# characterizing slope of an implicit function

• December 2nd 2010, 07:07 AM
bkguler
characterizing slope of an implicit function
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 ( $u^{\prime }>0$, and $u^{\prime \prime}<0$). Moreover, $-\frac{u^{\prime \prime}(x)}{u^{\prime
}(x)}$
is decreasing in x. I need to characterize the slope
of an implicit function, y(x) , which solves the following equation:

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

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

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

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