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Title: What is Money?

Authors: Joseph Mark Haykov (with Nathan and Phillip as interns)

Abstract

In this paper, we extend the Arrow-Debreu formal system by introducing the Labor-For-Goods Model to explore Pareto Efficiency. We propose the equation U = S + E, which redefines the concept of money within the Arrow-Debreu framework. In this equation, U represents the unit of account, now expanded to incorporate its roles as both a store of value (denoted S) and a medium of exchange (denoted E). This adjustment aligns the formal, axiomatic definition of money with its empirically observed functions, as recognized by the U.S. Federal Reserve and standard economic theory. Across all real-world economies, both historical and contemporary, money consistently serves three key functions: as a medium of exchange, a unit of account, and a store of value. In our model, a monetary unit functions as a store of value when it is not immediately used for exchange—for example, funds held in a bank account over an extended period.

Keywords: Labor-For-Goods Model, Pareto Efficiency, Dynamic Equilibrium, Arrow-Debreu, Game Theory, Rent-Seeking, Property Rights, Arbitrage-Free Pricing

JEL Codes: C70, C72, C73, C78, D72, D82, D83, D84, D93, D94

Introduction

In this paper, we introduce the Labor-For-Goods Mathematical Game Theory Model, a dynamic equilibrium framework designed to complement the traditional Arrow-Debreu model in mathematical economics. While the Arrow-Debreu framework establishes Pareto efficiency through static competitive equilibria—grounded in axioms such as local non-satiation, convex preferences, and complete markets—our model elucidates the dynamic processes through which Pareto-efficient outcomes, as predicted by the First Welfare Theorem, are realized in practice.

The Labor-For-Goods model focuses on exchanges of labor for goods and services as mutually beneficial, Pareto-improving trades. Using money as a medium of exchange, unit of account, and store of value (expressed through the equation U = S + E), the model demonstrates the emergence of a Pareto-efficient Nash Equilibrium. This result reinforces the First Welfare Theorem by showing how equilibrium is achieved through arbitrage-free trades.

Central to our approach is the integration of game theory with general equilibrium concepts, providing a more realistic depiction of how equilibrium emerges from continuous trade interactions. The model emphasizes the critical roles of well-defined property rights, symmetric information, and arbitrage-free exchange rates in ensuring market efficiency and preventing rent-seeking behavior. We introduce the Rent-Seeking Lemma, which highlights the tendency of rational, utility-maximizing agents to exploit opportunities for unearned wealth extraction. This underscores the necessity of robust property rights and appropriate market conditions to sustain Pareto efficiency.

Further, while some game-theoretic ideas within our model may resemble concepts from Marxian economics, we explicitly distance ourselves from Marxist formal systems. Through a sound critique, we illustrate how axiomatic contradictions in Marxist economics, when tested against empirical realities, lead to inefficiencies and systemic failures. In contrast, the Labor-For-Goods Model aligns with real-world economic behaviors, maintaining soundness and completeness by incorporating the full definition of money as a unit of account, medium of exchange, and store of value.

Through formal proofs, we demonstrate that under conditions of well-defined property rights, complete markets, symmetric information, voluntary exchange, local non-satiation, and arbitrage-free pricing, the Labor-For-Goods Model reliably achieves global Pareto efficiency. This dynamic framework not only enhances the explanatory power of mathematical economics but also provides actionable insights for policy-making. Its alignment with the Federal Reserve's use of general equilibrium models in guiding economic policy further demonstrates its practical relevance.

This paper is organized as follows:

• Part I formally defines the Labor-For-Goods exchange model, ensuring consistency with the existing axioms and theorems of mathematical economics, particularly the First Welfare Theorem. This formal foundation is necessary for a rigorous discussion of the U = S + E model of money.

• Part II explores how the U = S + E extended definition of money (as a unit of account, medium of exchange, and store of value) operates within this dynamic exchange framework, both theoretically and practically.

• Part III examines how the U = S + E definition of money within a formal system aligns with historical economic evidence. It explains what makes money “USE-able”.

• Part IV presents macroeconomic predictions based on the U = S + E model. Given the formal nature of the system, all predictions can be independently verified, from short-term to long-term forecasts. This verifiability is a key strength of formal systems: all propositions can be tested for accuracy.

For readers less interested in formal systems, we have organized the paper so that Part I can be skipped without losing the core insights. Starting with Part II, readers will be guided on which sections to bypass if they wish to focus solely on the economic conclusions. This structure allows readers to engage directly with the results in Parts III and IV without needing to review the formal derivations and proofs, while still ensuring that everything claimed as true is mathematically sound and verifiable.

Part I: The Labor-For-Goods Dynamic Equilibrium Model within Mathematical Economics

Mathematical economics operates as a formal system, in which theorems—such as the First Welfare Theorem—are derived from a set of foundational axioms and formal inference rules. These axioms include key assumptions such as local non-satiation, convex preferences, and the existence of complete markets. From these premises, the First Welfare Theorem is derived, establishing that any competitive equilibrium is Pareto efficient. This theorem, along with others like the Second Welfare Theorem, forms the foundation of the Arrow-Debreu model, which is central to mainstream mathematical economics. For instance, the Federal Reserve Bank of the United States employs general equilibrium models based on the Arrow-Debreu framework to inform critical policy decisions, such as the setting of interest rates.

While the conclusions drawn from the Arrow-Debreu axioms—such as rational, utility-maximizing representative agents—are theoretically robust within the model's idealized conditions (such as perfect markets), our paper introduces a dynamic alternative. Specifically, we present a model that demonstrates how Pareto-efficient Nash equilibria, as predicted by the First Welfare Theorem, can be achieved through dynamic processes rather than static ones. Our model, centered on exchanges of labor for goods and services, illustrates how a series of mutually beneficial, Pareto-improving trades leads to the same Pareto-efficient Nash equilibrium predicted by the First Welfare Theorem, but through a dynamic mechanism. We call this framework the Labor-For-Goods Game Theory Model, which formalizes the existence of a Pareto-efficient Nash Equilibrium through ongoing trade interactions.

This dynamic model is central to our paper, as all claims and assertions are developed within its framework. Our model does not contradict the Arrow-Debreu framework; instead, it leverages specific axioms to reflect the dynamic processes observed in real-world markets. While the Arrow-Debreu model focuses on static equilibrium, our model emphasizes how Pareto-efficient outcomes emerge through continuous, mutually beneficial trades. This approach offers a more nuanced understanding of equilibrium, not as a static state, but as an emergent property of ongoing trade interactions.

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Why We Use Mathematical Economics

There are various schools of economic thought¹, but any rigorous approach to modeling real-world economics relies on a formal system, commonly referred to as mathematical economics. The primary reason for this is that formal systems provide a sound framework that distinguishes axioms from hypotheses, a critical safeguard against the inaccuracies that can arise in less formal approaches.

Conflating Hypotheses and Axioms: Implications for Modeling Central Banking and Economic Crises

A dominant hypothesis in mainstream economic theory, first proposed by Friedman and Schwartz in their 1963 work A Monetary History of the United States, 1867–1960, posits that the primary cause of the Great Depression was the central bank’s failure to act during the late 1920s and early 1930s. Specifically, the Federal Reserve did not provide sufficient support to individual banks facing closures due to bank runs. These runs were triggered by the banks' inability to convert representative money (e.g., checking and savings account balances) into commodity money (such as gold coins). While this hypothesis remains influential, alternative explanations suggest that other factors—such as structural economic weaknesses, trade policies, and psychological factors—also played significant roles in causing the Great Depression.

This highlights the importance of formal systems in economic modeling: they ensure soundness by preventing the inclusion of assumptions that may later prove false. If we were to accept Friedman’s hypothesis as an axiom—that is, as a foundational, self-evident truth—our formal system would become unsound. This is because, if the hypothesis were later disproven, the formal system would effectively misrepresent reality. A sound formal system, when constructed with proper inference rules, does not generate false conclusions from false premises.

Formal systems are particularly useful for testing hypotheses, where we evaluate whether empirical evidence supports or contradicts a given hypothesis (e.g., whether the Federal Reserve’s actions—or inaction—caused the Great Depression). However, a hypothesis cannot be treated as an axiom within a formal system without introducing the risk of unsoundness. By definition, axioms must be consistent with real-world facts, serving as self-evident truths, while hypotheses are conjectures requiring empirical validation. Unlike axioms, hypotheses are not self-evidently true and may ultimately prove false.

The Difference Between Hypotheses and Axioms

In a sound formal system, such as those used in mathematical economics, axioms are assumed to be indisputably true, ensuring that all theorems derived from them are also true. Hypotheses, on the other hand, are conjectures—claims about reality that may or may not be true and thus require empirical testing. For instance, Friedman’s hypothesis regarding the Federal Reserve’s role in the Great Depression is still subject to debate and empirical scrutiny. This is analogous to the Riemann Hypothesis in mathematics: widely believed to be true, but still unproven, and therefore not yet a theorem. Similarly, Euler’s conjecture, proposed in 1769, was considered plausible for nearly two centuries until it was disproven by a counterexample in 1966. Hypotheses remain open to falsification, whereas axioms in a formal system must be irrefutable.

As such, hypotheses cannot serve as the foundation for a formal system unless proven beyond doubt. Assuming a hypothesis to be true without proof and treating it as an axiom introduces the risk of error, rendering the system unsound. This is one reason why Marx’s economic theory became unsound: his assumption regarding agency costs flowing from agents to principals did not align with empirical reality, leading to flawed conclusions.

Modeling Money and Central Banking with Sound Axioms

To accurately model money and central banking within a formal system, such as the Labor-For-Goods Pareto Efficiency Model, it is essential to avoid assumptions that could later be disproven. For instance, while Friedman’s hypothesis suggests that the central bank’s inaction caused the Great Depression, using this hypothesis as an axiom would be unsound, as it remains subject to empirical validation and potential falsification. Instead, a sound approach must focus on facts that are irrefutable.

One such fact is that rapid deflation was a key feature of the Great Depression. This is not a hypothesis—it is an empirical reality. While the specific causes of this deflation are debated, its occurrence is undeniable. From this, we can derive a self-evident axiom: volatility in the money supply, whether through inflation or deflation, is harmful to economic growth. This is a universally observed phenomenon across real-world economies, with no empirical evidence contradicting it. Moreover, no responsible economist disputes this claim. This is evident in the real-world behavior of central banks, which treat deflation as a dire threat and actively combat inflation to stabilize prices. Therefore, this principle can safely serve as an axiom in a formal system to model the effects of monetary policy on economic outcomes.

