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Money

The Unit of Account Axiom: A Unified Theory of Money from Gold to Bitcoin
by Joseph Mark Haykov
July 1, 2025

Abstract
Axiom: In all functioning economies, a single, unique Unit of Account (UoA) constitutes the essential and defining function of money, necessary for coherent, arbitrage-free valuation, pricing, and exchange. To date, no real-world counterexamples have emerged.

This paper develops a unified informational theory of money’s origins, integrating two influential yet traditionally opposing traditions:

• Austrian (Mengerian): Markets spontaneously select the most marketable commodity as a medium of exchange (MoE), implicitly assigning it monetary functions.

• Chartalist/MMT: States explicitly impose a monetary standard top-down through taxation and legal tender laws, formally aligning prices within a unified numéraire.

We demonstrate that these traditions describe complementary, concurrently operating mechanisms within real-world monetary systems. Specifically, decentralized market optimization toward the most credible informational anchor (commodity money such as gold or Bitcoin, particularly in international trade contexts) and deliberate state enforcement of an informational standard (fiat currencies like the U.S. dollar or the Soviet ruble) coexist, mutually reinforcing the informational necessity of a stable, unique UoA.

This unified informational criterion—stability and predictability of the UoA—bifurcates into two complementary roles:

1. Spot-Price Anchor (UoA → MoE): A stable and universally recognized UoA facilitates widespread acceptance as an immediate medium of exchange, ensuring coherent spot valuations.

2. Forward-Price Anchor (UoA → SoV): A predictable and transparent UoA enables reliable intertemporal valuations, supporting the effectiveness of money as a store of value (SoV).

By explicitly defining monetary emergence as decentralized optimization against informational constraints, we elucidate the historical ascendancy of commodity money, its transition to ledger-based fiat systems, and the contemporary emergence of decentralized digital bearer assets like Bitcoin.

This unified axiom fully reconciles with foundational mathematical economics—including Arrow-Debreu general equilibrium and canonical game-theoretic models—and accurately predicts empirical phenomena ranging from hyperinflation-driven currency substitution (e.g., Argentina, Soviet Union) to spontaneous parallel numéraire adoption during institutional breakdowns.

Keywords: Unit of Account, Medium of Exchange, Store of Value, Monetary Theory, Austrian Economics, Chartalism, General Equilibrium, Bitcoin, Commodity Money, Fiat Currency, Informational Economics, Monetary Stability

JEL Classification Codes:
E40 (Money and Interest: General),
E42 (Monetary Systems; Standards; Regimes; Government and the Monetary System),
B53 (Austrian Economics),
D51 (Exchange and Production Economies),
G15 (International Financial Markets)

Introduction

A central question in monetary theory—still debated after centuries—is why money exists at all, and what function makes it indispensable to markets. One of the most rigorous and widely-cited modern attempts to answer this question is Abhijit V. Banerjee and Eric S. Maskin’s seminal work, “A Walrasian Theory of Money and Barter” (Banerjee & Maskin, 1996). Their model, grounded in general equilibrium theory, provides a powerful and clear explanation of how a common medium of exchange can emerge endogenously in competitive markets—without institutional design or fiat decree.

Their central insight is that informational asymmetry—specifically, agents’ inability to verify the quality of unfamiliar goods—naturally leads markets to select a single good as the preferred medium of exchange. This selection arises from several key factors:

• Goods exist in both high- and low-quality forms.

• Agents can only reliably verify quality in goods that they personally produce or consume.

• Adverse selection discourages trade in unfamiliar goods, creating demand for a universally trusted asset.

• Consequently, the good with the smallest quality gap becomes the medium of exchange, as it carries the least risk and is easiest to verify.

Their model yields several significant results:

• Emergence: A unique medium of exchange arises spontaneously from market dynamics, independent of government fiat.

• Inefficiency: This chosen medium is produced in excess of its intrinsic consumption value due to demand driven by its exchange role.

• Role of Fiat: Introducing verifiable, costless fiat money can correct this inefficiency and enhance welfare.

• Instability: Excessive inflation or loss of verifiability pushes markets back toward commodity money—a phenomenon mirrored in real-world episodes of demonetization.

The theoretical innovation in Banerjee and Maskin’s framework is its emphasis on verifiability and price stability, rather than simply liquidity or transactional convenience. Thus, it provides a compelling refinement to the classic “double coincidence of wants” narrative, demonstrating that quality uncertainty—not merely search costs—can independently give rise to monetary exchange.

However, a fundamental limitation exists within their otherwise rigorous framework:

Their analysis presumes—rather than explains— the existence of a unit of account (UoA). Prices are assumed to already be quoted in a single metric, leaving only the question of which good emerges as the medium for settling trades. Consequently, their model exclusively addresses the medium-of-exchange (MoE) function, overlooking the logically prior and necessary unit-of-account (UoA) function.

This oversight is not trivial. As argued in this paper, the UoA is not a derivative property arising from exchange; rather, it is the informational foundation upon which all monetary systems fundamentally depend—whether these systems emerge bottom-up, as described in Menger’s spontaneous order theory, or top-down, as articulated in Chartalist or Modern Monetary Theory (MMT) frameworks. Without a unit of account, no coherent system of prices is possible, valuation becomes meaningless, and consequently, market exchange itself collapses.

