Cobb douglas regression and interaction

Question

Consider a Cobb-Douglas model :

$$ y = \beta_0x_1^{\beta_1}x_2^{\beta_2} $$

I would like to know if we can talk about interaction in this model concerning the variables $x_1$ and $x_2$. When thinking of this with $x_1$ the labour, $x_2$ the capital and $y$ the the total production we like to think of $\beta_1$ and $\beta_2$ as output elasticities and at first I thought this gives us nothing in terms of interaction. One way to see interactions is to talk about the fact that when we have two variables $x_1$ and $x_2$ that influences another variable $y$, we could see the interaction between $x_1$ and $x_2$ as follows : a change in $x_1$ depends of $x_2$ and so the impact of a change in $x_1$ on $y$ will not be the same given different levels of $x_2$. We can formalize this by saying that the partial derivatives of $y$ with respect to $x_1$ depends on $x_2$. From this, we see that in a Cobb-Douglas model, there are interactions between explanatory variables.

Here is my thoughts, I would like to know if it is correct ?

Thank you a lot !

Answer 1

This is more a matter of terminology than of substance. Yes, in the form you show, there is a product between $x_1$ and $x_2$. One might consider such a product to represent an "interaction," as the association between one predictor and outcome depends on the value of the other predictor.

But if you do a log transformation, you have:

$$\log y = \log\beta_0 + \beta_1 \log x_1+ \beta_2 \log x_2, $$

which is an equation for a linear (in the coefficients) regression without an interaction term. The association of neither log-transformed predictor with outcome depends on the value of the other. So I'm not sure that it's helpful to think of an "interaction" here.