NI_11th Edtion (1) (1)-79

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SUMMARY
In this chapter we examined some models that focus on pric-
ing in the labor market. Because labor demand was already
treated as being derived from the profit-maximization hy-
pothesis in Chapter 11, most of the new material here focused
on labor supply. Our primary findings were as follows.
• A utility-maximizing individual will choose to supply
an amount of labor at which his or her marginal rate of
substitution of leisure for consumption is equal to the
real wage rate.
• An increase in the real wage creates substitution and
income effects that work in opposite directions in
affecting the quantity of labor supplied. This result can
be summarized by a Slutsky-type equation much like
the one already derived in consumer theory.
• A competitive labor market will establish an equilibrium
real wage at which the quantity of labor supplied by
individuals is equal to the quantity demanded by firms.
• Wages may vary among workers for a number of rea-
sons. Workers may have invested in different levels of
skills and therefore have different productivities. Jobs
may differ in their characteristics, thereby creating
compensating wage differentials. And individuals may
experience differing degrees of job-finding success.
Economists have developed models that address all of
these features of the labor market.
• Monopsony power by firms on the demand side of the
labor market will reduce both the quantity of labor
hired and the real wage. As in the monopoly case, there
will also be a welfare loss.
• Labor unions can be treated analytically as monopoly
suppliers of labor. The nature of labor market equilib-
rium in the presence of unions will depend importantly
on the goals the union chooses to pursue.
PROBLEMS
16.1
Suppose there are 8,000 hours in a year (actually there are 8,760) and that an individual has a potential market wage of $5 per
hour.
a. What is the individual’s full income? If he or she chooses to devote 75 percent of this income to leisure, how many hours
will be worked?
b. Suppose a rich uncle dies and leaves the individual an annual income of $4,000 per year. If he or she continues to devote
75 percent of full income to leisure, how many hours will be worked?
c. How would your answer to part (b) change if the market wage were $10 per hour instead of $5 per hour?
d. Graph the individual’s supply of labor curve implied by parts (b) and (c).
16.2
As we saw in this chapter, the elements of labor supply theory can also be derived from an expenditure-minimization approach.
Suppose a person’s utility function for consumption and leisure takes the Cobb–Douglas form U(c, h) ¼ cah1–a. Then the
expenditure-minimization problem is
minimize c% wð24% hÞ s.t. Uðc, hÞ ¼ cah1%a ¼ U :
The inefficiency of the labor contract in this two-stage game is similar to the inefficiency of
some of the repeated Nash equilibria we studied in Chapter 15. This suggests that, with
repeated rounds of contract negotiations, trigger strategies might be developed that form a
subgame-perfect equilibrium and maintain Pareto-superior outcomes. For a simple example,
see Problem 16.10.
QUERY: Suppose the firm’s total revenue function differed depending on whether the economy
was in an expansion or a recession. What kinds of labor contracts might be Pareto optimal?
Chapter 16: Labor Markets 601
a. Use this approach to derive the expenditure function for this problem.
b. Use the envelope theorem to derive the compensated demand functions for consumption and leisure.
c. Derive the compensated labor supply function. Show that @l c /@w > 0.
d. Compare the compensated labor supply function from part (c) to the uncompensated labor supply function in Example 16.2
(with n ¼ 0). Use the Slutsky equation to show why income and substitution effects of a change in the real wage are precisely
offsetting in the uncompensated Cobb–Douglas labor supply function.
16.3
A welfare program for low-income people offers a family a basic grant of $6,000 per year. This grant is reduced by $0.75 for
each $1 of other income the family has.
a. How much in welfare benefits does the family receive if it has no other income? If the head of the family earns $2,000 per
year? How about $4,000 per year?
b. At what level of earnings does the welfare grant become zero?
c. Assume the head of this family can earn $4 per hour and that the family has no other income. What is the annual budget
constraint for this family if it does not participate in the welfare program? That is, how are consumption (c) and hours of
leisure (h) related?
d. What is the budget constraint if the family opts to participate in the welfare program? (Remember, the welfare grant can
only be positive.)
e. Graph your results from parts (c) and (d).
f. Suppose the government changes the rules of the welfare program to permit families to keep 50 percent of what they earn.
How would this change your answers to parts (d) and (e)?
g. Using your results from part (f), can you predict whether the head of this family will work more or less under the new rules
described in part (f )?
