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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!

Fića
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1 Answers1

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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 $y$.

Hans Lundmark
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  • Thanks very much for that. I will go away and have a think about which conditions would need to apply for this to be true. – Fića May 04 '19 at 04:25