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?