  We discuss the instabilities appearing in the cosmological model with a quasi
de Sitter phase following from a fourth-order gravity theory. Both the
classical equation as well as the quantization in form of a Wheeler - De Witt
equation are conformally related to the analogous model with Einstein's theory
of gravity with a minimally coupled scalar field. Results are: 1. In the
non-tachyonic case, classical fourth-order gravity is not more unstable than
Einstein's theory itself. 2. The well-known classically valid conformal
relation is also (at least for some typical cases) valid on the level of the
corresponding Wheeler - De Witt equations, which turns out to be a non-trivial
statement. (to appear in: Proc. Sem. Relativistic Astrophysics Potsdam 1994, Ed
.: J. M\"ucket)
