•We provide a model of an endowment economy with two competing, but intrinsically worthless currencies (Dollar, Bitcoin) serving as medium of exchange.•We show a fundamental pricing equation, which in its simplest form implies that Bitcoin prices form a martingale.•“Mutual impatience” rules out Bitcoin speculation. Price volatility does not invalidate the medium-of-exchange function.•The Bitcoin block rewards are not a tax on Bitcoin holders: they are financed with a Dollar tax.•We discuss monetary policy implications, Bitcoin production via the proof-of-work competition, taxation of Bitcoin production, welfare implications and entry of new cryptocurrencies. We characterize the range of equilibria and provides specific examples. We provide a model of an endowment economy with two competing, but intrinsically worthless currencies (Dollar, Bitcoin). Dollars are supplied by a central bank to achieve its inflation target, while the Bitcoin supply grows deterministically. Our fundamental pricing equation implies in its simplest form that Bitcoin prices form a martingale. “Mutual impatience” implies absence of speculation. Price volatility therefore does not invalidate the medium-of-exchange function. Bitcoin block rewards are not a tax on Bitcoin holders: they are financed with a Dollar tax. We discuss monetary policy implications, Bitcoin production, taxation, welfare and entry, and characterize the range of equilibria.