At Marginal Revolution, Tyler Cowen has a post on the New Monetary Economics. He says,
If you're looking for a definition of the NME, I would say it is the study of unusual monetary arrangements involving either explicit prices for monetary media of exchange (i.e., separating money's medium of exchange function from its medium of account function), and/or paying interest on currency or bank reserves held at the Fed.
If you are ever tempted to call something "the New X," remember that eventually it will cease to be new. The New Monetary Economics now qualifies under "what's old," as I defined it in an earlier post, that is, something those who are interested in the subject have known about for at least 15 years.
Work on free banking since Lawrence H. White's Free Banking in Britain (first edition 1984) has been strongly influenced by historical cases of the system, in which the three textbook functions of money as medium of exchange, unit of account, and store of value have been combined into one. As White discussed in another 1984 publication ("Competitive Payments Systems and the Unit of Account" – JSTOR, requires subscription), the combination is quite powerful and durable. That is why, to my knowledge, all historical cases sometimes adduced as being examples of what the New Monetary Economics talks about have been at the edge of conventional finance — LETS (local exchange trading systems) and the like — rather than at the center, like the commercial banks in historical free banking systems.
Even so, the New Monetary Economics helps to imagine one possible path for the future. Several economists, including Kevin Dowd, Tyler Cowen, Randall Kroszner, and Leland Yeager, have contributed both to it and to research on free banking.
Cowen mentions, without endorsing it, the idea of the late Fischer Black that paying interest on deposit reserves that commercial banks hold at the Fed could cause the Fed to lose control of the price level. I fail to see why; the Fed could always switch from paying interest to charging a fee (negative interest), as Sweden's central bank did in 2009-10.