This paper studies estimates of the form $| y_1 -widehat y_1 | le arphi (hu )|y_0-widehat y_0|$, where $y_1,widehat y_1$ are the numerical solutions of a Runge-Kutta method applied to a stiff differential equation satisfying a one-sided Lipschitz condition (with constant $u$). An explicit formula for the optimal function $arphi (x)$ is given, and it is shown to be superexponential, i.e., $arphi (x_1) arphi (x_2) le arphi (x_1+x_2)$ if $x_1$ and $x_2$ have the same sign. As a consequence, results on asymptotic stability are obtained. Furthermore, upper bounds for $arphi (x)$ are presented that can be easily computed from the coefficients of the method.