \\ \hline \end{array} \end{array}$$ (a) Graph the three indifference curves. (b) You have 100 hours a week available for work and leisure combined, and you earn $$\$ 10 /$$ hour. Write an equation in terms of \(l\) and \(s\) which represents this constraint. (c) On the same axes, graph this constraint. (d) Estimate from the graph what combination of leisure hours and income you would choose under these circumstances. Give the corresponding number of hours per week you would work.

Step-by-Step Solution

Answer

Graph indifference curves and budget; choose combo where constraint is tangent to an indifference curve. Subtract chosen leisure from 100 to find work hours.

Step 1: Understanding Indifference Curves

An indifference curve shows combinations of income and leisure hours where a person is equally satisfied. For example, if you are indifferent between 0 leisure hours with $1125/week and 10 leisure hours with $750/week, these points lie on the same curve.

Step 2: Graphing Indifference Curves

Plot the given data points for Indifference Curves I, II, and III. Each curve connects points such as (0, 1125), (10, 750), etc. for Curve I. Use different colors or styles for each curve on the graph.

Step 3: Understanding the Budget Constraint

You have 100 hours a week for work and leisure, and you earn $10/hour. Thus, your income constraint equation is: \[ s = 10(100 - l) \] Here, \(s\) is income, and \(l\) is leisure.

Step 4: Graphing the Budget Constraint

On the same axes as the indifference curves, graph the equation \(s = 10(100 - l)\). This is a straight line starting at (0, 1000) on the y-axis and (100, 0) on the x-axis.

Step 5: Estimating Optimal Leisure and Income

Identify the point where your budget constraint is tangent to one of the indifference curves. This intersection suggests the optimal combination of leisure and income.

Step 6: Calculate Working Hours

Once the point of tangency on the graph is found, use it to determine the number of leisure hours. Subtract leisure hours from 100 to find how many hours you'd work per week.

Key Concepts

Income-Leisure Tradeoff

Everyone has a limited amount of time each week to allocate between work and leisure. This allocation creates a tradeoff because less time at work means a lower income. The **income-leisure tradeoff** illustrates the balance that individuals must strike between the desire for leisure and the need for income.

As you work fewer hours, your income decreases, and consequently, you have to make decisions that maximize your overall satisfaction given this tradeoff. An indifference curve comes into play here, as it graphically represents combinations of leisure hours and income where a person remains equally satisfied.

• The main idea is to find the perfect mix of working hours (which brings income) and leisure hours (which brings rest and personal time).
• This tradeoff emphasizes the choice between what you earn and the time you have for leisure activities.

Understanding this tradeoff is critical to making informed choices about how many hours to work and how many to dedicate to leisure activities.

Budget Constraint

When planning your week, the **budget constraint** becomes a key component. It signifies the boundary of possible combinations of leisure and income, dictated by the number of hours available and your earning rate.

In the given problem, you have 100 hours a week for both work and leisure, with an earning rate of \(10 per hour. This leads to the budget constraint equation: \[ s = 10(100 - l) \] Here, \(s\) is your weekly income, and \(l\) are your leisure hours.

• This equation shows how each additional hour of leisure affects your income by decreasing it by \)10.
• It provides a simple method to calculate potential income given any number of leisure hours.

By understanding this constraint, you can easily find how many hours you need to work to achieve your desired level of income.

Graphing Indifference Curves

**Graphing indifference curves** provides a visual representation of preferences between income and leisure. Each curve depicts a series of combinations where the individual feels equally satisfied.

To graph such curves, you'll plot points as given in the problem, like (0, 1125) or (10, 750) for Curve I. Each point represents a unique combination of leisure and income levels.

• Multiple indifference curves can exist, each on a different level of utility, representing different levels of satisfaction.
• On the same graph, plot the budget constraint, which will likely intersect the indifference curves.

The key point of interest on the graph is where one indifference curve just touches your budget constraint, suggesting the best feasible mix of leisure and income for the highest satisfaction given your constraints.