(d) Between 1890 and 1909, wage rates rose by
39
percent, but weekly earnings rose by only20
percent. For this period, thegain in earnings (overstates, understates)
understates
the gain in worker’s wealth, since they chose to take (more, less)more
leisure in 1909 than they took in 1890.9.11 (0) Professor Mohamed El Hodiri of the University of Kansas, in a classic tongue-in-cheek article “The Economics of Sleeping,” Manifold, vol. 17, 1975, o�ered the following analysis. “Assume there are 24 hours in a day. Daily consumption being x and hours of sleep s, the consumer maximizes a utility function of the form u = x2s, where x = w(24
− s), with w being the wage rate.”
(a) In El Hodiri’s model, does the optimal amount of sleeping increase, decrease, or stay the same as wages increase?
Stays the same.
(b) How many hours of sleep per day is best in El Hodiri’s model?
8.
9.12 (0) Wendy and Mac work in fast food restaurants. Wendy gets $4 an hour for the �rst 40 hours that she works and $6 an hour for every hour beyond 40 hours a week. Mac gets $5 an hour no matter how many hours he works. Each has 80 hours a week to allocate between work and leisure and neither has any income from sources other than labor. Each has a utility function U = cr, where c is consumption and r is leisure. Each can choose the number of hours to work.
(a) How many hours will Mac choose to work?
40.
(b) Wendy’s budget “line” has a kink in it at the point where r =
40
and c =160.
Use blue ink for the part of her budget line where she would be if she does not work overtime. Use red ink for the part where she would be if she worked overtime.0 20 40 60 80 100 200 300 Leisure Consumption 400
Blue part of line Red part of line
(c) The blue line segment that you drew lies on a line with equation
c + 4r = 320.
The red line that you drew lies on a line with equationc + 6r = 400.
(Hint: For the red line, you know one point on the line and you know its slope.)(d) If Wendy was paid $4 an hour no matter how many hours she worked, she would work
40
hours and earn a total of$160
a week. On your graph, use black ink to draw her indi�erence curve through this point.(e) Will Wendy choose to work overtime?
Yes.
What is the best choice for Wendy from the red budget line? (c, r) =(200, 33.3).
How many hours a week will she work?46.6.
(f) Suppose that the jobs are equally agreeable in all other respects. Since Wendy and Mac have the same preferences, they will be able to agree about who has the better job. Who has the better job?
Mac.
(Hint: Calculate Wendy’s utility when she makes her best choice. Calculate what her utility would be if she had Mac’s job and chose the best amount of time to work.)9.13 (1) Wally Piper is a plumber. He charges $10 per hour for his work and he can work as many hours as he likes. Wally has no source of income other than his labor. He has 168 hours per week to allocate between labor and leisure. On the graph below, draw Wally’s budget set, showing the various combinations of weekly leisure and income that Wally can a�ord.
0 120 160 200 240 400 800 1200 1600 2000 2400 40 Income 80 Leisure 128 Red budget line
(a) Write down Wally’s budget equation.
I + 10R = 1, 680.
(b) While self-employed, Wally chose to work 40 hours per week. The construction �rm, Glitz and Drywall, had a rush job to complete. They o�ered Wally $20 an hour and said that he could work as many hours as he liked. Wally still chose to work only 40 hours per week. On the graph you drew above, draw in Wally’s new budget line.(c) Wally has convex preferences and no kinks in his indi�erence curves. On the graph, draw indi�erence curves that are consistent with his choice of working hours when he was self-employed and when he worked for Glitz and Drywall.
(d) Glitz and Drywall were in a great hurry to complete their project and wanted Wally to work more than 40 hours. They decided that instead of paying him $20 per hour, they would pay him only $10 an hour for the �rst 40 hours that he worked per week and $20 an hour for every hour of
“overtime” that he worked beyond 40 hours per week. On the graph that you drew above, use red ink to sketch in Wally’s budget line with this pay schedule. Draw the indi�erence curve through the point that Wally chooses with this pay schedule. Will Wally work more than 40 hours or less than 40 hours per week with this pay schedule?
More.
9.14 (1) Felicity loves her job. She is paid $10 an hour and can work as many hours a day as she wishes. She chooses to work only 5 hours a day. She says the job is so interesting that she is happier working at this job than she would be if she made the same income without working at all. A skeptic asks, “If you like the job better than not working at all, why don’t you work more hours and earn more money?” Felicity, who is entirely rational, patiently explains that work may be desirable on average but undesirable on the margin. The skeptic insists that she show him her indi�erence curves and her budget line.
(a) On the axes below, draw a budget line and indi�erence curves that are consistent with Felicity’s behavior and her remarks. Put leisure on the horizontal axis and income on the vertical axis. (Hint: Where does the indi�erence curve through her actual choice hit the vertical line l = 24?)
Income
Leisure 50
240
24
9.15 (2) Dudley’s utility function is U(C, R) = C − (12 − R)2, where R
is the amount of leisure he has per day. He has 16 hours a day to divide between work and leisure. He has an income of $20 a day from nonlabor sources. The price of consumption goods is $1 per unit.
(a) If Dudley can work as many hours a day as he likes but gets zero wages for his labor, how many hours of leisure will he choose?
12.
(b) If Dudley can work as many hours a day as he wishes for a wage rate of $10 an hour, how many hours will he choose to work? (Hint: Write down Dudley’s budget constraint. Solve for his labor supply. Remember that the amount of labor he wishes to supply is 16 minus his demand for
leisure.)
9.
(c) If Dudley’s nonlabor income decreased to $5 a day, how many hours
would he choose to work?
7.
(d) Suppose that Dudley has to pay an income tax of 20 percent on all of his income, and suppose that his before-tax wage remained at $10 an hour and his before-tax nonlabor income was $20 per day; how many