In contrast, Friedman’s hypothesis about central banking cannot serve as an axiom because it remains subject to empirical validation and may be disproven. In any sound formal system, only axioms that are self-evidently true can be accepted—by definition of what constitutes an axiom—in order to preserve the system’s soundness. While influential, Friedman’s hypothesis does not meet this standard, unlike the consistently observed effects of monetary volatility, which are universally supported by empirical evidence. This distinction is critical for maintaining the integrity of mathematical economics as a reliable and robust formal system for modeling real-world phenomena.

It is this commitment to sound axiomatic foundations that has made the Arrow-Debreu framework so impactful. Its rigor and consistency have earned it multiple Nobel Prizes and solidified its position as a cornerstone of mainstream economic theory. The framework's strength lies in its soundness, which is why it continues to be widely adopted in both academic research and policy-making.

Explanation: Labor-For-Goods (and Services) Setup

In the Labor-For-Goods (and Services) framework, we model Pareto-efficient outcomes using game theory as group-optimal Nash equilibria. Unlike in the Prisoner’s Dilemma, where individual incentives lead to suboptimal outcomes, rational, utility-maximizing agents in this model exchange their labor for goods and services produced by the labor of other agents to achieve a group-optimal, Pareto-efficient result. This is made possible by the assumption of symmetric information, similar to that used in the First Welfare Theorem, but with the added constraint of no arbitrage.

In this system, money is dually defined as all sound definitions must be in dually-consistent formal systems, and it serves two key functions:

• First, money acts as a unit of account, measuring both wages and prices of goods and services relative to one another.

• Second, money functions as a medium of exchange, with which wages are paid and goods and services are purchased.

The Nash equilibrium in this setup results in a Pareto-efficient allocation, meaning that no agent can be made better off without making another worse off. While not all Nash equilibria are Pareto efficient—as exemplified by the Prisoner’s Dilemma—our model is specifically designed to ensure that the Nash equilibrium leads to a Pareto-efficient outcome. This is achieved by maximizing mutual benefits through trade under three key assumptions:

1. Arbitrage-free prices,

2. Symmetric information about the goods and services being exchanged, and

3. Unfettered voluntary trade in an open market.

These assumptions—(1) arbitrage-free prices, (2) symmetric information, and (3) voluntary exchanges driven by rational agents seeking to improve their individual welfare—ensure that all trades are ex-ante mutually beneficial (before the trade) and ex-post mutually beneficial (after the trade). The absence of information asymmetry is crucial for preserving this mutual benefit.

By eliminating information asymmetry, which could otherwise distort trade outcomes, this setup guarantees at least a locally Pareto-efficient allocation of resources. These conditions create an ideal environment where agents engage in trades that enhance the welfare of all parties. As a result, the model upholds both the rational decision-making of individual agents and the collective welfare of the economy.

The Economic Model and Collective Costs

Mathematically, this economic model—understood as a formal system of real-world interactions—holds because the only net collective costs involved in producing real GDP are:

• The labor contributed by individuals, and

• Negative externalities, such as pollution and resource depletion, which affect society as a whole.

Externalities are costs imposed on third parties not directly involved in a transaction, making them collective costs. Similarly, labor constitutes a collective cost because every agent in the economy contributes labor in some form, except for those engaged in non-productive or harmful activities, such as theft or economic exploitation. A sound formal system must account for all agents, including those who do not contribute positively to the economy.

While firms and individuals incur private costs for inputs such as raw materials, capital, or technology, these are not collective costs in the same way that labor and externalities are. For example, the ownership of raw materials used for intermediate consumption does not directly affect final consumption (i.e., GDP), which ultimately determines collective welfare. Although intermediate goods contribute to final GDP through production processes, the mere transfer of ownership (e.g., through stock market trading) reflects a redistribution of wealth rather than a contribution to productive activity. Such ownership transfers do not influence Pareto efficiency unless externalities are involved.

However, externalities related to ownership changes—such as positive externalities from more efficient capital allocation when stock prices are accurately established—fall outside the primary scope of this model and would require separate analysis. Nonetheless, our dynamic model offers insights into both positive and negative externalities related to ownership changes, which can be further explored in future layers of analysis.

Private vs. Collective Costs: Ownership’s Role in Pareto Efficiency

Negative externalities—such as pollution or resource depletion—are collective costs borne by society as a whole, whereas the ownership of capital is a private cost that does not directly influence collective welfare. In contrast, labor represents a net contribution by everyone, making it a universal collective cost in this framework. Therefore, negative externalities and labor are the primary collective costs considered in our model.

To illustrate this, consider Bob and Alice on a deserted island. Their collective costs and benefits are optimized through mutually beneficial trades, leading to a Pareto-efficient outcome, where neither can be made better off without making the other worse off.

However, when defining Pareto efficiency, the concept of ownership becomes irrelevant. Whether Bob "owns" the banana tree or Alice "owns" the water spring has no direct impact on the outcome. What matters is the exchange of resources in a mutually beneficial way. For example, even if Bob claims ownership of the banana tree and Alice claims ownership of the water spring, they can still achieve a Pareto-efficient outcome through trade. The perception of ownership is irrelevant as long as resources are allocated in a way that prevents either party from improving their welfare without reducing the welfare of the other.

In simpler terms, Pareto efficiency concerns not what resources agents think they own, but what they actually exchange. By trading the fruits of their labor, Bob and Alice maximize collective welfare, aligning with Adam Smith’s principle from The Wealth of Nations—that mutually beneficial trade improves overall welfare by maximizing labor productivity, thus reducing the time spent on labor. This principle, self-evident since 1776, serves as a foundational axiom in our formal system, where the fruits of one’s labor, measured by wages, are exchanged for the fruits of another’s labor, measured by price.

No sound formal system, based on such self-evident axiomatic assumptions, can contradict real-world facts. In this sense, Pareto efficiency pertains to how resources are allocated through trade, not to who claims ownership of them. Once mutually beneficial trades cease (i.e., when no further Pareto improvements can be made), the economy has reached an efficient state—regardless of resource ownership.

Conclusion: The Universal Role of Labor and Externalities

From a macroeconomic perspective, labor and negative externalities are the primary collective costs that impact everyone in the economy. This holds true both in practical reality and in the mathematical foundation of our model. These core principles regarding collective costs are not only empirically testable but also logically consistent within the model's mathematical structure, built on reasonable economic assumptions. By integrating these collective costs, the model provides a robust framework for understanding how they shape economic outcomes, influence Pareto efficiency, and ultimately affect collective welfare.

Pareto Efficiency and Gradient Descent: The Role of Money and Arbitrage-Free Exchanges

In this model, Pareto efficiency is achieved in a manner analogous to gradient descent optimization. The process unfolds through a series of Pareto-improving exchanges between rational, utility-maximizing agents in the economy. Each unfettered exchange is akin to a step in a gradient descent algorithm, where participants trade goods, services, or labor in ways that improve collective welfare—just as each step in gradient descent reduces a cost function.

Money plays two key roles in this process:

• As a unit of account, money allows participants to measure and compare the value of goods and services, facilitating fair exchanges.

• As a medium of exchange, it enables transactions to occur smoothly, allowing the economy to "move" through the gradient of mutually beneficial trades.

Additionally, money functions as a store of value when it is not actively used for exchanges, such as when funds are held in a bank account for extended periods.

This aligns with empirical data from the Federal Reserve Bank of the United States, which identifies money’s three key functions: medium of exchange (E), unit of account (U), and store of value (S). These functions are universally observed in real-world economies. Any formal model that ignores them would not only be inconsistent with empirical reality but also mathematically unsound, as it would contradict the key definitions of how money operates in economic systems.

We also assume the no-arbitrage principle—the idea that no risk-free profit opportunities exist. All trades are mutually beneficial and reflect fair value, with no possibility of risk-free profit. This corresponds to the "no free lunch" concept in gradient descent, where the algorithm progresses naturally toward an optimal solution without shortcuts. This assumption is crucial for the model to align with reality. As the economy progresses through a series of these mutually beneficial, arbitrage-free exchanges, it converges toward Pareto efficiency, much like gradient descent iteratively approaches the minimum of a cost function.

Each exchange nudges the economy closer to a state where no further Pareto improvements can be made. In gradient descent, optimization stops when the gradient of the cost function reaches zero—indicating that the minimum has been reached. Similarly, in our model, Pareto efficiency is achieved when no additional mutually beneficial trades are possible. At this final state, no one can be made better off without making someone else worse off—just as gradient descent halts once it reaches an optimal point.

Conditions and Axioms

Our core axiom of human behavior is the principle of rational utility maximization, a fundamental assumption in both mathematical economics and game theory. This axiom posits that individuals act to maximize their utility or wealth, subject to the constraints they face.

To more accurately reflect observed economic realities, we introduce the Rent-Seeking Lemma: the rational, utility-maximizing agent is prone to fraudulent or opportunistic behavior when the perceived costs of engaging in such actions are sufficiently low. This lemma acknowledges that agents will exploit opportunities for personal gain if the penalties or risks of such behavior are minimal, deviating from the idealized assumption that all agents always act in a socially optimal manner.

Rent-Seeking Lemma

The Rent-Seeking Lemma posits that rational, utility-maximizing agents are prone to opportunistic behavior when the perceived costs of exploiting such opportunities are low. This behavior leads to inefficiencies and underscores the importance of robust property rights and well-functioning markets to mitigate these tendencies.

This phenomenon is well-documented in Agency Theory, particularly in Jensen and Meckling’s 1976 paper, Theory of the Firm, which introduced the principal-agent problem. In this framework, managers (agents) may act in their own interests rather than in the best interests of the owners (principals). Their 1994 work, The Nature of Man, further formalized the axiomatic structure of economic systems based on rational, utility-maximizing agents, aligning closely with the Rent-Seeking Lemma. It illustrates how rational agents, given the opportunity, may exploit commercial transactions for personal gain, even at the expense of overall market efficiency.

Further evidence of rent-seeking behavior is found in George Akerlof’s 1970 paper, The Market for Lemons, which illustrates how information asymmetries in markets can lead to exploitation. Better-informed agents extract value from less-informed counterparts—a form of wealth-extracting behavior described by the Rent-Seeking Lemma. This erodes market efficiency by redistributing wealth without corresponding productive contributions, aligning with both Agency Theory and public choice theory.

Interestingly, both Marxist theory and free-market economics acknowledge the tendency for unearned wealth-seeking. Vladimir Lenin criticized the nonproductive bourgeoisie as "economic parasites," accusing them of consuming valuable goods and services without contributing to real GDP. This critique mirrors rent-seeking behavior described in public choice theory, as developed by Gordon Tullock and James Buchanan (the latter receiving the 1986 Nobel Prize for his work). In public choice theory, successful rent-seekers—akin to Lenin’s "economic parasites"—extract wealth without contributing to productivity.

