<p>Caution: Heavy citing</p>

<p>"In his doctoral dissertation <em>Revankar</em> (1967) expounded his generalized
production functions that permit variability to <em>returns-to-scale</em> as well as <em>elasticity of substitution</em>. In contrast with the production functions that (rather unrealistically) assume the same returns to scale at all levels of output, <strong>Zellner and Revankar (1969)</strong> found a procedure to generalize <strong>any given (neoclassical)</strong> production function with specified <em>constant</em> or <em>variable elasticities of substitution</em> such that the resulting production function <strong>retains its specification as to the elasticities of substitution</strong> all along but <strong>permits returns- to-scale to vary</strong> with the scale of output. Their Generalized Production Function (GPF) is given as
\begin{equation}
\ Pe^{\theta P} = c^h f^h
\end{equation}
where f is the basic function (e.g. <strong>Cobb-Douglas, CES</strong>, etc) as the
object of generalization, c is the constant of integration and <em>θ, h</em>
relate to parameters
associated with the returns-to-scale function. In particular, if the Cobb-Douglas production function is generalized, we have
\begin{equation}
\ Pe ^{\theta P}= AK^{\rho\alpha} L^ {\rho (1- \alpha )} 
\end{equation}. This function is <em>interesting</em>
from the viewpoint of <strong>estimation</strong> also. It has to be estimated so as to <em>maximize</em> the <strong>likelihood function</strong> since the Least Squares and Max Likelihood estimators of parameters do not coincide. The return to scale function is given by 
\begin{equation}
\rho (P) = \rho /(1 + \theta P) 
\end{equation} . Depending on θ the sign of θ , the returns-to-scale function monotonically increases or decreases with
increase in P. However, as we know, the returns to scale first increases with output, remains more or less constant in a domain and then begins falling. This fact is not captured by the Zellner-Revankar function since it gives us a linear returns-to-scale function."</p>

<p>from <strong>A Brief History of Production Functions</strong> by <em>SK Mishra</em> 2007</p>

<p>So, I would guess, the  \begin{equation}  \alpha  \end{equation} parameter 
in your log-linearized function would have something to do with returns-to-scale.</p>

<p>Regarding estimation, I'm way too much of a newb. But this might be something at least similar to what you are looking for: <a href="https://link.springer.com/chapter/10.1007/3-540-28556-3_2" rel="nofollow noreferrer">https://link.springer.com/chapter/10.1007/3-540-28556-3_2</a> 
<strong>Variable Elasticity of Substitution and Economic
Growth: Theory and Evidence</strong> by 
<em>Giannis Karagiannis,
Theodore Palivos,
and Chris Papageorgiou</em>,
2005</p>

CC BY-SA 3.0
2018-02-10T23:43:10.247Z
2018-02-11T05:35:53.660Z
2018-02-11T05:35:53.660Z
Rokis1990
<p>I took out a book from my university library called Econometric Modelling with Time Series: Specification Estimation and Testing in an attempt to understand the importance of MLE in Econometrics.</p>

<p>There is a small note mentioning a production function I've never seen before called the <strong>Zellener-Revankar Production Function (ZRPF)</strong>. Its a non-linear production function relating to output, capital and labor defined as:</p>

<p>$$\ln y_t+\alpha y_t=\beta_0+\beta_1 \ln k_t+ \beta_2 \ln l_t + u_t$$</p>

<p>The left hand side makes sense, however the right hand side seems odd. What kind of production it is trying to represent and more fundamentally, how this is mathematically considered a function give that there are two dependents on the left hand side?</p>

<p>Ive looked at the relevant paper work namely:</p>

<blockquote>
  <ul>
  <li><p><a href="http://www.accessecon.com/pubs/EB/2007/Volume3/EB-06C60008A.pdf" rel="noreferrer">Estimation of Zellner-Revankar production function revisited<br>
  SK Mishra,
  Department of Economics,
  North-Eastern Hill University<br>
  Shillong (India) </a></p></li>
  <li><p><a href="http://ageconsearch.umn.edu/record/198658/files/agecon-cal-832.pdf" rel="noreferrer">ALTERNATIVE FUNCTIONAL FORMS FOR PRODUCTION,
  COST AND RETURNS TO SCALE FUNCTIONS
  by
  Arnold Zenner and Hang Ryu</a></p></li>
  </ul>
</blockquote>

<p>Any help would be appreciated in terms of understanding what type of process this is coming to represent would be helpful.</p>

CC BY-SA 3.0
2018-02-05T07:31:03.117Z
"there are two dependents on the left hand side" I only see $y_t$, what is the other dependent variable?
Giskard
CC BY-SA 3.0
2018-02-05T20:31:50.857Z
To me, it's the RHS that makes sense: it's just the usual log-linearized Cobb-Douglas form. However, the LHS is a bit baffling. Are you sure there's no stated interpretation of the LHS? E.g. what does the parameter $\alpha$ capture?
Herr K.
CC BY-SA 3.0
2018-02-05T22:41:43.187Z
@HerrK thats what I'm trying to figure out.
EconJohn
CC BY-SA 3.0
2018-02-05T22:43:09.720Z
@denesp I probably used the wrong term, but from what I know about functions, this would not be defined as one.
EconJohn
CC BY-SA 3.0
2018-02-05T22:58:47.307Z
@EconJohn Why not? For any positive $\alpha$ the function $f(y) = \ln y + \alpha y$ is a strictly increasing and hence invertible function. So if you say $f(y) = g(k,l)$ then you are implicitly defining $y = f^{-1} \circ g(k,l)$.
Giskard
CC BY-SA 3.0
2018-02-09T03:54:34.970Z
@denesp thank you for the explanation on how this works. Now how does one estimate this?
EconJohn
CC BY-SA 3.0
2018-02-09T05:46:33.027Z
@EconJohn I have no clue. If that is your actual question I recommend making a significant edit to your question. It has no answers yet, so large edits are still fine.
Giskard
<econometrics><mathematical-economics><production-function>
Understanding the Zellener-Revankar Production Function
EconJohn
# Understanding the Zellener-Revankar Production Function

