# The CES Production Function – Addendum

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

$\sigma = \dfrac{1}{1 - \rho}$

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:

$\displaystyle\binom{n}{2} = \dfrac{n!}{2(n - 2)!}$

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

$\sigma_{ij} =\dfrac{\partial \ln (X_{i}/X_{j}) }{ \partial \ln (f_j/f_i)}$

where $f = f(X_1, X_2, ..., X_n)$ is the production function, $f_i = \dfrac{\partial f}{\partial X_i}$ and $f_j = \dfrac{\partial f}{\partial X_j}$. First, we calculate the first degree partial derivatives, for all $1 \leq k \leq n$:

$\dfrac{\partial f}{\partial X_k} =\gamma \alpha_{k}^{\rho} X_{k}^{\rho - 1} \left(\displaystyle\sum_{i=1}^{n}\alpha_{i}^{\rho}X_{i}^{\rho}\right)^{\frac{\gamma}{\rho} -1}$

Hence, the ratio of the first degree partial derivatives, for $i \neq j$ and $1 \leq i,j \leq n$:

$\dfrac{f_j}{f_i} = \dfrac{\alpha_{j}^{\rho} X_{j}^{\rho - 1}}{\alpha_{i}^{\rho} X_{i}^{\rho - 1}} \iff \dfrac{f_j}{f_i} =\left(\dfrac{\alpha_{j}}{\alpha_{i}}\right)^{\rho} \left(\dfrac{X_{i}}{X_{j}}\right)^{1 - \rho} \iff$

$\ln \dfrac{f_j}{f_i} = \rho\ln\dfrac{\alpha_i}{\alpha_j} + (1 - \rho)\ln \dfrac{X_i}{X_j}$

From this yields the result, which is now obvious:

$\sigma_{ij} =\dfrac{\partial \ln (X_{i}/X_{j}) }{ \partial \left(\rho\ln\dfrac{\alpha_i}{\alpha_j} + (1 - \rho)\ln \dfrac{X_i}{X_j}\right)} = \dfrac{1}{1 - \rho}\dfrac{\partial \ln (X_{i}/X_{j}) }{\partial \ln (X_{i}/X_{j})} \iff$

$\sigma_{ij} = \dfrac{1}{1 - \rho}$

QED.

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### One Comment on “The CES Production Function – Addendum”

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