Some Simple Models of Labor Market Equilibrium
1. Monopsony and Minimum Wages.
Let’s consider an industry in which a single firm employs all the labor.
w(L)
is the labor supply curve facing the firm (and industry)
MFC
(marginal factor cost) is the derivative of total labor costs (
w
(
L
)∙
L
) wrt
L
.
VMP
is the marginal revenue from another unit of
L
.
The monoponist maximizes profits at point a, where
VMP=MFC
.
It pays a wage of
w
0
and employs
L
0
units of labor. This is less than the socially efficient level,
L
*.
Imposing a binding minimum wage at
w
min
changes the
MFC
curve to the bold line.
The
profit-maximizing firm again sets
VMP=MFC
, which now occurs at point
b
.
The firm
now employs
L
1
units of labor, which is more than before.
So both wages and
employment rise as a result of the minimum wage law.
Note that, at least in the case
where
L
is the firm’s only input, output =
F
(
L
) will rise as well.
pF′(L) = VMP
w(L)
(labor supply)
MFC = w(L) + L∙w′(L)
L
w
min
w
0
a
L
0
L
1
L*
b

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*Sign up*2. Competitive Industry Labor Demand
The simplest way to move from the firm and household level to the market is to imagine a
fixed
number
of firms in an industry (endogenous entry decisions of firms can complicate matters), but
large enough in number so that each firm takes factor and product prices as given.
In this case, it
is well known that we can derive an industry level factor demand curve by horizontally summing
(i.e. summing over quantities demanded) the demand curves of the individual firms.
This yields a
market level demand curve of the form
L
D
= L
D
(
w
) when we are thinking of a single factor,
L
in
isolation.
Theory says it must be downward sloping.
More generally, the industry is
characterized by a system of factor demand equations of the form:
x
1
=
D
1
(
p, w
1
,
w
2
, … w
n
)
x
2
=
D
2
(
p, w
1
,
w
2
, … w
n
)
(1)
.
=
…….
.
…….
x
n
=
D
n
(
p, w
1
,
w
2
, … w
n
)
where the
x
’s are industry level input demands, the
w
’s are input prices, and
p
is the price of the
industry’s output.
The derivatives of (1) wrt
p
and
w
must satisfy the properties derived for the
individual firm’s labor demand (e.g. negative definiteness and symmetry).
In sum, the industry-
level factor demand relationship
maps prices into quantities
and has the same properties as
firm-level demand.
Definition
(Hamermesh 1993):
factors
i
and
j
are
p-complements
iff
∂x
i
/∂w
j
< 0 in (1).
Otherwise they are
p-substitutes
.
In other words,
x
i
and
x
j
are
p
-complements if an increase in the
price of factor
j
leads firms to use less of factor
i
.
Note two things about this:
First, we could just
as well define
p
-complements as occurring when “an increase in the price of one input reduces
the demand for the other” because (recall our notes on single-firm factor demand) factor demand
responses are predicted to be symmetric. Second, since own factor demand effects must be
negative,
p
-complementarity means the quantities of
x
i
and
x
j
both fall
when the price of

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