We examine a decentralized monetary economy in which households can use a means of exchange (barter or gold) other than fiat money. The alternative means of exchange may drive out money even if monetary exchange Pareto dominates. Legal restrictions prohibiting other means of exchange may therefore be necessary. With stochastic preferences, households may use barter to supplement monetary purchases when they have an unexpectedly high demand. However, this may drive down the value of money (in all states) so low that households are again better off with fiat money alone. The paper provides both stochastic and nonstochastic examples in which eliminating markets for goods or assets that compete with fiat money improves welfare.