  The focus of the present work is on the Cauchy problem for the quadratic
gravity models introduced in \cite{stelle}-\cite{stelle2}. These are
renormalizable higher order derivative models of gravity, but at cost of
ghostly states propagating in the phase space. A previous work on the subject
is \cite{noakes}. The techniques employed here differ slightly from those in
\cite{noakes}, but the main conclusions agree. Furthermore, the analysis of the
initial value formulation in \cite{noakes} is enlarged and the use of harmonic
coordinates is clarified. In particular, it is shown that the initial
constraints found \cite{noakes} include a redundant one. In other words, this
constraint is satisfied when the equations of motion are taken into account. In
addition, some terms that are not specified in \cite{noakes} are derived
explicitly. This procedure facilitates application of some of the mathematical
theorems given in \cite{ringstrom}. As a consequence of these theorems, the
existence of both $C^\infty$ solutions and maximal globally hyperbolic
developments is proved. The obtained equations may be relevant for the
stability analysis of the solutions under small perturbations of the initial
data.
