# Does multicolinearity of variables imply complementary inputs?

I've been thinking about how to answer this question How to econometrically identify perfect complements in production? and may think that multicollinearity has something to do with identifying such a process.

If I have a regression such that:

$$y=\beta_0+\beta_1x_1+\beta_2x_2+\mu$$

where $x_1$ and $x_2$ are correlated with each other.

does that imply that some function of the nature $\min\{x_1,x_2\}$ is present in the determination of $y$?

However, (perfect) multicollinearity does not necessarily imply non-substitutability. For instance, the Cobb-Douglas production technology of the form $y=x_1^{0.5}x_2^{0.5}$ is also consistent with a firm choosing inputs in fixed proportions. So the answer to your question is no (from a theoretical perspective).
• But in a case of fundamentally different $x_1$ and $x_2$ where they are not a linear combination of one another is this contradiction still true