Notes on Labor Demand Under A Cobb-Douglas Technology
R.L. Oaxaca
University of Arizona
1
Cobb-Douglas Production Function
Q = Aegt Lα K β
or
ln(Q) = ln(A) + gt + αln(L) + βln(K)
for g > 0, 0 < α, β < 1, and 0 < α + β < 1. These restrictions describe a CD technology with neutral
technological change and decreasing returns to scale. The marginal product expressions are given by
M PL =
∂Q
∂L
= αAegt Lα−1 K β
=α
Q
L
= α (APL ) ,
and
M PK =
∂Q
∂K
= βAegt Lα K β−1
=β
Q
K
β (APK ) .
2
Conditional Input Demand
Conditional input demand functions are obtained from cost minimization. Let w be the marginal cost of
an additional unit of labor (e.g. the hourly wage), and let r be the marginal cost/user cost (rental rate on
capital) of an additional unit of the non-labor input. The economic problem is formally stated as
min C = wL(Q) + rK(Q) s.t. Q = Aegt Lα K β .
(L,K)
1
The cost minimizing solutions for L and K are more easily obtained from
M PL
w
=
M PK
r
⇒
w
α K
=
β
L
r
⇒
K
=
L
β
w
.
α
r
This latter result describes the optimal capital(non-labor input) - labor ratio for a given wage rental rate
ratio. It follows that
K=
β
w
L.
α
r
This expression can be substituted for K in the production function to solve for L. One can then solve for
K. In terms of logs, the conditional input demand functions can be shown to be described by
−1
β
w
ln(L) =
ln(A) + βln
+ βln
+ gt − ln(Q)
α+β
α
r
−1
β
w
ln(K) =
ln(A) − αln
− αln
+ gt − ln(Q) .
α+β
α
r
The Cobb-Douglas production function is a special case of the Constant Elasticity of Substitution (CES)
production technology. To see this, note that the optimal capital/labor ratio may be expressed in logs as
ln
K
L
β
w
= ln
+ ln
.
α
r
In general a two-input CES technology implies
ln
K
L
w
= b + σln
.
r
β
In the special case of a CD technology, b = ln
and σ = 1 for a unitary elasticity of substitution.
α
3
Input Demand Under Long-Run Profit Maximization
Input demand functions under long-run profit maximization with decreasing returns to scale can be derived.
Let p be the price of each unit of input sold. In the simplest case one assumes price equals marginal revenue,
2
p = M R. The formal maximization problem is usually stated in terms of choosing the optimal output to
maximize profits:
max π = pQ − wL(Q) − rK(Q).
Q
The maximization problem can be stated in terms of optimal inputs:
max π = pQ(L, K) − wL − rK.
L,K
The input demand functions under LR profit maximization can be solved from the marginal revenue product
(MRP) conditions:
M RPL = w
M RPK = r,
where M RPL = M RxM PL , and M RPK = M RxM PK . Upon substituting p for M R and the expressions
derived earlier for M PL and M PK , one obtains two equations in two unknowns (L and K) . Upon solving
these equations and taking logs, one obtains the following input demand functions under LR profit max:
−1
β
w
r
ln(L) =
−ln(αA) − βln
+ (1 − β) ln
+ βln
− gt
1−α−β
α
p
p
−1
β
w
r
ln(K) =
−ln(βA) + αln
+ αln
+ (1 − α) ln
− gt .
1−α−β
α
p
p
Note: profit max ⇒ cost min because
M PL
w
M RxM PL
=
= .
M RxM PK
M PK
r
However, cost min ;profit max because
4
M PL
w
=
; M RxM PL = w and M RxM PK = r.
M PK
r
Input Demand Under Short-Run Profit Maximization
Labor demand under short-run profit max involves a single profit maximizing condition:
M RPL = w
3
⇒
M RxM PL = w
⇒
pxM PL = w
⇒
M PL =
w
.
p
Upon substitution for M PL and solving for L, we can obtain the demand for labor under SR profit max.
Expressed in logs, the labor demand function is given by
1
1−α
ln(L) =
w
ln(αA) − ln
+ βln(K) + gt .
p
In this case K is being held constant. It is possible to imagine cases in which the labor inputs cannot be
changed in the short-run so that only the non-labor inputs are variable. In this scenario the profit maximizing
condition and the labor demand function are given by
M PK =
r
p
and
ln(K) =
5
1
1−β
r
ln(βA) − ln
+ αln(L) + gt .
p
Input Demands For a Public Agency
Input demands for a public agency can be arrived at in an analogous fashion to consumer utility maximization. One can think of a public agency as tasked with maximizing the amount of goods or services provided
to the public subject to a budget constraint. In the present context, the decision problem is
max Q = Aegt Lα K β s.t. C = wL + rK,
L,K
where C is the public agency’s budget. This condition implies efficiency, so we may use the cost minimization
condition
M PL
w
=
M PK
r
4
which implies
w
β
L.
K=
α
r
Upon substitution for K in the cost equation C = wL + rK, we can solve for L. We can also solve for K in
an analogous fashion, so that the resulting input demand functions are given by
α C
α+β w
β C
K=
.
α+β r
L=
In terms of logs, the input demand functions are expressed by
C
α
+ ln
ln (L) = ln
α+β
w
β
C
ln (K) = ln
+ ln
.
α+β
r
Note that there is no effect of non-neutral technological change on the input demands of a public agency.
6
Output Supply
The cost function corresponding to a Cobb-Douglas technology can be obtained by substituting the conditional input demand functions L(w, r, Q, t) and K(w, r, Q, t) into the cost equation:
C(w, r, Q, t) = wL(w, r, Q, t) + rK(w, r, Q, t)





