Chapter 16
Labor Markets
Time Allocation
• Income is fixed when allocating income among goods
that provide utility.
• Time is fixed when allocating time between leisure and
working to earn income for consumption of goods, and
time must be used as it passes.
• Time can be spent
– Working (earning income for consumption),
– Consuming goods,
– Maintaining ones self, (not earning income for consumption)
– Sleeping.
• Understanding this time allocation decision helps
understand labor supply.
Two-Good Model
•
•
•
•
•
•
•
•
•
Individuals spend time in leisure or working at a real wage rate of w,
which allows consumption of goods, and they receive utility from
both leisure and consumption.
Utility = U(c,h), where c is consumption and h is hours of leisure
(two composite goods).
Constraint 1 is l + h = 24, where l is hours worked.
Constraint 2 is c = wl, or consumption equals income.
Combining Constraints 1 and 2 gives c = w(24 – h),
or c + wh = 24w; where 24w equals maximum possible real
consumption of goods per day.
24w is a person’s full income.
A person’s full income (24w) can be spent by working for
consumption of goods (c = wl) or not working (wh) and enjoying
leisure.
The opportunity cost of leisure is w per hour, or earnings forgone by
not working an hour. The real wage rate (w) is the price of leisure.
The price of $1 of consumption is $1.
Utility Maximization Model
Max U = U(c,h)
St. 24w – c – wh = 0
Lagrangian expression
  U (c, h)  λ(24w  c  wh)
FOCs
 U
1.

 λ  0 3.   24w  c  wh  0
c c

 U
To maximize utility, the individual
2.
 wλ  0

should work up to the point where
h h
the marginal rate of substitution of
Dividing 2 by 1 gives
leisure (h) for consumption (c) is
U h MU h

 MRS h for c  w
U c MU c
equal to the real wage rate (w),
provided that the SOC are satisfied;
MRSh for c is diminishing.
Income and Substitution Effects
The real wage rate increases from w0 to w1 (the price of leisure increases). The
constraint rotates to the right because the individual can consume more for the same
amount of labor, ie., w1l > w0l. The individual is made better off by an increase in w
because the individual is the supplier of labor.
Consumption=c
This example assumes the
individual will not reduce
consumption when income
increases in going from point B
to point C (although consumption
could increase because
C consumption is a normal good).
Consumption=c
c1
C
c = w0(24)
B
A
c0
0
h1 h0
Substitution Effect
Income Effect
c1
c
=
w
(24
–
h)
0
U1
c = w1(24 – h) ); slope=-w1
c0
c = w0(24 – h); slope=-w0
U0
0
Leisure=h
h = 24,
so c = 0.
B
A
h0 h1
U1
c = w1(24 – h)
U0
Leisure= h
Substitution Effect
Income Effect
The substitution effect is always negative; w
h in going from point A to point B. The income
effect will be positive because leisure is a normal good; w
h in going from point B to point C.
The income and substitution effects work in opposite directions because h is a normal good. In the
left graph, the substitution effect from an increase in the real wage rate outweighs the income
effect, so h declines from h0 to h1 (l increases) as w increases. In the right graph, the income effect
outweighs the substitution effect, so h increases from h0 to h1 (l decreases) as w increases. This
example suggests that the supply of labor could be backward bending (negatively sloped) as in the
right graph.
Mathematical Analysis of Labor Supply
Change the constraint to add nonlabor income (n).
c = wl + n
n shifts the budget constraint out in a parallel manner, and if leisure is a
normal good (∂h/∂n > 0), so ∂l/∂n < 0. An increase in n will increase the
demand for leisure and reduce the supply of l. Thus, we have the
individual’s labor supply function as l(w,n); the number of hours of labor
supplied is a function of the real wage rate and nonlabor income. n plays
the role of nominal income in the typical utility maximization problem.
Look at the dual problem, which is
Min E = c – wl Choose values of c and h that minimize additional
St U0 = U(c,h) expenditures (n=c-wl) to achieve a given level of utility.
Solve the dual for minimum E and apply the envelop theorem to get,
∂E/∂w = -lc = hc. This shows that a labor supply function can be calculated
from the expenditure function by partial differentiation, but this is a
compensated labor supply function because utility is held constant: lc(w, U).
The compensated labor supply function is different from the uncompensated
labor supply function: Max U=U(c,h); S.T. 24w-c-wh=0; Solve for c and h;
h(w,n)=-l(w, n) is the uncompensated labor supply function.
Slutsky Equation for Labor Supply
Explore the substitution and income effects of a change in w.
First, expenditures minimized in the Dual [E(w, U)] are like nonlabor income (n)
in the Primal.
At the optimal point
lc(w,U) = l(w,n) = l[w,E(w,U)].
Partially differentiate both sides with respect to w to get,
l c l l E Then realizing that ∂E/∂w = -lc and substituting n for E,