The goal of this paper is therefore to demonstrate that the unit of account is the true universal axiom of money—one underpinning and unifying every known monetary system, admitting no real-world counterexamples. By explicitly establishing this axiom, we can systematically explain not only why various forms of money historically emerged but also why they all ultimately serve this same informational function, from ancient commodity money (gold), through contemporary fiat systems (the U.S. dollar), to modern decentralized digital assets (Bitcoin).

What’s the Problem?

Banerjee and Maskin’s model is analytically rigorous within its stated parameters: it convincingly demonstrates how a single good can emerge as a medium of exchange (MoE) under conditions of quality uncertainty and informational asymmetry—analogous to how CarFax reports or lemon laws resolve informational frictions in Akerlof’s famous "market for lemons." Yet, despite its mathematical elegance, the model contains a foundational conceptual oversight that limits its explanatory power regarding the true origin and nature of money.

The crux of the problem is as follows:

The Banerjee-Maskin model assumes from the outset that money’s core and primary function is as a medium of exchange. All other monetary properties, including the unit of account (UoA) and store of value (SoV), are treated as secondary consequences derived from this MoE function. This assumption is not merely a neutral modeling convenience; it is the explicit foundation of their analytical framework. Their analysis explains convincingly how one particular good begins to circulate more widely (thus becoming “money”) due to informational frictions, but it does not justify why that good should simultaneously serve as a universal pricing standard.

In real-world economies, as Banerjee and Maskin themselves implicitly acknowledge, people exchange goods and services in markets—placing what they produce “in the pile” and withdrawing what they consume—all at consistent, arbitrage-free (one-price-per-asset) exchange ratios. Achieving this coherent and universal pricing system necessarily requires a pre-existing unit of account—a single, stable standard in which all prices are quoted—long before any specific good emerges as the preferred medium of exchange.

This point is not merely semantic, but rather fundamentally logical:

• A unit of account is a prerequisite for rational valuation.

• Rational valuation, in turn, is a prerequisite for coherent and efficient exchange.

Thus, exchange necessarily presupposes valuation, and valuation necessarily presupposes a common metric—a universal unit of account. To model the emergence of a medium of exchange while implicitly assuming a functioning pricing system is to erroneously treat an effect (MoE) as though it were the fundamental cause.

Therefore, the correct logical hierarchy is explicitly:

1. Unit of Account enables

2. Valuation, which subsequently enables

3. Exchange.

Just as physics requires a fundamental unit of length before measuring distances, economics similarly requires a fundamental unit of account before measuring value. This essential logic holds true even for "near-money" assets such as casino chips—assets which might never function as traditional media of exchange or stores of value yet still serve as units of account within their specific contexts. Thus, the unit-of-account role represents money’s foundational and universal function, one that all forms of money—near, full, far, perfect, or otherwise—must inherently fulfill.

Historical Clarification

Concrete historical examples reinforce the logical primacy of the unit of account (UoA):

German Deutsche Mark Example:
The Deutsche Mark (DM) developed its store-of-value (SoV) function explicitly because it had gained widespread acceptance as a reliable medium of exchange (MoE). When its transactional acceptance declined—particularly during Germany’s transition to the Euro—its utility as a dependable store of value correspondingly vanished. This vividly illustrates that SoV functionally depends upon MoE. Crucially, however, both MoE and SoV fundamentally rely upon the prior existence of a stable and widely recognized unit of account. The correct and complete monetary hierarchy is therefore explicitly:

UoA→MoE→SoV

Silver and Gold Historical Example:
A similar dynamic emerges historically with gold and silver. Both metals were historically recognized as monetary units, stabilizing their relative price ratio at approximately 15:1 for centuries. The gradual demonetization of silver—initiated in the 1850s and completed by the late 1890s—removed its monetary function, causing its relative value to sharply decline. Today (June 2025), the gold-to-silver price ratio exceeds 90:1. This dramatic divergence clearly illustrates that the loss of the unit-of-account function directly undermines a commodity’s capacity to function as a stable store of value.

Rai Stones Example (Island of Yap):
The Rai Stones of Yap provide a profound validation of the primacy of the unit-of-account function. Although physically immovable, these massive stone discs served effectively as money precisely because Yapese society enforced a unified UoA, denominating value explicitly in standardized stone-units. This UoA enabled an abstracted medium of exchange: transactions were settled not by physically moving stones, but rather by socially acknowledging changes of ownership within the collective ledger of communal memory. This abstraction precisely mirrors how the U.S. dollar functions as the numéraire in global FX markets (e.g., JPY/GBP trades), where cross-rates derive from dollar-pair benchmarks without physical dollar transfers.

This is not an exception to the hierarchy UoA → MoE—it is, in fact, its purest expression. When physical transfer is impossible (as with Rai Stones) or inefficient (as in global forex), the UoA becomes the necessary informational scaffold enabling an abstracted MoE to function effectively. Remove the UoA, and neither system remains viable: Yapese trade collapses into inefficient barter, while forex markets dissolve into arbitrage chaos. The Rai Stones thus conclusively demonstrate that MoE emerges as a consequence of a pre-existing UoA—not its prerequisite.

Banerjee and Maskin are not fundamentally incorrect; their theory is simply incomplete. Their model implicitly presumes a functioning unit of account and focuses solely on how a medium of exchange subsequently emerges under informational frictions. However, the deeper foundational question remains unanswered:

How does a unit of account itself emerge, particularly under real-world informational asymmetries? And why does one specific good or standard preemptively become the universal numéraire?

This is the fundamental question the present paper explicitly addresses—by articulating the unit-of-account axiom and demonstrating its comprehensive explanatory power across all known monetary systems, supported by the absence of any real-world counterexamples.

Money as an Informationally Essential Unit of Account

In mathematical economics, precision is paramount. Foundational frameworks such as the Arrow-Debreu general equilibrium model—upon which the First and Second Welfare Theorems depend—require explicit definitions. Within this framework, the numéraire is chosen to express all prices in consistent relative terms.

In purely theoretical terms, it might seem that the Arrow-Debreu numéraire could be an arbitrary good selected solely for mathematical convenience. However, this overlooks a crucial logical constraint: In real-world economies, the numéraire cannot remain arbitrary. Instead, market participants inevitably select as their numéraire the asset that best fulfills two essential informational roles:

1. Immediate (spot) price stability and verifiability: ensuring consistent valuations across diverse transactions.

2. Future (forward) price predictability: ensuring reliable valuation over time, essential for saving and investment decisions.

Because only assets meeting these dual informational criteria can logically serve as effective numéraires under practical conditions, the theoretical possibility of arbitrarily selecting a numéraire (as suggested by certain interpretations of Arrow-Debreu) is invalidated by real-world informational constraints. Thus, in practical terms, the numéraire role is necessarily occupied by a monetary asset.

In other words, the informational structure required by coherent market economies logically transforms the numéraire—initially appearing merely as an arbitrary mathematical normalization—into the very definition of money itself. Therefore, the definition proposed here—money explicitly as the informationally essential unit of account—is fully consistent with classical first-order logic (CFOL). No known real-world counterexamples exist precisely because the informational logic underpinning economic exchange cannot be satisfied by any arbitrary good. It must be money.

Simply put, just as physics requires a standard unit such as the meter to measure distance, a coherent economic model requires a unit of account to measure and compare value. Without it, the arbitrage-free, single-price condition necessary for a Pareto-efficient equilibrium becomes impossible. Consequently, the root function of money—axiomatically—is to serve as the unit of account, from which all other monetary functions, including medium of exchange (MoE) and store of value (SoV), logically follow.

This formal hierarchy is not merely theoretical; it is consistently demonstrated empirically, especially in monetary systems under stress.

Consider nations experiencing hyperinflation and institutional instability, such as Venezuela or the former Soviet Union. Although governments may force the population to use a local fiat currency as a medium of exchange, citizens spontaneously adopt a more stable asset—like the U.S. dollar or gold—as their practical unit of account. Prices are mentally denominated in dollars, even if transactions settle in the rapidly devaluing local currency. This fracturing of monetary roles critically reveals the primacy of the UoA function above all others.

The same dynamic is evident in international trade. Because no global authority enforces a single fiat currency, "bottom-up" monetary assets—historically silver and gold, and more recently the U.S. dollar—must act as the common unit of account facilitating cross-border valuation and settlement.

These examples substantiate a universal monetary hierarchy:

UoA→(MoE,SoV)

This model robustly explains all observed monetary phenomena, and, crucially, no real-world counterexamples have emerged.

Real-World FX Markets: The Dollar as Unit of Account

The hierarchical structure of money—unit of account (UoA) as foundation, with medium of exchange (MoE) and store of value (SoV) as derived roles—is not merely a theoretical construct. It is directly observable in the empirical structure of modern foreign exchange (FX) markets. Here, all instruments are ultimately quoted and valued in terms of a common unit of account. This informational function anchors the global price system, eliminates arbitrage opportunities, enforces valuation symmetry, and ensures the economic meaning of every transaction.

Whenever this anchor is absent or misapplied, markets become vulnerable to "free-lunch" arbitrage, where agents extract riskless profit by exploiting inconsistencies in quoted rates—without contributing to productive value.

In global FX markets, exchange rates among major currencies can be expressed as a matrix, providing a structured view of the system. A fundamental condition of market efficiency is the absence of arbitrage — the impossibility of generating risk-free profits without contributing to output. Competitive pressures adjust exchange rates to eliminate these inefficiencies, preserving allocative efficiency and informational integrity.

The first and simplest requirement for an arbitrage-free system is reciprocal consistency: the exchange rate from currency i to currency j must be the reciprocal of the rate from j to i.

e_ij​×e_ji​=1

For example, if one U.S. dollar buys 0.80 euros, then one euro must buy exactly 1.25 dollars. Any deviation allows for an arbitrage loop — cycling through currencies to return with more than one started — which distorts resource allocation and violates market discipline.

This principle generalizes to matrix form. Let E be the exchange rate matrix where each element eij​ denotes the amount of currency j obtained for one unit of currency i. Two conditions must be satisfied for an arbitrage-free system:

1. Symmetry Condition (Reciprocal Consistency):

E=(E^T)^∘−1

Here, E^T is the transpose of E, and ^∘−1 indicates element-wise inversion (Hadamard inverse). This ensures all pairwise rates are reciprocal.

2. Rank-1 Condition (Global Consistency):

rank(E)=1

This condition implies that all exchange rates are determined by a single vector of relative values. In other words, every row and column in the matrix is a scalar multiple of every other, enforcing full internal consistency and eliminating the possibility of arbitrage cycles across multiple currencies.

Computational Illustration in R

We can illustrate this structure computationally. First, consider a matrix that satisfies the symmetry condition but not the rank-1 condition:

x <- matrix(c(1, 2, 3,

1/2, 1, 4,

1/3, 1/4, 1), nrow = 3, byrow = TRUE)

print(x)

Output:

[,1] [,2] [,3]

[1,] 1.00 2.00 3.00

[2,] 0.50 1.00 4.00

[3,] 0.33 0.25 1.00

This matrix is symmetric in the reciprocal sense, but not rank-1 — the rows are linearly independent. This means the system allows for arbitrage (e.g., cycling from currency 1 → 2 → 3 → 1 with a profit).

Now contrast this with a properly constructed rank-1 matrix, derived from a vector of relative values:

r <- c(1, 0.8, 150) # e.g., USD, EUR, JPY

E <- outer(1/r, r)

print(E)

Output:

[,1] [,2] [,3]

[1,] 1.0000000 0.80000 150.00

[2,] 1.2500000 1.00000 187.50

[3,] 0.0066667 0.00533 1.00

This matrix is both symmetric and rank-1. It enforces internal consistency, eliminating the possibility of arbitrage across any exchange cycle.

The Dollar as a Global Unit of Account

This mathematical structure underpins modern FX markets: the U.S. dollar acts as a de facto global unit of account. Most currency pairs are quoted directly against the dollar, approximating a rank-1 matrix. Even in direct cross-pair trades (EUR-JPY, GBP-EUR), the dollar remains the informational reference that determines the correct cross-rate—anchoring valuation, even when not directly involved in the transaction.

Crucially, the dollar is not always the medium of exchange—it is the unit of account that enables valuation. This is the informational infrastructure that makes other functions of money possible.

Summary

Both in theory and in practice—most obviously in the global FX market—money’s role as the unit of account is the indispensable foundation of market efficiency. Arbitrage-free pricing and coherent exchange require a single, consistent unit of account. The U.S. dollar’s structural dominance in global valuation is not a historical fluke or simply a matter of political power; it is a mathematical and informational necessity.

Objection: Multi-Currency or Multi-Crypto Systems

A potential objection to the unit of account axiom arises in environments where multiple currencies or digital assets circulate in parallel. One might ask: Do such systems represent exceptions to the necessity of a unique unit of account?

In fact, even in the presence of multiple circulating currencies or crypto-tokens, the mathematical logic of arbitrage-free pricing dictates that all prices are referenced to a single effective numéraire. As demonstrated above, the exchange rate matrix is arbitrage-free if and only if it is rank-1—meaning all relative prices can be derived from a single vector of values. This condition is equivalent to the existence of a unique unit of account.

In practice, whether in hyperinflationary economies (such as Zimbabwe), dual-currency environments, or on cryptocurrency exchanges listing hundreds of tokens, a dominant unit of account always emerges. Goods, services, and assets may be transacted using different instruments, but valuation consistently collapses to a single numéraire: USD in Zimbabwe, USDT or BTC on exchanges, etc.

Attempts to maintain more than one simultaneous unit of account create pricing incoherence and arbitrage opportunities, which market competition rapidly eliminates. Thus, the rank-1 matrix condition formalizes and reinforces the unit of account axiom, even in the most complex multi-currency or multi-token environments.

But why was the U.S. dollar selected as the FX market’s global unit of account?

The answer is pragmatic: among competing alternatives, the dollar is simply the “least worst” option. Principals and agents converge upon the unit of account (UoA) that best meets real-world legal, informational, and institutional constraints—including inflation risk, legal certainty, transaction infrastructure, reputational reliability, and global liquidity.

The U.S. dollar benefits from deep liquidity, broad integration into global trade, and—compared to alternatives—responsible monetary governance, robust institutional backing, and wide accessibility. Dollar-denominated assets (e.g., deposits at J.P. Morgan or holdings of U.S. Treasuries) offer lower risk and higher credibility than most competitors. Existing global payment infrastructures (e.g., SWIFT, EFT, checks) provide cost-effective, rapid, and reliable transactions unavailable to many competing monetary units. Even promising alternatives like the euro or Bitcoin still fail to match the dollar’s comprehensive balance of stability, accessibility, legal certainty, and low transaction costs.

A similar phenomenon can be consistently observed in nations experiencing hyperinflation or institutional instability—such as the Soviet Union, Venezuela, Argentina, Zimbabwe, and others. In these economies, agents spontaneously shift to alternative money-units (e.g., dollars, gold, euros) because their domestic fiat currency fails as a credible unit of account due to extreme strategic uncertainty, specifically caused by unpredictability in the future spendable money supply (e.g., M2).

Critically, this shift aligns precisely with our unified theory’s predictions: agents naturally seek alternative informational anchors that better serve as stable spot-price and forward-price anchors, enabling coherent valuation despite economic instability.

In short:

The U.S. dollar’s continued global dominance as a numéraire persists not because it is ideal, but because it consistently emerges as the best available informational anchor given existing real-world constraints. As soon as a superior alternative arises—one offering greater stability, transparency, and lower friction—global markets will organically select it, in alignment with the fundamental informational axiom that underpins all monetary systems:

The informational necessity of a single, credible unit of account.

Why Gold Is No Longer Used as Money After Thousands of Years

The historical retreat of gold from center stage in the global monetary system was not the result of any failure as a unit of account. For centuries, gold effectively fulfilled this role—and in many ways continues to do so as an international benchmark of value. Gold’s persistent role as a reference point is exemplified by initiatives such as the Central Bank Gold Agreement (CBGA), first signed in 1999 and renewed several times thereafter, aimed at stabilizing gold markets through coordinated central bank sales. The eventual cessation of these agreements—triggered by a shift from net selling to net buying, particularly by China and Russia—underscores gold’s enduring credibility as a stable, universally recognized standard.

The real reason gold no longer functions as daily “money” is not conceptual, but physical.

Gold’s limitations lie in its practical usability as a medium of exchange (MoE)—not in its role as a unit of account (UoA) or a store of value (SoV). As a tangible asset, gold excels in face-to-face transactions where physical verification and exchange are straightforward. However, as global commerce accelerated and expanded geographically, these very physical attributes became liabilities. Gold is heavy, expensive to transport and safeguard, and inherently vulnerable to theft or loss.

Early attempts to overcome these constraints—such as bank vault storage and paper-based gold certificates—abstracted gold ownership further, allowing claims to be traded while the metal remained immobile. Yet even these innovations faced an insurmountable final obstacle: a physical object, regardless of abstraction, cannot be moved, verified, or settled across global distances and time zones at the instantaneous speed required by modern real-time commerce.

The modern economy demands settlement at the speed of information.

The rise of digital ledger-based money—building upon historical precedents like medieval tally sticks in England and Venetian double-entry bookkeeping (13th century)—allowed value to be represented without physically transporting commodities, enabling global, instantaneous settlement. Unlike earlier ledger systems limited by geography or manual reconciliation delays, digital systems update balances seamlessly and immediately across continents. Today, trillions of dollars move in milliseconds—speeds entirely unattainable by gold-based systems, no matter how cleverly abstracted.

Gold’s obsolescence as everyday money thus stems not from conceptual failure but from the fundamental physics of settlement: atoms cannot move at the speed of light. The informational demands of contemporary global markets mandated dematerialization.

In short:

Gold did not “fail” as money. Rather, the infrastructure supporting global trade outpaced the settlement limits of commodity-based monetary systems. The needs of a digitized, high-velocity economy require a dematerialized monetary layer capable of settling value at the speed of thought, not at the speed of ships.

A Modern Taxonomy of Money: From Bearer Instruments to Digital Assets

As Milton Friedman famously observed, the central lesson from monetary history is empirical: money is what money does. Across centuries and civilizations, an extraordinary diversity of objects have functioned as money—from cattle and salt to tea bricks and Rai Stones. This historical diversity underscores a crucial fact: the monetary function is fundamentally independent of physical form.

To clearly understand both monetary evolution and the emergence of contemporary assets like Bitcoin, a precise classification system is essential. Despite their apparent variety, all monetary and financial instruments can be categorized into two mutually exclusive types based on a single criterion: whether ownership requires identity verification:

1. Bearer Instruments: Anonymous Ownership by Possession

Ownership is established solely through physical or cryptographic control. Whoever holds the asset is its rightful owner—no recorded identity or verification is required. Transfers are permissionless, anonymous, and immediately final.

• Ownership Mechanism: "What you hold is yours."

• Examples:

  • Physical cash

  • Gold coins held personally

  • Self-custodied Bitcoin (UTXOs)

  • Bearer bonds

2. Registered Instruments: Identity-Dependent Ownership

Ownership is explicitly linked to identity verification and validated through an external system or ledger. Mere possession is insufficient; transfers require implicit or explicit authorization.

• Ownership Mechanism: "Who you are determines what you own."

• Registry Types:

  • Centralized: Bank ledgers (SWIFT/KYC systems)

  • Decentralized: Blockchain-based custodial wallets

  • Social Consensus: Rai Stones validated by community memory

• Examples:

  • Bank deposits (custodial)

  • Bitcoin held on exchanges (Coinbase wallets)

  • Stocks and land titles

  • Rai Stones (ownership validated socially)

This binary classification—anonymous bearer assets versus identity-registered systems—provides the clearest possible lens through which to analyze the evolution of monetary systems. Underlying both paradigms is the invariant foundational function of money as a Unit of Account (UoA).

Gold: The Historical Champion of Bearer Money

For thousands of years, gold served as humanity’s most successful bearer asset and global monetary standard. This dominance was no accident; gold offered unique advantages:

• Predictable and Stable Supply: Gold’s exceptionally high stock-to-flow ratio ensured a stable supply. Even significant historical discoveries—such as the Alaskan and Australian gold rushes of the late 19th century—barely affected total supply or value, allowing gold’s purchasing power and UoA role to remain stable over long periods.

• Resistance to Dilution: Unlike perishable goods or politically manipulated commodities, gold’s stable scarcity fostered long-term trust across borders and generations.

Yet despite these significant advantages, gold was ultimately undermined by two critical flaws:

1. Physically Vulnerable Store of Value: Although gold reliably preserved purchasing power, its tangible form posed logistical problems. Heavy, costly to transport and store, and perpetually vulnerable to theft or seizure, these physical weaknesses encouraged the gradual abstraction of gold into paper claims—the first significant step away from physical bearer money.

2. Failed Medium of Exchange at Scale: Gold’s tangible nature rendered it impractical for high-speed, global transactions. Effective in local face-to-face markets, gold could not sustain the demands of a modern, borderless, instantly settled economy.

Bitcoin: The Digital Successor to Gold

Bitcoin can be best understood as a direct response to gold’s limitations—a digital upgrade of the bearer asset for a high-velocity, global economic system. Bitcoin’s core innovations address gold’s weaknesses directly:

1. Remote Transferability: As a natively digital bearer asset, Bitcoin can be transmitted globally, instantly, and permissionlessly—without vaults, borders, physical handling, or delay.

2. Improved Physical Security: Ownership protection shifts from safeguarding physical assets to securing cryptographic private keys, backed by a decentralized Proof-of-Work network. This cryptographic foundation arguably makes Bitcoin the most secure bearer asset ever created.

In essence, Bitcoin does not simply replicate gold; it expands and enhances the bearer asset paradigm. By shedding the physical constraints that limited its predecessor, Bitcoin creates a new class of digital money explicitly consistent with the foundational axiom that all money, from gold to cryptocurrencies, fundamentally serves the informational role of a unit of account.

Concluding Thoughts: Bitcoin's Promise and the Path Forward

Bitcoin’s rise as a globally recognized monetary asset stems from a unique convergence of essential properties. It delivers the sovereignty, finality, and scarcity historically associated with gold—without the burdens of physical custody or atomic limitations. By abstracting scarcity into code and embedding it within a decentralized, cryptographically verifiable protocol, Bitcoin completes the monetary transformation that gold initiated. It represents a bearer asset tailored for an economy moving at the speed of information rather than the speed of physical transport.

Yet, Bitcoin does not merely replicate gold; it significantly enhances and finalizes gold’s fundamental monetary innovation by solving the critical problem of secure custody without centralized trust.

The Unfinished Job: Bitcoin’s Design Dilemma

Despite this breakthrough, Bitcoin’s innovation comes at substantial computational and environmental costs. Its underlying Proof-of-Work (PoW) consensus mechanism demands extensive resources, rendering Bitcoin inefficient as a practical medium of exchange—slow, costly, and environmentally taxing.

Bitcoin is inherently constrained by a fundamental trade-off:

• Lower transaction costs imply reduced network security.

• Greater security (enhanced store-of-value properties) implies increased transaction costs, limiting Bitcoin's effectiveness as a medium of exchange.

Importantly, this trade-off is not dictated by fundamental physics or economic necessity but arises explicitly from Bitcoin’s specific game-theoretic design choices. As Bitcoin strengthens its position as a secure store of value, its viability as a day-to-day medium of exchange diminishes. Each incremental gain in security demands greater resource consumption—manifesting directly in increased electricity usage, specialized mining hardware, and environmental impact.

This security/usability trade-off is not immutable or universal; it results directly from intentional design choices. As history has consistently demonstrated, monetary architectures evolve to optimize beyond existing constraints. New systems can—and inevitably will—emerge to resolve this fundamental limitation.

The Informational Essence of Money

The central lesson is clear: The defining essence of money—across all historical eras, technological platforms, and institutional frameworks—is the informational function of the unit of account (UoA). Bitcoin, gold, and every monetary system past and future must ultimately satisfy this informational axiom.

The challenge facing future monetary designs is not simply to preserve monetary sovereignty and transactional finality but to achieve informational credibility without unnecessary environmental and computational waste. Secure custody and regulatory compliance can indeed be achieved without centralized intermediaries—and without expending gigawatts of energy on computational puzzles lacking intrinsic economic purpose.

A superior monetary framework is not merely achievable—it is necessary, urgent, and overdue.

Strategic Uncertainty and the Path to Monetary Stability

Ultimately, the core argument presented here, dear reader, is unequivocal: strategic uncertainty inevitably undermines economic efficiency.

From Akerlof’s "Market for Lemons," where informational asymmetry generates adverse selection, to the Prisoners' Dilemma, where mutual distrust locks participants into suboptimal outcomes, the root cause of inefficiency is always strategic uncertainty. Money, a cornerstone of economic efficiency, is no exception.

A robust monetary system must successfully resolve two critical informational challenges:

• Immediate verifiability of monetary quality: Agents must trust the legitimacy and authenticity of monetary assets at the point of exchange.

• Predictability of future money supply: Agents must be confident their purchasing power will not be unexpectedly eroded by arbitrary monetary expansion.

Indeed, it is precisely the second criterion—predictability of future supply—that fundamentally distinguishes robust monetary systems (e.g., gold historically and Bitcoin digitally) from weaker ones (e.g., fiat currencies vulnerable to hyperinflation). While Bitcoin emerges as the digital successor to gold by ensuring mathematical scarcity, it does so at a considerable computational and ecological cost that limits practical usability.

Therefore, the next chapter in monetary innovation must aim explicitly at delivering informational credibility—both verifiable authenticity and predictable supply—without incurring wasteful inefficiencies. This monetary evolution is not merely possible; it is essential, urgent, and overdue.

The future of monetary design remains unwritten—but the blueprint for this next generation of money can begin here:

👉 haykov.substack.com/p/collusion

Markowitz and CAPM: The Minimum Risk Market Portfolio

Modern asset pricing frameworks—such as the Capital Asset Pricing Model (CAPM), Fama-French, and Arbitrage Pricing Theory (APT)—are built on a foundational empirical insight: investors cannot monetize idiosyncratic (diversifiable) risk. Activities like running a casino, operating a private lottery, or even counting cards in a casino are either illegal or actively discouraged. Only systemic (undiversifiable) risk is compensated in expected returns, both in theory and in practice—a point on which there is universal agreement. From this, it can be shown that the market portfolio is the minimum risk portfolio.

Beyond statistical measures, there is a deeper, practical reason why the market portfolio is considered the lowest true-risk portfolio, and it has little to do with variance per se. The ultimate risk in investing is the loss of purchasing power: the inability to consume tomorrow what one could consume today.

But what can investors collectively consume? Not all economic output is ultimately available for consumption. Total economic output—referred to as gross output—comprises both intermediate consumption (such as lumber used in furniture manufacturing) and final consumption (such as a finished dining set, actually consumed). By accounting identity:

  Gross Output = Intermediate Consumption + Final Consumption
  where Final Consumption = GDP − I − G − (X − M)

Here, we extend the definition of intermediate consumption to include all non-final production uses—such as gross value added and indirect fiscal adjustments (e.g., taxes and subsidies)—so that gross domestic product (GDP) reflects the upper bound from which final consumption can be accurately isolated.

This decomposition highlights a critical constraint: investors, as a group, can only consume what is left as final consumption. GDP must therefore be adjusted to exclude business investment, government outlays (such as defense spending), and net exports (which benefit foreign consumers). Although not identical, real GDP is closely correlated with final consumption in most economies and serves as a practical proxy.

From this perspective, real GDP is the only meaningful constraint on future consumption possibilities. It is the sole source of consumable output in the economy. This reframing imposes a hard boundary: in the aggregate, investors cannot consume more than the total real GDP produced, net of labor’s share. In this sense, what investors consume is, in effect, the cost of capital. Since labor compensation claims a relatively stable proportion of output, the residual—available to capital holders—defines the upper limit of investable and consumable economic value.

In short: it is not nominal account balances but real GDP that determines a society’s capacity for future consumption. An investor’s purchasing power is ultimately constrained not by the quantity of currency held or promised, but by the productive capacity of the real economy.

That GDP, in turn, is created through the productive activity of firms that comprise the investable market portfolio. Even government expenditures—such as contracts with defense firms—are routed through transactions with private-sector entities listed in major indices (e.g., the S&P 500), and paid for using the circulating money supply.

While non-tradable (privately held) firms introduce undiversifiable risk due to incomplete markets—a well-established feature of all real-world economies—the core principle remains: ownership of the total market portfolio confers proportional exposure to the productive base of the economy, that is, to the real GDP that ultimately supports final consumption.

More precisely, such ownership provides access to the residual economic output available to capital providers after labor compensation. Though basic, this insight is often overlooked: consumption is bounded by net production. This is not a theoretical claim—it is an accounting identity grounded in the arithmetic of real economies. Real GDP is the definitive upper limit of consumable output. Purchasing power, in this context, is defined not by the quantity of nominal currency held, but by the capacity to claim a share of that finite, real output.

Thus, all financial assets—bonds, equities, derivatives—are ultimately claims on future GDP. Their value derives from their ability to provide access to real consumption. From this vantage point, a “safe” asset must offer guaranteed, proportional access to future GDP. Anything less is a nominal construct—a safety illusion.

Fixed-income instruments, by their very design, fail to meet this threshold. Because they are denominated in nominal terms and subject to future monetary uncertainty, they cannot guarantee real purchasing power. Their supposed “safety” is more marketing narrative than economic fact.

Only the total market portfolio offers a truly safe position: it grants the holder a proportionate claim on the economy’s future production and consumption. All other instruments—fixed income, leveraged trades, synthetic assets—are nominal overlays that lack real guarantees.

Not Ideology: Arithmetic

When investors hold the entire market portfolio, they acquire a proportional claim on the economy’s real productive output. This is why broad market exposure—across all tradable debt and equity instruments—is the most economically “safe” investment. It secures access to future GDP, which, by definition, constitutes the total pool of goods and services available for future consumption.

So, What’s the Problem with CAPM?

The fundamental flaw in the Capital Asset Pricing Model (CAPM) lies in its core assumption that the risk-free rate (r_f​)—the rate at which investors can borrow or lend—is truly “risk-free.” In reality, this is a significant oversimplification. In practice, the so-called risk-free rate is typically represented by the borrow/lend rate in margin accounts offered by brokers, usually benchmarked to short-term government securities (such as 4-week Treasury bills or the federal funds rate). While these instruments are generally considered free from default risk, they are not protected against inflation and often deliver negative real (inflation-adjusted) returns. As a result, what is labeled “risk-free” in theoretical models can, in practice, expose investors to significant purchasing power risk—undermining the very premise of a truly riskless asset.

The consequence is that the supposed “risk-free” asset is not only exposed to purchasing power risk, but its real yield is often systematically negative. When this reality is properly incorporated, it becomes clear that leveraging or deleveraging—borrowing or lending at the so-called risk-free rate—actually reduces the Sharpe ratio and increases risk per unit of expected return. In other words, the lowest true-risk portfolio remains the broad market portfolio itself. Borrowing at the risk-free rate exposes the investor to the risk of a margin call, with the possibility of losing 100% of the investment, while lending at the risk-free rate (as CAPM suggests) does not improve risk-adjusted returns, since the real r_f​ is consistently below zero after accounting for inflation.

According to the Modigliani–Miller theorem and Sharpe’s identity, the market’s beneficial owners, in aggregate, receive the full spectrum of operating cash flows. This makes the total market portfolio not only comprehensive but also the lowest-risk portfolio in economic terms: it guarantees a fractional claim on real GDP—the maximum available pool of future consumable output. By contrast, any departure from broad market exposure—through selective positioning in either debt or equity—introduces idiosyncratic or structural risk.

This framework leads to a critical and uncomfortable implication: debt instruments often function less as tools for investor protection and more as mechanisms for rent extraction. Financial intermediaries capitalize on widespread misconceptions—particularly the confusion between nominal stability and real security—to market fixed-income products as inherently “safe.” In doing so, they reallocate economic value away from uninformed investors under the guise of risk management.

This pattern echoes a broader trend in financial innovation. Just as trading platforms like Robinhood encourage behaviors that systematically degrade long-term investor performance, bond markets frequently act as conduits of value transfer—from risk-averse households to financially sophisticated institutions.

This is not a function of portfolio optimization.
It is a function of structural design.
In many cases, it is not financial prudence, but systemic predation.

The Illusion of Safety: Bonds and Monetary Uncertainty

To understand why fixed-income instruments are not inherently safe, one must first consider the medium in which they are denominated and ultimately repaid: the economy’s spendable money supply. In contemporary systems, this is most often measured by aggregates such as M2—which includes cash, checking and savings accounts, and other liquid instruments available for immediate spending.

Historically, this monetary function was served by tangible commodities such as gold. For example, in the Roman Empire, the aureus coin functioned as both a medium of exchange and a store of value. In both ancient and modern systems, a bond represents a claim on a future portion of this monetary base—whether in coins or digital account balances.

The key distinction lies in the predictability of that monetary base over time. As explained above already, in theory, technological advances (e.g., extracting gold from ocean water) could expand the gold supply, yet in practice, gold’s defining economic strength has always been its stability. Its high stock-to-flow ratio—where the above-ground supply vastly exceeds annual production—ensures that even major mining surges (such as the Alaskan or Australian gold rushes) barely alter its overall supply. This inherent resistance to supply shocks gave gold enduring value as a unit of account.

By contrast, modern fiat currencies like the U.S. dollar (as measured by M2) are subject to significant uncertainty. Their supply is shaped not by geological constraints, but by political decisions, macroeconomic objectives, and central bank operations. Unlike gold, fiat money can expand or contract at the discretion of monetary authorities—a fact that introduces systemic unpredictability. This uncertainty is not theoretical; it is empirically verifiable. For example, the U.S. M2 money supply has exhibited rapid, discretionary expansions in recent years (see FRED: M2 Money Stock).

Therefore, the common axiom in financial economics—that fixed-income instruments are the safest possible investment—is fundamentally flawed. To see why, consider both commodity-based and fiat monetary regimes:

• Under a commodity-based system, the fixed size of the monetary base imposes a hard constraint: bonds represent nominal claims on future GDP (i.e., real consumable output), but the supply of redeemable currency is finite. This creates arithmetic default risk, not just probabilistic uncertainty. When aggregate claims exceed the amount of currency available for redemption, default becomes not merely possible, but inevitable. Historically, even sovereign governments have defaulted on gold-backed bonds—not always from fiscal irresponsibility, but because of this intrinsic mismatch between nominal claims and a hard-capped monetary base. No entity—not even a central government—can override this constraint without debasing the currency itself. This is the essence of counterparty risk under a commodity standard: it arises not from institutional weakness, but from mathematical contradiction.

• Under a fiat regime (such as the current M2-based U.S. dollar system), the constraint shifts. The primary risk becomes inflation. The money supply is discretionary—subject to the policies of central banks and fiscal authorities—introducing strategic uncertainty. Whether through credit expansion, monetary stimulus, or bank runs, the real purchasing power of future fixed-income payments remains fundamentally unpredictable. As in commodity systems, fixed-income instruments under fiat regimes fail to guarantee real purchasing power—undermining their claim to “safety.”

• In a commodity-backed system with fractional reserve banking—whether centrally regulated or operating in a free-banking environment—an additional layer of fragility emerges: the risk of bank runs. Deposits in such systems are nominally backed by commodities (e.g., gold) but only partially so. The mismatch between redeemable claims and actual reserves creates both liquidity risk and counterparty risk: the system can remain solvent only as long as not all claimants demand redemption at once. When that confidence fails, cascading institutional defaults become mathematically unavoidable.

In all three cases—pure commodity, fiat, and fractional commodity-backed banking (regulated or otherwise)—the belief that fixed-income instruments (including non-interest-bearing cash) are the lowest-risk investment is categorically false.

The only strategy that consistently minimizes structural, monetary, and counterparty risk is to hold the entire market portfolio, encompassing both equity and debt. This approach secures a proportional claim on the economy’s real output—GDP—and, by extension, on society’s future consumable goods and services. As previously demonstrated, the form of ownership—debt versus equity—is immaterial at the aggregate level. What matters is holding claims on the full productive base.