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The Phillips curve has generally been estimated in a linear framework. This paper investigates the possibility that the Phillips curve is indeed a curve, and shows that a convex short-run Phillips curve may be a more accurate representation of reality than the traditionally used linear specification. The paper also discusses the policy implications of convexity in the Phillips curve. These include the need for policy to be forward looking and to act pre-emptively. Convexity provides a strong rationale for stabilization policy, and it reinforces the need for policy makers to proceed cautiously. It also implies that deep recessions may have only a marginally greater disinflationary impact than shallower ones, unless they induce large credibility bonuses.