Thus, the Rent-Seeking Lemma captures a universal phenomenon: in both free-market and Marxist critiques, a subset of agents exploits systemic opportunities to accumulate wealth without producing value, distorting economic efficiency and fairness. However, this does not imply that Marx’s broader conclusions were correct. Quite the opposite—his ideas were fundamentally flawed. Marx’s error lay in his belief that the bourgeois principals could extract unearned wealth from the, by definition, better-informed agents (the workers). This contradicts Agency Theory, which shows that unearned wealth typically flows in the opposite direction: from less-informed principals to better-informed agents.

These contradictions with empirical truths render Marxism an unsound formal system. The tragic consequences of relying on such flawed theories were starkly demonstrated during the Holodomor in Ukraine, where Soviet collectivization efforts led to widespread famine and even instances of real-world cannibalism—a historical fact of the twentieth century. This empirical reality underscores the dangers of relying on unsound formal systems, where theoretical errors can lead to catastrophic real-world outcomes.

By contrast, on Wall Street, we avoid such fundamental mistakes. The use of rigorous formal systems is essential for making real, reliable profits, ensuring that decisions are grounded in sound, empirically tested models rather than flawed theoretical assumptions. As the movie says, those of us who actually make money on Wall Street don’t "throw darts at the board"—we bet on sure things by applying formal systems in mathematical arbitrage, much like Jim Simons and his team at Renaissance Technologies. If you’re curious, it’s worth looking up what they do.

Soundness, Completeness, and Consistency in Formal Systems

We raise the issue of the unsoundness of the Marxist formal system of economics to illustrate that for any formal system to be considered sound, none of its axioms or definitions can contradict empirical, objective, real-world facts. In a sound system, all conclusions must logically follow from its axioms, and those axioms must align with observable reality—defined as being self-evidently true—if the system is intended to model the real world.

To illustrate this, consider Peano's axioms in mathematics, where 2 + 2 = 4 is universally true within the system of arithmetic. Peano’s second axiom states that every natural number n has a successor, denoted n′. This is a mathematical truth, accepted within the formal system without needing empirical proof. However, if we misapply this formal system to real-world objects, contradictions can arise.

For instance, claiming that "2 moons of Mars + 2 moons of Mars = 4 moons of Mars" is nonsensical because Mars has only two moons: Phobos and Deimos. Applying Peano’s abstract successor function—used to prove that 2 + 2 = 4 is correct—to count the moons of Mars misapplies a mathematical concept in an empirical context where only two moons exist. This creates a contradiction between the formal system and observable reality, rendering the system unsound when applied to this real-world scenario.

This example emphasizes an important point: while mathematical systems like Peano’s axioms are consistent and valid within their own framework, they become unsound when applied in ways that contradict real-world facts. This principle explains why communism, derived from Marxist economic formal systems, has consistently failed in practice, despite being implemented multiple times. The unsoundness arises because the system’s axioms—such as its assumptions about agency costs and the flow of wealth—contradict observable economic behaviors and incentives. Just as a mathematical system becomes unsound when its axioms contradict facts, any economic formal system that violates empirical truths will fail to produce reliable models of reality, leading to systemic collapse and widespread failure.

Maintaining soundness in a formal system is, therefore, crucial for reliably modeling and predicting real-world outcomes.

This brings us to the Arrow-Debreu framework, which, while sound, is incomplete. In this model, money is primarily defined as a unit of account, which works well in equilibrium because that’s the role money plays once the system has reached a steady state. However, the other functions of money—store of value and medium of exchange—become essential during the dynamic process of achieving equilibrium in real-world economies. By focusing solely on equilibrium, the Arrow-Debreu model does not explain how the economy dynamically reaches that equilibrium, leaving the model incomplete.

Our Labor-For-Goods Game Theory model complements the Arrow-Debreu framework by explaining how equilibrium is dynamically achieved. It incorporates the full definition of money as it operates in the real world—serving as a unit of account, store of value, and medium of exchange—thus completing the model. By accounting for the dynamic process through which economies reach equilibrium, our model maintains both soundness and completeness, while ensuring consistency with real-world economic behavior.

The Gradient Descent Process: Arbitrage-Free Exchange Rates

To recap: each exchange in the economy brings it closer to a more efficient allocation of resources, much like how each step in gradient descent moves toward an optimal solution. In this analogy, each mutually beneficial trade is a step toward an economy-wide Pareto-efficient allocation. These trades improve general welfare by enabling participants to exchange in ways that benefit both parties, without creating arbitrage opportunities. Eventually, the process of mutually beneficial exchanges reaches a point where no further improvements can be made—similar to reaching the maximum or minimum of a function where the gradient becomes zero. At this point, Pareto efficiency is achieved: no one can be made better off without making someone else worse off, and no more mutually beneficial trades are possible.

The arbitrage-free exchange rates condition in this model follows the same no-arbitrage principle that governs exchange rates in the foreign exchange (Forex) market. Let the exchange rate matrix E represent the rates between approximately 30 major currencies, where the element e_ij represents the exchange rate from currency i to currency j. The no-arbitrage condition requires that the exchange rate from currency i to j is the reciprocal of the exchange rate from currency j to i (i.e., e_ij=1÷e_ji).

For example, if 1 USD buys 0.5 GBP, then 1 GBP must buy 2 USD. This condition eliminates arbitrage opportunities by enforcing symmetry and reciprocity in exchange rates. Mathematically, this is expressed by stating that matrix E is equal to the transpose of E after taking its element-wise reciprocal, also known as the Hadamard inverse.

In practice, the no-arbitrage condition in the Forex market is enforced using the US dollar as the unit of account to determine cross rates for currency pairs like JPY/EUR or GBP/EUR. In these cases, the dollar is not used as a medium of exchange but serves purely as a unit of account to ensure consistent pricing and prevent arbitrage opportunities.

In the foreign exchange market, where goods (represented by currencies) are exchanged directly without using money as a medium of exchange, we can clearly see that the primary role of money—aligned with the Arrow-Debreu framework—is that of a unit of account. This role is necessary to enforce the no-arbitrage condition on the exchange rate matrix by quoting prices in a consistent unit of account, such as the US dollar’s role in the Forex market.

Mathematically, arbitrage—such as profiting from trading currencies in the FX market—represents unearned wealth obtained through superior information. This is similar to how a used car dealer in a "lemon" market extracts unearned wealth from an uninformed buyer. An economic parasite, or arbitrageur, can gain wealth by exploiting currency discrepancies without contributing to productivity.

This is akin to finding $100 on the street—the person who found the money can use it to purchase goods and services, consuming resources without making any reciprocal contribution to productivity. This behavior aligns with Lenin’s definition of economic parasites and corresponds to successful rent-seekers in public choice theory, who gain wealth through manipulation or exploitation rather than through productive activities.

In public choice theory, rent-seeking includes opportunistic behavior such as arbitrage. To prevent such behavior, prices are structured relative to a unit of account, ensuring consistency across markets. By maintaining uniform pricing, this structure eliminates inconsistencies that could otherwise be exploited for arbitrage. As a result, the behavior of economic parasites—who would otherwise capitalize on price discrepancies—is effectively precluded.

Thus, it becomes clear that the primary function of money is as a unit of account. It only serves as a medium of exchange secondarily, facilitating transactions for goods and services. Given that most money today is digital, its role as a unit of account is paramount. However, we will explore this in greater detail in the main section of the paper.

The Role of Property Rights and Arbitrage-Free Pricing

Although the First Welfare Theorem assumes ideal market conditions, such as voluntary trade and symmetric information, it does not explicitly address the need for well-defined property rights. However, both the Rent-Seeking Lemma and the principal-agent problem illustrate that clear and enforceable property rights are essential for market efficiency. Without these rights, agents who fail to perform their fiduciary duties—often described as economic parasites—can exploit their positions in any organization, including government, to extract unearned wealth in the form of agency costs or economic rents. This behavior introduces significant inefficiencies, which can be severe enough to prevent Pareto efficiency in real-world economic systems.

The importance of property rights becomes even more apparent when considering that, under the Rent-Seeking Lemma and the principal-agent problem, the only individuals whose incentives are truly aligned with maximizing labor productivity are the beneficial owners. These owners directly reap the rewards of productivity improvements. By contrast, workers compensated with fixed wages are more likely to prioritize their own self-interest, which may not necessarily align with maximizing labor productivity. Under this framework, the principal-agent problem persists in most commercial, arms-length transactions, though it may not apply in cases where personal relationships, such as family, are involved (e.g., if a family member is running the business). Nevertheless, the principal-agent problem remains pervasive within the broader axiomatic framework, highlighting the crucial role of property rights in maintaining market efficiency.

Furthermore, to align with the concept of no unearned wealth, a market must also satisfy the no-arbitrage condition. Exchange rates between goods and services must remain consistent across markets to prevent arbitrage, where wealth-maximizing rational agents exploit price discrepancies for risk-free profits. Arbitrage disrupts market efficiency by enabling wealth extraction without productive contribution, similar to rent-seeking behavior. Without consistent pricing across markets, wealth can be unfairly redistributed through these exploitations, undermining both efficiency and fairness.

Implications of Opportunism: First Welfare Corollary

The tendency toward opportunistic behavior, as predicted by the Rent-Seeking Lemma under our core "opportunistic nature of man" axiom, implies that for trade to be genuinely mutually beneficial, two essential conditions must be met. This is referred to as the First Welfare Corollary of the Rent-Seeking Lemma of rational behavior:

• Unfettered Markets: Traders must be free to engage in voluntary exchanges without undue restrictions. This freedom maximizes the opportunity for Pareto-improving trades, where at least one party benefits without making the other worse off.

• Symmetric Information: To prevent exploitation, information symmetry is crucial. When one party possesses more information than the other, it can lead to rent-seeking behavior or the extraction of unearned wealth, undermining the fairness and efficiency of exchanges. Asymmetric information, as described by George Akerlof in The Market for Lemons, creates opportunities for opportunistic agents—sometimes referred to as "economic parasites" (a term borrowed from Lenin)—to extract value without contributing productively. This undermines the potential for mutually beneficial exchanges.

To maintain fairness and efficiency, markets must promote both information symmetry and unfettered voluntary exchange. However, while these conditions—unfettered trade and symmetric information—are required by the First Welfare Theorem and are key elements of the First Welfare Corollary, they are not sufficient on their own. Additional ideal market conditions are necessary for both the First Welfare Theorem and more complex models, such as the Labor-for-Goods model, to function effectively within a sound formal system that accurately reflects economic reality.

Market Conditions for Pareto Efficiency: Labor-For-Goods Game Theory Model

The following conditions are essential for achieving Pareto efficiency in the Labor-For-Goods Game Theory Model:

1. Well-Defined Property Rights: Clear and enforceable property rights prevent resource misallocation and promote optimal resource use. Agents can only trade goods they legitimately own, reducing the risk of rent-seeking through ambiguous claims or exploitation.

