# 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?

• 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

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.
• the qualification $\mu_k+\mu_L\neq1$ is redundant. Dec 21, 2020 at 22:03