In this paper, we consider various types of models where fiat money is used to facilitate trades, and investigate whether models with fiat money have the same equilibrium allocations. In particular, we investigate whether or not such models have a continuum of stationary equilibria. We first present a dynamic centralized auction market model with money, which has not been investigated in the literature. We compare its outcome with that of Walrasian models with cash-in-advance constraints, and show that the outcomes are very different: the Walrasian markets model has determinate stationary equilibria and the auction markets model has a continuum of stationary equilibria. Moreover, we also build models on decentralized auction markets and on decentralized markets with bargaining, and obtain results similar to those of the dynamic centralized auction markets model.