Download Profit Maximization : Cobb–Douglas Example

Profit Maximization : Cobb–Douglas
Example
f (x) = Πni=1xai i
(1)
where
a ≡ a1 + a2 + · · · + an < 1
ai
∂f
= f (x)
∂xi xi
(2)
first–order condition for cost minimization
fi
wi
=
fj
wj
from equation (2),
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1
ai x1
wi
=
w1 a1 xi
(3)
or
xi =
ai w1
x1
wi a1
i = 1, 2, · · · , n
(4)
to produce output of y,
Πni=1xai i = y
(5)
substituting from (4),
w1 a1 n ai n
−ai a
y=
Πi=1ai Πi=1wi
x1
a1
(6)
w1 a1 A a
y=
x1
a1
W
(7)
or
where
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2
A ≡ Πni=1aai i
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;
W ≡ Πni=1wiai
3
conditional input demands
re–arranging (6) yields
xc1(w, y)
a1 W 1/a 1/a
y
=
w1 A
(8)
and substituting back into (4)
xci(w, y)
ai W 1/a 1/a
=
y
wi A
(9)
cost function : cost of conditional input demands
W 1/a 1/a
C(w, y) = a
y
A
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(10)
4
profit maximization
maximize
py − C(w, y)
(11)
where C(w, y) is defined by equation (10)
first–order condition
W 1/a 1/a−1
y
=0
p−
A
(12)
second–order condition
1 − a W 1/a 1/a−2
y
>0
a
A
(13)
holds if a < 1
equation (12) can be re–written
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5
y=p
a/(1−a)
W −1/(1−a)
A
(14)
which is the supply curve for the firm
profit function
π(p, w) is the maximized value of py − C(w, y),
or
π(p, w) = py(p, w) − C(w, y(p, w)
(15)
Substituting from (14), π(p, w) equals
pp
a/(1−a)
W −1/(1−a)
W 1/a a/(1−a) W −1/(1−a)1/a
−a
p
A
A
A
(16)
which simplifies to
π(p, w) = (1 − a)p
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1/(1−a)
W −1/(1−a)
A
(17)
6
unconditional input demands
one way : substitute from supply function (14)
into conditional factor demands (9)
xui(p, w) = xci(w, y(p, w))
here that implies
xui(p, w)
ai W 1/a
[y(p, w)]1/a
=
wi A
(18)
or (substituting from (14)),
xui(p, w)
ai W 1/a a/(1−a) W −1/(1−a)1/a
=
p
wi A
A
(19)
which can be simplified to
xui(p, w)
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ai 1/(1−a) W −1/(1−a)
= p
wi
A
(20)
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alternate method : use Hotelling’s Lemma (2) ;
the unconditional factor demands are the negatives
of the partial derivatives of the profit function (17)
with respect to the input prices wi
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