2. Voluntary Exchange: Voluntary exchange ensures that all trades are mutually beneficial. When agents freely engage in exchanges that improve or maintain their utility, the market moves toward Pareto improvements—trades where at least one party benefits without making the other worse off.

3. Symmetric Information: Symmetric information guarantees that all agents have access to the same information, preventing exploitation due to information asymmetry. With equally informed participants, the market functions more fairly, reducing opportunities for unearned wealth extraction and ensuring efficient resource allocation.

4. Arbitrage-Free Exchange Rates: Arbitrage-free exchange rates maintain price consistency across markets, preventing distortions caused by price discrepancies. The absence of arbitrage ensures that prices reflect the true value of goods and services, preventing agents from profiting without contributing productively to the economy.

5. Local Non-Satiation: Local non-satiation assumes that agents prefer more of a good to less, meaning they will continue trading until no further utility improvements are possible. This drives the market toward optimal resource allocation, as agents pursue mutually beneficial trades until no gains are left.

6. Perfect Competition: Perfect competition ensures that prices accurately reflect supply and demand. In a perfectly competitive market, no single agent can manipulate prices, resulting in fair and optimal pricing across goods and services. This facilitates efficient resource distribution by guiding agents' decisions in line with market conditions.

7. Complete Markets: Complete markets ensure that all potential trades can occur, eliminating unexploited gains from trade. When markets are complete, the exchange of all goods and services is possible, leaving no valuable trades unrealized.

8. No Externalities: The absence of externalities ensures that all social costs and benefits are reflected in market prices. When external costs (such as pollution) or benefits (such as public goods) are excluded from pricing, inefficiencies arise, distorting resource allocation. Proper pricing of externalities ensures the market reflects the true social value of goods and services.

9. Rational Behavior: Rational behavior assumes that agents act to maximize their utility or wealth, contributing to overall market efficiency. As part of the core axiom of utility maximization, rational behavior ensures that agents' decisions align with broader market outcomes.

Conclusion:

For the Labor-For-Goods model to function optimally and achieve Pareto efficiency, the market must not only ensure unfettered trade and symmetric information but also satisfy the additional conditions outlined above. Together, these conditions guarantee efficient resource allocation, prevent unearned wealth extraction through rent-seeking or arbitrage, and ensure that all potential gains from trade are realized. By adhering to these principles, the market reaches an allocation where no agent can be made better off without making someone else worse off.

Labor-For-Goods Game Theory Model: Formal Proof of Pareto Efficiency Under Assumed Conditions

We demonstrate that, under the assumptions of well-defined property rights, complete markets, symmetric information, voluntary exchange, local non-satiation, and arbitrage-free exchange rates, a competitive market will result in a Pareto-efficient allocation of resources.

We begin by establishing a local Pareto optimum through mutually beneficial trades, where at least one party benefits without making others worse off. This local result shows that no further improvements can be made at the individual level. We then extend this result to a global Pareto optimum by introducing additional conditions that eliminate inefficiencies, ensuring that no further improvements can be made without making other agents worse off.

Part 1: Local Pareto Optimum Through Mutually Beneficial Trade

Assumptions for Local Pareto Optimum:

1. Symmetric Information: All agents have equal access to relevant information about the goods or services being traded.

2. Voluntary Exchange: Agents engage in trade only if both parties expect to benefit from the exchange.

3. Local Non-Satiation: Agents prefer more of any good to less, ensuring they continuously seek out and engage in beneficial trades.

Proof:

1. Symmetric Information and Voluntary Exchange: With symmetric information, no agent can exploit hidden knowledge to take advantage of another. Each trade is mutually beneficial, as both parties are fully aware of the value of the goods or services being exchanged. Since voluntary exchange implies that agents only trade when they expect to improve or maintain their utility, each exchange results in a Pareto improvement.
   Key Result: Each trade improves or maintains utility for both parties, meaning no one is made worse off, and at least one party is better off.

2. Local Non-Satiation: Given that agents prefer more of a good to less, they will continue to trade as long as opportunities for mutually beneficial exchanges exist. This process pushes the market toward a local Pareto maximum, where all possible gains from trade have been realized, and no further mutually beneficial trades are possible.
   Key Result: At the local market level, all mutually beneficial trades have been exhausted, and no agent can improve their position without making someone else worse off.

Conclusion (Local Pareto Maximum):

At this stage, no agent can further improve their welfare through additional mutually beneficial trades within the local market. Thus, a local Pareto optimum is achieved, where no further Pareto-improving trades are possible within the given set of exchanges.

Part 2: From Local Pareto Optimum to Global Pareto Efficiency

To extend the local Pareto optimum to the entire economy and ensure global Pareto efficiency, we introduce additional assumptions that eliminate inefficiencies beyond the local context. These conditions guarantee that every possible beneficial trade is realized across the entire economy.

Additional Assumptions for Global Pareto Efficiency:

• Well-Defined Property Rights: Clear and enforceable property rights prevent resource misallocation and ensure that all trades occur with legitimate ownership.

• Complete Markets: All goods and services can be traded, meaning no beneficial trade is blocked due to missing markets.

• No Externalities: The costs and benefits of each agent’s actions are fully internalized, so prices reflect the true social value of goods and services.

• Perfect Competition: Agents are price-takers, and market prices accurately reflect supply and demand, guiding resources to their most efficient use.

• Arbitrage-Free Exchange Rates: Prices or exchange rates are consistent across markets, preventing agents from exploiting price discrepancies for risk-free profits.

Proof of Global Pareto Efficiency:

1. Well-Defined Property Rights:
   Clear property rights ensure agents can only trade goods they legitimately own. This eliminates inefficiencies from rent-seeking or resource misallocation.
   
   Key Result: Legitimate ownership ensures resources are allocated efficiently, preventing rent-seeking and ensuring all trades are efficient.

2. Complete Markets:
   Complete markets ensure that all potential goods and services can be traded, removing any barriers to beneficial trade.
   
   Key Result: Complete markets ensure every possible mutually beneficial trade occurs, leaving no gains from trade unrealized.

3. No Externalities:
   The absence of externalities ensures that the prices of goods and services reflect their true social costs and benefits, preventing inefficiencies caused by unaccounted external costs or benefits.
   
   Key Result: Prices reflect true social value, ensuring efficient resource allocation.

4. Perfect Competition:
   In a perfectly competitive market, prices are determined by supply and demand, and no agent can manipulate prices. This ensures prices guide resources efficiently.
   
   Key Result: Prices allocate resources efficiently, aligning with market conditions.

5. Arbitrage-Free Exchange Rates:
   The assumption of arbitrage-free exchange rates ensures that exchange rates—represented by relative prices—are quoted in a single unit of account, thereby precluding opportunistic arbitrage opportunities. This prevents rent-seeking agents from exploiting price discrepancies for risk-free profit. By maintaining consistent pricing across markets, this condition eliminates inefficiencies arising from arbitrage.
   
   Key Result: Consistent pricing across all markets eliminates distortions caused by arbitrage opportunities, ensuring efficient resource allocation.

Conclusion (Global Pareto Efficiency)

With these additional conditions, we extend the local Pareto optimum to a global Pareto optimum. When the following conditions hold:

1. Well-defined property rights,

2. Complete markets,

3. No externalities,

4. Perfect competition, and

5. Arbitrage-free pricing,

all potential Pareto improvements across the economy are realized. No agent can improve their welfare without making another agent worse off, confirming that the market is globally Pareto efficient.

Final Conclusion: Part I

The proof above demonstrates that local Pareto efficiency is achieved through mutually beneficial trade under the assumptions of symmetric information and voluntary exchange, consistent with the First Welfare Corollary and the additional assumption of local non-satiation. This ensures that agents are self-motivated to engage in mutually beneficial trades, consistent with the rational, opportunistic, utility-maximizing representative agent axiom.

By introducing further conditions—well-defined property rights, complete markets, no externalities, perfect competition, and arbitrage-free exchange rates—we extend this result to the entire economy, ensuring global Pareto efficiency. While this framework achieves a high level of Pareto efficiency, there may still be other unidentified conditions that could preclude mutually beneficial trade. As with any theory, there is no claim to a universal global maximum of efficiency. However, this represents the highest level of Pareto efficiency achievable within this theory and, to our knowledge, in reality.

Therefore, under these conditions, the market achieves a Pareto-efficient allocation of resources, where no agent can be made better off without making someone else worse off. With this understanding of the axioms and definitions provided, and recognizing that we are discussing the U = S + E model, which captures the real-world use-value and exchange-value of money within the framework of a formal system, we can now proceed with our discussion about money, with everyone aligned on the precise meanings of the terms we are using.

Another purpose of this preface is to clarify that if the predictions of both the First Welfare Theorem (within the Arrow-Debreu framework) and the Labor-for-Goods Game Theory model—both being fully sound and consistent with reality—fail to align with real-world outcomes, such as Pareto efficiency and high, growing real GDP per capita, it conclusively indicates that one or more of the axioms or perfect market conditions are violated in practice. Identifying and addressing the violated condition(s) will enable us to improve real GDP growth.

With this understanding, dear reader, you are now well-prepared for the mathematical economics discussion about money in Part II that follows. Engaging in this discussion ensures that all our predictions hold true, not only in theory but also in reality—barring the violation of one of our self-evidently true axioms. Recall what happens when Peano’s second axiom is violated: 2 + 2 = 4 no longer holds when counting Mars’s moons. Otherwise, by using a formal system, all our conclusions are guaranteed to hold true in reality. That’s how we don’t lose money trading on Wall Street, in practice.

Part II: The U=S+E of Money in Labor-For-Goods Exchanges

Contrary to common misconceptions, there is a well-agreed-upon definition of money in economics. The Federal Reserve of the United States, which issues the US dollar—the most globally recognized and used currency—explicitly defines money on its website². According to the Federal Reserve, money consistently fulfills three key roles across all economies, both historically and currently: as a unit of account, a medium of exchange, and a store of value. This definition is universally acknowledged and uncontested.

In the Arrow-Debreu model—a cornerstone of mainstream economic thought and the basis for general equilibrium models used by institutions like the Federal Reserve to set interest rates—money functions as a unit of account, in which prices are expressed in a Pareto-efficient general equilibrium. The debate on the nature of money, as summarized in A Walrasian Theory of Money and Barter (Banerjee-Maskin, 1996), can be traced back to the 1870s, with the seminal contributions of early mathematical economists like Jevons, Menger, and Walras. In the late 19th century, these economists identified a key function of money as a medium of exchange, addressing the double coincidence of wants problem inherent in direct barter.

