Question

Question: A worker views leisure and income as “goods” and has an opportunity to work at an hourly wage of \(15 per hour.

a. Illustrate the worker’s opportunity set in a given 24-hour period.

b. Suppose the worker is always willing to give up \)11 of income for each hour of leisure. Do her preferences exhibit a diminishing marginal rate of substitution? How many hours per day will she choose to work?

Short answer

Answer:

1. The worker’s opportunity set is $360 for 0 leisure and $0 for 24 hours of leisure.
2. Yes, her preferences exhibit a diminishing marginal rate. She will choose 11 hours to work per day

Step-by-step solution

Illustrating the worker’s Opportunity Set

Workers consider income and leisure to be goods and substitute them at a diminishing rate.

1. $\text{Earnings}\left(E\right)=15\left(24-L\right)$
   

Here, E is the income or earnings and L is the leisure.

If the worker considers working 24 hours without any leisure,

For ,

$$\begin{gathered} \text{Earnings}\left(E\right)=15\left(24-0\right) \\ =15\times 24 \\ =360 \end{gathered}$$

Hence, the total income of the worker would be $360 without any leisure.

Now, if the worker considers not working and append 24 hours in leisure,

For ,

$$\begin{gathered} \text{Earnings}\left(E\right)=15\left(24-24\right) \\ =15\times 0 \\ =0 \end{gathered}$$

Hence, the worker if chooses to spend 24 hours in leisure will receive an income of $0. Thus, the opportunity set in 24 hours period is $360 and $0.

Illustrating the worker’s Opportunity Set if she is willing to give up $11 for each leisure hour

1. If the worker is willing to give up $11 of income for 1 leisure

Then, for L number of leisure workers will give up 11L amount of income. So, the new equation becomes,

$$\begin{gathered} \text{Earnings}\left(E\right)=15-11L\left(24-L\right) \\ =360-15L-264L+11L^2 \end{gathered}$$

If L = 0,

Then the earning will be $360 which is the opportunity cost of leisure and the opportunity cost of work would be $0 at L = 24 hours. If the person chooses to take leisure of 1 hour,

$$\begin{gathered} \text{Earnings}\left(E\right)=15-11\times 1\left(24-1\right) \\ =4\times 23 \\ =92 \end{gathered}$$

then its earning becomes $92 which decreases with increase in leisure. Hence it shows diminishing marginal rate.

So, to maximize the earnings,

$$\begin{gathered} \frac{\partial E}{\partial L}=0 \\ -15-264+22L=0 \\ 22L=279 \\ L\simeq 13 \end{gathered}$$

Hence, the worker will choose 13 hours of leisure.

The hours the worker will choose to work is (24-13) = 11 hours.