Interpretation of OLS: Scale dependent variable by independent variable instead of including it in the regression model

Question

How does the interpretation on the coefficient of X in this OLS model:

Y(Profit in €) = b0 + b1 X + b2 X2 (Firm Size in €) + e

change if I rewrite the model as

Y(Profit in €) / X2(Firm Size in €) = b0 + b1 X + e.

In other words, is easy way to understand what using "Firm Size" in the regression model vs. using it to scale Y is doing to the coefficient estimate?

Answer 1

In the first case you are controlling for "Firm Size", therefore you would estimate $b_1$ as the effect on "Profits" of a change in "X" given the same "Firm Size".

In the second case you would estimate $b_1$ as, for example, the effect of "X" on the ROE (= Net Income/Equity, as explained in comments).