Over time, the characterization of money as a medium of exchange has become somewhat axiomatic in mainstream mathematical economics, almost taken as dogma. However, accepting such conjectures without scrutiny can be precarious. It is essential to remember that axioms—even the most obvious, such as the Euclidean postulate that the shortest distance between two points is a straight line—are, by definition, assumptions introduced without proof. While they may seem self-evident, such axioms can, and often do, prove inapplicable in certain contexts. For example, the 'straight line' axiom does not hold in non-Euclidean geometries, such as those employed by GPS systems, which adjust for the curvature of space-time using Riemannian geometry, as described by Einstein.

All axioms are assumptions—educated guesses—often considered 'self-evidently true' by those who formulated them, sometimes centuries ago. Over time, axioms can become conflated with facts, transforming into dogma. This becomes problematic when such assumptions are not universally applicable, as they may prove false in specific circumstances. For example, the axiom of separation in Zermelo-Fraenkel (ZF) set theory posits that any set of two elements can be separated into two subsets, each containing one element. While this axiom seems intuitively self-evident, it is logically false when applied to entangled elementary particles like electrons or photons, as demonstrated in quantum mechanics. The limitations of traditional axiomatic approaches become evident when faced with quantum entanglement and the empirical falsification of Bell’s Inequality³.

Despite ongoing debates—particularly regarding whether assets like gold, which central banks use as reserves but not as mediums of exchange, should be considered 'real' money—the fundamental facts remain uncontroversial: Money has taken various forms throughout history. From cattle and tobacco leaves to cowrie shells, gold, and silver coins, to today's fiat currencies, all have served as money. Within the Arrow-Debreu framework, these diverse forms of money are categorized as units of account or simply ‘money-units,’ facilitating global trade. This diversity is exemplified by the approximately 30 different currencies actively traded on the Forex market, each recognized as valid fiat currency within its respective country or regional economy, such as the Euro (EUR).

Drawing insights from Bertrand Russell’s 1959 interview⁴ on the legacy of future generations, it becomes clear that there is an imperative to transcend existing dogmas and focus on factual evidence. This approach shows that the views of Jevons, Menger, Walras, and even Arrow and Debreu, are not in conflict with one another, nor with the empirically observed roles and functions of money as detailed by the Federal Reserve Bank of the United States. In theory and in practice, there is agreement on the three indispensable functions any currency must serve within an economy: acting as a unit of account (U), a store of value (S), and a medium of exchange (E).

The Three Roles of Money Equality: U = S + E

We begin by examining money’s role as a medium of exchange ('E'), a concept extensively explored by Jevons, Menger, and Walras. The proverb "measure twice, cut once" aptly applies to monetary transactions, capturing the essence of using money as a medium of exchange to pay for goods and services. Before making a purchase, rational economic agents engage in comparative price evaluations—whether assessing different car models or comparing other goods and services—considering both price and how it aligns with their financial status. In this sense, money as a medium of exchange also measures the exchange rates between all traded goods and services, including wages, which are derived from the time spent working. These wages, representing the fruits of a representative agent’s labor, are used to purchase goods and services—the fruits of others’ labor. Money as a unit of account relates wages to prices, defining its 'U + E' (Unit of Account + Medium of Exchange) role.

The use of money as a unit of account ('U') measures the exchange rates of goods and services relative to one another and to wages (such as the relative costs of consumption options or the price of an apartment compared to monthly wages). This role operates ex-ante in the decision-making process. At the point of purchase, money acts as a medium of exchange ('E'), facilitating the transaction. This dual functionality allows for a comprehensive appraisal of purchasing options within one’s economic means.

Although the Arrow-Debreu model primarily focuses on general equilibrium without explicitly integrating money, it implicitly requires money as a unit of account to articulate equilibrium prices. This highlights the importance of the total spendable money supply in facilitating complex economic calculations and supporting market operations. In today’s real-world economy, the U.S. dollar's total spendable supply—measured by the M2 money supply, as estimated by the Federal Reserve—includes all readily available funds such as cash, balances in checking and savings accounts, and money market funds, all denominated in U.S. dollars. This parallels ancient economies like Rome, where the total quantity of minted aureus gold coins represented a comparable measure of spendable wealth.

Defining 'U' as units of money available for payment concurrently (all at once) illustrates the dual functionality of money as both a medium of exchange and a store of value ('S'). Unspent money, whether it’s gold coins stored in a vault or funds saved in a bank, serves as a store of value. This concept aligns with Keynes’s liquidity trap from his 1936 work, The General Theory of Employment, Interest, and Money, where money operates as 'U + S' (Unit of Account + Store of Value), representing generalized purchasing power.

We assert the principle of exclusive dual-use, stating that money can either be saved or spent, but not both simultaneously—much like the adage, "you cannot have your cake and eat it too." Thus, at any given time, money functions exclusively as either a store of value ('S') or a medium of exchange ('E')—but not both simultaneously, as indicated by XOR. By combining 'S' XOR 'E', which is mathematically equivalent to simply adding the E and S usages, we arrive at the spendable money supply, such as M2 or minted aureus coins, denoted as 'U'. This integrated view, represented by 'U', merges the roles of 'U + E' and 'U + S', linking overall purchasing power with the specific prices of goods and services. Therefore, money’s dual role—transitioning between a unit of account, a medium of exchange, and a store of value—encapsulates its comprehensive function within the economic system.

In conclusion, money is mathematically defined by its usage, represented as U = S + E. This foundation prepares us to explore the deeper mathematical underpinnings of our money definition, aligning it with the first welfare theorem of mathematical economics through the labor-for-goods game-theoretic approach outlined in Part I. The upcoming technical sections aim to reinforce this discussion. Readers less inclined toward mathematical detail may skip this section and proceed to "What Makes Money 'USE'-able?", which focuses on the practical aspects of money’s utility.

The Mechanism of Welfare Optimization in Perfect Markets: Exploring Mutual Benefit and Pareto Efficiency

The Arrow-Debreu model mathematically proves that a Pareto-efficient equilibrium, defined by maximum overall welfare, inevitably results under a set of strict assumptions. These include local nonsatiation and marginal diminishing utility of consumption, which ensure that a maximum exists and is sought. Additional assumptions, such as the absence of externalities and the presence of competitive markets (meaning no monopolies), are necessary to ensure that the maximum overall welfare is indeed the global maximum. This represents the maximum theoretical happiness achievable under existing production constraints. Yet, as George Orwell famously wrote, “All animals are equal, but some are more equal than others.” In this context, the key assumptions that ensure at least a local, if not global, maximum welfare are the dual requirements of unfettered and symmetrically informed exchange.

Unfettered exchange means fully voluntary transactions by each counterparty, excluding theft and robbery. Symmetrically informed exchange means that each counterparty has equal knowledge about the goods and services being exchanged, precluding the possibility of fraud. Together, these conditions ensure the generation of both producer and consumer surplus.

Fraud facilitated by asymmetric information is an extensively studied topic in mathematical economics. This type of fraud manifests as the difference between ex-ante consumer surplus (represented by the expected use value of purchased eggs, thought to be fresh) and ex-post consumer surplus (defined, for example, by the lack of use value upon realizing the eggs are rotten).

The first welfare theorem of mathematical economics asserts that unfettered and symmetrically informed exchange is guaranteed to be mutually beneficial. The reason is simple: no rational individual, under any definition of rationality, would willingly engage in an arm's length transaction unless they perceived a net benefit beforehand, thereby guaranteeing ex-ante surplus.

This guarantee of ex-ante surplus translates, under the condition of symmetric information, into a realized ex-post surplus, barring acts of God, such as dropping purchased eggs on the way home from the supermarket. Both counterparties to the trade being equally well-informed about the goods and services being traded ensures mutual benefits in reality under the condition of unfettered trade. For example, when someone buys a bottle of Coca-Cola from Walmart, the seller knows exactly how much money they are receiving, and the buyer knows exactly what they are buying.

Because such unfettered, symmetrically informed trade is guaranteed to be mutually beneficial, in the context of this theorem, it is referred to as Pareto-improving, and represents real-world gradient descent optimization. Mutual benefits incrementally drive the economy towards a state of Pareto-efficiency, where the gradient of the objective function becomes zero in a theoretical equilibrium. Vilfredo Pareto elegantly described this as a situation where no one can be made better off without making someone else worse off.

To mathematically represent consumer and producer surplus, the Arrow-Debreu model introduces the concept of money as a unit of account in which prices are measured. Prices are used to define consumer surplus as the difference between the maximum price a consumer is willing to pay and the actual market price. Similarly, a producer's surplus is the difference between the sale price and the total costs of production, including opportunity costs. However, all these costs and benefits are measured in money.

Moreover, the maximum price a consumer is willing to pay is dually defined by the opportunity cost as a producer, specifically the time necessary to spend working to earn enough wages to purchase the desired goods and services. Wages, also measured in money, make money a crucial unit of measure within the Arrow-Debreu framework. It is used not only to compare the prices of competing goods and services, such as different cars we might consider purchasing, but also to relate these prices to our income. This allows us to compare the subjective costs and benefits of consuming goods and services versus producing the necessary wages to engage in further consumption.

This comparison motivates us to improve labor productivity in a free market. In effect, this proves mathematically that at least a local maximum (though perhaps not a global one) in labor productivity results from “honest” trade. This naturally motivates us to engage in labor specialization to maximize productivity, exactly as Adam Smith posited in The Wealth of Nations.

Assuming no robbery, theft, cheating, or fraud facilitated by asymmetric information about the goods and services being traded, the Arrow-Debreu framework provides a comprehensive system to measure and optimize economic welfare. And the U = S + E equation makes the role of money clear.

Quantifying the subjective benefits of experiences poses a significant challenge due to their inherently unmeasurable nature. Within this economic framework, money plays a crucial role as a unit of account, facilitating trade and optimizing collective welfare by converting subjective utilities into monetary terms. Consumers and producers engage in mutually beneficial trade, exchanging the fruits of their labor, quantified as monetary wages, for goods and services similarly priced in monetary terms. Thus, money serves a dual function: firstly, as a medium of exchange that enables transactions; and secondly, as a unit of account, quantifying the subjective expected benefits of purchases against their subjective costs, often assessed in terms of the time required to earn the necessary funds. Many individuals may evaluate the subjective cost of an apartment based on the time it takes to earn the income necessary to cover the rent. Additionally, in scenarios where money is not immediately utilized for payments, it seamlessly transitions into its defined dual role as a store of value, thereby encapsulating its universally recognized functions as a unit of account, medium of exchange, and store of value.

