  We show that within the framework of the minimal $SU(5)$ supergravity model,
radiatively-induced electroweak symmetry breaking and presently available
experimental lower bounds on nucleon decay, impose severe constraints on the
available parameter space of the model which correspond to fine-tuning of the
model parameters of over two orders of magnitude. Furthermore, a
straightforward calculation of the cosmic relic density of neutralinos ($\chi$)
gives $\Omega_\chi h^2\gg1$ for most of the allowed parameter space in this
model, although small regions may still be cosmologically acceptable. We
finally discuss how the {\it no-scale flipped $SU(5)$ supergravity model}
avoids naturally the above troubles and thus constitutes a good candidate for
the low-energy effective supergravity model.
