The Friedman rule, a widely studied prescription for monetary policy, is optimal in Townsend's turnpike model of money; it is not so in the overlapping generations version of his stochastic relocation model of money. We investigate these monetary models in the light of this disparity. To that end, we create a modified version of the turnpike model that generates the same stationary monetary equilibria as does the two-period overlapping generations model with random relocation. We exploit this equivalence to explain the aforementioned disparity. We also discuss the importance of whether or not the economy has an initial date.