Unraveling the Optimization of Welfare in Perfect Markets: A Comprehensive Analysis

In a perfect market, welfare maximization results from assured mutual gains from trade, operating without restrictions and based on symmetric information. This principle ensures that every participant, whether consumer or producer, benefits from their transactions. Effectively, the Arrow-Debreu model conceptualizes welfare maximization as a real-world gradient descent optimization, where mutual benefits drive the system toward a global maximum. Here, maximum welfare is achieved when the gradient of this objective function approaches zero—a condition characterized by Pareto efficiency. This state, where no further mutually beneficial trades are possible because improving one party's welfare would necessitate diminishing another's, is eloquently defined by Vilfredo Pareto. He described it as a situation where "no one can be made better off without making someone else worse off," saying that "the gradient of the objective function becomes zero" in more intuitive language for describing economic equilibria.

To ensure mutual benefit, buyers must achieve a consumer surplus, which represents the difference between the maximum amount they are willing to pay and the actual price they pay. Similarly, sellers must realize a producer surplus, defined as the difference between the price they receive and their cost of production. These surpluses are essential for defining Pareto-improving outcomes, where the welfare of one participant can be enhanced only alongside the benefit of another, ensuring a situation where everyone wins.

Such outcomes inevitably lead to Pareto efficiency, a state in which it is impossible to improve the economic welfare of one party without reducing the welfare of another. This state marks the culmination of optimal resource distribution and maximum welfare under perfect market conditions.

Producer surplus is defined as the difference between the price at which goods are sold and their total economic cost of production. Measuring this surplus is relatively straightforward, encompassing not only direct expenses such as materials and labor but also the opportunity costs associated with the production process. In mathematical economics, opportunity costs are considered the value of the next best alternative that is forgone when a decision is made. For example, in the real estate industry, opportunity costs might include expenses related to advertising an apartment, the time spent by owners or managers in collecting rent, performing repairs, and managing tenant relations, including the eviction of non-paying tenants. Each of these elements represents an investment of resources that could have been allocated elsewhere. Accurately accounting for these costs is crucial for assessing the economic profitability of property management comprehensively. It provides a more detailed view of the economic sacrifices made in the pursuit of business activities, highlighting the real costs involved in maintaining and enhancing the value of real estate properties.

Unlike producer surplus, consumer surplus cannot be directly quantified due to its reliance on the subjective perceptions of benefit and cost to the individual. This is precisely why money plays a crucial role in defining consumer surplus in mathematical economics. Consumer surplus is defined as the difference between the maximum price a consumer is willing to pay for an item—based on their personal valuation—and the market price at which the item is actually purchased. Money, in its role as a medium of exchange, naturally becomes a unit of account in which the exchange rates of all goods and services in the economy are measured, including the exchange rate between our wages and the goods and services we purchase and consume. Money transforms such subjective valuations into quantifiable terms. This allows individuals to compare their willingness to pay with actual market prices and to assess the surplus they derive from transactions. For instance, the consumer surplus on an iPhone would be the difference between what a consumer is prepared to pay for it, reflecting their personal valuation of its features and utility, and the price they actually pay at the store. This surplus represents the perceived value gained by purchasing the product at a lower price than what was internally deemed acceptable.

Quantifying Consumer Surplus: The Role of Money in Valuing Subjective Benefits

Money plays a pivotal role in quantifying the subjective costs of items to individuals, serving as a unit of account for the ‘true’ opportunity cost in economic transactions. Individuals, in their dual roles as both producers and consumers, typically measure the subjective cost of an item in terms of the time required to earn the income necessary for its purchase, often calculated in wages. This measurement captures not only the financial expense but also the effort and time invested, which individuals equate with the value of the product. Thus, consumer surplus provides valuable insights into the additional value individuals derive from their purchases relative to the labor they expend to earn the necessary income. This concept is fundamental in the context of mathematical economics, highlighting the intricate relationship between subjective valuations and economic decision-making.

Although consumer surplus is subjective, it can be quantified in monetary units because the amount a consumer is willing to pay for a product reflects their perceived benefit from the purchase. It is impossible for a third-party observer to determine whether an individual consumer derives twice the subjective benefit from buying Product A (e.g., a Bentley) as from Product B (e.g., a Ford) simply because the price of A is twice that of B. However, we can assert with absolute certainty that the consumer perceives a relatively greater benefit from Product A than from Product B, as evidenced by their willingness to pay a higher price for it. This willingness to pay more provides a 'rank-order' measure of utility. Money does not measure subjective benefit directly, but it is 100% rank-order correlated with subjective utility to the consumer, as more money is guaranteed to provide more utility by imposing fewer constraints on what the consumer is able to consume. Thus, money serves as a unit of account for the rank-order value of subjective utility.

Direct measurement of subjective utility is challenging due to its dual nature: expected subjective utility of a product (ex-ante, or before the purchase) can drastically differ from the realized utility (ex-post, or after the purchase), as illustrated by George Akerlof in the market for “Lemons.” For example, the expected utility of eggs might be high before purchase, but this can change significantly if the eggs turn out to be rotten. Measuring subjective utility is inherently difficult due to its abstract nature. However, an approximation can be achieved by observing the rate at which goods and services are consumed over time. This rate of consumption can be mathematically expressed in monetary units as ‘subjective-rank-order(dU/dT),’ where dU/dT represents the rate of change in subjective utility with respect to time.

In this formulation, money plays a crucial role not only in quantifying consumption rates but also as a critical unit of account and a measure of purchasing power. It is continuously used to compare the costs of different potential purchase options to the hourly wages, serving as a unit of measure before being utilized as a medium of exchange to facilitate trade. Furthermore, its function as a unit of account is intertwined with its capacity as a store of wealth, effectively modulating consumption rates. This underscores the essential function of money as a unit of account: it quantifies and ranks subjective utilities over time before making purchases, providing a practical framework for economic analysis and decision-making. By moderating consumption according to financial resources, money fulfills its dual role as both a measure of value and a store of value, helping preserve purchasing power over time, stabilize economic interactions, and promote a sustainable economic environment.

Unfettered and Symmetrically Informed Exchange: The first Welfare Corollary

It is an evidence-based claim, independently verifiable for accuracy, and therefore one that cannot turn out to be false, that any parasitic infestation, such as locusts or vermin like rats eating grain stored in a warehouse, directly reduces efficiency. In reality, the consumption of real-world goods and resources by “economic parasites” often results from involuntary exchanges, such as robbery, theft, extortion, and kidnapping. All such obviously criminal activities are universally punishable by imprisonment due to the fact that “unearned wealth extraction” by “economic parasites” inevitably reduces economic efficiency. For example, due to lawlessness, the per capita GDP in Haiti is five times lower than that in the neighboring Dominican Republic. This real-world inefficiency, as measured by a fivefold difference in real-world per capita GDP, results from the direct violation of the unfettered trade assumption, a condition necessary for achieving Pareto efficiency.

According to the first welfare corollary of mathematical economics, any violations of two key conditions of unfettered (meaning fully voluntary) and symmetrically informed exchange inevitably result in real-world inefficiencies. George Akerlof pointed this out in "The Market for Lemons" in 1970, where he showed that the presence of asymmetric information in real-world trade inevitably results in market inefficiencies. This is exemplified by the unearned wealth extraction by a fraudulent used car dealer sticking a less-informed (or asymmetrically informed) buyer with a broken-down car, known as a “lemon.” This means that in order to achieve any kind of real-world efficiency, trade must be fully voluntary and symmetrically informed.

Limitations of the Arrow-Debreu Model: Rational Utility Maximization and Real-World Application

The Arrow-Debreu model accurately describes real-world human behavior using a praxeological method rooted in ancient Greece, where praxeology posited as an axiom that human action was not random, but goal-driven. This goal, both in mathematical economics and game theory, has been identified as ‘winning’. At the core of this model is the axiom of the 'rational utility maximizer representative agent'. This axiom posits that individuals—or players in a game—are rational, utility-maximizing actors. These actors behave in ways that maximize their welfare within the constraints of the game’s rules, employing strategies that yield the highest possible payoff.

This characterization logically implies that a representative agent in the economy aims to maximize utility not only by maximizing labor productivity but also by minimizing prices/costs, even if it means resorting to unethical behaviors when they prove profitable. For instance, in San Francisco, the policy decision not to prosecute thefts under $950 as criminal offenses has led to predictable outcomes. The minimal enforcement of this threshold has spurred such a high rate of theft that many stores have been forced to close or relocate. This situation illustrates a significant deviation from theoretical economic models, highlighting a stark contrast when foundational assumptions—such as effective legal deterrence—are not met in practice.

Another notable deviation from the Arrow-Debreu conditions is highlighted by George Akerlof’s study on the market for "lemons," where asymmetric information—not direct theft like in San Francisco—leads to suboptimal outcomes. An even more striking example of the violation of assumptions is observed in the stark disparity in per capita GDP between Haiti and the Dominican Republic, attributable to widespread involuntary exchanges in Haiti. These conditions violate the unfettered trade assumption posited by the first welfare theorem of mathematical economics and the Arrow-Debreu model, underscoring the critical importance of adhering to these theoretical prerequisites for achieving efficient market conditions. Such discrepancies between axiomatic assumptions and reality emphasize the inherent limitations of applying theoretical economic models without accommodating the variability and unpredictability of real-world economic environments.

The Impact of Asymmetric Information on Economic Efficiency

Violations of the two key Arrow-Debreu conditions—voluntary exchange and symmetric information—always undermine economic efficiency, both in theoretical models and real-world scenarios. The reason is obvious: such violations prevent the achievement of even a local, let alone a global, maximum. George Akerlof, along with Michael Jensen and William Meckling, has profoundly enhanced our understanding of how asymmetric information can lead to market inefficiencies. In their seminal work "The Theory of the Firm," Jensen and Meckling introduce the concept of "agency costs," which arise from information asymmetry between agents and principals.

Expanding on this dynamic, Jensen's influential 1986 paper "Agency Costs of Free Cash Flow, Corporate Finance, and Takeovers," discusses the agency conflicts between managers and shareholders that influence corporate decisions, such as dividend policies. Jensen argues that managers might prefer to retain earnings rather than distribute them as dividends, particularly when they control investment decisions. This preference can disadvantage shareholders by restricting their potential gains from dividends, thus illustrating the practical implications of asymmetric information on dividend policy. This discussion also highlights the broader consequences of information gaps on corporate governance and shareholder value, demonstrating the extensive reach of asymmetric information in affecting economic outcomes.

Mitigating Market Inefficiencies: The Role of Money in Reducing Asymmetric Information and Arbitrage

As we delve into the concept of asymmetric information, it becomes clear how this phenomenon not only facilitates fraud—characterized by exchanges that are not mutually beneficial—but also underscores the crucial role of money as a unit of account in preventing arbitrage. Arbitrage involves the simultaneous purchase and sale of the same asset in different markets and is a clear indicator of market inefficiency. Within the Arrow-Debreu model, market inefficiency, or failure, is defined as the ability to earn 'economic rents.' These can be metaphorically likened to goods pilfered by rodents or other vermin in a warehouse, who consume without producing—aligning with Gordon Tullock’s definition of rent-seeking, which describes the pursuit of wealth without a reciprocal contribution to productivity.

