# Cobb douglas regression and interaction

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 !

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.

• Thank you for your answer ! It is true that a log change annihilates this interaction... I would like to be able to explain why but I am not aha. Beyond that, I'm interested in interactions because I'm trying to represent them (and ideally quantify them, which is going to be complicated). Commented Jun 8, 2023 at 6:39
• @coboy the simplest way to represent the apparent "interaction" in the Cobb-Douglas formula is to work in the log-transformed scale where they disappear. In this case, the fundamental information is the same in either situation. The question is how well the Cobb-Douglas formula will actually represent the data.
– EdM
Commented Jun 8, 2023 at 13:31
• Yes I had the same thoughts on the invariance of "fundamental information" by a log transformation, the model is pretty relevant for my purpose. Commented Jun 8, 2023 at 14:43