Inflation inertia may be quite tenacious because of the simultaneous interaction between policy actions and inflationary expectations under imperfect credibility. This result is particularly relevant for understanding some of the failed efforts to stabilize inflation in South America. This article deals with the issue of inertia in the framework of imperfect information about the type of the policymaker and extends the existing models to an infinite horizon. Because policymakers do not have perfect control of inflation, a "frivolous stabilizer" may deviate from the policies of a "serious stabilizer" without necessarily being unmasked immediately. When the difference in the ability of "strong" and "weak" policymakers to control inflation is large, unexpected inflation may be persistently negative for quite a while, thus causing reduced economic activity and giving the indication that credibility is low. If the policymaker persists with the stabilization, this pattern gradually disappears as his reputation rises. But before this final stage the serious policymaker has to compromise his inflation objective in view of adverse expectations about his type and pay the cost of imperfect credibility.