Rhoda takes a job with a construction company. She earns $5 an hour for the first 40 hours of each week and then gets "double-time" for overtime. That is, she is paid $10 an hour for every hour beyond 40 hours a week that she works. Rhoda has 70 hours a week available to divide between construction work and leisure. She has no other source of income, and her utility function is U = cr, where c is her income to spend on goods and r is the number of hours of leisure that she has per week. She is allowed to work as many hours as she wants to. How many hours will she work?

Question

Quizplus · Accepted Answer

To maximize her income, Rhoda should work as many hours as possible until she reaches the point where the marginal utility of leisure equals the marginal utility of income. The marginal utility of leisure is simply 1, as each additional hour of leisure is valued the same. The marginal utility of income can be found by calculating the additional utility gained from earning an additional dollar. Since her utility function is U = cr, then the marginal utility of income (MUc) is simply r, her leisure hours.

To determine Rhoda's optimal work schedule, we need to calculate her hourly wage under different scenarios. Rhoda earns $5 an hour for the first 40 hours, so her hourly wage is $5 for those hours. For hours worked beyond 40, her hourly wage increases to $10. Let x be the number of hours she works beyond 40 hours.

Her weekly income (I) can be expressed as:

I = 5(40) + 10x

Her leisure hours can be found by subtracting her work hours (W) from 70:

r = 70 - W

We can rearrange these equations to solve for W in terms of r:

W = 40 + (1/2)(r - 30)

Now, we can substitute this expression for W into Rhoda's income equation to get:

I = 5(40) + 10(1/2)(r - 30)

Simplifying yields:

I = 200 + 5(r - 30)

Taking the derivative of I with respect to r gives:

dI/dr = 5

Since the derivative of I is positive, Rhoda's income is increasing as her leisure hours decrease. However, the marginal utility of income is decreasing as she works more hours, while the marginal utility of leisure is constant. At some point, the marginal utility of income will be less than the marginal utility of leisure, and Rhoda will no longer want to work additional hours.

Setting the marginal utility of income equal to the marginal utility of leisure gives:

r = 5

Substituting this value of r into the expression we derived for W gives:

W = 45

Therefore, Rhoda should work 45 hours and have 25 hours of leisure each week.

Rhoda Takes a Job with a Construction Company