1
Lecture 14: Cost Functions
1
Contingent Demand for Inputs
z
Contingent demand functions for all of the
firms inputs can be derived from the cost
function
2
–
Shephard’s lemma
z
the contingent demand function for any input is given by
the partial derivative of the totalcost function with
respect to that input’s price
Contingent Demand for Inputs
Shephard’s lemma:
q
w
v
L
q
w
v
C
q
w
v
k
c
∂
=
∂
=
)
,
,
,
(
)
,
,
(
)
(
λ
3
Intuition:
If price of labor (w) increases by 1, by how
much will cost increase if nothing else changes?
→
by l (the number of workers)
v
v
∂
∂
,
,
w
q
w
v
L
w
q
w
v
C
q
w
v
l
c
∂
∂
=
∂
∂
=
)
,
,
,
(
)
,
,
(
)
,
,
(
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Contingent Demand for Inputs
z
Example: Production of automobiles
s
= A(r
l
)
0.5
4
The Lagrangian expression for cost minimization of
producing
s
is
L
=
vr
+
w
l
+
λ
[
s

A(r
l
)
0.5
]
The firstorder conditions for a minimum are
∂
L
/
∂
r
=
v

λ
0.5 A
r
0.5
l
0.5
= 0
∂
L
/
∂
l
=
w

λ
0.5 A
r
0.5
l
0.5
= 0
∂
L
/
∂λ
=
s

A(r
l
)
0.5
= 0
Contingent Demand for Inputs
z
Dividing the first equation by the second gives
us
r
w
w
5
l
v
=
Hence
s = A (w/v)
0.5
l
l= s /
A (w/v)
0.5
=
s
A
1
(v/w)
0.5
r= s /
A (v/w)
0.5
=
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 Spring '08
 Akbulut
 Economics, Shephard, contingent demand

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