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Static Labor Demand Theory
1.
Simplest Case:
Single Competitive Firm, One Factor of Production (Labor).
Choose
L
to maximize:
wL
L
pF

=
Π
)
(
FOC:
0
)
(
=

′
=
Π
w
L
F
p
dL
d
; or, VMP=wage
SOC:
0
)
(
2
2
<
′
′
=
Π
L
F
p
dL
d
, or, diminishing returns to labor
Totally differentiating the FOC to get comparative statics:
0
)
(
=

′
+
′
′
dw
dp
F
dL
L
F
p
Rearranging:
(holding dw=0):
0
=
′
′
′

=
neg
neg
F
p
F
dp
dL
(higher output prices lead to increased labor demand).
(holding dp=0):
0
1
1
<
=
′
′
=
neg
F
p
dw
dL
Therefore, labor demand curves are unambiguously downwardsloping.
2.
Single Competitive Firm, Multifactor Labor Demand
Now, choose
x
1
, …
x
n
to maximize:
∑

=
Π
i
i
i
n
x
w
x
x
pF
)
,...,
(
1
According to Varian’s graduate micro text, the solution to this problem can be represented by the
profit
function
,
)
,...,
;
(
1
n
w
w
p
Π
which gives the
maximized
level of profits as a function of all the exogenous parameters.
Varian also
shows that:
y
p
=
∂
Π
∂
,
where
)
,...,
(
1
n
x
x
F
y
=
, i.e. output supplied
i
i
x
w

=
∂
Π
∂
, i.e. the demand for input
i
.
These results are sometimes known as
Hotelling’s lemma
.
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View Full DocumentFinally, note that (purely because it represents the
maximized
value of a function) that the profit function
must be convex in its arguments (
p
and the vector of
w
’s).
Applying Hotelling’s lemma to the Hessian (i.e. the matrix of second derivatives) of the profit function
yields:






=
n
n
n
n
n
n
nn
n
np
n
p
pn
p
pp
dw
dx
dw
dx
dp
dx
dw
dx
dw
dx
dp
dx
dw
dy
dw
dy
dp
dy
...
...
...
...
...
...
...
...
...
...
...
...
...
...
1
1
1
1
1
1
1
1
11
1
1
π
Convexity of Π implies positive definiteness of the above matrices, which in turn implies:
1.
dy
/
dp
>0.
An upwardsloping supply curve.
2.
dx
i
/
dw
i
<0, for all
i.
Downwardsloping “own” demand curves for every factor
i
.
In contrast to the static
labor supply case, there is
no
ambiguity here.
3.
“Crossdemand” effects,
dx
i
/
dw
j
, can in general be either positive or negative.
These elasticities
sometimes matter a lot for policy purposes; a great deal of empirical work in labor economics has been
devoted to estimating them in various contexts.
4. The most surprising result derives not from convexity but from the fact that the matrix of supply and
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 Fall '09
 Kuhn

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