On Money as a Medium of Exchange

The paper analyzes a simple general-equilibrium matching economy in which “the objects that become media of exchange will be determined endogenously as part of the non-cooperative equilibrium” (Sec. I, p. 928), in contrast to the two dominant paradigms at the time – overlapping-generations models, which “basically ignore the medium of exchange role,” and cash-in-advance models, which “simply impose the medium of exchange role by an ad hoc restriction” (Sec. I, n.1, pp. 928-929). There is a continuum of infinitely-lived agents of three types, equal in proportion, who specialize in production and consumption (type i consumes only good i and produces only one other good), so that, following Jevons, direct barter runs into the “double coincidence of wants” problem (Sec. I, p. 929). Goods are indivisible, storable only at a strictly positive cost that differs by good, and each agent can hold only one unit at a time; agents meet randomly in pairs each period and trade only if mutually agreeable, since “there is no credit since a given pair will meet again with probability zero” (Sec. II, p. 931).

A steady-state Nash equilibrium is a set of trading strategies, one per type, together with a steady-state distribution of inventories, such that each agent’s strategy maximizes his expected utility given others’ strategies and the distribution, and the distribution is itself the one implied by those strategies (rational expectations) (Sec. II, Definition, p. 932). Because a trading strategy reduces to a ranking of the value “indirect utilities” V_i(j) of holding each good j, and there are only finitely many possible rankings, the authors can exhaustively search the finite set of candidate strategy profiles, check which ones are mutually best responses, and thereby completely characterize the full equilibrium set rather than merely exhibiting one equilibrium (Sec. II, pp. 932-933). A good that is accepted in trade “not to be consumed or used in production, but to be used to facilitate further trade” is defined as a medium of exchange, called commodity money if it has intrinsic value and fiat money if it does not (Sec. II, p. 932).

In model A (where type 1 produces good 2, type 2 produces good 3, and type 3 produces good 1), Theorem 1 shows that if the storage-cost gap c13 - c12 exceeds a threshold .5*beta*u1, there is a unique “fundamental” equilibrium in which every agent prefers lower-storage-cost goods to higher ones (except for their own consumption good) and good 1 alone serves as the economy’s commodity money (Sec. III, Theorem 1a, pp. 934-937). If instead the storage-cost gap is small enough, a unique “speculative” equilibrium obtains in which type-1 agents deliberately trade their low-storage-cost production good for a higher-storage-cost good because it is more marketable, so that goods 1 and 3 both serve as media of exchange (Theorem 1b, pp. 937-938); for storage-cost gaps in between, “no pure strategy, steady-state equilibri[um]” exists at all (p. 939). In the speculative case, a type-1 agent holds the objectively lower-return good 3 over good 2 purely because of its superior marketability – “an example of an object being used as a medium of exchange in spite of the fact that it is dominated in rate of return by another object” (p. 938).

In model B (a different assignment of who produces what), Theorem 2 shows a fundamental equilibrium always exists, with goods 1 and 2 serving as commodity money, and for a further range of parameters a second, speculative equilibrium also exists in which goods 2 and 3 instead serve as commodity money (Sec. IV, Theorem 2, pp. 939-941). Critically, the paper notes that “in some nonempty region of parameter space these two equilibria coexist,” so that “[e]ither goods 1 and 2 or goods 2 and 3 may end up as commodity monies, depending solely on extrinsic beliefs and in spite of the fact that fundamentals suggest that good 1 is a much better medium of exchange than good 3” (p. 940-941). The authors are explicit that this is not a novel philosophical point in itself, but that “we have simply shown that such self-fulfilling prophecies are consistent with our notion of equilibrium” (p. 941).

