Wendy and Mac work in fast-food restaurants. Wendy is paid $4 an hour for the first 40 hours a week that she works and $6 an hour for every hour beyond 40 hours per week. Mac gets $5 an hour no matter how many hours he works. Each has 110 hours per week to allocate between work and leisure. Each has a utility function U = cr, where c is expenditure per week on consumption and r is hours of leisure per week. Each can choose the number of hours to work. If Wendy works W hours and Mac works M hours, then

Question

Quizplus · Accepted Answer

To maximize their utility, both Wendy and Mac will choose to work some hours and have some leisure time. Let's first consider Mac's situation. He gets paid $5 per hour regardless of how many hours he works, so his income is simply $5 multiplied by the number of hours he works. Let's denote his income by I and the number of hours of leisure he chooses by L. Then we have:

I = 5M
L = 110 - M

His expenditure on consumption is simply his income, since he has no other source of wealth. Therefore:

c = I = 5M

His utility function is U = cr, which becomes:

U = 5M (110 - M)

To maximize U with respect to M, we take the derivative of U with respect to M and set it equal to zero:

dU/dM = 5(110 - 2M) = 0
110 - 2M = 0
M = 55

Therefore, Mac should work 55 hours and have 55 hours of leisure. His income is I = 5M = $275, and his expenditure on consumption is c = I = $275. His utility is:

U = 5M (110 - M) = 5(55)(55) = 15,125

Now let's consider Wendy's situation. She gets paid $4 per hour for the first 40 hours and $6 per hour for every hour beyond 40 hours. Let's break her hours of work into two parts: the first 40 hours and the remaining W - 40 hours. Her income is then:

I = 4(40) + 6(W - 40) = 240 + 6W - 240 = 6W

Her leisure time is:

L = 110 - W

Her expenditure on consumption is:

c = I = 6W

The utility function is the same as before, so we have:

U = 6W (110 - W)

Taking the derivative with respect to W and setting it equal to zero, we get:

dU/dW = 6(110 - 2W) = 0
110 - 2W = 0
W = 55

Therefore, Wendy should work 55 hours, the same as Mac. Her income is I = 6W = $330, and her expenditure on consumption is c = I = $330. Her utility is:

U = 6W (110 - W) = 6(55)(55) = 18,150

Therefore, Wendy and Mac should both work 55 hours and split their remaining 55 hours between leisure and consumption. Their choices are not identical, since their pay structures are different, but they both end up with the same utility-maximizing choice of hours worked. The correct answer is (C).

Wendy and Mac Work in Fast-Food Restaurants