  .
w w E w
l c l
l l
l
l c l

l

 l . Now let

U U0 and rearrange terms to get,
w w E w n
w w
h (leisure) is a normal good
(∂h/∂n > 0), so ∂l/∂n is <0.
l l
l


l
.
U U0
w w
n
(+or-) = (+)
+ (+)(-)
The change in labor supplied as w changes equals the
change in labor supplied from the substitution effect when U is
held constant plus the change in labor supplied from the
income effect resulting from an appropriate change in
nonlabor income).
Thus, labor supply could be backward bending if the income effect outweighs the substitution
effect. If l is larger (eg., the person is working 18 hr/day), the negative income effect may be
greater in absolute value than the positive substitution effect.
Market Supply Curve for Labor
The market supply curve for labor is the horizontal sum of the amounts of
labor supplied by all individuals in the labor market.
A higher real wage rate might cause each person in the market to work more
hours and it might induce more individuals to enter the labor market; thus,
total labor might increase.
Real wage
S
w*
D
0
l*
Quantity of labor
D is demand for labor by firms and S is supply of
labor by individuals.
If w were above w*, involuntary unemployment
would exist and individuals would bid down the
real wage. If w were below w*, firms would want
to hire more labor than was available, so firms
would bid up the real wage.
Mandated Labor Benefits
Such as health insurance and paid leave
Suppose labor supply is lS = a + bw and labor demand is lD = c – dw and that lS = lD.
Solve for w to get,
w* = (c – a)/(b + d).
Now suppose government mandates a labor benefit that costs firms t per unit of
hired labor. Labor now costs firms w + t per unit. Suppose that the monetary value
of the benefits to workers is k per unit of labor supplied. Workers now receive w + k
per unit of labor supplied.
lS = lD now requires a + b(w + k) = c – d(w + t), and the net wage is
w** = (c – a)/(b + d) – (bk + dt)/(b + d) = w* – (bk + dt)/(b + d).
If k = 0 (the benefit is 0) and t is the tax on employment faced by firms, the demand
for labor decreases (shifts left). Supply does not shift, so the equilibrium amount of
labor hired decreases. The employee’s share of the tax is given by the ratio d/(b +
d). As long as k < t, the equilibrium amount of labor hired will decrease because
demand shifts left by more than supply shifts right.
If k = t, the new equilibrium wage falls by t because w** = w* – t and the equilibrium
amount of labor hired does not change because demand and supply shifts offset
each other.
Causes of Wage Variation
•
Human capital. A firm’s demand for labor depends on the worker’s
marginal productivity, so differences in productivity should lead to different
wages.
– Education
– Experience
– On-the-job training
•
Compensating Wage Differentials. Some jobs are more desirable than
others (Assistant Professor at UH), so workers are willing to receive a lower
real wage. Other jobs are less desirable (truck driver in Iraq), so workers
must be compensated for the unpleasantness.
•
Monopsony. The firm is not a price taker for the inputs it buys.
Monopsony in Labor Purchase
Many employers are not perfect competitors in purchasing inputs. In fact, most large
firms are not. They are not price takers for w. The supply curve faced by the firm is the
market labor supply curve. S is the “average cost” of labor equal to w = wl/l. The
Marginal Expense of labor (MEL) is the “marginal cost” of
Labor Supply
labor equal to ∂wl/∂l.
 wl
 MEl = marginal
w
l
cost of labor
Thus, the ME curve for labor is above the supply curve for
labor because the firm must pay all workers more (not just the
last one hired) to get one more worker.
wl
 S = average
Using the product rule
wl
w
l
cost of labor
MEl 
 wl
because w = f(l).
l
l
l
Supply of labor is
positively sloped.
Similar to
P
PQ
on the output side.
 MR  P  Q
Q
Q
Demand for the good is
negatively sloped.
If the firm were a perfectly competitive buyer of labor, w/l = 0 and MEl = w. This
means the elasticity of supply for labor equals positive infinity (perfectly elastic supply of
labor; horizontal labor supply curve).
We have already stated that profit maximization requires MEl = MRPl. Therefore,
the monopsonist purchases l1 labor at w1 price. This equilibrium is determined at
point A where MEl = MRPl.
w
w*
w1
MEl

A


C
Sl average cost of labor
curve = market supply of
labor
B
MRPl = P(MPl)
l1
l*
l
The perfectly competitive purchaser would purchase l* at w* (determined by
point C where w = MRPl.
The relationship between l and w is determined by the MRPl curve. However,
there is actually no such thing as the monopsonist’s demand curve for labor, just
as there was no monopolist’s supply curve for output. MRPl is not a
monopsonist’s demand curve for labor, but it does help determine one optimum
point of w1 l1. This point represents the firm’s demand for labor given P and
MPPl.
Bilateral Monopoly
A seller of an input may act as a monopolist if it is the only
potential supplier of the input. This may occur in the cases of
labor unions or where a particular input is available from a single
seller. The seller of the input acts just as our earlier monopolist
acted.
If buyers of the input are perfect competitors, the monopolist may
choose to operate anywhere on its demand curve as shown below.
Monopolist selling labor (eg., labor union)
w
MCl of seller
w2
w**
0
l2
MRl faced by seller Dl faced by seller = MRPl of buyers
l**
l
When a monopolistic seller faces a monopsonistic
buyer, a Bilateral Monopoly exists.
The Monopolist (seller) wants to operate at
MRS=MCS at point E1 (P1, Q1), while the
Monopsonist (buyer) wants to operate at
MRPB=MEB at point E2 (P2, Q2). They must
negotiate a price and quantity somewhere
between P1 and P2 and between Q1 and Q2.
The seller’s MC curve
is not the seller’s
supply curve for labor,
but it is the supply
curve faced by the
buyer (no S curve for
monopoly).
B
B
S
S
= MRPB
S
The buyer’s MRP
curve is not the
buyer’s demand curve
for labor, but it is the
demand curve faced
by the seller (no D
curve for monopsony).