# CES production function with non constant returns to scale

In the equation $$\begin{equation} Y=\left[ aK^{\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\frac{\sigma -1% }{\sigma }}\right] ^{\frac{\mu \sigma }{\sigma -1}} \label{ces_pf} \end{equation}$$ if $$\mu \ne 1$$ we have non constant returns to scale (RTS).

This is the only way I can see to get non constant RTS into a CES production function. One cannot have factors which are raised to exponents which sum to above or below one since that violates the form of the CES function.

Is there some other way to get non unitary RTS into the CES function?

## 1 Answer

Another possibility is $$\begin{equation} Y=\left[ aK^{\mu_K\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\mu_L \frac{\sigma -1% }{\sigma }}\right] ^{\frac{ \sigma }{\sigma -1}}, \end{equation}$$ with $$\mu_K+\mu_L \neq 1$$.

Edit: The function is not homothetic in $$(K,L)$$, and as mentioned by Giskard, the elasticity of substitution is not longer constant, unless $$\mu_K=\mu_L$$, in which case we obtain the specification proposed by cel.

• This does not seem to be a CES function? Or is elasticity of substitution still constant and I am missing something? – Giskard Sep 14 '20 at 20:01
• Interesting. Have you seen this function used in some context, some paper? – cel Sep 14 '20 at 20:05
• Well, there is a quite long controversy about the correct specification of CES production functions, see for instance Sato and the literature cited therein. Sato, R., 1975, "The Most General Class of CES Functions," Econometrica, 43, 999-1003. – Bertrand Sep 14 '20 at 20:16
• the qualification $\mu_k+\mu_L\neq1$ is redundant. – Konstantin Dec 21 '20 at 22:03