Suppose we have functions $f(p_1,p_2,w_1)$ and $g(p_1,p_2,w_2)$. Under what condition do we have a function $h(p_1,p_2,w)=f(p_1,p_2,w_1)+g(p_1,p_2,w_2)$ where $w=w_1+w_2$?
Or if we generalise this to higher dimension, $f(p_1,...,p_l,w_1)$ and $g(p_1,...,p_l,w_2)$. Under what condition do we have a function $h(p_1,...,p_l,w)=f+g$ where $w=w_1+w_2$?
This question is actually from economics, and it says to have such a result, we must have something related to homogeneous of degree 0 of $f,g$. I don't understand where it comes from.
Feel free to assume all functions here are perfectly smooth.