Monetary Economics at 30: A Reexamination of the Relevance of Money in Cashless Limiting Monetary Economies

That artificial economies without money provide an adequate framework for analysing monetary policy — a view the paper traces to a single result and to its textbook transmission. The paper opens with Galí’s (2008, p. 34) statement that the favoured modelling approach “does not incorporate money explicitly in the analysis” and that such economies “can be viewed as a limiting case (the cashless limit) of an economy in which money is valued and held by households,” with Woodford (2003) providing “a detailed discussion and a forceful defense of that approach.” It then quotes Woodford’s (2003, p. 32) own framing, that the basic model “abstracts from monetary frictions, in order to focus attention on more essential aspects of the monetary transmission mechanism.” The consequence the paper reports is that applied monetary work “now typically abstracts from money altogether, or if money is present, it is treated as a redundant asset, and the incentives that underlie money demand are left out of the analysis.” The target is narrow and explicit: the cashless-limiting approximation result of Woodford (1998).

A two-stage-per-period model with three classes of agents, whose first stage contains three simultaneous competitive markets — cash goods, credit goods, and loans — so that canonical monetary frameworks appear as special cases rather than as rivals. Each period has two stages: in the first, producers turn labour into good 1, which only consumers consume; in the second, everyone can produce and consume good 2, and trade labour and money competitively. Money is durable and intrinsically useless; the government pays a nominal interest rate on money holdings and injects or withdraws money through lump-sum transfers, so monetary policy is the pair of paths for the money supply and for the interest on money — the latter described as “a rudimentary approximation of a monetary operating framework with interest-bearing reserves, akin to the contemporary system employed by the Federal Reserve.” Agents are sorted each period into trust types (on-trust consumers can commit to honouring deferred payment; no-trust consumers cannot) and, among no-trust agents, into market-access types: cash agents, who reach only the cash-goods market, and banked agents, who also reach the loan market. Bankers can commit and enforce, and so intermediate between no-trust consumers and no-trust producers. The nesting is the design goal: shutting down both the bond and credit-goods markets gives Lucas (1980) cash-in-advance; shutting down only the bond market gives an explicit microstructure for Lucas and Stokey’s (1983) cash/credit generalisation; shutting down the credit-goods market and giving all consumers loan access gives Lagos and Zhang (2022); and the second stage plays the role of the securities-trading stage that makes portfolio choice independent of trading histories, as in the literature building on Lagos and Wright (2005).

Through bilateral Nash bargaining between producers and bankers, with producer bargaining power θ. Banked consumers trade competitively in the loan market, but banked producers reach it only indirectly: each producer–banker pair negotiates the quantities of bonds and money the banker will buy or sell on the producer’s behalf, plus an intermediation fee paid in good 2 in the second stage, with terms set by Nash bargaining and producer bargaining power θ ∈ [0,1]. Perfect competition in credit intermediation is the single point θ = 1. The paper offers several readings of the bond market — literally bonds issued by consumers and bought by producers through fee-collecting broker-bankers, in the spirit of Duffie et al. (2005); rudimentary credit-card companies; or deposit-taking institutions that take producer deposits, lend to consumers, trade in an interbank market and earn fees from depositors, implying deposit rates below the interbank rate.

Deliberately, to pre-empt the objection that the results depend on search and bargaining. The paper notes that, strictly speaking, the environment does not nest the canonical Lagos–Wright model, because the cash-goods market here is competitive rather than bilateral: “I adopt a competitive cash-goods market to highlight that the main results do not rely on search and bargaining—features typically assumed in the literature on the microfoundations of money demand.” Versions of the Lagos–Wright framework with competitive pricing, such as Rocheteau and Wright (2005), are nested.

Woodford’s case is the special case with no bond market at all, and his cashless limit is the fraction of on-trust consumers going to one. With no loan market, the first stage reduces to the Lucas–Stokey cash/credit structure. As the share of on-trust consumers converges to one, the fraction of cash consumers converges to zero and the monetary economy approaches a limiting economy in which all consumers finance purchases with Lucas–Stokey trade credit; that limiting economy coincides with the limit of the nonmonetary economy, which is the Arrow–Debreu frictionless benchmark in which every consumer affords and consumes the efficient quantity. The paper labels this Result 1: “As velocity diverges, the equilibrium allocation of an economy where money is used as a medium of exchange converges to that of the corresponding nonmonetary economy.” Its companion parts confirm the limit really is cashless — real money balances go to zero while nominal prices and velocity diverge. The paper also supplies the interpretation: the fraction of on-trust consumers being below one “amounts to imposing a ‘cash-in-advance’ constraint on the Arrow-Debreu economy,” which distorts the allocation whenever it binds, and taking the limit “effectively undoes the cash-in-advance constraint and restores the efficient Arrow-Debreu allocation.”

