<p>Another possibility is
<span class="math-container">\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}</span>
with <span class="math-container">$\mu_K+\mu_L \neq 1$</span>.</p>
<p>Edit: The function is not homothetic in <span class="math-container">$(K,L)$</span>, and as mentioned by Giskard, the elasticity of substitution is not longer constant, unless <span class="math-container">$\mu_K=\mu_L$</span>, in which case we obtain the specification proposed by cel.</p>

CC BY-SA 4.0
2020-09-14T20:01:44.603Z
This does not seem to be a CES function? Or is elasticity of substitution still constant and I am missing something?
Giskard
CC BY-SA 4.0
2020-09-14T20:05:04.093Z
Interesting. Have you seen this function used in some context, some paper?
cel
CC BY-SA 4.0
2020-09-14T20:16:44.303Z
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
CC BY-SA 4.0
2020-12-21T22:03:37.993Z
the qualification $\mu_k+\mu_L\neq1$ is redundant.
Konstantin
CC BY-SA 4.0
2020-09-14T20:00:06.850Z
2020-09-14T20:36:37.100Z
2020-09-14T20:36:37.100Z
Bertrand
<p>In the equation
<span class="math-container">\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}</span>
if <span class="math-container">$\mu \ne 1$</span> we have non constant returns to scale (RTS).</p>
<p>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.</p>
<p>Is there some other way to get non unitary RTS into the CES function?</p>

CC BY-SA 4.0
2020-09-15T18:11:27.613Z
Indeed, in the CES function it is the outer exponent that determines the degree of homogeneity, see https://economics.stackexchange.com/a/399/61
Alecos Papadopoulos
<production-function>
CES production function with non constant returns to scale
cel
# CES production function with non constant returns to scale

(Asked by: username_0 on 2020-09-14)

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?

**Tags:** production-function
> **Comments:**
> **username_1**: Indeed, in the CES function it is the outer exponent that determines the degree of homogeneity, see https://economics.stackexchange.com/a/399/61  

---

## 1 answers

### Answer by username_2 on 2020-09-14. Score: 1

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.

> **Comments:**
> **username_3**: This does not seem to be a CES function? Or is elasticity of substitution still constant and I am missing something?  
> **username_0**: Interesting. Have you seen this function used in some context, some paper?  
> **username_2**: 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.  
> **username_4**: the qualification $\mu_k+\mu_L\neq1$ is redundant.  

---