(Asked by: username_0 on 2018-02-04)

I took out a book from my university library called Econometric Modelling with Time Series: Specification Estimation and Testing in an attempt to understand the importance of MLE in Econometrics.

There is a small note mentioning a production function I've never seen before called the **Zellener-Revankar Production Function (ZRPF)**. Its a non-linear production function relating to output, capital and labor defined as:

$$\ln y\_t+\alpha y\_t=\beta\_0+\beta\_1 \ln k\_t+ \beta\_2 \ln l\_t + u\_t$$

The left hand side makes sense, however the right hand side seems odd. What kind of production it is trying to represent and more fundamentally, how this is mathematically considered a function give that there are two dependents on the left hand side?

Ive looked at the relevant paper work namely:

> * [Estimation of Zellner-Revankar production function revisited  
>   SK Mishra,
>   Department of Economics,
>   North-Eastern Hill University  
>   Shillong (India)](http://www.accessecon.com/pubs/EB/2007/Volume3/EB-06C60008A.pdf)
> * [ALTERNATIVE FUNCTIONAL FORMS FOR PRODUCTION,
>   COST AND RETURNS TO SCALE FUNCTIONS
>   by
>   Arnold Zenner and Hang Ryu](http://ageconsearch.umn.edu/record/198658/files/agecon-cal-832.pdf)

Any help would be appreciated in terms of understanding what type of process this is coming to represent would be helpful.

**Tags:** econometrics, mathematical-economics, production-function

> **Comments:**
> **username_1**: "there are two dependents on the left hand side" I only see $y_t$, what is the other dependent variable?  
> **username_2**: To me, it's the RHS that makes sense: it's just the usual log-linearized Cobb-Douglas form. However, the LHS is a bit baffling. Are you sure there's no stated interpretation of the LHS? E.g. what does the parameter $\alpha$ capture?  
> **username_0**: @HerrK thats what I'm trying to figure out.  
> **username_0**: @denesp I probably used the wrong term, but from what I know about functions, this would not be defined as one.  
> **username_1**: @username_0 Why not? For any positive $\alpha$ the function $f(y) = \ln y + \alpha y$ is a strictly increasing and hence invertible function. So if you say $f(y) = g(k,l)$ then you are implicitly defining $y = f^{-1} \circ g(k,l)$.  
> **username_0**: @denesp thank you for the explanation on how this works. Now how does one estimate this?  
> **username_1**: @username_0 I have no clue. If that is your actual question I recommend making a significant edit to your question. It has no answers yet, so large edits are still fine.  

---

## 1 answers

### Answer by username_3 on 2018-02-10. Score: 3

Caution: Heavy citing

"In his doctoral dissertation *Revankar* (1967) expounded his generalized
production functions that permit variability to *returns-to-scale* as well as *elasticity of substitution*. In contrast with the production functions that (rather unrealistically) assume the same returns to scale at all levels of output, **Zellner and Revankar (1969)** found a procedure to generalize **any given (neoclassical)** production function with specified *constant* or *variable elasticities of substitution* such that the resulting production function **retains its specification as to the elasticities of substitution** all along but **permits returns- to-scale to vary** with the scale of output. Their Generalized Production Function (GPF) is given as
\begin{equation}
\ Pe^{\theta P} = c^h f^h
\end{equation}
where f is the basic function (e.g. **Cobb-Douglas, CES**, etc) as the
object of generalization, c is the constant of integration and *θ, h*
relate to parameters
associated with the returns-to-scale function. In particular, if the Cobb-Douglas production function is generalized, we have
\begin{equation}
\ Pe ^{\theta P}= AK^{\rho\alpha} L^ {\rho (1- \alpha )}
\end{equation}. This function is *interesting*
from the viewpoint of **estimation** also. It has to be estimated so as to *maximize* the **likelihood function** since the Least Squares and Max Likelihood estimators of parameters do not coincide. The return to scale function is given by
\begin{equation}
\rho (P) = \rho /(1 + \theta P)
\end{equation} . Depending on θ the sign of θ , the returns-to-scale function monotonically increases or decreases with
increase in P. However, as we know, the returns to scale first increases with output, remains more or less constant in a domain and then begins falling. This fact is not captured by the Zellner-Revankar function since it gives us a linear returns-to-scale function."

from **A Brief History of Production Functions** by *SK Mishra* 2007

So, I would guess, the \begin{equation} \alpha \end{equation} parameter
in your log-linearized function would have something to do with returns-to-scale.

Regarding estimation, I'm way too much of a newb. But this might be something at least similar to what you are looking for: <https://link.springer.com/chapter/10.1007/3-540-28556-3_2>
**Variable Elasticity of Substitution and Economic
Growth: Theory and Evidence** by
*Giannis Karagiannis,
Theodore Palivos,
and Chris Papageorgiou*,
2005


---

