Fig. 8

The surface \(g(\alpha ,b)=b^{1{-}\alpha }{\cdot }\big (\alpha {\cdot }\ln (b){-}1\big ){+}1\) (blue) is plotted as a function of \(\alpha\) and b and intersected by the plane at 0 (blue) (left panel). The curve \(\hat{\alpha }(b)\) as the implicit function \(g(\alpha ,b){=}0\) is plotted as the intersection of the surface \(g(\alpha ,b)\) with the plane, demonstrating that \(\hat{\alpha }(b){>}0.5\) for all b and that \(\hat{\alpha }\) is a strictly decreasing function of b for \(0{<}b{<}1\) (right panel)