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We propose a new framework for understanding the effectiveness of central bank announcements when firms have heterogeneous inflation expectations. Expectations are updated through social dynamics and, with heterogeneity, not all firms choose to operate, putting downward pressure on realized inflation. Our model rationalizes why countries stuck at the zero lower bound have had a hard time increasing inflation without being aggressive. The same model also predicts that announcing an abrupt target to disinflate will cause inflation to undershoot the target, whereas announcing gradual targets will not. We present new empirical evidence that corroborates this prediction.