This paper looks at the implications of heterogeneous beliefs for inflation dynamics. Following a monetary policy shock, inflation peaks after output, is inertial, and can be characterized by a Hybrid Phillips Curve. It presents a novel channel through which systematic monetary policy can affect the degree of inflation persistence. It does so by altering the effective extent of strategic complementarities in pricing, and hence the role of higher-order expectations in the equilibrium. In particular, stronger inflation targeting reduces the impact of uncertainty on the economy and therefore the degree of inertia. It is possible to calibrate at around 25 percent the fraction of relevant information processed every period by the private sector. The imperfect common knowledge framework does not require any exogenous shocks to create heterogeneity. Despite the fact that prices can be adjusted at no cost in every period, there are nominal rigidities, and monetary policy has real effects.