So far I have not discussed what commodity a private system should base its money on. Historically, the most common standards were probably gold and silver. They were well suited for the purpose since they have a high value to weight ratio, making them portable, are easily subdivided and recombined and relatively easy to measure and evaluate.
In a modern society none of these characteristics is important, since the circulating medium is not the commodity itself but claims upon it. The disadvantage of silver and gold is that they have very inelastic supplies and relatively inelastic demands; judging by recent history the value of both in terms of most other commodities can and does vary erratically even without the additional instabilities that might be introduced by a fractional reserve system based on them.
The ideal commodity backing for a modern system would not be a single commodity but rather a commodity bundle. The bank would guarantee to provide anyone bringing in (say) a hundred thousand of its dollars with a bundle consisting of a ton of steel of a specified grade, a hundred bushels of wheat, an ounce of gold, and a number of other items. The goods making up the bundle would be chosen to make the value of the total bundle correlate as closely as possible with the general price level. While a change in production technology or non-monetary demand might alter the value of one good in the bundle, it would have only a small effect on the value of the bundle as a whole. Since the quantity of such goods being used for monetary purposes would be a tiny fraction of the total quantity of steel, wheat, gold, etc., changes in monetary demand would have a negligible influence on the value of the bundle. So the value of such a money should be stable against both monetary and non-monetary changes.
Such a system would work in practice very much like an ideal fiat system in which the monetary authority manages the money supply to maintain a stable price level. If the money supply increased to the point where the bundle was worth more than 100,000 dollars, holders of dollars would turn them in for commodities, bringing the money supply and the price level back down. If the money supply fell so that the commodities were worth less than the money, banks would find that they could issue additional money without any of it being turned in for commodities and the money supply would rise. The system as a whole would therefore stabilize prices in such a way as to make the price of the bundle, a crude price index, stable at its face value.
The advantage of this system over a government-run fiat system is that it does not rely on the wisdom or benevolence of the people appointed to manage the money supply. It provides a mechanism for making it in the interest of the private people controlling the money supply to behave in exactly the way we would want the officials controlling a government fiat system to behave. Since the nature of the reserves in this system makes it unnecessary for the banks to hold any significant quantity of them, such a system is, in effect, a fiat system in which the obligation to redeem the currency in commodities forces the people controlling the money supply to maintain stable prices.