How do Bitcoin prices evolve? What are the consequences for monetary policy? We answer these questions in a novel, yet simple endowment economy. There are two types of money, both useful for transactions: Bitcoins and Dollars. A central bank keeps the real value of Dollars constant, while Bitcoin production is decentralized via proof-of-work. We obtain a “fundamental condition, which is a version of the exchange-rate indeterminacy result in Kareken-Wallace (1981), and a “speculative” condition. Under some conditions, we show that Bitcoin prices form convergent supermartingales or submartingales and derive implications for monetary policy.