Suppose u\left( x\right) =k\left( \rho +\tau x\right) ^{-\frac{1}{\tau }+1}
for some constant real numbers k and \rho, and \tau >0. Suppose \phi is defined by the following implicit function:
u\left( x+\phi \right) -u\left( x+b\right) =\alpha \int_{\phi }^{\infty }\left[ u\left( x+y\right) -u\left( x+\phi \right) \right] dF\left( y\right)
for a given continuous cumulative distribution function F and a constant real number b. I need to characterize the slope of \phi , which can be written as (using implicit function theorem)
\frac{d\phi }{dx}=-1+\frac{\alpha \int u^{\prime }\left( x+y\right)dF\left( y\right) +u^{\prime }\left( x+b\right) }{u^{\prime }\left( x+\phi\right) \left( 1+\alpha \left( 1-F\left( \phi \right) \right) \right) } =-1+\frac{\alpha \int u^{\prime }\left( x+y\right) dF\left( y\right)+u^{\prime }\left( x+b\right) }{\alpha \int u^{\prime }\left( x+y\right)\frac{\rho +\tau \left( x+y\right) }{\rho +\tau \left( x+\phi \right) }dF\left( y\right) +u^{\prime }\left( x+b\right) \frac{\rho +\tau\left(x+b\right) }{\rho +\tau\left( x+\phi \right) }} =-1+\frac{\alpha \int u\left( x+y\right) \frac{\rho +\tau \left( x+\phi\right) }{\rho +\tau \left( x+y\right) }dF\left( y\right) +u\left(x+b\right) \frac{\rho +\tau \left( x+\phi \right) }{\rho +\tau \left(x+b\right) }}{\alpha \int u\left( x+y\right) dF\left( y\right) +u\left(x+b\right) }
Second and third lines are derived using the first equation and the functional form of u. Is it possible to show \frac{d\phi }{dx}\in \left( 0,1\right) $ if $\tau >0?