16.4
Suppose demand for labor is given by
l ¼ %50wþ 450
and supply is given by
l ¼ 100w,
where l represents the number of people employed and w is the real wage rate per hour.
a. What will be the equilibrium levels for w and l in this market?
b. Suppose the government wishes to increase the equilibrium wage to $4 per hour by offering a subsidy to employers for
each person hired. How much will this subsidy have to be? What will the new equilibrium level of employment be? How
much total subsidy will be paid?
c. Suppose instead that the government declared a minimum wage of $4 per hour. How much labor would be demanded at
this price? How much unemployment would there be?
d. Graph your results.
16.5
Carl the clothier owns a large garment factory on an isolated island. Carl’s factory is the only source of employment for most of
the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by
l ¼ 80w,
where l is the number of workers hired and w is their hourly wage. Assume also that Carl’s labor demand (marginal revenue
product) curve is given by
l ¼ 400% 40MRPl:
a. How many workers will Carl hire to maximize his profits, and what wage will he pay?
b. Assume now that the government implements a minimum wage law covering all garment workers. How many workers will
Carl now hire, and how much unemployment will there be if the minimum wage is set at $4 per hour?
602 Part 7: Pricing in Input Markets
c. Graph your results.
d. How does a minimum wage imposed under monopsony differ in results as compared with a minimum wage imposed
under perfect competition? (Assume the minimum wage is above the market-determined wage.)
16.6
The Ajax Coal Company is the only hirer of labor in its area. It can hire any number of female workers or male workers it
wishes. The supply curve for women is given by
lf ¼ 100wf
and for men by
lm ¼ 9w2
m,
where wf and wm are the hourly wage rates paid to female and male workers, respectively. Assume that Ajax sells its coal in a
perfectly competitive market at $5 per ton and that each worker hired (both men and women) can mine 2 tons per hour. If the
firm wishes to maximize profits, how many female and male workers should be hired, and what will the wage rates be for these
two groups? How much will Ajax earn in profits per hour on its mine machinery? How will that result compare to one in which
Ajax was constrained (say, by market forces) to pay all workers the same wage based on the value of their marginal products?
16.7
Universal Fur is located in Clyde, Baffin Island, and sells high-quality fur bow ties throughout the world at a price of $5 each.
The production function for fur bow ties (q) is given by
q ¼ 240x % 2x2,
where x is the quantity of pelts used each week. Pelts are supplied only by Dan’s Trading Post, which obtains them by hiring
Eskimo trappers at a rate of $10 per day. Dan’s weekly production function for peltsis given by
x ¼
ffiffi
l
p
,
where l represents the number of days of Eskimo time used each week.
a. For a quasi-competitive case in which both Universal Fur and Dan’s Trading Post act as price-takers for pelts, what will be
the equilibrium price (px) and how many pelts will be traded?
b. Suppose Dan acts as a monopolist, while Universal Fur continues to be a price-taker. What equilibrium will emerge in the
pelt market?
c. Suppose Universal Fur acts as a monopsonist but Dan acts as a price-taker. What will the equilibrium be?
d. Graph your results, and discuss the type of equilibrium that is likely to emerge in the bilateral monopoly bargaining
between Universal Fur and Dan.
16.8
Following in the spirit of the labor market game described in Example 16.6, suppose the firm’s total revenue function is given by
R ¼ 10l % l2
and the union’s utility is simply a function of the total wage bill:
Uðw, lÞ ¼ wl:
a. What is the Nash equilibrium wage contract in the two-stage game described in Example 16.6?
b. Show that the alternative wage contract w0 ¼ l 0 ¼ 4 is Pareto superior to the contract identified in part (a).
c. Under what conditions would the contract described in part (b) be sustainable as a subgame-perfect equilibrium?
Chapter 16: Labor Markets 603
Analytical Problems
16.9 Compensating wage differentials for risk
An individual receives utility from daily income (y), given by
Uð yÞ ¼ 100y % 1
2
y2:
The only source of income is earnings. Hence y ¼ wl, where w is the hourly wage and l is hours worked per day. The individual
knows of a job that pays $5 per hour for a certain 8-hour day. What wage must be offered for a construction job where hours of
work are random—with a mean of 8 hours and a standard deviation of 6 hours—to get the individual to accept this more
‘‘risky’’ job? Hint: This problem makes use of the statistical identity
Eðx2Þ ¼ Var x þ Eðx2Þ:
16.10 Family labor supply
A family with two adult members seeks to maximize a utility function of the form
Uðc, h1, h2Þ,
where c is family consumption and h1 and h2 are hours of leisure of each family member. Choices are constrained by
c ¼ w1ð24% h1Þ þ w2ð24% h2Þ þ n,
where w1 and w2 are the wages of each family member and n is nonlabor income.
a. Without attempting a mathematical presentation, use the notions of substitution and income effects to discuss the likely
signs of the cross-substitution effects @h1 /@w2 and @h2 /@w1.
b. Suppose that one family member (say, individual 1) can work in the home, thereby converting leisure hours into consump-
tion according to the function
c1 ¼ f ðh1Þ,
where f 0 > 0 and f 00 < 0. How might this additional option affect the optimal division of work among family members?