The prevalent issue of real-world arbitrage is particularly troubling as it allows arbitrageurs to gain purchasing power—represented by money—without enhancing productivity. This scenario critically undermines market efficiency, emphasizing the importance of money in maintaining economic stability and fairness.

Information asymmetry exists not in space, but in time; it manifests ex-post when the true value of goods and services obtained in a trade becomes apparent. This realization often does not align with the ex-ante expectations of utility or use value that the buyer had before purchasing items like rotten eggs or a "lemon" car. This temporal discrepancy between expected use value ex-ante and actual use value ex-post leads to dissatisfaction or perceptions of unfairness in the exchange value paid or received.

Arbitrage opportunities are often facilitated by temporal asymmetric information between buyers and sellers about prices, a concept vividly illustrated by Michael Lewis in Flash Boys: A Wall Street Revolt (Lewis, 2014). The absence of real-time knowledge regarding all active pending bids and offers allows high-frequency trading (HFT) firms, such as Citadel and Virtu Financial, to amass substantial profits through straightforward arbitrage strategies. These firms exploit the information gap by accessing data on different prices for the same asset more quickly than others. In an environment devoid of such information asymmetry, transactions that these intermediaries facilitate would instead occur directly between buyers and sellers across markets. In scenarios where information asymmetry is eliminated, all parties would have complete visibility and understanding of market conditions and prices, thereby rendering the arbitrageur's role obsolete. This ideal situation underscores the significant impact of timely and transparent information in ensuring market efficiency. The profits that arbitrageurs derive from these informational gaps underscore the necessity for mechanisms that can bridge these asymmetries.

Of course, a clear violation of real-world market efficiency is the existence of arbitrage in the foreign exchange, or Forex, market. This type of arbitrage allows one to earn wealth simply by trading currencies – which today occurs simply by pressing buttons on a computer and doing nothing else – without participating in the production of goods and services consumed with this wealth. It is the very definition of unearned wealth extraction through the use of asymmetric information about currency prices at different banks.

No-Arbitrage Constraint on E: The Transpose of Its Own Hadamard Inverse

Our discussion begins with an analysis of the Forex market. In the real-world FX market, approximately n=30 of the most actively traded currencies are exchanged, and their exchange rates can be mathematically represented as an exchange rate matrix, denoted as E. In this matrix, the value in row i and column j represents the exchange rate from currency i to currency j. This matrix serves as a model for understanding how exchange rates between not only currencies but also all goods and services that can be bought or sold in an arm's length commercial transaction in an economy are structured to prevent arbitrage—the very definition of a market failure.

Arbitrage, by its very nature, is not possible if a uniform price is maintained for an asset in different markets. Specifically, in the foreign exchange market, if the exchange rate of currency A to B is set, then it must be the reciprocal of the exchange rate of B to A. For example, if $1 buys £0.50, then £1 should buy $2. This reciprocal relationship is crucial to eliminating arbitrage opportunities arising from exchange rate discrepancies. Let the matrix E represent a set of exchange rates, such as those observed between the 30 or so currencies in the FX market, where n, the number of rows and columns, would be 30. The no-arbitrage condition dictates that E must be equal to its own transpose once the Hadamard inverse is applied. Mathematically, the exchange rate matrix is constrained as follows: . This condition, where the matrix becomes the reciprocal of its own transpose under the E=E_T “no-arb” constraint, is very similar to the property of a matrix being involutory, where an involutory matrix is its own inverse, A=A^-1.

Let us formally refer to such matrices as ‘evolutory’, rather than involutory. An evolutory matrix E, is equal to its own reciprocal transpose, E_T constrained such that e_i,j=1÷e_j,i. The reason for the distinction is that while A·A^-1=I (the identity matrix), 𝐸⋅𝐸_𝑇=E²=𝑛⋅𝐸=(𝐸^𝑇⋅𝐸^𝑇)^𝑇. We note in passing that 𝐸⋅𝐸^𝑇, and 𝐸^𝑇⋅𝐸, do not multiply to form 𝑛⋅𝐸, but result in two other matrices.

As we can see, the evolutory matrix, when multiplied by its-own reciprocal transpose, results not in the identity matrix but rather a scalar multiple of E scaled by its row count, n effectively becoming E². The reason for this is that the constrained matrix E=E_T has a single eigenvalue, which is also its trace, and is invariably equal to n, due to the fact that the exchange rate of a currency with itself is, by definition, always 1.

By imposing the E=E_T condition, the matrix E simplifies, having only a single eigenvalue, n, and reducing to a vector-like structure. This simplification occurs because each row or column of E can define the entire matrix, dramatically reducing the dimensionality of the information required to quote exchange rates. For example, the entire matrix E is equal to the outer product of its first column and its first row, which also happens to be the reciprocal of the first column, producing the full matrix. Consequently, each row or column of E is proportional to the others, meaning that all rows or columns are scalar multiples of one another. This characteristic renders E a rank-1 matrix, indicating that all of its information can be captured by a single vector.

What is interesting here is that in theory, an unbounded matrix raised to the fourth power should have four roots, but in reality, because of the E=E_T constraint, the matrix E has only two such fourth roots, E and E^T because E⁴=E·E·E·E=(E^T·E^T·E^T·E^T)^T. We believe it is worthwhile to highlight what might be an obvious connection. In the context of Einstein's equation E=mc², if E is a bounded E= E_T matrix, then E⁴=n²·E=m·c². Here, mass is simply the fourth root of energy.

However, while E theoretically has four roots, in reality, only two roots exist due to the E=E_T evolutory constraint imposed on E via quantum entanglement. Thus, under this evolutory constraint on E, mass is equivalent to energy but exists as a strictly constrained subset of all possible energy states, restricted by the E=E_T condition imposed on E.

While this remains purely conjectural, it intriguingly aligns not only with the principle of supersymmetry in theoretical physics but also, surprisingly, with the ancient Hermetic axiom 'as above, so below.' This concept also echoes the geometry of the Egyptian pyramids and is reminiscent of the notion that '42' is the 'answer' to the ultimate question of life, the universe, and everything, as humorously proposed in The Hitchhiker's Guide to the Galaxy. Although this reference is not directly related to quantum physics, it touches on probability.

At this point, we must caution that our expertise in theoretical physics is limited to interactions with physicist colleagues during our tenure managing the stat-arb book at RBC on Wall Street. Therefore, please treat our comments about physics with considerable skepticism, particularly the ideas about quantum set theory outlined below, which are purely speculative. This may assist a physicist who is not on Wall Street, unlike those making real money at hedge funds such as Renaissance, founded by the late Jim Simons.

In a matrix that simplifies to a vector-like structure, the entirety of the matrix can be described by any of its rows or columns. Here’s what happens in such a scenario:

1. Instead of needing to know all elements of a matrix (which in a full matrix would be 𝑛×𝑚 values), you only need to know the elements of a single vector (either 𝑛 or 𝑚 values, depending on whether it's a row or a column vector). This drastically reduces the dimensionality of the information required.

2. This vector represents a form of data compression, where instead of storing or processing multiple independent pieces of information, one vector informs the entire structure. This simplification could improve the efficiency of computations and analyses involving 𝐸.

3. Extending this idea to a theoretical framework, especially in contexts like quantum mechanics, can lead to intriguing possibilities:

4. In quantum mechanics, states can be superposed and entangled. A matrix that simplifies to a vector-like structure might analogously suggest a system where states are not independently variable but are intrinsically linked—a form of quantum entanglement at a mathematical level.

A new set theory that models such matrices could consider sets where elements are fundamentally interconnected. Traditional set theory deals with distinct, separate elements, but this new theory could focus on sets where elements are vector-like projections of one another.

Such a theory could be useful in fields like quantum computing or quantum information, where understanding entangled states in a compressed, simplified form could lead to more efficient algorithms and a better understanding of quantum systems.

By utilizing a matrix that reduces to a vector-like structure as a basic element, we could potentially model a system where traditional notions of independence between elements are replaced by a more interconnected, entangled state representation. This could open new avenues in both theoretical and applied physics, especially in handling complex systems where interdependencies are crucial.

We note in passing, as illustrated here in this video from MIT online lectures⁵, the axiom of separation from ZF set theory is used to derive Bell’s Inequality. At approximately the 1 hour and 15 minute mark, the lecturer uses the axiom of separation, for example, to split up the set 𝑁(𝑈,¬𝐵) into 𝑁(𝑈,¬𝐵,¬𝑀) and 𝑁(𝑈,¬𝐵,𝑀) In this particular case, when set elements are pairs of entangled particles, the axiom of separation does not work, simply because such a set cannot be split up into two separate subsets. That is what “entangled” means in reality – “inseparable.” However, if we replace set elements with vectors that are all entangled on account of being constrained by 𝐸=𝐸_𝑇, we may—with hard work that no one in their right mind would do for free—develop a better set theory that will more accurately model quantum entanglement, akin to the way Riemannian geometry was derived from a set of axioms that more accurately reflect the reality of how curved space-time actually operates.

The Role of Linear Algebra in Market Efficiency

As mathematical economists, we find that the linear algebra formulation captures the essential idea that in an arbitrage-free market, the reciprocal relationships between exchange rates of different currencies, as well as all goods and services, must be consistent. An arbitrage-free exchange rate matrix E, such that E_T=E (since it is equal to its reciprocal transpose), imposes constraints on exchange rates, eliminating opportunities for arbitrage by simply transposing and reciprocating the exchange rate matrix.

In this framework, prices represent the exchange rates of all goods and services relative to a single specific row or column in the full exchange rate matrix E, chosen as the unit of account. This framework supports the theories of Arrow and Debreu and is even, surprisingly, consistent with the ideas of Marx. Indeed, the key role of money is to regulate markets by preventing arbitrage, by acting as a single unit of account in which the prices of all other goods and services available for sale are expressed, precluding the existence of multiple prices for the same asset, which inherently facilitates arbitrage.

This is vividly illustrated by the real-world practice of quoting all currencies in the foreign exchange (FX) market against a single standard currency, currently the U.S. dollar, which plays a pivotal role in reducing the scope for arbitrage, thereby nudging the market toward an ideal no-arbitrage condition. By standardizing currency pairs relative to the dollar, there is greater predictability and consistency in exchange rates. This systemic approach effectively minimizes the discrepancies and gaps that arbitrageurs typically exploit, leading to a more stable and equitable trading environment.

