Differential Models of Production: Change in the Marginal Cost and the
Multi-Product Firm
Lecture XXVI
I.
Change in the Marginal Cost
A.
Shares of Marginal Cost
1.
Since both total and marginal cost depend on output levels and input
prices, we start by considering marginal share of each input price
(
)
1,
i
i
i
p q
z
i
n
C
z
θ
∂
∂
=
=
∂
∂
L
.
2.
Based on this definition, we define a Firsch price index for inputs as
(
)
(
)
(
)
(
)
1
ln
ln
n
i
i
i
d
P
d
p
θ
=
′
=
∑
B.
Completing the single output model
( )
(
)
ln
1
ln
ln
C
d
d
z
z
γ
ψ
⎛
⎞
⎛
∂
⎞
⎛
⎞
d
P
′
=
−
+
⎜
⎟
⎜
⎟
⎜
⎟
∂
⎝
⎠
⎝
⎠
⎝
⎠
(
)
( )
( )
2
2
ln
1
1
1
ln
ln
C
z
z
ψ
γ
∂
=
+
∂
∂
II.
Multiproduct Firm
A.
Expanding the production function to a multiproduct technology
(
)
,
0
h q z
=
(
)
(
)
1
,
1
ln
m
r
r
h q z
z
=
∂
= −
∂
∑
B.
Expanding the preceding proof
(
)
(
1
,
,
n
i
i
i
)
L q
p q
h q
ρ
ρ
=
=
−
∑
z
1.
Computing the first-order conditions
(
)
(
)
(
)
(
)
,
,
0
1,
ln
ln
i
i
i
i
L q
h q z
p q
i
n
q
q
ρ
ρ
∂
∂
=
−
=
=
∂
∂
K
which implies that the optimum values of each input can be formulated
as a function of the output level and input prices:
(
)
,
1,
i
i
q
q
z p
i
=
=
K
n
p
and the optimal cost function:
(
)
(
)
1
,
,
n
i
i
i
C
p q
z p
C z
=
=
=
∑
1

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