The concept of money often plays an important role in social ontology literature; both Searle and Guala treat it as a paradigmatic institution or 'institutional fact', and a great deal of the social ontology literature argues whether this or that account of institutional facts can do proper justice to the institutional character of money. I think this is entirely based on a mistake; money is not an institution or institutional fact. Currency and bills are, of course, as are bank drafts and so forth, which are instituted in order to play money-roles, but money itself is an abstract feature of certain kinds of exchanges. Money doesn't need to be instituted, because it is a natural phenomenon that occurs in exchanges when you have a lot of exchanges of broadly similar kinds that can be compared to each other. It doesn't need to be instituted by collective intentionality (in the Searlean fashion), and in fact it's fairly easy to find cases in which people are using money without conceptualizing it as money, or (as happened with early banking) only realizing later that it was money. It doesn't need to be instituted as an equilibrium solution to a coordination problem (in the Gualan fashion) because it's in fact the pre-existence of money that makes it possible for money to be a solution to any coordination problems, and we can be using money without having any definite regulative rules for how to do so. It might sound a little weird to say that money is a natural phenomenon; but that's because we are often thinking 'currency' when we think money.
It is clear that currency is instituted. A dollar, properly speaking, is a piece of paper that is printed and issued as a Federal Reserve Note under the authority of the Bureau of Engraving and Printing; it presupposes the existence of the Federal Reserve and the Bureau of Engraving and Printing, which presuppose the existence of the U. S. Constitution and the Congress of the United States of America and Congress's statutory laws, all of which are themselves obvious institutions. But none of this makes the dollar money. A thousand years from now, it's unlikely that a dollar bill printed today will still be money. And of course, money pre-existed the dollar -- the dollar bill was instituted in order to serve an already existing role of money, which previously was filled by different things but most notably by the Spanish dollar, which filled a money-role previously filled by the thaler (among other things), which filled a money-role previously filled by the guldiner (among other things), which filled a money-role previously filled by etc., etc., etc. The origin of this traditional chain is lost in time, but there is good reason to think that currency developed to fill an already-recognized money-role. The usual story is that currency began as a token-receipt for stored goods, especially grain, when the receipt itself began to be traded; if that's the case, measures of stored grain (and occasionally other goods) pre-exist currency as a form of money. Money itself has no identifiable origin; as far as we can tell, it rises spontaneously as soon as any extensive kind of trade exists. And of course my claim is that the reason for this is that it is a natural phenomenon; it presupposes certain very general kinds of exchanges, and it seems to presuppose there being enough exchanges of sufficiently similar kinds that they are easily compared, but given those you get money without any need to institute it at all.
This is tied to the Aristotelian idea, which is still the primary theory of money today, that money is fundamentally a medium of exchange. Properly speaking, I think this is the genus of money; there are media in exchanges that are not monetary (ritual formalities like promises or favors are a very common case). Money is more like a medium commensurable across exchanges that is indifferent to the content of the exchange. But we can still call this 'medium of exchange' for short. Human beings, interacting with each other, exchange things in lots of ways; some of these exchanges involve indirectness and thus a medium, and when there are enough of these exchanges going on, some of these media are found not to be limited to their particular exchange but to be 'detachable' and usable in lots of exchanges, and that is a particular form money, just in and of itself. This then gets 'locked in' as a stable feature as we recognize that we can use it to solve certain kinds of problems, and over time as we endeavor to make the problem-solving easier, we develop conventions in which certain things stand in this role. By trial and error, some of those conventions win out, until eventually you are in the present year in the history of money. But the money-role is just part of the exchanges themselves; all the rest is trying to work out the most useful ways to take advantage of it.
From money being a medium of exchange in this way, we get the other Jevonsian functions. Jevons famously identifies four functions of money, which eventually were put into the mnemonic rhyme:
Money's a matter of functions four,
A Medium, a Measure, a Standard, a Store.
Because money is a medium in similar exchanges, it can therefore be used as a store of value and as a unit of account. Money as a store of value arises from the fact that the similar exchanges are diachronic; to be a store of value is just to be a medium of exchange stable across exchanges across time. Money as a unit of account (or measure of value) arises from record-keeping; to be a unit of account is to be something recordable for a wide set of synchronic and diachronic exchanges. And a unit of account that can play a role in diachronic planning on the basis of how the records work can be standard of value (or standard of deferred payment), so that one can trade with future as well as present units of account. These four are mutually reinforcing. (There are theories of money that take value storage rather than mediation of exchange as the fundamental function of money, but all the versions I have ever seen shipwreck on the question, "Value for what and as what?")
From the four functions, we get the various desiderata for successful institutions to serve in the role of money. A medium of exchange is best if it is fungible and portable (and therefore easy to exchange); a store of value is best if it is durable and difficult to replicate (and therefore easy to keep and difficult to fake); a unit of account is best if it is verifiable (and therefore easy to record); a standard of deferred payment is best if it is stable (and therefore easy to plan around); and so forth. But all of these desiderata are just success-conditions for the functions of money; you can perfectly well have money that does poorly on all these grounds, and (unsurprisingly) things that spontaneously take on the character of money in exchanges are often rather poor fits. All of the institutional facts concerned with money arise not in order to have money but in order to use money more easily. Money itself is not instituted; it just happens, spontaneously and often without people even recognizing it.
I think, incidentally, that this is also true of some other things that are often seen in social ontology literature as 'paradigmatic institutional facts' -- I think language is a plausible case. Grammar is an institutional fact -- grammar is the intentional use of language to organize language, so that language can be used more successfully -- and, given language, one can institute language (just as, given money, one can institute money), but language itself just happens naturally if you have enough people thrown together, and it likely pre-exists both our collective intentions about communication and any of the definable coordination problems that it actually solves. There are a lot of institutions within language, but language is not an institution, unless you count human beings coming to exist in proximity as an institution. Society is a more obvious case; marriage a more debatable case. But in any case, the result is that I think a lot of social ontology literature goes wrong by assuming that non-institutions are institutions, or (perhaps more commonly) confusing institutional facts with more basic natural facts that they presuppose.
