# Understanding the Zellener-Revankar Production Function

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:

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

• "there are two dependents on the left hand side" I only see $y_t$, what is the other dependent variable? – Giskard Feb 5 '18 at 7:31
• 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. Feb 5 '18 at 20:31
• @HerrK thats what I'm trying to figure out. – EconJohn Feb 5 '18 at 22:41
• @denesp I probably used the wrong term, but from what I know about functions, this would not be defined as one. – EconJohn Feb 5 '18 at 22:43
• @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 Feb 5 '18 at 22:58

"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 $$\ Pe^{\theta P} = c^h f^h$$ 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 $$\ Pe ^{\theta P}= AK^{\rho\alpha} L^ {\rho (1- \alpha )}$$. 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 $$\rho (P) = \rho /(1 + \theta P)$$ . 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."
So, I would guess, the $$\alpha$$ parameter in your log-linearized function would have something to do with returns-to-scale.