The literature on the time inconsistency of optimal monetary policy puts forward the idea that a central bank may strategically exploit the first mover advantage against the privat sector, manipulating expectations so as to achieve a higher level of employment and output. We argue that this view is largely ill-founded. We show that the dynamic version of the basic model used in the literature is an optimal control model yealding a time consistent and stable solution to the central banker's problem, where prices are stable and the output reaches the full employment level in steady state. Then we extend it to include a strategic privat sector, which transforms the initial setup into a different game. We prove that such a game has a strongly time consistent open-loop Nash equilibrium, as well as a time consistent Stackelberg open-loop equilibrium with the bank leading,where, however, the bank cannot gain as compared to the simultaneous game. With the privat sector leading, inflation may arise in equilibrium if output is below the full employment level.