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5.5 THE CHOICE OF LABOR AND LEISURE 191
LEARNING-BY-DOING EXERCISE 5.10
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The Demand for Leisure and the Supply of Labor
Problem Jan’s utility for leisure (L) income she loses when she enjoys an extra hour of leisure
and a composite good (Y ) is U LY. The marginal util- instead of working for that hour. Thus, at an optimum
ity of leisure is MU L Y, and the marginal utility of the
MU L MU Y
composite good is MU Y L. The price of the compos-
ite good is $1. When she enjoys L hours of leisure per w 1
day, Jan works (24 L) hours per day. Her wage rate is The tangency condition tells us that Y/w L, or that
w, so her daily income is w(24 L). Show that, for any Y wL.
positive wage rate, the optimal number of hours of Jan must also satisfy her budget constraint. She re-
leisure that Jan enjoys is always the same. What is the ceives an income equal to the wage rate times the num-
number of hours of leisure she would demand, and how bers of hours she works; she therefore earns an income
many hours of labor will she supply each day? equal to w(24 L). She buys Y units of the composite
good at a price of $1; she therefore spends $Y. So her
Solution With the Cobb–Douglas utility function,
there will be an interior optimum, with positive values of budget constraint is just w(24 L) Y.
Together the tangency condition and the budget
Y and L. Once we find Jan’s optimal choice for leisure line require that w(24 L) wL. In this example, Jan’s
each day (L), we know she will work (24 L) hours. optimal demand for leisure is L 12 hours per day, and
At her optimal choices of Y and L, Jan will need to she will supply 12 hours of labor per day, independent of
satisfy two conditions. First, the tangency condition re- the wage rate. Of course, for many other utility func-
quires that the ratio of the marginal utility of leisure to the tions, her demand for leisure (and thus, her supply of
price of leisure must equal the ratio of the marginal util- labor), will depend on the wage rate.
ity of the composite good to the price of that good. The
price of leisure is the wage rate; that represents how much Similar Problems: 5.29, 5.30, 5.31
APPLICA TION 5.7
The Backward-Bending Supply
for labor. Using data from 2000, an academic study
of Nursing Services estimated the labor supply of nurses in the United
States. 18 The study concluded that the short-run labor
Medical groups and hospitals have long had difficulty supply curve was backward bending. It appears that
attracting enough workers. In response, they have the labor market for nurses may continue to experi-
often increased the pay of medical workers, but this ence the short-run problems it suffered from in 1991.
may not always increase the amount of labor supplied. Since wage increases alone do not always attract
In 1991 the Wall Street Journal described some of more workers, employers have resorted to other
these difficulties in an article titled “Medical Groups strategies. For example, the article in the Wall Street
Use Pay Boosts, Other Means to Find More Workers.” Journal states that the M.D. Anderson Cancer Center
According to the article, the American Hospital at the University of Texas gave employees a $500
Association concluded that “Pay rises may have wors- bonus if they referred new applicants who took
ened the nursing shortage in Massachusetts by en- “hard-to-fill” jobs. The Texas Heart Institute in
abling nurses to work fewer hours.” 17 Houston recruited nurses partly by showcasing
Why might this have happened? As we saw in our prospects for promotion. The University of Pittsburgh
discussion related to Figure 5.26, a higher wage may Medical Center started an “adopt-a-high-school” pro-
induce a consumer to pursue more leisure and thus gram to encourage students to enter the health care
supply less labor. In other words, many nurses may be sector, and reimbursed employees’ tuition fees when
on the backward-bending region of their supply curve they enrolled in programs to increase their skills.
17 Albert R., Karr, “Medical Groups Use Pay Boosts, Other Means to Find More Workers,” The Wall Street
Journal, August 27, 1991, p. A1.
18 Lynn Unruh and Joanne Spetz, “Can Wage Increases End Nursing Shortages? A Reexamination of the Supply
Curve of Registered Nurses.” Academy of Health Meetings Abstracts, 2005: vol. 22, abstract no. 4480.