Some prominent economists have begun to analyze formally the market for a privately issued outside money that they associate with Bitcoin. Rodney Garratt and Neil Wallace (2018) (ungated version here) model the relative values of (exchange rates between) “Bitcoin 1” and other hypothetical cryptoassets (“Bitcoin 2,” etc.). Linda Schilling and Harald Uhlig (2018) take a related approach to the exchange rate between “Bitcoin” and “the US dollar.” I use quotation marks here to indicate that the authors’ subjects are modeling entities, named after but not the real things. Their correspondence to the real things should not be taken for granted.
Both pairs of authors draw on a well-known theoretical result by Kareken and Wallace (1981): when two fiat currencies are perfect substitutes, the equilibrium exchange rate between them is indeterminate. To get the intuition, imagine that any payment made in US dollars can equally be made in Canadian dollars valued according to the going exchange rate. Nobody then has a reason to swap one currency for an equivalent amount of the other. No matter the level of the exchange rate, there is no pressure for it to change. Only the combined real money supply and demand matter, and any exchange rate is consistent with combined real combined real supply equaling combined real money demand. The combined real money stock could be 99% US dollars (at an exchange rate making one USD worth many CAD) or 99% Canadian dollars. Either rate is compatible with monetary equilibrium.