That monetary-fiscal policy can pin down any desired finite price level in the cashless limit — needed because the elementary version of the limit sends the price level to infinity. Much of Woodford (1998), the paper observes, is devoted to amending the infinite discontinuity in the price level at the limit, for two stated motives: dissatisfaction that the cashless-limit price level cannot be approximated by the price level of nearby economies, and the conflict between a price level that must “cease to be defined” and the goal of establishing that a central bank can control the price level in a model without money. The fix is to replace the constant policy parameter with a function of the price level, which yields Result 2: “The government can use monetary-fiscal policy to implement any desired equilibrium price level… in the cashless-limiting economy. Moreover, this price level approximates the equilibrium price level corresponding to an economy in which transaction velocity is arbitrarily large.” Lagos registers a prior objection of his own — that it is unclear why an infinite price level should be a concern when every variable relevant to agents’ decisions converges and the level of every nominal variable becomes irrelevant, and that approximation is available anyway if one focuses on real money balances — and a pointed characterisation of the exercise: Woodford “theorizes on the determination of X in the context of a model in which X is irrelevant for agents’ decisions and welfare.”

That the framing “misconstrues the two approaches as being fundamentally different,” because a money-growth rule can deliver Result 2 just as an interest-rate rule can. The key observation is that the policy function delivering Result 2 is only a relabelling of the normalised opportunity cost of holding money, which can be written as a function of either the administered rate on money or the illiquid-bond rate that a money-growth rule influences — so “any Ω that delivers Result 2 can be implemented through many combinations of monetarist and Wicksellian policies.” Woodford’s “Wicksellian regime” is the special case in which the administered rate on money rises when real balances fall, matching Woodford’s reading of one of Wicksell’s proposals. But a monetarist regime with a constant rate on money and a money-growth rule that increases in real balances also delivers Result 2 — contradicting Woodford’s (2003, p. 82) claim that “in a cashless economy, a money-growth target will not succeed in determining an equilibrium price level.”

Because Woodford imposes an ad hoc restriction that the money supply stay bounded away from zero, and the money-growth rules that would work violate it. With no interest paid on money, the monetarist implementation prescribes a gross money-growth rate equal to the discount factor, so the money supply shrinks every period — which “violates Woodford’s assumption that the money supply must be bounded away from zero.” The paper states the consequence bluntly: “This restriction plays no role in Woodford’s analysis, other than effectively assuming away money-growth rules” of the kind that would deliver the result. It further shows that the feedback-rule aspect is not essential either: a constant money-growth target set to the discount factor plus the cash-consumer share times a constant — a rule independent of endogenous variables — delivers a finite price level in the limit. The intuition offered is that “as the economy becomes cashless, simply allowing for a secular downward drift in the money-growth rule allows for a standard monetarist money-demand-equals-money-supply determination of a finite price level.” The conclusion is that “the only reason to favor Wicksellian over monetarist policy is that the former allows proponents of the moneyless approach to formulate and discuss central-bank policy without referencing money demand or supply.”

Because with a bond market there is a second route to a cashless economy, and it leads somewhere different. In the Lucas–Stokey structure Woodford assumed, the only path to a cashless economy is to send the share of on-trust consumers to one. With bankers and a bond market, one can instead send the share of cash agents to zero while holding the on-trust share fixed strictly between zero and one. That limiting economy still has real money balances going to zero, prices and velocity diverging — it is genuinely cashless — but it “features both on-trust consumers and producers who settle purchases of good 1 with zero-interest deferred-payment loans, and no-trust consumers and producers who settle using interest-bearing claims that are intermediated by bankers.” To let the two cash shares vanish at different rates, the paper replaces them with strictly decreasing functions converging to zero. This is the setup for the paper’s main corollary.

