If I have a function f(x,y) which is increasing in x but decreasing in y, could I in general, rewrite that function as g(x/y)?
For context, I have an objective function F(x,y)*H(x,y) which is increasing in x for F and decreasing for H, and increasing in y for H and decreasing for F. If I could rewrite my objective function V(p)*W(1/p), where p=x/y, I would only be doing maximization in one variable. This would greatly simplify my problem.
Many thanks!
No.
If $f(x,y)=g(x/y)$, then $f$ is constant along straight lines $y=kx$, and that need not be the case just because $f$ is increasing in $x$ and decreasing in $y$.