  We consider piecewise linear discrete time macroeconomic models, which
possess a continuum of equilibrium states. These systems are obtained by
replacing rational inflation expectations with a boundedly rational, and
genuinely sticky, response of agents to changes in the actual inflation rate in
a standard Dynamic Stochastic General Equilibrium model. Both for a
low-dimensional variant of the model, with one representative agent, and the
multi-agent model, we show that, when exogenous noise is absent from the
system, the continuum of equilibrium states is the global attractor. Further,
when a uniformly bounded noise is present, or the equilibrium states are
destabilized by an imperfect Central Bank policy (or both), we estimate the
size of the domain that attracts all the trajectories. The proofs are based on
introducing a family of Lyapunov functions and, for the multi-agent model,
deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in
the space of discrete time inputs and outputs.
