  The supersymmetric extension of "Starobinsky" $R+\alpha R^2$ models of
inflation is particularly simple in the "new minimal" formalism of
supergravity, where the inflaton has no scalar superpartners. This paper is
devoted to matter couplings in such supergravity models. We show how in the new
minimal formalism matter coupling presents certain features absent in other
formalisms. In particular, for the large class of matter couplings considered
in this paper, matter must possess an R-symmetry, which is gauged by the vector
field which becomes dynamical in the "new minimal" completion of the $R+\alpha
R^2$ theory. Thus, in the dual formulation of the theory, where the gauge
vector is part of a massive vector multiplet, the inflaton is the superpartner
of the massive vector of a nonlinearly realized R-symmetry. The F-term
potential of this theory is of no-scale type, while the inflaton potential is
given by the D-term of the gauged R-symmetry. The absolute minimum of the
potential is always exactly supersymmetric, so in this class of models if
realistic vacua exist, they must be always metastable. We also briefly comment
on possible generalizations of the examples discussed here and we exhibit some
features of higher-curvature supergravity coupled to matter in the "old
minimal" formalism.
