In models of money with an infinitely-lived representative agent (ILRA models), the optimal monetary policy is almost always the Friedman rule. Overlapping generations (OG) models are different: in this paper, we study how they are different, and why. We investigate the welfare properties of monetary policy in a simple OG model under two different types of money demand specifications and under two alternative assumptions about the generational timing of taxes for money retirement. We find that the Friedman rule is generally not the policy that maximizes steady-state utility. We conclude that the key difference between ILRA and OG monetary models is that in the latter, the standard method for constructing a monetary regime causes transactions involving money to become intergenerational transfers. Overlapping generations are different in this regard; we study how they are different and why.