# CES production function with non constant returns to scale

In the equation $$$$Y=\left[ aK^{\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\frac{\sigma -1% }{\sigma }}\right] ^{\frac{\mu \sigma }{\sigma -1}} \label{ces_pf}$$$$ if $$\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?

• Indeed, in the CES function it is the outer exponent that determines the degree of homogeneity, see economics.stackexchange.com/a/399/61 Sep 15, 2020 at 18:11

## 1 Answer

Another possibility is $$$$Y=\left[ aK^{\mu_K\frac{\sigma -1}{\sigma }}+\left( 1-a\right) L^{\mu_L \frac{\sigma -1% }{\sigma }}\right] ^{\frac{ \sigma }{\sigma -1}},$$$$ with $$\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? Sep 14, 2020 at 20:01
• Interesting. Have you seen this function used in some context, some paper?
– cel
Sep 14, 2020 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. Sep 14, 2020 at 20:16
• the qualification $\mu_k+\mu_L\neq1$ is redundant. Dec 21, 2020 at 22:03