In this paper, I consider a simple model in which agents learn about the inflation target of a central bank over time by observing the policy instrument or inflation outcomes. Measuring credibility as the distance between the perceived target and the actual target, an increase in credibility is beneficial to the central bank because it brings the policy consistent with attaining the inflation target closer to that required to attain the output target. In this model, the crucial assumptions are that (1) the central bank knows what its target is, but lacks the means to credibly communicate this to agents; and (2) observed changes in the policy instrument are not perfectly informative to agents as to the objective of the central bank. Optimal monetary policy therefore entails endogenising the learning process of agents and solving the resultant "optimal control" problem. I show that a linear approximation to the optimal control problem is observationally equivalent to a "conservative central banker" in the sense of Rogoff (1985), and results in most of the gains that are available from pursuing a higher order approximation, and may actually be preferable if agents cannot determine the exact weights with which to update their view of the target. A conservative central banker is especially beneficial if society places a high weight on output deviations from target. I then illustrate the impact of other factors on credibility formation, including choice of monetary policy instrument, being transparent about the central bank's view of the economy, and publishing forecasts.
