Any money model should address the most important phenomenon of a monetary economy, which is the phenomenon of the rate of return dominance. Even if the holding returns on financial or nonfinancial assets are higher than the rate of return on fiat money holding, which is typically zero, people still hold and use money. In a period of accelerating inflation, number of dominating assets increases continuously, yet people continue to hold and use money. Wallace's (1980) overlapping generations model cannot address the rate of return dominance phenomenon. His model does not capture the medium of exchange role of fiat money. In this paper, an overlapping types model of fiat money is constructed, in which different types of consumers have different preferences on different types of goods, are endowed with different types of goods, are located at separated regions, and live for only two periods. In this model, people hold and use money despite the dominating assets, even if inflation accelates. Money in this case serves as a pure medium of exchange, whereas in Wallace's model, money serves as a pure store of value, and money disappears if a dominating asset exists. An interesting feature of the overlapping types model presented in this paper is that money does not provide a cheap approximation to an idealized and efficient real allocation. A monetary economy is always superior to a nonmonetary economy, because money helps overcome the incompleteness of the overlapping types friction. In a monetary economy, however, a pareto optimal allocation cannot always be achieved, because money cannot always overcome the overlapping types friction itself. Therefore, with the criterion of optimality of real allocations, the monetary economy is more optimal than a nonmonetary economy but less optimal than a complete Arrow Debreu economy. This feature has important implications on macro modelling. Because of the difficulty in introducing money into a macro model in an essential and endogenous manner as in the overlapping types model of this paper, a macro model typically ignores money and studies real allocations without the money factor. The possible inefficiencies of a monetary economy, relative to a complete real Arrow-Debreu economy, may indicate differences in real allocations between the two models.