16.11 A few results from demand theory
The theory developed in this chapter treats labor supply as the mirror image of the demand for leisure. Hence, the entire body
of demand theory developed in Part 2 of the text becomes relevant to the study of labor supply as well. Here are three examples.
a. Roy’s identity. In the Extensions to Chapter 5 we showed how demand functions can be derived from indirect utility func-
tions by using Roy’s identity. Use a similar approach to show that the labor supply function associated with the utility-
maximization problem described in Equation 16.20 can be derived from the indirect utility function by
lðw, nÞ ¼ @Vðw, nÞ=@w
@Vðw, nÞ=@n
:
Illustrate this result for the Cobb–Douglas case described in Example 16.1.
b. Substitutes and complements. A change in the real wage will affect not only labor supply, but also the demand for specific
items in the preferred consumption bundle. Develop a Slutsky-type equation for the cross-price effect of a change in w on
a particular consumption item and then use it to discuss whether leisure and the item are (net or gross) substitutes or com-
plements. Provide an example of each type of relationship.
c. Labor supply and marginal expense. Use a derivation similar to that used to calculate marginal revenue for a given
demand curve to show that MEl ¼ w(1 þ 1/el, w).
604 Part 7: Pricing in Input Markets
16.12 Intertemporal labor supply
It is relatively easy to extend the single-period model of labor supply presented in Chapter 16 to many periods. Here we look at
a simple example. Suppose that an individual makes his or her labor supply and consumption decisions over two periods.14
Assume that this person begins period 1 with initial wealth W0 and that he or she has 1 unit of time to devote to work or
leisure in each period. Therefore, the two-period budget constraint is given by W0 ¼ w1(1 % h1) % c1 þ w2(1 % h2) % c2, where
the w’s are the real wage rates prevailing in each period. Here we treat w2 as uncertain, so utility in period 2 will also be
uncertain. If we assume utility is additive across the two periods, we have E[U(c1, h1, c2, h2)] ¼ U(c1, h1) þ E[U(c2, h2)].
a. Show that the first-order conditions for utility maximization in period 1 are the same as those shown in Chapter 16; in par-
ticular, show MRS(c1 for h1) ¼ w1. Explain how changes in W0 will affect the actual choices of c1 and h1.
b. Explain why the indirect utility function for the second period can be written as V(w2,W)), whereW) ¼W0þ w1(1% h1)%
c1. (Note that because w2 is a random variable, V is also random.)
c. Use the envelope theorem to show that optimal choice of W) requires that the Lagrange multipliers for the wealth con-
straint in the two periods obey the condition l1 ¼ E(l2) (where l1 is the Lagrange multiplier for the original problem and
l2 is the implied Lagrange multiplier for the period 2 utility-maximization problem). That is, the expected marginal utility
of wealth should be the same in the two periods. Explain this result intuitively.
d. Although the comparative statics of this model will depend on the specific form of the utility function, discuss in general
terms how a governmental policy that added k dollars to all period 2 wages might be expected to affect choices in both
periods.
SUGGESTIONS FOR FURTHER READING
Ashenfelter, O. C., and D. Card. Handbook of Labor Eco-
nomics, 3. Amsterdam: North Holland, 1999.
Contains a variety of high-level essays on many labor market
topics. Survey articles on labor supply and demand in volumes
1 and 2 (1986) are also highly recommended.
Becker, G. ‘‘A Theory of the Allocation of Time.’’ Economic
Journal (September 1965): 493–517.
One of the most influential papers in microeconomics. Becker’s
observations on both labor supply and consumption decisions
were revolutionary.
Binger, B. R., and E. Hoffman. Microeconomics with Calcu-
lus, 2nd ed. Reading, MA: Addison-Wesley, 1998.
Chapter 17 has a thorough discussion of the labor supply model,
including some applications to household labor supply.
Hamermesh, D. S. Labor Demand. Princeton, NJ: Princeton
University Press, 1993.
The author offers a complete coverage of both theoretical and
empirical issues. The book also has nice coverage of dynamic
issues in labor demand theory.
Silberberg, E., and W. Suen. The Structure of Economics: A
Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill,
2001.
