Can a monetary system in which privately issued cryptocurrencies circulate as media of exchange work? Is such a system stable? How should governments react to digital currencies? Can these currencies and government-issued money coexist? Are cryptocurrencies consistent with an efficient allocation? These are some of the important questions that the sudden rise of cryptocurrencies has brought to contemporary policy discussions. To answer these questions, we construct a model of competition among privately issued .at currencies. We .nd that a purely private arrangement fails to implement an efficient allocation, even though it candeliver price stability under certain technological conditions. Currency comptition creates problems for monetary policy implementation under conventional methods. However, it is possible to design a policy rule that uniquely implements an efficient allocation by driving private currencies out of the market. We also show that unique implementation of an efficient allocation can be achieved without government intervention if productive capital is introduced.