concavity

# concavity - Proving that a Cobb-Douglas function is concave...

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Proving that a Cobb-Douglas 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 Cobb-Douglas 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 quasi-concave. To show this, we make use of the fact that any monotone increasing transformation of a concave function is quasi-concave. Lemma 2.

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concavity - Proving that a Cobb-Douglas function is concave...

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