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Proving that a CobbDouglas function is concave
if the sum of exponents is no bigger than 1
Ted Bergstrom, Econ 210A, UCSB
If you tried this problem in your homework, you learned from painful experi
ence that the Hessian conditions for concavity of the CobbDouglas function
F
(
x
1
,...x
n
) =
n
Y
i
=1
x
α
i
i
from
<
n
+
to
<
are cumbersome to work with, when
n
≥
3. Maybe you were
thinking “There must be an easier way.” Well, there is. Indeed there is more
than one other way to skin this cat, but the way that I will show you here
is instructive and in the process you will pick up a couple of useful tools.
It turns out that the function
F
is a concave function if
α
i
≥
0 for all
i
and
∑
n
i
=1
α
i
≤
1.
I propose the following road to a proof.
We ﬁrst note the following:
Lemma 1.
The function deﬁned by
F
(
x
1
,...x
n
) =
n
Y
i
=1
x
α
i
i
is homogeneous of degree
∑
n
i
=1
α
i
.
You should be able to supply the proof of this lemma.
We next note that
F
is quasiconcave. To show this, we make use of the
fact that any monotone increasing transformation of a concave function is
quasiconcave.
Lemma 2.
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 Fall '09
 Bergstrom

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