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                                                         5.5 THE CHOICE OF LABOR AND LEISURE                    191

                                 LEARNING-BY-DOING EXERCISE 5.10
                          S
                          D
                        E
                                 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.
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