free banking

Bitcoin and, to a lesser extent, Ethereum are pretty well-known cryptocurrencies at this point; only slightly behind them in scale is something called Ripple.  My understanding, though, is that Ripple started as something else, though not a lot different.  A cryptocurrency system is essentially a system for indelibly publishing statements of the form "I Alice have 42.3 units of currency from Source X of which I hereby give 24.9 units to Bob and retain 17.3 units for myself," and I won't here go into why the numbers don't quite add up.  My understanding is that Ripple collected statements of the form "I Alice owe Bob 24.9 units of currency," with some capacity for detecting cycles, so that if Bob owed Carol 24.9 or more units and Carol owed Alice 24.9 or more units those might be cancelled against each other, and if Bob wanted to transfer the debt that Alice owed him to Carol (so that now she owes Carol the money instead of owing it to Bob), there might have been a capacity for that, too.

If I'm wrong about Ripple's historical origins, I don't really care.   Let's discuss such a system anyway.

The bitcoin blockchain has a mechanism by which a positive number of bitcoins actually exist — in fact, it's an important part of the way the blockchain works — but we don't really need that to be the case.  As long as there are a lot of people who are each willing to lend some amount of money to each of several other people, you can run the system entirely on credit, where the total amount of currency in circulation is zero.  If I'm willing to lend Bob up to 25 units, I can "sell" him a good for up to that amount of money, with his payment going into the system; I've sold it for the IOU.  If Carol is willing to lend Bob up to 15 units, and has an item I wish to buy (and she wishes to sell) in exchange for 15 units worth of debt from Bob, we can use Bob's debt to mediate our exchange.

Of course, if Bob and Carol and I don't really know each other, there's an issue.  If there's someone we all trust for some reason — let's call him Uncle Sam — and I (by some means or another) am owed money by him, then we're in a fine (and very familiar) place; I can buy things from Bob and Carol by novating to them the debt from Sam.[1]

At this point, I've perhaps worked backward a bit; my point is that the supply of money doesn't depend on any monetary base, and certainly not any commodity (or other) backing.  "Medium of exchange" is essentially created wherever credit is extended.[2]  We can, in principal, have "net cash" of zero in an economy, with every unit of currency representing the entirely unsecured liability of a private individual.

In closing, I'll note that there are many measures of "money supply", the broadest of which include commercial paper; I think it probable that there are purposes for which lines of credit, credit card credit limits, and pledgeable assets should be counted as contributing to the supply of money.

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[1] Hyman Minsky is reported to have observed, "Anyone can create money; the problem lies in getting it accepted."

[2] This is more or less one of the best points that the adherents of "Modern Monetary Theory" have made.