Can competition among privately issued fiat currencies such as Bitcoin or Ethereum work? Only sometimes. To show this, we build a model of competition among privately issued fiat currencies. We modify the current workhorse of monetary economics, the Lagos-Wright environment, by including entrepreneurs who can issue their own fiat currencies in order to maximize their utility. Otherwise, the model is standard. We show that there exists an equilibrium in which price stability is consistent with competing private monies, but also that there exists a continuum of equilibrium trajectories with the property that the value of private currencies monotonically converges to zero. These latter equilibria disappear, however, when we introduce productive capital. We also investigate the properties of hybrid monetary arrangements with private and government monies, of automata issuing money, and the role of network effects.