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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.
Schilling et al. (Mon,) studied this question.