In this paper, I consider the policy implications of two alternative structural interpretations of observed inflation persistence, which correspond to two alternative specifications of the new Keynesian Phillips curve (NKPC). The first specification allows for some degree of intrinsic persistence by way of a lagged inflation term in the NKPC. The second is a purely forward-looking model, in which expectations farther into the future matter and coefficients are time-varying. In this specification, most of the observed inflation persistence is attributed to fluctuations in the underlying inflation trend, which are a consequence of monetary policy rather than a structural feature of the economy. With a simple quantitative exercise, I illustrate the consequences of implementing monetary policy, assuming a degree of intrinsic persistence that differs from the true one. The results suggest that the costs of implementing a stabilization policy when the policymaker overestimates the degree of intrinsic persistence are potentially higher than the costs of ignoring actual structural persistence; the result is more clear-cut when the policymaker minimizes a welfare-based loss function.