I have been working on the following problem for a long time, and got stuck. I really appreciate if someone can help.

Let u be a twice differentiable, strictly increasing and concave function

defined over positive real numbers ($\displaystyle u^{\prime }>0$, and $\displaystyle u^{\prime \prime}<0$). Moreover,$\displaystyle -\frac{u^{\prime \prime}(x)}{u^{\prime

}(x)}$ is decreasing in x. I need to characterize the slope

of an implicit function, y(x) , which solves the following equation:

$\displaystyle u(x+y) -u(x+b) =\alpha \int_{y}[u(x+x^{\prime}) -u( x+y)] dG(x^{\prime })$

where b is positive constant, and G is a continuous cumulative distribution function.

I need to show that $\displaystyle y^{\prime }(x) <1$.

So far I can show $\displaystyle y^{\prime }(x) >0$: