Money function

From Numeraire to Unit of Measurement: The Core Function of Money

Equilibrium, Implementation, and the Functional Necessity of the Unit of Account

Abstract

Standard economic theory draws a sharp distinction between the existence of equilibrium outcomes and the mechanisms by which those outcomes are realized in the world. Arrow–Debreu general equilibrium theory establishes that prices are defined only up to scale and therefore require a numeraire for representation, but it does not explain how a price system is implemented or maintained in practice. Subsequent work—most notably Banerjee and Maskin (1996)—shows that even once a Walrasian price system exists, decentralized trade under informational frictions requires a medium of exchange, and that monetary equilibria fail both when goods are not fungible and when the unit of account itself becomes unstable due to unpredictable money supply growth (e.g., high inflation).

In this paper, we unify these strands by making explicit a distinction that is standard in game theory and mechanism design but rarely applied in monetary economics: the distinction between equilibrium logic (G) and real-world generating mechanisms (F). We show that if a Walrasian equilibrium exists in the real world—not merely as a theoretical fixed point—then the generating mechanism F must enforce a unit of account in order to maintain an arbitrage-free price system. The unit-of-account role is therefore not merely a representational convenience, but a functional necessity for real equilibria. Banerjee and Maskin then explain why, conditional on such a unit of account, a medium of exchange is still required, and why fiat money fails precisely when the unit-of-account role loses credibility. Taken together, these results yield a clean, non-ideological sequencing of monetary functions: unit of account first, medium of exchange second, store of value third.

1. Introduction

Much confusion in monetary theory arises from the persistent conflation of three distinct questions:

1. When do equilibrium outcomes exist?

2. Which objects can function as money?

3. How are equilibrium outcomes implemented in the real world?

Arrow–Debreu general equilibrium theory answers the first question.

Banerjee and Maskin (1996) answer the second.

This paper addresses the third—and shows how all three fit together.

The key distinction organizing our analysis is between equilibrium logic (G)—the abstract characterization of stable outcomes—and implementation (F)—the real, executable rules that generate and sustain those outcomes in practice. This distinction is standard in game theory and mechanism design, yet it is rarely applied explicitly in monetary economics. Making it explicit resolves long-standing confusions about the roles of the unit of account and the medium of exchange, and clarifies how canonical equilibrium theories relate to real markets.

2. G: Equilibrium Logic

2.1 Arrow–Debreu

Arrow–Debreu theory establishes two foundational results: competitive equilibria exist under standard assumptions, and equilibrium prices are defined only up to a positive scalar. Because prices are only defined up to scale, a numeraire is introduced to represent the price vector.

Crucially, in Arrow–Debreu:

• the numeraire is purely representational;

• it fixes scale but does not explain market operation;

• it does not generate a medium of exchange.

Arrow–Debreu is intentionally silent on implementation. It characterizes what equilibrium is, not how it is produced or sustained in the world.

2.2 Banerjee–Maskin

Banerjee and Maskin (1996) take Arrow–Debreu prices as given and ask a different question:

Given a Walrasian price system, why does barter fail and money emerge—and why does fiat money sometimes fail entirely?

They identify two distinct and conceptually separate failure modes.

First, medium-of-exchange failure (E-failure): when goods are not fungible or identifiable, adverse selection—the lemons problem—causes agents to refuse most goods in payment. Barter fails to scale even though prices exist.

Second, unit-of-account failure for fiat money (U-failure): when money supply growth becomes sufficiently unstable or unpredictable—most starkly under high inflation—the monetary equilibrium ceases to exist. Agents abandon fiat money and revert to commodity exchange or barter. This is explicit demonetization, not merely reduced efficiency.

Throughout this analysis, Banerjee–Maskin assume a functioning unit of account as part of the Walrasian background. Their contribution explains why money fails once that unit-of-account role loses credibility, and why decentralized trade requires a medium of exchange even when prices exist. They do not ask why the unit of account exists in the first place, because that question lies outside equilibrium logic.

3. The Missing Question

Neither Arrow–Debreu nor Banerjee–Maskin ask the following question:

Why must a unit of account exist functionally in the first place for a real equilibrium to exist at all?

They do not need to ask it, because both operate entirely within G. But once we ask whether an equilibrium exists in the world, not merely on paper, this question becomes unavoidable.

4. F: Real-World Generating Mechanisms

Let F denote the real, executable generating mechanism that produces observed behavior. This includes quoting conventions, accounting systems, enforcement mechanisms, arbitrage discipline, and institutional rules.

A basic but often implicit fact follows immediately:

If a real-world equilibrium exists, then some executable F exists that sustains it.

Recognizing this fact imposes a concrete constraint on F in any price-mediated system.

5. The Functional Necessity of the Unit of Account

A system of bilateral exchange rates can be arbitrage-free if and only if it can be generated from a single underlying price vector. Equivalently, the exchange-rate matrix must be rank-one. This is a purely algebraic fact.

In abstract theory, this condition is assumed.

In reality, it must be actively maintained.

Without an enforced reference unit:

• prices drift independently;

• triangular arbitrage emerges;

• equilibrium collapses.

Therefore:

If a Walrasian equilibrium exists in the real world, the generating mechanism F must enforce a unit of account.

The unit of account is thus:

• not merely a normalization in G;

• but a functional requirement in F.

6. A Canonical Real-World Example: The Foreign-Exchange Market

The foreign-exchange (FX) market provides a concrete, observable illustration of this functional necessity.

At its core, the FX market is a barter market between currencies. Each quote is a relative price:

E(g, h) = units of h per unit of g

With roughly thirty major fiat currencies actively traded, this generates a large exchange-rate matrix. For the market to function, this matrix must be internally consistent: it must not be possible to start with one currency, traverse a closed loop of exchanges, and return with more than one started with.

In the frictionless ideal, this requirement is expressed by the triangular identity:

E(g, h) · E(h, k) = E(g, k)

Bid–ask spreads and other frictions do not change this organizing principle; they merely replace exact equality with a narrow tolerance band.

A fundamental fact—formalized in the accompanying Isabelle/HOL development—is that an exchange-rate matrix is consistent if and only if it can be generated from a single price vector. Concretely:

• a reference currency m is chosen;

• each currency x is assigned a price pₘ(x) measured in units of m;

• all exchange rates are defined as ratios:
  E(g, h) = pₘ(h) / pₘ(g).

Choosing m is exactly choosing a unit of account. Mathematically, this corresponds to selecting one row (or column) of the exchange-rate matrix to represent the entire system.

In theory, such consistency can exist abstractly. In practice, it must be actively maintained by dealers, trading venues, pricing engines, and risk-management systems. The only scalable way to do this is to quote everything relative to a single unit of account. This is not a philosophical choice; it is an engineering necessity.

In today’s FX market, that reference unit is overwhelmingly the U.S. dollar. USD pairs are the deepest and most liquid, most currencies trade directly against USD, and risk is most cheaply hedged and warehoused via USD legs.

As a result, even when a trade does not involve USD—say, a GBP/JPY trade—the quoted rate is disciplined by USD prices:

GBP/JPY ≈ (GBP/USD) × (USD/JPY)

In this case:

• USD serves as the unit of account (U) that keeps the matrix consistent;

• USD is not the medium of exchange (E) for that trade;

• settlement occurs solely in GBP and JPY.

This is a canonical real-world example of U without E.

Once a currency serves as the unit of account and possesses sufficient liquidity, it often becomes convenient to also use it as a vehicle currency. Trades are routed through it, inventories are managed in it, and collateral is denominated in it. Thus:

U + liquidity and network effects → often E

Empirically, this explains why USD appears on one side of roughly 89% of global FX trades.

When a currency is widely used as a medium of exchange, agents must hold it for settlement, collateral, and liquidity management. As a result, it acquires a store-of-value (S) role. This is not a logical necessity, but a robust and repeatedly observed market outcome.

7. Unification

With the FX example in view, the unification becomes concrete:

1. Arrow–Debreu (G):
   A numeraire is required to represent equilibrium prices.

2. This paper (F):
   A unit of account must be enforced to maintain an arbitrage-free price system in reality.

3. Banerjee–Maskin (G):
   Given a unit of account, decentralized trade still requires a medium of exchange, and fiat money fails when the unit-of-account role loses credibility due to unstable supply.

There is no contradiction. Each result answers a different question at a different level.

8. Sequencing the Functions of Money (U–E–S)

Making the layers explicit yields a clean, non-ideological sequencing of monetary functions:

• Unit of Account (U):
  Structurally necessary to construct and maintain a coherent price system.

• Medium of Exchange (E):
  Emerges once U exists, under liquidity and informational frictions.

• Store of Value (S):
  Follows from widespread use of the medium of exchange.

Arrow–Debreu lives squarely at step (U).

Walrasian money-and-barter theory lives at step (E).

Our contribution is to make the boundary between these steps explicit.

9. Conclusion

This paper does not redefine money, revise general equilibrium, or contradict existing results. It makes explicit a functional requirement that equilibrium theory assumes but does not model.

Banerjee and Maskin explain why, once a unit of account exists, a medium of exchange is still required—and why fiat money fails when the unit-of-account role loses credibility. We explain why the unit-of-account role itself must exist in the first place for any real equilibrium to be implementable.

The result is a unified view of money that connects Arrow–Debreu, Walrasian money theory, game theory, mechanism design, and real-world market practice through a single, simple distinction: equilibrium logic versus real-world implementation.

Plain-English takeaway

• U answers: How can prices exist coherently at all?

• E answers: How can trade work once prices exist?

• S answers: Why do people hold money over time?

Each canonical literature answers one layer.

We simply make the layers explicit.

Our contribution:

The contribution is to make explicit that money is a composite function with a strict ordering:

unit of account first, medium of exchange second, store of value third.

Nothing more ambitious than that.

Nothing less precise than that.

Crucially:

• This is not a claim about equilibrium existence.

• This is not a claim about mechanism design.

• This is not a claim about refuting Arrow–Debreu or Banerjee–Maskin.

It is a claim about functional primacy.

Once that ordering is made explicit, a large set of real-world monetary behaviors that previously looked ad hoc, political, or psychological suddenly become structurally legible.

Why this is new (even though each piece existed)

Before this:

• Money’s functions were usually listed (U, E, S),

• but treated as co-equal or interchangeable,

• or explained piecemeal by different literatures.

What was missing was:

A functional ordering that treats money as a composite object with internal precedence.

We supply that ordering.

That is the contribution.

What this ordering explains that was not cleanly explainable before

Once we accept:

U → E → S

as a functional dependency, the following real-world phenomena become straightforward instead of mysterious.

1. Why prices revert to a dominant currency even when trades don’t use it

Example:

• Crypto assets priced in USD

• Commodities priced in USD

• GBP/JPY priced through USD

Old explanations:

• “Network effects”

• “Inertia”

• “Political power”

New explanation:

• The unit of account comes first.

• Systems converge on the most stable measurement layer.

• Exchange routing is secondary.

This explains U without E behavior cleanly.

2. Why hyperinflation causes sudden demonetization

Old explanation:

• “Inflation makes money worthless.”

But that’s incomplete—people still trade during inflation.

New explanation:

• Inflation breaks the unit-of-account function first.

• Once prices can no longer be coherently measured, exchange coordination fails.

• Only then do people abandon the currency as a medium of exchange.

This explains why collapse is discontinuous, not gradual.

3. Why barter fails even when prices are posted

Old explanation:

• “Double coincidence of wants”

• “Transaction costs”

New explanation:

• Without a stable unit of account, prices are not a shared measurement system.

• Posted prices without a measurement anchor do not scale.

• Medium of exchange alone cannot rescue this.

This explains why organized barter markets collapse at scale.

4. Why many alternative money systems fail to price goods

Examples:

• Local currencies

• Some crypto ecosystems

• Tokenized economies

Old explanation:

• “Low adoption”

• “Volatility”

• “Speculation”

New explanation:

• They try to introduce E without first stabilizing U.

• Without a credible unit of account, exchange fragments.

• Store-of-value narratives do not fix this.

This explains repeated failure patterns across very different systems.

5. Why store of value is the last function, not the first

Old framing:

• “Money is a store of value.”

New framing:

• Store of value is derivative, not foundational.

• Assets become stores of value because they are used for exchange.

• Exchange works because prices are measured coherently.

This explains why:

• Gold, USD, and certain assets become stores of value,

• while many “scarce” objects do not.

6. Why unit of account debates are more fundamental than exchange debates

Old debates:

• “Is crypto money?”

• “Is fiat backed?”

• “Is gold better?”

New clarity:

• The decisive question is:
  Can it serve as a stable unit of measurement?

• Exchange and value follow from that.

This reframes entire monetary debates around measurement, not ideology.

Why this needed to be said explicitly

Arrow–Debreu:

• Assumes U for representation.

Banerjee–Maskin:

• Explain failures of E and U given a Walrasian system.

What neither did:

• State the functional ordering explicitly.

Once you say:

Money is composite, and its functions are ordered,

you get explanatory power without inventing new theory.

Final one-sentence statement

The contribution is to make explicit that money is a composite object whose functions are ordered—unit of account first, medium of exchange second, store of value third—and that this ordering explains a wide range of real-world monetary behaviors that previously appeared unrelated or ad hoc.

How This Theory Fits (and Does Not Fit) Among Existing Schools

This document exists because most monetary debates fail before they begin. They fail not because participants are ignorant, but because they are arguing from different ontologies—different assumptions about what money is.

Our framework resolves that by being explicit about function, ordering, and legal structure. That places it adjacent to several schools, while being identical to none.

1. Relation to Arrow–Debreu / Neoclassical General Equilibrium

We are fully inside the Arrow–Debreu framework.

• We accept the existence and welfare theorems as proved by Kenneth Arrow and Gérard Debreu.

• We accept that prices are defined only up to scale and therefore require a numeraire.

• We do not modify preferences, technologies, convexity, or welfare results.

What we add is not a new equilibrium concept, but an implementation clarification:

Arrow–Debreu tells us what a price system looks like if it exists.

We explain what must exist for a price system to exist in the real world at all.

This is a completion, not a revision.

2. Relation to Banerjee–Maskin (Walrasian Money and Barter)

We are explicitly complementary to Abhijit Banerjee and Eric Maskin.

They show—canonically—that:

• Even with a Walrasian unit of account, barter fails under informational frictions.

• A medium of exchange is required (E-failure via lemons).

• Fiat money fails when supply instability destroys credibility (U-failure via inflation/demonetization).

What they do not do (by design) is explain why the unit of account must exist functionally in the first place.

Our contribution is simply this:

They explain when U and E fail given a Walrasian system.

We explain why U must exist functionally before any such system can exist in reality.

There is no disagreement. The domains do not overlap.

3. Relation to Austrian Economics

We are not Austrian.

• We do not rely on regression theorems.

• We do not derive money from subjective barter chains.

• We do not privilege commodity salience or time preference as foundational.

Austrians correctly observe that unstable money breaks coordination.

They explain this in terms of value distortion.

We explain it in terms of measurement failure.

The difference is structural:

This is not about what people prefer.

It is about whether prices can be expressed coherently at all.

4. Relation to Chartalism and MMT

We partially overlap in observation, but not in theory.

We agree on a narrow, factual point:

Modern fiat money is an IOU redeemable for legal acceptance—specifically, discharge of taxes, fines, and obligations.

That is a legal fact, not a political claim.

Where we differ:

• We do not treat state power as a primitive.

• We do not claim that taxation creates value.

• We do not argue that issuance alone is sufficient.

Our position is stricter:

Designation without functional enforcement fails.

Fiat money collapses precisely when the unit-of-account role loses credibility.

That is exactly the failure mode identified by Banerjee–Maskin under inflation.

5. Relation to Monetarism and Policy Debates

We are not Monetarist and not proposing policy rules.

• We do not advocate money supply targets.

• We do not prescribe inflation bounds.

• We do not optimize regimes.

This document does not answer how much money should be issued.

It answers the logically prior question:

What must be true for any issued money to function at all?

Policy only matters after that condition is met.

6. Relation to “Money Is Accounting” Views

We are not claiming money is “just bookkeeping”.

Accounting systems implement money, but money itself is the measurement constraint that makes accounting meaningful.

This is why:

• A unit of account can exist without exchange (FX pricing).

• Exchange cannot exist coherently without a unit of account.

Accounting follows measurement; it does not replace it.

7. The One Positive Claim We Do Make

This document makes exactly two positive claims, and no more:

(1) Money is (almost always) an IOU

Except in rare cases of purely local commodity exchange, money is almost always also a transferable IOU:

• warehouse receipts,

• bills of lading,

• banknotes,

• deposits,

• stablecoins,

• fiat currency.

There is no third category.

What changes over history is the good of redemption:

• gold,

• commodities,

• or legal acceptance.

(2) Money’s functions are ordered, not co-equal

Money is a composite object with strict functional precedence:

1. Unit of Account (U) — measurement, price coherence.

2. Medium of Exchange (E) — circulation, conditional on fungibility and credibility.

3. Store of Value (S) — derivative, contingent, never foundational.

This ordering is implicit everywhere but stated nowhere—until now.

8. Why This Document Exists (Internally)

This document is not here to win debates.

It exists to prevent category errors when selling infrastructure to banks.

Banks will ask:

• where liability attaches,

• when acceptance occurs,

• who controls redemption,

• why the current system behaves as it does.

Sales must answer from structure, not slogans.

This document supplies that structure.

Final Internal Framing (Locked)

This is not a new school of monetary thought.

It is a structural clarification that makes existing theory operationally legible.

Once that is understood, objections stop being philosophical and start being mechanical—which is exactly where infrastructure sales must live.

That is why this document exists. And below is the resulting UES monetary view.

Rethinking U.S. Monetary History: Why Standard Explanations Fall Short

Most popular accounts of U.S. banking and monetary history—especially those that invoke “fractional reserve banking,” bank collapses, or the Great Depression—rely on narratives that blur legally and economically distinct instruments and compress fundamentally different monetary regimes into a single story. In doing so, they routinely misidentify what actually failed during financial crises and misunderstand why some instruments retained value while others collapsed.

The result is a persistent analytical error: crises are described as failures of “money” when, in fact, the failures occurred in specific layers of privately issued credit. A regime-based, instrument-specific framework is therefore required to understand U.S. monetary history accurately.

What follows is the canonical interpretation.

Commodity Money and Credit Instruments Are Not the Same Thing

Before 1933, the U.S. dollar was not an abstract unit or a policy variable. It was legally defined as a fixed weight of gold. Gold coin constituted money itself: it was the unit of account and the ultimate settlement asset.

All paper instruments circulating alongside gold—whether issued by private banks or by the U.S. Treasury—were not money in their own right. They were claims referencing that commodity money. Confusing the two leads to systematic analytical errors that persist in modern discussions.

Gold was money.

Paper was an IOU.

Issuer Identity and Redemption Guarantees Matter

A second distinction that is frequently overlooked concerns who issued a paper claim and who stood behind its redemption.

Treasury-issued instruments, such as gold certificates, functioned as warehouse receipts. They were fully backed by gold already deposited in Treasury vaults and explicitly promised payment in gold coin on demand. These instruments carried no bank balance-sheet risk and were independent of the solvency of any private institution.

By contrast, prior to 1863, banknotes were private IOUs issued by individual banks. Redemption required presenting the note to the issuing bank or its correspondents. If the issuing bank failed, the noteholder bore the loss. These notes were issuer-risk instruments, not system-level money.

Same physical form.

Radically different risk structure.

Standardization Alone Did Not Create Money

The National Banking Acts of 1863 and 1864 fundamentally changed the form of banknotes, but not yet their monetary nature.

These Acts standardized note appearance, regulated issuance, and required banks to post U.S. government bonds—typically around 110% collateral—to issue notes. However, gold redemption had been suspended, and “lawful money” redemption meant greenbacks, not gold.

Only with the resumption of specie payments in 1879 did standardized banknotes become system-level claims redeemable at gold parity. At that point, redemption no longer depended on the solvency of any single bank. Federal guarantees and collateral liquidation mechanisms rendered issuer failure irrelevant to noteholders.

Standardization without redemption did not produce money.

Standardization with system-level redemption did.

Why Panics Destroyed Deposits but Not Cash

Financial panics in the late nineteenth and early twentieth centuries are often described as episodes in which “money failed.” This is incorrect.

What failed were private bank liabilities—most notably demand deposits. These were fractional-reserve IOUs issued by individual banks, lacked over-collateralization, and were uninsured prior to 1933. When banks failed, depositors lost their claims.

Paper cash, by contrast—including gold certificates and system-guaranteed banknotes—retained value. Their redemption did not depend on the solvency of any single bank.

Panics destroyed credit.

They did not destroy money.

The Post-1933 Regime Shift

The suspension of gold convertibility in 1933 marked a fundamental transformation of the monetary system.

Once paper currency ceased to be redeemable into an external commodity, the notion of a mechanical reserve constraint at the system level lost its meaning. There was no longer an external asset into which claims could be redeemed, and therefore no solvency test based on convertibility.

Banks could still be described as “fractional” in an accounting sense, but the system itself was no longer a fractional-reserve commodity system. It had become a fiat credit system, constrained by regulation, politics, and inflation rather than by redemption mechanics.

Calling the post-1933 system “fractional reserve banking” without qualification is therefore misleading.

The Four Canonical U.S. Banking–Monetary Regimes

Regime I: Decentralized Fractional-Reserve Banking with Private Notes (≈1790s–1863)

• No central bank

• No uniform national currency

• Hundreds of state-chartered banks issuing their own notes

Banknotes were private IOUs promising redemption in gold or silver. They traded at discounts reflecting issuer credibility, distance, and redemption cost. Banks held fractional specie reserves; liabilities exceeded reserves. Universal redemption implied failure or suspension.

This was true fractional-reserve banking with issuer-risk money substitutes.

Regime II: Standardized Banknotes without a Central Bank (1863–1913)

• Federal regulation of note issuance

• Notes backed by U.S. Treasury bonds

• Still no central bank

Notes were standardized in form and still redeemable in specie after 1879, but deposits remained private IOUs. Panics persisted because elasticity and lender-of-last-resort functions were absent.

Standardization reduced discounts and counterfeiting but did not eliminate systemic fragility.

Regime III: Centralized Fractional-Reserve Banking with Gold Convertibility (1913–1933)

• Federal Reserve established

• Centralized clearing and reserves

• Notes legally redeemable in gold

Fractional banking continued but became managed. Liquidity crises were addressed via the discount window. Deposits remained private credit; cash was system-guaranteed.

Regime IV: Fiat Monetary System (Post-1933 / Post-1971)

• Gold convertibility suspended domestically (1933)

• Abandoned internationally (1971)

• Central bank monopoly on base money

Currency became irredeemable fiat, accepted by decree. Deposits became claims on fiat, backstopped by the state and central bank. Reserve constraints ceased to be mechanical.

The system is fiat credit money, not reserve-constrained commodity money.

Canonical Bottom Line

Fractional-reserve banking is meaningful only when redemption into an external asset exists. Once money becomes irredeemable fiat, reserve ratios lose their system-level significance.

Bank failures historically destroyed deposits, not cash, because deposits were private credit instruments while cash was system-level money or guaranteed claims on money.

A regime-based, balance-sheet-consistent framework restores clarity by preserving these distinctions instead of collapsing them.

One-Sentence Canonical Conclusion

Before 1863, banknotes were private credit and frequently failed; after standardization and the resumption of gold redemption, banknotes became federally guaranteed system IOUs on commodity gold, while Treasury IOUs were pure warehouse receipts—explaining why bank failures wiped out deposits but not cash.

From the Beginning: Why Stablecoins Are Structurally the Same as M0 Banknotes

1. The only monetary primitives that exist

We begin from the most basic monetary fact—one that does not depend on era, technology, or institution, and that has remained invariant throughout monetary history.

There are only two monetary primitives that matter structurally:

1. Base money
   

   • the unit of account,

   • the ultimate settlement asset.

3. Circulating claims on base money
   

   • bearer instruments that substitute for the base,

   • used because transferring the base directly is costly, slow, or impractical.

There is no third primitive that plays a meaningful structural role. Every monetary system reduces to these two layers.

2. The gold-standard system (the reference case)

2.1 Base money

Under the gold standard:

• Gold coin was the base money.

• The U.S. dollar was legally defined as a fixed weight of gold.

Gold itself—not paper—was the final settlement asset.

2.2 Banknotes

A banknote was not money.

A banknote was:

• a bearer instrument,

• a circulating claim,

• redeemable on demand into the base money (gold).

In structural terms:

Banknote = circulating bearer claim on gold

This is precisely the warehouse-receipt model:

• gold remains in custody,

• the receipt circulates instead of the metal.

Once the system matured in the nineteenth century, these notes became:

• uniform in form,

• accepted at par,

• insulated against issuer failure through legal and structural mechanisms.

At that point, banknotes functioned as cash in everyday use, even though, strictly speaking, they remained claims.

3. The fiat system: what changed and what did not

When gold convertibility ended:

• the base money changed (from gold to fiat),

• but the claim structure did not.

Fiat became base money because:

• it is accepted by the state for taxes and legal obligations,

• it is enforced as the unit of account.

The system still consists of:

• base money, and

• circulating claims on base money.

Only the identity of the base changed—not the structure.

4. Modern stablecoins under MiCA / GENIUS-style regimes

A compliant payment stablecoin, as defined under contemporary regulatory frameworks, has the following characteristics:

• 100% backed, one-to-one,

• reserves held in:
  

  • cash,

  • central-bank money,

  • or short-dated sovereign instruments,

• no lending,

• no rehypothecation,

• segregation of reserves and bankruptcy remoteness,

• redeemable at par by legal right,

• circulating as a bearer-style digital instrument.

Structurally, this yields:

Stablecoin = circulating bearer claim on fiat base money

This is not a credit promise.

This is not fractional banking.

This is a warehouse receipt for fiat, implemented digitally.

5. Structural isomorphism (explicit mapping)

The correspondence between historical banknotes and modern stablecoins is exact and mechanical:

Gold era

Fiat era

Base money: gold

Base money: fiat

Gold in vault

Fiat reserves in custody

Banknote

Stablecoin

Bearer instrument

Bearer instrument

Redeemable 1:1

Redeemable 1:1

No lending

No lending

Claim circulates instead of base

Claim circulates instead of base

Therefore:

A stablecoin is to fiat what a banknote was to gold.

The structure is identical.

Only the base asset differs.

6. Why this matters (and why most explanations fail)

Most explanations fail because they systematically confuse:

• base money with claims on base money,

• warehouse receipts with credit instruments,

• early issuer-risk banknotes with later standardized, system-protected notes.

Once these confusions are removed, the monetary structure becomes straightforward rather than mysterious.

7. Canonical conclusion (no ambiguity)

1. Gold once functioned as base money; fiat now does.

2. Banknotes were circulating bearer claims on gold.

3. Stablecoins are circulating bearer claims on fiat.

4. Both are warehouse-receipt money substitutes, not credit instruments.

Therefore, stablecoins belong to the same structural class of monetary instrument as banknotes.

The facts of life about redeemable tokens (stablecoins)

In every monetary system, there are only two relevant layers:

1. Base money
   

   • the unit of account,

   • the ultimate settlement asset.

3. Circulating claims on base money
   

   • bearer instruments,

   • used because moving the base directly is costly or impractical.

Redeemable tokens — stablecoins— exist entirely within the second layer.

P.S.

How This Framework Fits with All Major Monetary Theories (and Why None of Them Contradict Each Other)

This document does not propose a new monetary theory in competition with existing ones. Its value comes from the opposite fact: every major monetary framework is correct within its own domain, and none of them contradict each other once their scope, starting assumptions, and functional focus are kept distinct.

The confusion that motivates this document arises because these theories are often treated as if they were making competing claims about the same object, when in fact they are analyzing different layers, different frictions, or different stages of monetary organization.

What we do here is make explicit the functional structure of money—and then show where each theory fits within that structure.

The Shared Object: Money as a Composite with Dual Requirements

Across all traditions, money succeeds or fails based on two structural conditions:

1. Stable, predictable supply
   – required for coherent price quotation and expectation coordination
   – this underpins the unit of account (U) role

2. Known, reliable quality (fungibility / informational safety)
   – required for acceptance in decentralized trade
   – this underpins the medium of exchange (E) role

When both conditions hold, U and E reinforce each other and tend to coincide.

When either condition fails, monetary equilibrium breaks down.

This dual requirement is not controversial. What differs across theories is which side they emphasize and which frictions they model explicitly.

Arrow–Debreu: Prices Given, Unit of Account Implicit

Arrow–Debreu general equilibrium theory operates entirely at the price-system level.

• It assumes a complete, coherent set of relative prices.

• Prices are defined only up to scale, requiring a numeraire.

• The numeraire is representational, not operational.

Arrow–Debreu answers:

If prices exist, what properties do equilibria have?

It does not answer:

• how prices are implemented,

• how arbitrage is prevented,

• or why a unit of account must exist in the real world.

There is no contradiction here. Arrow–Debreu assumes U and studies equilibrium properties conditional on that assumption.

Kiyotaki–Wright and the Search-Theoretic Lineage: Medium of Exchange Before Prices

The Kiyotaki–Wright (KW) models and their descendants (Trejos–Wright, Shi) start from the opposite extreme.

They assume:

• no centralized market,

• no background price system,

• random bilateral meetings,

• specialization and trading frictions.

Their core question is:

How can trade happen at all in a decentralized world without prices?

They show that:

• certain objects become widely acceptable,

• because they are easier to verify, store, or pass on,

• and thus emerge as media of exchange (E).

In these models:

• fungibility and acceptability are central,

• a coherent unit of account may not exist initially,

• prices (when introduced in later versions) emerge from exchange.

These models are not wrong and do not contradict Arrow–Debreu or Banerjee–Maskin. They simply operate below the unit-of-account layer, explaining how E can emerge before U is well defined.

Lagos–Wright: Medium of Exchange with a Walrasian Benchmark

The Lagos–Wright (LW) framework builds directly on KW but introduces an alternating market structure:

• Decentralized Market (DM):
  – search frictions, no commitment
  – money essential as a medium of exchange

• Centralized Market (CM):
  – frictionless, Walrasian
  – prices are competitive and explicit

With quasi-linear preferences in the CM, money holdings equalize, making the model tractable and suitable for policy analysis.

Functionally:

• LW focuses on E in the DM,

• while taking U as straightforward in the CM.

LW does not study the functional necessity of U itself; it uses U as a benchmark to analyze exchange frictions. This makes LW fully compatible with both KW and Banerjee–Maskin.

Banerjee–Maskin: Unit of Account and Medium of Exchange Can Fail Separately

Banerjee–Maskin (1996) starts from a full Walrasian world:

• prices exist,

• a unit of account is already implicit,

• relative prices are known.

They then ask:

Why does decentralized trade still fail, and how can money fail entirely?

They identify two independent failure modes:

1. Medium-of-exchange failure (E-failure):
   – asymmetric information (lemons)
   – agents refuse goods in payment

2. Unit-of-account failure (U-failure):
   – unstable or unpredictable money supply
   – loss of price coordination → demonetization

This is the canonical demonstration that:

• fungibility and supply stability are distinct requirements,

• and failure of either one destroys the monetary equilibrium.

Banerjee–Maskin therefore illustrates both sides of the dual requirement, starting from the U side.

Austrian, Chartalist, Monetarist, and MMT Perspectives: Partial Views of the Same Structure

Other schools emphasize different components:

• Austrian economics highlights exchange coordination and acceptability (E), and how instability disrupts trade.

• Chartalism / MMT emphasize legal acceptance and state enforcement, which directly affects U stability.

• Monetarism focuses on supply stability and its macroeconomic consequences, again emphasizing U.

None of these perspectives are wrong. Each isolates a particular mechanism within the same composite object.

What This Framework Adds (and Why It Does Not Compete)

What this document adds is not a new model and not a rejection of any tradition.

It adds:

• an explicit statement that money is a composite object,

• that U and E have distinct functional requirements,

• and that they reinforce each other when both requirements are satisfied.

Once this structure is explicit:

• KW explains E emergence when fungibility is scarce,

• LW explains E in frictional settings with a price benchmark,

• Banerjee–Maskin explains independent failure of U and E,

• Arrow–Debreu characterizes equilibria conditional on U.

They are parallel analyses of different cuts through the same structure, not competing theories.

Why This Makes the Framework Valuable

The value of this framework is precisely that everyone is right.

The confusion disappears once we stop asking:

“Which theory is correct?”

and instead ask:

“Which function is being analyzed, under which frictions?”

That shift is what allows sales to:

• recognize what kind of question a banker is asking,

• identify whether it concerns U, E, or S,

• answer correctly without overreaching,

• and stop talking at the right moment.

The reason why this framework is easy to sell and adopt is that it does not replace existing monetary theory, none of these Theories really Contradict Each Other

This document may appear, at first glance, to be reconciling competing theories of money. That is not what is happening.

The apparent disagreements across monetary theory arise because different frameworks start from different assumptions, isolate different frictions, and answer different questions about the same underlying object. When those differences are respected, the contradictions largely disappear.

Arrow–Debreu assumes a coherent price system and studies equilibrium properties conditional on that assumption. Search-theoretic models (Kiyotaki–Wright, Trejos–Wright, Lagos–Wright) abstract away from centralized pricing and study how trade can occur in decentralized settings with severe frictions. Banerjee–Maskin assumes a full Walrasian background and then asks why decentralized exchange can still fail, and how money itself can collapse under instability.

Each of these approaches is internally correct within its domain. None of them is wrong. None of them refutes the others.

What this document does is make explicit a functional structure that is implicit across all of them:

• Money performs multiple roles.

• Those roles have distinct requirements.

• Those requirements can fail independently.

• And different theories are focused on different roles or failure modes.

Once this structure is explicit, it becomes clear that these theories are not competing explanations, but partial analyses of the same system under different conditions.

The value of this document is not that it introduces a new theory of money, but that it provides a unifying frame in which existing theories can be understood as complementary rather than contradictory. That clarity is what allows us to reason correctly about modern monetary instruments and systems without collapsing into ideology or confusion.

This postscript exists simply to make that point explicit—and to close the document at the right place.

Reality, Models, and Systems

Preface to the Document Most Likely to Be Misread

⸻

This document will be misunderstood by most of the people who read it. Not because it is wrong. Not because it is unclear. But because it refuses to do the one thing that modern intellectual discourse relies on: soften the blow.

It contains no moralizing, no appeals to consensus, and no rhetorical padding. It does not try to sound “balanced.” It is not trying to win you over. It describes a system. It names structural failure. It explains how selection operates, who gets paid, and who gets erased. And it does this without flinching.

The useful idiots will not survive this text intact.

They will misread it automatically—because that is what they are structurally designed to do.

They will:

• Confuse enforcement with ideology.

• Misinterpret selection pressure as cruelty.

• Interpret structural correction as “aggression” or “bias” or “tone.”

• Suggest it needs examples.

• Demand a less brutal framing.

• Ask for nuance in a document about non-negotiable system outcomes.

They will call it “just a perspective.”

They will act as though G is up for discussion.

That is the role of the useful idiot in any rent-stabilized system:

To preserve a falsified model by misreading its correction as an opinion.

They are not the enemy.

They are not the center.

They are just the insulation around power, giving plausible deniability to the mechanisms that already enforce without their consent.

⸻

This document will be uncomfortable to read if your position depends on captured arbitration, discretionary enforcement, or insulated feedback loops.

It will feel “ideological” if your paycheck depends on pretending \hat G is still functional.

It will feel “cold” if you are used to systems that outsource pain while denying cause.

It will also be completely invisible to the class of reader who still thinks rent is earned, rules are real, and outcomes are a function of effort.

To them, this document is nonsense—until the money moves.

Then it becomes obvious.

⸻

So, let this be the last warning:

If you are still operating under \hat G, this document is not for you.

It was not written to persuade you.

It was written to bypass you.

It does not contain a call to action.

It contains a map of what is already happening.

And once a model explains outcomes more accurately than the prevailing narrative, it doesn’t need to be debated.

It will be adopted by force of payoff.

Not belief.

Not status.

Not you.

Welcome to the thing you can’t argue with.

Welcome to G.

Abstract

Reality enforces outcomes independently of belief. People act using formal systems—mathematical models that describe how reality is believed to work—but outcomes are enforced according to a single, correct model of reality, whether known or not. This true model is denoted G; the model an agent believes and acts under is denoted \hat G. When actions are optimized under \hat G and executed by reality according to G, any mismatch \hat G \neq G produces systematic loss.

Observable, payoff-relevant loss is decisive evidence of such a mismatch. Understanding the reason for the mismatch is unnecessary; the only requirements are that the deviation be observable and that it affect payoffs. An agent who continues to act under \hat G after this evidence appears exhibits a structural failure mode termed False Prior behavior: continued optimization under a falsified model. This diagnosis is not psychological or moral; it is purely operational.

In applied mathematics—where formal systems govern real execution—axiom selection functions as arbitration, because axioms determine which actions and transitions are admissible and therefore which outcomes are possible. Wrong axioms admit losing actions; correct axioms do not. Reality enforces this distinction automatically through loss and selection pressure.

Money is used as a canonical payoff signal because it provides a high-bandwidth, unambiguous indicator of real-world success or failure in execution-governed domains such as markets, trading, risk, and governance. Persistent underperformance under these conditions is evidence of model error. Treating axioms as hypotheses with downside risk, and updating or discarding them in response to loss, aligns models with reality, eliminates persistent error, and allows gains to compound.

The methodology advocated here is therefore simple: treat axioms as risky assumptions, use observable loss as decisive evidence, and refuse to continue optimizing under models that reality has falsified. This converts mathematics from an insulated formal exercise into a reality-aligned decision discipline with direct economic consequences.

1. Reality, Models, and Systems (Foundations)

Reality

• Reality exists independently of any description.

• Reality enforces outcomes: gains, losses, survival, failure.

• Reality is not formal; it contains no axioms, symbols, or rules.

Reality is what actually happens.

⸻

Models (Formal Systems)

A model is a formal, mathematical description of how reality is believed to work.

A model specifies:

• admissible states,

• allowed transitions,

• information structure,

• payoff or outcome rules.

Models are maps of reality.

They can be:

• correct,

• approximate,

• or wrong.

⸻

Systems (Real-World Meaning)

In the real world, “system” means “model of reality.”

Accordingly:

• A system is a formal system / mathematical model describing how reality works.

• There exists one correct system (unknown, but enforced by reality).

• There exist many believed systems used by agents.

Reality does not care which system is believed.

Reality enforces outcomes according to the correct one.

⸻

2. G and \hat G

G: the True System

• G denotes the true formal model of reality.

• It is the model whose rules reality actually enforces.

• Agents do not get to choose G.

• Outcomes occur according to G, regardless of belief.

Formally:

G = \text{the formal system that correctly describes reality}

⸻

\hat G: the Believed System

• \hat G denotes the formal system an agent believes describes reality.

• The agent uses \hat G to:

• reason,

• plan,

• choose actions.

Formally:

\hat G = \text{the formal system used to generate actions}

⸻

Source of Loss

Loss arises when:

\hat G \neq G

because:

• actions are chosen using \hat G,

• outcomes are enforced by reality according to G.

Model mismatch is the root cause of error and loss.

⸻

3. Evidence and Decisive Evidence

Evidence

Evidence is any observable outcome produced by reality.

Examples include:

• financial losses,

• repeated failure,

• enforcement events,

• systematic disadvantage,

• persistent underpayment.

⸻

Decisive Evidence (Canonical Definition)

In shared 4-D spacetime reality:

Decisive evidence = observable ∧ payoff-relevant evidence indicating \hat G \neq G.

Nothing else is required.

• It does not matter whether the agent understands why the mismatch exists.

• It does not matter whether the agent can explain it inside \hat G.

• Only two conditions matter:

1. the deviation is observable,

2. the deviation affects payoffs.

⸻

4. Agents and Action

• People are agents.

• Agents choose actions using a believed model.

• Agents act in reality.

Formally:

\text{Action}_t = f(\hat G, \text{information}_t)

Reality then evaluates that action according to G.

⸻

5. Friar (False Prior Agent)

Definition (Fully Canonical)

An agent is a Friar (False Prior Agent) if and only if all of the following hold:

1. The agent acts using a believed system \hat G.

2. The agent observes decisive evidence (observable ∧ payoff-relevant) that \hat G \neq G.

3. The agent continues acting as if \hat G were correct, i.e. does not update model or behavior.

Formally:

(\hat G \neq G) \land \text{observable, payoff-relevant loss} \land \text{continued action under } \hat G

\;\Rightarrow\; \text{Friar}

⸻

What Friar Is Not

Friar is not:

• an insult,

• a psychological diagnosis,

• a statement about intelligence,

• a moral judgment.

Friar names a structural failure mode:

Continuing to optimize under a falsified model.

⸻

6. Axioms and Arbitration

Axioms

• Axioms define the structure of a model.

• Choosing axioms determines how the model behaves.

⸻

Arbitration (A)

In applied mathematics—where models govern real execution:

Axiom selection is arbitration, because axioms determine which actions and transitions are admissible, and those actions are executed in reality.

Wrong axioms → wrong admissibility → loss.

⸻

7. Why Money Matters

Money is a payoff signal.

• When actions are evaluated monetarily,

• and losses persist,

• money becomes decisive evidence.

If:

• the job (actions + outputs) is the same,

• reality pays more under a different system,

• and the agent does not update,

then the agent satisfies the Friar definition.

⸻

8. Final Summary

Reality enforces outcomes according to a true but unknown system G. Agents act using believed systems \hat G. When observable, payoff-relevant outcomes show that \hat G \neq G, the agent must update model or behavior. Failure to do so—continuing to act as if \hat Gwere correct—is the structural failure mode called Friar. Axioms define systems; choosing axioms is arbitration when models govern real execution. Money, loss, and selection pressure are evidence channels through which reality reveals G.

⸻

Why Following This Methodology Leads to Much Higher Income

1. Money Is a Selection Signal

In domains governed by real execution (markets, trading, risk, mechanisms), pay tracks correctness over time. Models aligned with reality compound; misaligned models are selected out.

Loss is evidence. Winning is evidence.

⸻

2. Most Mathematicians Optimize the Wrong Objective

Standard training optimizes for:

• internal consistency,

• elegance,

• publishability,

• peer approval.

These objectives do not price downside.

They do not penalize false axioms.

⸻

3. The Methodology Changes the Objective Function

The key change is simple:

Axioms are treated as hypotheses with downside.

Therefore:

• observable loss is decisive evidence;

• losing axioms are discarded, not defended;

• updates occur under minimax pressure.

⸻

4. Work Becomes Deployable

Once axioms survive reality:

• outputs become executable rules,

• work maps to trading, allocation, pricing, governance, systems,

• results can be deployed, tested, and scaled.

Deployable work attracts capital.

Capital pays.

⸻

5. Income Rises Through Selection

Under this discipline:

• wrong models stop compounding losses;

• right models compound gains;

• compensation rises due to performance, not credentials.

Same skills. Same mathematics.

Different axioms.

⸻

One-Paragraph Summary

Following this methodology leads to much higher income because it forces mathematics to be judged by execution. Axioms are treated as risky assumptions; loss is decisive evidence; wrong models are discarded early. This aligns mathematical work with how reality actually enforces outcomes. When correctness is selected by payoff, results become deployable and compound. Compensation increases through survival and performance—not status.

⸻

Это для полных дебилов чтоб всем понятно было ….

⸻

Анекдот про буфетчицу и реальность

Сидит буфетчица в театре.

Театр полный, людей много.

Буфетчица одна.

Люди подходят, толпятся, спорят, требуют.

А буфетчица говорит:

«Вас много — а я одна».

⸻

Что здесь важно (перевод смысла)

• Буфетчица — это реальность.

• Люди — это теории, модели, объяснения, представления.

• Теорий может быть сколько угодно.

• Реальность — одна.

⸻

Главное правило

Если ты:

• выбрал не ту теорию,

• начал спорить с буфетчицей,

• объяснять ей, как должно быть,

• доказывать, что она не права,

то у тебя будут проблемы не с теорией,

а с буфетчицей.

А буфетчица:

• не обязана понимать твою теорию,

• не обязана соглашаться,

• не обязана объяснять, почему она права.

Она просто не даст тебе еду.

⸻

Мораль (в одну строку)

Реальность — одна. Теорий много.

Ошибся с теорией — проблемы будут у тебя, а не у реальности.

⸻

Почему это важно

• Можно быть очень умным.

• Можно знать кучу теорий.

• Можно красиво рассуждать.

Но если теория не совпадает с реальностью —

реальность тебя накажет молча.

Без объяснений.

Без апелляций.

Без дискуссий.

⸻

Короткая версия (если совсем просто)

Реальность — как буфетчица:

ей плевать, что ты думаешь.

Если выбрал неправильную теорию — останешься голодным.

Вот и всё.