Imagine that there are two types of people in the world — recognizable and unrecognizable. Recognizable people can develop reputations, which can have either positive or negative effects. If a recognizable person develops a reputation for being trustworthy, then he or she will likely be able to issue debt to finance the purchase of goods and services. If the person is not trustworthy, but is easily recognized, then he or she will not be able to issue debt. All else equal, people who are recognizable will have an incentive to be trustworthy so that they can issue IOUs to pay for stuff. People who are not recognizable don’t have the same incentives. Since nobody can recognize them, they will never be able to issue IOUs.
Let’s think about this in the context of general equilibrium theory (without a Walrasian auctioneer and without any exogenously specified thing called “money”). In the absence of some auctioneer to price and distribute goods and in the absence of money, people must meet with every other person in the market to determine whether trade is possible. Recognizable people will be able to issue IOUs in order to trade as long as they are trustworthy and there is a mechanism to punish them if they do not repay their debts (e.g., exclude them to a world of autarky if they don’t repay their debts). But how can the non-recognizable people trade? Well, they could sell their goods to recognizable people in exchange for an IOU, but then all they have is an IOU. And what exactly does the IOU provide?
Assuming that each recognizable person can produce some good, the IOU would represent a promise to produce some quantity of the good in the future. A non-recognizable person could then present the IOU for this good at some future date or the person could turn around and use this IOU to purchase some other good. The seller in this circumstance would be willing to accept a third party IOU if (a) they want the good that IOU promises, or (b) they think they can pass on the IOU to someone else. Whether or not condition (b) is satisfied depends on the good the IOU-issuer is promising. Ultimately, what happens in this scenario is that certain goods will be found to satisfy condition (b) and those IOUs will start to circulate as a medium of exchange.
What this example resembles is a sort of free banking regime. People with good reputations are able to issue IOUs that can end up circulating like bank notes — these are banks. Eventually an IOU can be redeemed by whoever is holding it for some fixed quantity of a good that the IOU-issuer promised.
This implies that there is a key feature of free banking regimes. In actual free banking regimes, bank notes could be redeemed for one particular good, gold. The value of one bank note, consistent with our example, was defined as a particular quantity of gold. This implies that the price of gold in terms of bank notes is necessarily fixed. However, the relative price of gold to an index of all other prices is not fixed. In a growing economy, under these conditions, prices would decline on average with increases in productivity. As a result, the promise to pay a fixed amount of gold at a future date would actually entail a positive rate of return.
Thus, under a free banking regime, if (1) productivity is growing, (2) the decline in prices due to rising productivity completely offset the real interest rate, then a free banking regime naturally reproduces the Friedman rule.