Result 1. The limiting economy is cashless, but its allocation does not coincide with the nonmonetary economy’s. In the generalised limit the equilibrium loan rate for banked consumers exceeds the rate on money and on zero-interest trade credit; the relative price of goods in the cash market falls below the relative price in the credit market; cash consumers’ consumption goes to zero while banked consumers consume a quantity determined by a cum-interest relative price that depends on the opportunity cost of money and on bargaining power. Aggregate consumption in the cashless limit of the monetary economy minus aggregate consumption in the corresponding limit of the nonmonetary economy is weakly positive, and the paper states the exact conditions for equality: it holds “only if” the on-trust share equals one (Woodford’s case), or the opportunity cost of holding money is infinite so no consumer is willing to hold money along the limit, or bankers have no market power. Outside those three cases — on-trust share below one, bargaining power strictly interior, finite opportunity cost of money — “output is strictly higher in the pure-credit (cashless) limit of the monetary equilibrium than in the corresponding limit of the cashless economy.” The gap is decreasing in the opportunity cost of money, so “despite real money balances being virtually zero in the cashless limit, the opportunity cost of holding money still influences the real allocation—much as it does in conventional monetary equilibria outside the cashless limit.” The paper also derives the elasticity of aggregate output with respect to the opportunity cost of money in the limit, factoring it into a microstructure term (in the on-trust share and bargaining power) and a preference term.

In the general microstructure the marginal value of a dollar in the cashless limit equals the net interest cost of borrowing, which is positive. The paper contrasts the two cases directly. Under Lucas–Stokey credit, “bringing an additional dollar into the credit-goods market does not increase a consumer’s purchasing capacity, since consumers have unlimited access to zero-interest deferred-payment credit,” so the marginal value of carrying money is zero in the pure-credit limit and the right side of the Euler equation collapses. With an endogenous bond rate, it does not collapse: “the marginal value of money as a medium of exchange remains equal to the net interest rate cost of borrowing money in stage 1, i.e., [the loan rate minus the rate on money], which is positive (or strictly positive away from the Friedman rule).” The behavioural statement is worth quoting because it is the paper’s answer to the intuition that money cannot matter when nobody needs it: “In the cashless limit, agents know they will never be constrained to pay with dollars. Nevertheless, they are willing to bring dollars into stage 1 because each dollar saves them the interest cost they would otherwise incur by borrowing to purchase good 1.” In the limiting equilibrium the bond rate adjusts to satisfy the Euler equation, so the opportunity cost of money does not drop out and neither do the monetarist channels. The mechanism is named in the conclusion as latent money demand: “the off-equilibrium threat to settle transactions directly with money… strengthens the stance of sellers relative to these intermediaries, and propagates policy-induced changes in the opportunity cost of holding money to the relative prices faced by all agents—even those who use credit instead of money to settle trades.”

Yes, with the same three-case structure, and with welfare monotone in the opportunity cost of money. Away from any limit, and assuming not all agents are on-trust and not all no-trust agents are banked, welfare in the stationary monetary equilibrium lies strictly between welfare in the nonmonetary economy and the first best for every strictly positive opportunity cost of money; it converges to the first best as that cost goes to zero, remains above the nonmonetary level even as the cost goes to infinity, and is strictly decreasing in it. In the cashless limits, the ordering is: nonmonetary limit ≤ monetary limit ≤ first best. If either the on-trust share equals one or bankers have no market power, all three coincide at the first best — Woodford’s case, in which “money is inessential because it no longer improves upon the allocation implemented by the barter equilibrium of the nonmonetary economy.” Otherwise the inequalities are strict for every finite positive opportunity cost of money, with the monetary limit converging to the nonmonetary limit as that cost goes to infinity and to the first best as it goes to zero, and decreasing in between. The paper draws the normative conclusion in the same form as the positive one: “from a normative standpoint, it does not follow that money, its opportunity cost, and medium-of-exchange considerations are irrelevant simply because aggregate real money balances are close to zero.”

