  We study the beta-function of the N=2 sigma-model coupled to N=2 induced
supergravity. We compute corrections to first order in the semiclassical limit,
$c \to -\infty$, beyond one-loop in the matter fields. As compared to the
corresponding bosonic, metric sigma-model calculation, we find new types of
contributions arising from the dilaton coupling automatically accounted for,
once the K\"ahler potential is coupled to N=2 supergravity.