1
−gt
α 
β




= (ψ) w α + β  r α + β  Q α + β  e α + β  ,
where


−1
α
−β
β α+β

 β α + β
ψ = A α + β  
+
.
α
α

5
The marginal cost function for the Cobb-Douglas technology is obtained as
∂C(w, r, Q, t)
∂Q





α 
β
1−α−β
−gt
ψ



w α + β  
=
r α + β  Q α + β  e α + β  .
α+β
M C(w, r, Q, t) =
In a competitive or price taking setting, M R = p = M C(w, r, Q, t). Therefore, the inverse supply function
would simply be
p=
ψ
α+β




β
1−α−β
−gt
α 



w α + β  
r α + β  Q α + β  e α + β  .

The output supply function is obtained by solving for Q as a function of p:
Qs =
α+β
ψ





α+β
−β
gt
α+β
−α
1 − (α + β)  1 − (α + β)   1 − (α + β)   1 − (α + β)   1 − (α + β) 
 r
 e
 p
.
w
In logs and after rearranging terms, the output supply function may be expressed as
s
ln(Q ) =
7
1
1−α−β
w
r
[ln(A) + αln(α) + βln(β)] − αln
− βln
+ gt .
p
p
Output Demand Function
To complete the market, we require an output demand function. A simple example that will suffice for
illustrative purposes is given by
p
+ θ2 t
ln(Qd ) = θ0 + θ1 ln
y
where y is some measure of consumer income and θ1 < 0.
8
Market Demand and Supply
We can solve for equilibrium market quantity and price by equating demand and supply:
ln(Qd ) = ln(Qs )
6
⇒
1−α−β
[−θ0 + θ1 ln(y) − θ2 t]
θ1 (1 − α − β) − (α + β)
1
1−α−β
[ln(A) + αln(α) + βln(β)]
+
θ1 (1 − α − β) − (α + β)
1−α−β
ln(p) =
−αln(wt ) − βln(rt ) + gt}
1−α−β
1
=
−θ0 +
[ln(A) + αln(α) + βln(β)]
θ1 (1 − α − β) − (α + β)
1−α−β
g
αln(w) + βln(r)
+
− θ2 t + θ1 ln(y) −
1−α−β
1−α−β
θ1 (α + β)
ln(A) + αln(α) + βln(β)
−
ln(Q) =
θ1 (1 − α − β) − (α + β)
(α + β)
θ1 (α + β)
αln(w) + βln(r)
−
θ1 (1 − α − β) − (α + β)
(α + β)
θ1 (α + β)
g
θ2
+
−
t
θ1 (1 − α − β) − (α + β)
α+β
θ1
θ1 (α + β)
+
ln(y)
θ1 (1 − α − β) − (α + β)
θ1
ln(A) + αln(α) + βln(β) −
=
θ1 (1 − α − β) − (α + β)
θ2
−αln(w) − βln(r) + g −
(α + β) t + (α + β) ln(y)
θ1
7
θ0
θ1
θ0
(α + β)
θ1