An email query from Brad DeLong reminds me of this old Bart Taub paper, “Private Fiat Money with Many Suppliers” (jstor):
A dynamic rational expectations model of money is used to investigate whether a Nash equilibrium of many firms, each supplying its own brand-name currency, will optimally deflate their currencies in Friedman’s (1969) sense. The optimal deflation does arise under an open loop dynamic structure, but the equilibrium breaks down under a more realistic feedback control structure.
There is also Marimon, Nicolini, and Teles (pdf) and the work of Berentsen., all building on Ben Klein’s piece from 1974. This literature has been read a few different ways, but I take the upshot to be that a) a monopolized private fiat money might be stable in supply, to protect the stream of future quasi-rents, and b) private competing fiat monies will not be stable in overall supply, for reasons of time consistency and also the competitive erosion of available rents. In other words, when it comes to the proliferation of cryptocurrencies, the more the merrier but not for those holding them.
I don’t agree with the modeling strategy of this 1981 Kareken and Wallace paper on the indeterminacy of equilibrium exchange rates, but still it is another useful starting point.
Addendum: Krugman adds a bit more on Bitcoin, from a friend of his, John R. Levine. Here is the final bit from Levine:
My current guess is that the Bitcoin bubble will collapse when there is some bad news, e.g., a regulator demands registration of Bitcoin wallets, people try and cash out, and find that that while it’s easy to buy bitcoins, it’s much harder to find people willing to buy back nontrivial amounts, very hard to collect the sales proceeds, and completely impossible without revealing exactly who you are.