Introducing a new, intrinsically worthless object (“good 0”) in fixed supply M, Theorem 3 gives explicit parameter conditions under which a steady-state equilibrium exists where all agents rank good 0 above every other good except their own consumption good, so that good 0 circulates as a genuine fiat currency alongside residual commodity money (Sec. V, Theorem 3, pp. 943-945). But the paper first establishes that an equilibrium with no one accepting fiat money always exists too: if an agent believes no one will ever accept good 0 from him, taking it yields zero continuation value, so he will refuse it, and “[i]f no one believes in fiat money, then it cannot get off the ground (like the Susan B. Anthony dollar)” (Sec. V, p. 943). The authors call this property “tenuousness,” note that it is “shared by the overlapping generations model, although not by the cash-in-advance model,” and state that they “think that it is a property that a good theory of money ought to have,” since “[t]he value of any medium of exchange…ultimately depends at least partially on faith” (p. 943).

The paper shows that velocity of circulation is “a very poor indicator of moneyness”: in the fiat-money equilibrium of model A, good 3 has the highest velocity of any object, “surpassing both goods 0 and 1 even though they are media of exchange, simply because the stock of good 3 is so low in equilibrium” (Sec. V, p. 947, incl. n.16). Acceptability – the probability a good is accepted when offered – performs much better: fiat currency has acceptability of exactly one (“which is what makes it a general medium of exchange”), and a commodity money’s acceptability falls as the stock of real balances S rises and its role as money diminishes (Sec. V, p. 947). Liquidity, defined for each type as the expected time to trade (directly or indirectly) into one’s own consumption good, rises in every good as real balances S increase – “a clear case of ’too much money chasing too few goods’” – and the ranking of liquidity across goods for each type exactly mirrors which goods that type is willing to speculate in (Sec. V, p. 948).

Equilibria are shown not to be generally Pareto optimal: comparing the equilibrium of model A’s fundamental strategies to the (non-equilibrium, non-implementable) strategy in which every agent always trades regardless of the match, the authors compute welfare gaps Delta_i that are strictly positive for all three types when consumption utility u_i is large, so “equilibria are not generally optimal,” even though the always-trade strategy “[is] not implementable” because rejecting high-storage-cost goods is individually rational even though collectively self-defeating (Sec. VI, pp. 948-949). Where fundamental and speculative equilibria coexist (model B), the two are generally not Pareto-comparable – some types are strictly better off under one, other types under the other (Sec. VI, pp. 949). Introducing fiat money into the fundamental commodity-money equilibrium of model A is shown, in the Appendix, to raise every type’s welfare at the margin (dW_i/dS > 0 at S=0) as long as consumption utilities are not too large relative to storage costs, “because using fiat money reduces the inefficient storage of real commodities,” but welfare is not monotonic in the real money stock – too much fiat money is always welfare-reducing (Sec. VI, p. 949).

The authors argue their matching framework captures a feature neither rival approach can: overlapping-generations models “basically ignore the medium of exchange role, concentrating on money’s store of value function,” while cash-in-advance models “simply impose the medium of exchange role by an ad hoc restriction that goods can be acquired only using money,” so that in Kareken and Wallace’s words (quoted in Sec. VII, p. 951), “no Clower constraint model determines what is used as a means of payment… there is never any switching from one thing to another” – whereas the present model lets different commodities (or fiat money) endogenously take over the medium-of-exchange role depending on parameters and beliefs. The paper notes its model does share the “tenuousness” of fiat-money value with the overlapping-generations model, a property the cash-in-advance model lacks by construction (Sec. V, p. 943).

The authors are explicit that this is “a simple model with a special explicit structure” (Sec. VII, n.20, p. 950) built around just three goods and three types – the minimum needed to generate indirect exchange at all – and that extending to more types or goods “should not change the message but would make things much messier.” They flag that the assumption of indivisible consumption goods (which keeps the state space finite and the model tractable) means relative prices are not determined the way they would be with divisible goods and bargaining, and that they deliberately “concentrated on storability exclusively because it seemed the easiest to handle technically,” leaving other intrinsic properties such as recognizability for future work under informational frictions (Sec. VII, pp. 950-951).