# 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