That it is not a high-velocity limit but a high-velocity-and-frictionless-credit limit, and the two are conceptually unrelated. The formulation: the limit “is not merely a high-velocity limit reflecting widespread substitution from money toward credit. Rather, it is a high-velocity limit reflecting widespread substitution from money toward perfectly competitive credit. In other words, the irrelevance of money in Woodford’s cashless limit is not a generic feature of economies where real balances are small because agents rely on liquidity-saving credit arrangements. It is a feature of a special limiting market microstructure in which those credit arrangements are also frictionless.” The corollary for the empirical motivation is stated directly: “The secular rise in velocity observed through the mid-1990s—often used at the time to motivate the cashless-limiting approach—has no bearing on whether the relevant pure-credit benchmark should assume perfect competition or retain frictions such as market power in credit markets.”

Not wrong mathematically; wrong as a warrant for the practice built on it. “Woodford’s approximation result is not wrong—at least not in the narrow mathematical sense. The problem, rather, is that the result implicitly presumes a frictionless market microstructure for credit and payments—an assumption conceptually unrelated to how far the economy is along the high-velocity limit.” The applied objection follows: “the assumption that credit markets are perfectly competitive, which was common in the 1980s and through the early 2000s, is now widely deemed ill-suited for empirical or quantitative applications of macro models with credit.” The final verdict is expressed as an answerable question rather than a theorem: “Can we be confident that ignoring money entails no significant loss for monetary policy analysis? My findings suggest the answer is no. The microstructure for which this result holds is too restrictive—and too detached from our current understanding of credit and money markets—to justify a three-decade detour of monetary economics away from money.”

That it went the other way — and the paper is explicit that this observation is separate from its theoretical argument. Woodford’s motivation quoted in the paper anticipated “further innovations, due to improvements in information processing and to increased creativity in the evasion of the remaining regulatory constraints, that will continually reduce the quantity of the monetary base that needs to be held (on average) to carry out a given volume of transactions.” Lagos notes that at the time of the 1996 keynote, measures of velocity based on M0, M1 or bank reserves were at historical highs or trending up. Against that: “Despite remarkable advances in information technology and no obvious decline in human creativity, money velocity has remained substantially below its 1960 level for the last eighteen years, and the optimal quantity of monetary base is currently the subject of active debate at the Federal Reserve.” The paper’s own framing of the weight this carries: “While this paper has focused on the theoretical fragility of the cashless-limit proposition, it is worth noting that Woodford’s moneyless prophecy has not materialized empirically.”

That the properties of money demand are artefacts of the assumed microstructure, so conclusions about money demand inherit the microstructure’s limitations. The paper’s diagnosis of why monetary economics resisted integration into macroeconomics is that Walrasian general equilibrium theory “cannot answer the most elementary question in monetary economics: Why does money sell at a positive price?”, that there is agreement trade under that paradigm is too seamless for money to matter, and long-standing disagreement about which frictions to add — leaving progress “siloed along methodological lines.” The paper is candid that this fragmentation is inconvenient for applied work and that the coordination benefit helps explain why “Woodford’s proposal to abandon money altogether gained traction, despite being grounded solely in the cashless limit of Woodford (1998) and a handful of numerical exercises based on models from the 1960s.” Its rejoinder is a rewriting of Keynes: “Practical monetary economists, who believe themselves to be quite exempt from the fine details of payments and sources of money demand, are usually slaves of some defunct market microstructure.” The constructive close resists prescribing a single framework: no general agreement on the right way to incorporate liquidity is likely, “in each application, the research question and relevant evidence must guide which microstructure features matter most,” one can always dig deeper, and “progress in the field will invariably turn the deep microfoundations of today into the reduced forms of tomorrow.”

It began as lecture notes for a plenary talk, and it is in part a first-person retrospective — which shapes the register but not the results. A footnote records that the paper “originates in lecture notes prepared for a plenary talk entitled ‘onetary Economics: How It Started, How It’s Going’, delivered at the Winter Meeting of the Society for Economic Dynamics in Buenos Aires, December 12-14, 2024.” The concluding section supplies the frame: Woodford delivered the keynote “Money and Prices in the Theory of Value” at the Society for Economic Dynamics annual meeting in Mexico City on 29 June 1996, and Lagos was in the audience as a first-year conference-going PhD student — “In time, I came to realize I had witnessed a turning point in the history of monetary economics—one that would shape the field for the next thirty years.” The label “onetary Economics” is the author’s own coinage for monetary economics done without money, and it is carried into the article’s deposited title, which renders the phrase in italics; the record here stores the title with the M restored, which is why the metadata carries a title-changed flag.