The welfare costs of inflation are studied in a fully specified international monetary general equilibrium model. We are particularly interested in studying the equilibrium level of inflation when each country chooses its policy while taking into account the monetary policy of the foreign monetary authority. We compute Nash equilibrium monetary policies and evaluate the welfare costs of the equilibrium inflation rates that result from those policies. The equilibrium inflation rate in this environment depends on the amount of currency that is held abroad. We also study the following issues: (1) the role of dominant international currency in the determination of the welfare costs, (2) the welfare gain of forming currency union and (3) the role of other distorting taxes in determining Nash inflation rates and the welfare costs in the various cases. The results show that when other taxes are present and when the trade sector is of modest size, there is little support for the notion that there is natural tendency to collect seignorage.