Jan Hagemejer
Advanced Microeconomics
Example firm problem: Cobb-Douglas technology
Based on example 5.C.1 from MWG
1
Cost minimization problem with a Cobb-Douglas function
Derive the cost function and conditional factor demands for the Cobb-Douglas utility function of the
form:
q = f (z1 , z2 ) = z1α z2β
The cost minimization problem is:
Min
w1 z1 + w2 z2 subject to q = f (z1 , z2 ),
z ,z
1
2
where q is an arbitrary output level. The Lagrange function is:
L = w1 z1 + w2 z2 − λ(f (z1 , z2 ) − q),
The first order conditions;
∂L
= w1 − λαz1α−1 z2β = 0
∂z1
(1)
∂L
= w2 − λβz1α z2β−1 = 0
∂z2
(2)
∂L
= z1α z2β − q = 0
∂λ
(3)
Dividing (1) by (2) we get:
w1
α z2
=
w2
β z1
Solve for z2 :
w1 β
z1 = z2
w2 α
Substitute to (3):
z1α
z1α+β
w1 β
z1
w2 α
!β
w1 β
w2 α
!β
1
=q
=q
!
z1 (w1 , w2 , q) =
w2 α
w1 β
!
z2 (w1 , w2 , q) =
w1 β
w2 α
β
α+β
1
q α+β
α
α+β
1
q α+β
The above are condtional (on the w1 , w2 , q) factor demands for z1 and z2 .
How to get the cost function? Algebra is a little nasty:
C(w1 , w2 , q) = w1 z1 (w1 , w2 , q) + w2 z2 (w1 , w2 , q)
= w1
w2 α
w1 β
!
β
α+β
q

=q
1
α+β
w1

=q
1
α+β

1
= q α+β w1


1
= q α+β w1

w1
w2 α
w1 β
!
w2 α
w1 β
!
w2β w2α
w1β w2α
α α
β β αα
w2α+β
w1β w2α
α+β
β
α
α
β β αα

=q
1
α+β
1
α+β
1
1
w1 w2 α
β α β α
w1 w2 β α
w1 β
w2 α
+ w2
β
α+β
=q
w1 w2 (α +
=q

!
+ w2
w1 β
w2 α
+ w2
w1α w1β
w1β w2α
β β
β β αα
α+β
+ w2
w1α+β
w1β w2α
1
! α+β
=

α
α+β

=

α β
β
β β αα
1
! α+β



1
! α+β


1
α+β

w α/(α+β) w β/(α+β) (α
1
2
1
α
α ββ
1
+ β)
α
α ββ
!
!



1
α+β
=
1
α+β
!
−β/(α+β) −α(α+β)
β)w1
w2
=

1
1
+ w1 w2 β
β α β α
w1 w2 β α

1
α+β
α
α+β
+ w2

1
α+β
1
q α+β =
!
β
α+β
1
α+β
α
α+β
w1 β
w2 α
1
! α+β
!
!
=


=
Or to get the same expression as in MWG:

=q
1
α+β

αα+β
w α/(α+β) w β/(α+β) 
1
2
αα β β

C(w1 , w2 , q) = q
1
α+β
α

β
!
β
α+β
!
α
+
β
or:
2
1
α+β
!
β α+β
+
αα β β
−α
α+β
!
1
α+β


=

 w α/(α+β) w β/(α+β)
1
2
=
1
α/(α+β)
C(w1 , w2 , q) = q α+β θw1
β
α+β
where θ = αβ
+
Some things to note:
α
β
−α
α+β
β/(α+β)
w2
,
and is just a parameter.
1
• In the cost function, only the q α+β is a function of q. The rest is a scalar if prices are fixed.
• α + β is the measure of returns to scale. If α + β = 1, then the function becomes (should remind
you the expenditure function for a Cobb-Douglas utility function):
C(w1 , w2 , q) = qθw1α w21−α ,
• with α + β = 1, M C = AC = θw1α w21−α . In this case, θ = α−α β −β = α−α (1 − α)−(1−α) .
• we can also rewrite the function as:
1
C(w1 , w2 , q) = q α+β θφ(w1 , w2 ),
α/(α+β)
where φ(w1 , w2 ) = w1
2
β/(α+β)
w2
Profit maximization using the derived cost function
We have can now state the profit maximization problem in the following way:
Max π(p, w1 , w2 , q) = pq − C(w1 , w2 , q)
q
1
= pq − q α+β θφ(w1 , w2 )
FOC:
∂π
1
=p−(
)q 1/(α+β)−1 θφ(w1 , w2 ) = 0
∂q
α+β
Solving for q from the above will give us the profit maximizing output (the second term after the minus
is the marginal cost). However, we have to check SOC:
∂π
1
1
= −(
− 1)(
)q 1/(α+β)−2 θφ(w1 , w2 ) ≤ 0
∂q
α+β
α+β
whenever 0 < α + β ≤ 1 so we need to have either decreasing or constant returns to scale. However,
with CRS the q dissapears from the FOC and we cannot determine the profit maximizing q (any q such
that P = M C is in fact profit maximizing).
3