Provides a nice discussion of the dual approach to labor supply
theory.
14Here we assume that the individual does not discount utility in the second period and that the real interest rate between the two periods is zero. Discounting in a
multiperiod context is taken up in Chapter 17. The discussion in that chapter also generalizes the approach to studying changes in the Lagrange multiplier over time
shown in part (c).
Chapter 16: Labor Markets 605
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C H A P T E R
S E V E N T E E N Capital and Time
In this chapter we provide an introduction to the theory of capital. In many ways that
theory resembles our previous analysis of input pricing in general—the principles of
profit-maximizing input choice do not change. But capital theory adds an important time
dimension to economic decision making; our goal here is to explorethat extra dimension.
We begin with a broad characterization of the capital accumulation process and the
notion of the rate of return. Then we turn to more specific models of economic behavior
over time.
Capital and the Rate of Return
When we speak of the capital stock of an economy, we mean the sum total of machines,
buildings, and other reproducible resources in existence at some point in time. These
assets represent some part of an economy’s past output that was not consumed but was
instead set aside to be used for production in the future. All societies, from the most
primitive to the most complex, engage in capital accumulation. Hunters in a primitive so-
ciety taking time off from hunting to make arrows, individuals in a modern society using
part of their incomes to buy houses, or governments taxing citizens in order to purchase
dams and post office buildings are all engaging in essentially the same sort of activity:
Some portion of current output is being set aside for use in producing output in
future periods. As we saw in the previous chapter, this is also true for human capital—
individuals invest time and money in improving their skills so that they can earn more
in the future. Present ‘‘sacrifice’’ for future gain is the essential aspect of all capital
accumulation.
Rate of return
The process of capital accumulation is pictured schematically in Figure 17.1. In both pan-
els of the figure, society is initially consuming level c0 and has been doing so for some
time. At time t1 a decision is made to withhold some output (amount s) from current
consumption for one period. Starting in period t2, this withheld consumption is in some
way put to use producing future consumption. An important concept connected with this
process is the rate of return, which is earned on that consumption that is put aside. In
panel (a), for example, all of the withheld consumption is used to produce additional out-
put only in period t2. Consumption is increased by amount x in period t2 and then
returns to the long-run level c0. Society has saved in one year in order to splurge in the
next year. The (one-period) rate of return from this activity is defined as follows.
607
If x > s (if more consumption comes out of this process than went into it), we would
say that the one-period rate of return to capital accumulation is positive. For example, if
withholding 100 units from current consumption permitted society to consume an extra
110 units next year, then the one-period rate of return would be
110
100
! 1 ¼ 0:10
or 10 percent.
In panel (b) of Figure 17.1, society takes a more long-term view in its capital accumula-
tion. Again, an amount s is set aside at time t1. Now, however, this set-aside consumption
is used to increase the consumption level for all periods in the future. If the permanent level
of consumption is increased to c0þ y, we define the perpetual rate of return as follows.
If capital accumulation succeeds in raising c0 permanently, then r1 will be positive.
For example, suppose that society set aside 100 units of output in period t1 to be devoted
to capital accumulation. If this capital would permit output to be increased by 10 units
for every period in the future (starting at time period t2), the perpetual rate of return
would be 10 percent.
In (a), society withdraws some current consumption (s) to gorge itself (with x extra consumption) in the
next period. The one-period rate of return would be measured by x/s ! 1. The society in (b) takes a more
long-term view and uses s to increase its consumption perpetually by y. The perpetual rate of return
would be given by y/s.
Consumption Consumption
Time Time
(a) One-period return (b) Perpetual return
c0
t1 t2 t3 t1 t2 t3
c0
x
s s
y
D E F I N I T I O N Single-period rate of return. The single-period rate of return (r1) on an investment is the extra
consumption provided in period 2 as a fraction of the consumption forgone in period 1. That is,
r1 ¼
x ! s
s
¼ x
s
! 1: (17:1)
D E F I N I T I O N Perpetual rate of return. The perpetual rate of return (r1) is the permanent increment to future
consumption expressed as a fraction of the initial consumption forgone. That is,
r1¼
y
s : (17:2)
FIGURE 17.1
Two Views of Capital
Accumulation
608 Part 7: Pricing in Input Markets
	PART SEVEN: Pricing in Input Markets
	CHAPTER 16 Labor Markets
	Summary��������������
	Problems���������������
	Suggestions for Further Reading��������������������������������������
	CHAPTER 17 Capital and Time
	Capital and the Rate of Return�������������������������������������