# The CES Production Function – Addendum

**Posted:**March 13, 2010

**Filed under:**Uncategorized 1 Comment

In the last post, I presented the *CES *production function and mentionned that its elasticity of substitution (which is constant), is:

I already have received a question about how to proove this so I thought it would be a good idea to share my answer with everyone. First of all, for a many-input production function,* *the elasticity of substitution must be measured between two inputs and, as such, it was imprecise of me to say that the elasticity of substitution is constant and I should’ve rather said that *all elasticities of substitutions are constant and equal. It is interesting to note that there are, in fact:*

distinct elasticities of substitution. We note the elasticity of substitution between the *i-th *and the *j-th *production factor:

where is the production function, and . First, we calculate the first degree partial derivatives, for all :

Hence, the ratio of the first degree partial derivatives, for and :

From this yields the result, which is now obvious:

QED.

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