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I ran into this function reading a macroeconomic paper" $ \psi (n)=G(1-n)-\beta \frac{G'(1-n)F(n)}{F'(n)}.$ The paper claims this function is monotonic decreasing and the central result depends on this claim. But I find this is may not be true. The first order of this function seems to change sign when $n$ varies between $0$ and $1$. Both $F$ and $G$ are concave functions.$n$ is in the close interval of $0$ and $1$ and $\beta$ is in the open interval of $0$ and $1$

Can anyone corroborate on this?

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    @oenamen I mean it's not increasing in its domain. sorry about the prevous mistake2012-06-28

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Here are two simple counterexamples to the claim.

(I) Let $F(x) = x$ and $G(x) = -x$. These functions are concave (and convex). But $\psi'(n) = 1+\beta > 0$.

(II) Let $F(x) = G(x) = -x^2$. If $n < \frac{2+\beta}{2(1+\beta)}$, then $\psi'(n) > 0$. For example, if $\beta = 1/2$, then $\psi'(n) > 0$ for $n<5/6$.