Are There 2 owners A and B of commodities x and y, respectively, of whom a Desires B and Y Desires x. With no Cash and no third Product, the only way for both owners to obtain their Desirable commodities is Straight from each other:
A –> y | B –> x
X _____ | y
Y _____ | x
Otherwise, B and A must assign their product possession to somebody who then redistributes them. But, this type of centralized solution would partly contradict the exact same possession, by partly shifting it from its rightful controls. Therefore, just a decentralized solution can conserve the entire product ownership inherent this market, by B and A exchanging x and y straight coinmarket.
However, direct commodity trade presents two issues, both of that alone is sufficient to block it. The primary problem has a abstract character:
In order exchangeable for one another, y and x must share the identical market value.
It may occur that each exchangeable amount of x has another exchange worth to that of almost any exchangeable amount of y.
The next problem comes with a goal nature rather. Permit (as under) A, B, and C possess products x, y, and z, respectively. If A wants y, B desires z C needs x, subsequently direct trade couldn’t offer those 3 owners their desirable commodities — as not one of them possesses the identical commodity needed by who possesses their desired one. Moneyless exchange today can only occur if one of these products becomes a multiequivalent: a simultaneous equal of both of the other commodities at least to the proprietor who needs nor possesses it — if both of the other owners also understand of the multiequivalence or never. As an Example, A can obtain interlocking in market for x with C just to provide it in exchange for y using B, that way which makes z a multiequivalent (as asterisked):
A –> y | B –> Query | C –> x
X _____ | y _____ | z*
z* ____ | y _____ | x
Y _____ | z _____ | x
However, this individually-handled multiequivalence introduces Another set of issues:
It empowers contradictory direct exchange plans. In this previous instance, A could attempt to get z in market for x with C (just to provide it in exchange for y with B) even with B concurrently hoping to get x in market for y using A (just to provide it in exchange for z with C).
It not only enables — — for many mutually exchangeable amounts of 2 commodities to have distinct trade worth, but also raises the chance of the mismatch, by determined by further exchanges between distinct pairs of goods.
Luckily, those issues have the sole and same way of one multiequivalent m getting sociable, or cash. Afterward, commodity owners are able to either give (sell) their products in exchange for give m in trade for (purchase) the products they desire. By way of instance, again allow A, B, and C own products x, y, and z, respectively. Still supposing A desires y, B desires z, and C needs x, if today they only exchange their products for that m societal multiequivalent — originally owned by A — afterward:
A — y | B –> Query | C –> x
x, m __ | y _____ | z
Y __ | m _____ | z
Y __ | z _____ | m
y, m __ | z _____ | x
With social (as opposed to individual) multiequivalence:
There are always two exchanges to the proprietor of every commodity (who buys or sells it before purchasing or later selling a different one, respectively), together with any number of these owners, either in a uniform series.
All commodity owners trade a common (societal) multiequivalent, which returns to its first owner.
Furthermore, using a societal multiequivalent (currency) divisible into little and similar enough components, even though all exchangeable amounts of 2 commodities have distinct exchange values, both of these commodities will stay mutually exchangeable. By way of instance, allow two commodities x and y be worth one and 2 components of a societal multiequivalent m, respectively — x(1m) and y(2m). After that, allow their owners A of B and x of y be the proprietors of three m components — 3m — every. If A and B need x and y, respectively, however constantly exchange their products for components — x to 1m and y for 2m — afterward:
A –> y _ | B –> x
x(1m), 3m | y(2m), 3m
Y(2m), 2m | x(1m), 4m
Finally, with societal multiequivalence thus earning, as just cash does, commodity trade always possible, each societal multiequivalent is cash, which can be conversely any kind of societal multiequivalence.
Money as Decentralization
Nevertheless, historically, even though maintaining the decentralized ownership of commodities throughout their trade, currency has itself become quite concentrated, by falling under the jurisdiction of governments. Really:
It has to represent the exact same decentralized ownership it keeps.
It has to be tangible for many commodity owners to discuss it.
Its concreteness to every one among those owners needs its personal control with a public authority — if selling, purchasing, making, or ruining it. 
Its then-centralized control partly prevents it from representing a decentralized product possession — hence defeating its initial function.
Luckily, despite always concrete to all folks, or socially tangible, a financial representation could be somewhat subjective to every individual, or independently subjective. By way of instance, cryptocurrencies — such as Bitcoin — utilize public-key cryptography to concurrently signify money as a personal key and also this private key as a public secret, so cash gets metarepresented, or metamoney. Afterward, despite staying socially tangible as a decentralized community, some such metarepresentation of cash gets independently subjective as a financial — meta — component, which averts its decentralization, by preventing any public authority from independently controlling it.