While the application of linear algebra might often seem excessive in financial contexts, its use in this scenario is particularly warranted. Viewing the prices of goods and services through an exchange rate matrix effectively underscores money’s role strictly as a unit of account. In the real-world FX market, where all currencies are traded in pairs, cross rates for pairs such as EUR/GBP or EUR/JPY are determined using the U.S. dollar solely as a unit of account. This approach not only emphasizes the functional use of money exclusively as a unit of account but also highlights the practical utility of quoting all prices relative to a single standard asset. Adopting this methodological choice significantly enhances market efficiency by increasing information symmetry among participants and reducing arbitrage opportunities, thereby establishing consistent prices for each asset across all markets.

Diminishing Productivity Through Arbitrage

Arbitrage diminishes productivity because it allows a non-producing arbitrageur, X, to consume goods and services produced by others without contributing to their production. X earns money by facilitating a trade between A and B that should have occurred directly between them in a more efficient market with symmetric information. The fact that both parties placed open orders to buy and sell, which did not cross due to X's speed, highlights this inefficiency. We posit that losing half the bid-ask spread is not worth having your trade executed 2 milliseconds earlier for 99.999% of the public.

Consider two scenarios: one with and one without the presence of arbitrageur X, who acts as an unwanted intermediary preventing A and B from trading directly. In the absence of X, all trades occur at the mid-quote. However, if X is present, some trades occur at the bid and others at the offer. In other words, the difference between E and E_T becomes greater in the presence of X, as this bid-ask bounce volatility represents the “alpha” that X earns.

This highlights that the root cause of market inefficiency, defined by arbitrage in terms of prices, is the existence of multiple prices for the same asset. This is exemplified by the ability to buy at the offer and sell at the bid, instead of consistently trading at the mid-quote. Within the framework of the exchange rate matrix E, this inefficiency can be quantified as the difference between E (ask or bid) and E_T (mid-quote), multiplied by the trading volume. In this scenario, the calculation equals the profits earned by the arbitrageur (half the bid-ask spread), down to the penny.

Not every trade in reality is facilitated by an arbitrageur. However, we can approximate E_T by the Volume Weighted Average Price (VWAP) for the period we are examining, subtracting it from the price of each executed trade, using the fill price as E (ask or bid). What we are doing here is “collapsing the wave function”—averaging out all the different E matrices over time, thereby producing an estimate of the value of E_T. When we measure Pareto efficiency this way, it becomes clear that the more volatile the prices, the greater the difference between E and E_T, and the less Pareto efficient the market.

Beyond arbitrage, other types of unearned wealth extraction occur in the economy, as exemplified by agency costs and economic rents in public choice theory. However, the role of profits in this context is often overlooked. While Karl Marx's theory correctly states that corporate profits measure Pareto efficiency, it also has a flaw: it equates corporate profits with "economic rents"—an idea that is both theoretically and factually inaccurate.

For Marx's theory to be true, the employer must have asymmetric information about the labor being provided to cheat the worker and extract economic rents. This would apply to extracting value from any counterparty in free trade, such as selling labor for wages in a free market economy. However, this is impossible. In reality, any such "surplus"—i.e., "unearned or fraudulent" extraction of value—can only flow in the other direction, from employer to employee, as illustrated by agency costs. This is because the employee is always better informed than the employer about the labor they produce.

Setting aside the bug that equates corporate profits with rent-seeking, we see that corporate profits measure the relative Pareto efficiency of an economy, given existing production constraints. In this sense, higher economic profits – net of all opportunity costs, including the opportunity cost of capital – indicate a less Pareto-efficient economy. While the bid-ask bounce for most commodities is relatively low, this is not the case for labor. The difference between the price at which you sell your labor to your employer and the price at which your employer effectively re-sells your labor to the end user (consumer) is measured by the economic profits the employer earns, adjusted for all opportunity costs. Here, E−E_T for labor measures the bid-ask spread on wages and is always rank-order correlated with “excess” corporate profits.

Moreover, if you think about it for a moment, it becomes clear without any math: price volatility is bad. Why do we still use the imperial system of units in the U.S., even though the metric system is easier to use, given that we use a base 10 system for math and the scales are better aligned than in the imperial system? The reason is obvious: once we get used to a system of imperial units as a unit of account, it becomes uniquely difficult to switch to metric.

Imagine how inconvenient and difficult it would be to measure exchange rates with a ruler whose length is constantly changing, as the spendable money supply, like M2, keeps fluctuating. Imagine switching from metric to imperial and back to metric, as the units in which prices are quoted and measured—both absolute and relative—keep changing. Additionally, some prices, like wages, tend to be stickier than others, such as gasoline prices, further destabilizing relative prices during inflationary or deflationary periods.

It's important to point out that money, as a unit of account, measures not only absolute prices but also relative prices. When the money supply becomes prone to inflation and unpredictable, it becomes a poor unit of account. No wonder price volatility is bad for market efficiency. This is why all central banks fear deflation worse than death and vigorously fight inflation at the same time, aiming to keep prices stable. We are not saying anything new here.

This mathematically illustrates, with absolute precision, exactly why price volatility, as measured by inflation and deflation, is detrimental to the economy—an obvious fact now mathematically proven, rather than merely a hypothesis. This is also why Bitcoin is worth over a trillion dollars.

Involuntary Exchanges and Their Economic Consequences

Violations of the voluntary trade assumption can have severe implications for economic efficiency, often more significant than those caused by information asymmetries. This phenomenon is vividly illustrated in the former Soviet Union republics, where real GDP growth has been notably hindered by numerous instances of involuntary exchanges, similar to those observed in Haiti. In these countries, involuntary exchanges often occur not primarily as acts of robbery or theft due to general lawlessness—as seen in Haiti—but rather through mechanisms such as bribes and asset expropriation by politically connected individuals, including FSB colonels.

These activities are frequently tolerated by governing coalitions as a means to generate wealth transfers or bribes to those in charge of law enforcement. This arrangement ensures the loyalty of law enforcement officials to the ruling government, allowing them to engage in corruption as a form of compensation for their support. The result is a sub-optimal form of a stable Nash equilibrium—reminiscent of the sub-optimal outcomes in the Prisoner’s Dilemma, where accomplices betray each other instead of cooperating.

This type of equilibrium, characterized by corruption, is prevalent across all former Soviet Union republics, from Ukraine to Russia. The widespread persistence of involuntary exchanges starkly contradicts the Arrow-Debreu model’s assumption of unhindered trade, leading naturally to inefficiencies that affect regions from Haiti to the former Soviet Union and beyond.

The term 'involuntary exchange' used here specifically refers to violations of the unfettered trade condition posited by the Arrow-Debreu model of mathematical economics, which is crucial for achieving Pareto efficiency. According to this model, all trade should be voluntarily entered into by both parties, without duress, entirely of their own free will, and unhindered. Clear examples of involuntary exchange include acts like robbery and theft—unequivocal crimes punishable by prison terms under a proper legal system.

The Role of Taxes and Market Efficiency

The impact of taxes on market efficiency presents a nuanced scenario. In a competitive, free-market economy, taxes do not inherently constitute an involuntary exchange, although they can lead to such outcomes. The crucial distinction lies in whether taxes induce market failures—defined as transactions that are not mutually beneficial. Taxes that result in involuntary exchanges ultimately reduce welfare and productivity efficiency by preventing trades from being Pareto-improving, thereby diminishing overall welfare.

From an economic perspective, owning property in a country like France is fundamentally no different from other forms of fractional ownership, such as owning a condominium. Just as a condominium association collects fees necessary to maintain the property and its common areas—such as gyms, restaurants, and front desks, all staffed by employees—governments collect taxes to fund services that benefit everyone. These services include police departments, the military, the legal system, and welfare payments aimed at minimizing police budgets. When taxes are effectively used to provide such services, they are generally considered mutually beneficial exchanges, akin to paying condominium fees or rent, which are compensations for services rendered and thus not classified as involuntary exchanges.

However, anyone familiar with condominium associations knows that there is often a propensity for corruption among the boards, typically manifesting as management misappropriating fees for personal benefits. This scenario aligns with the concept of agency costs, as described by Jensen and Meckling, and parallels public choice theory and rent-seeking behaviors, particularly in the context of government tax mismanagement. Notably, Gordon Tullock contributed significantly to these theories, but it was James Buchanan Jr. who was awarded the 1986 Nobel Prize in Economics for his work in developing public choice theory, which identifies rent-seeking as a market failure. This is comparable to market failures induced by asymmetric information, as identified by George Akerlof, or the theft and robbery scenarios in Haiti.

Indeed, a simple example of the imposition of taxes and regulations through rent-seeking—such as the prohibition of raw milk sales while allowing the sale of raw oysters and eggs—vividly illustrates how such policies can prevent voluntary exchanges, thus qualifying as market failures. This comparison highlights the similar challenges faced in both the private and public sectors regarding the management and allocation of collected funds. These inconsistencies in regulatory practices not only disrupt market efficiency but also raise questions about the equitable treatment of different goods within the same regulatory framework.

Intuitive Recognition of Market Failures and Economic Ideologies

It is fascinating to observe how individuals, even those without formal training in mathematical economics or game theory, often intuitively recognize market failures—scenarios characterized by the pursuit of wealth without corresponding contributions to productivity. This phenomenon, identified by Gordon Tullock as economic rents, echoes Lenin's principle of 'from each according to his ability, to each according to his contribution.' Within this ideological framework, Lenin labels economic parasites—those who consume goods and services without contributing to their production—as members of the capitalist class, accusing them of living off savings and consuming without producing.

However, it's crucial to understand that merely living off savings, provided the wealth was acquired legitimately without fraud, theft, or dishonesty, does not inherently introduce inefficiencies into the economy, either theoretically or practically. The absence of deceptive practices in the accumulation of wealth ensures that such savings do not disrupt economic efficiency, thereby separating ethical considerations from economic outcomes.

Lenin's principle, akin to the concept of ‘fairness in trade,’ reflects the notion of ‘no economic rents being earned’ as understood in modern mathematical economics. This finds a practical echo in the Arrow-Debreu model within a perfectly competitive market, where the marginal revenue of labor aligns with its marginal cost. This alignment aims to avoid the pitfalls of ineffective economic policies rooted in involuntary exchange—an approach widely regarded as fundamentally flawed. Observing violations of Arrow-Debreu assumptions in reality is undoubtedly fascinating, yet often acutely painful for the countries subjected to policies enacted by the mathematically illiterate. And as we are about to show yet again, logical deduction does not lie.

Petr Chaadaev's critical view of Russia's historical role, as articulated in his 'Philosophical Letters,' serves as a cautionary tale, urging us to consider the outcomes of economic and political experiments that have led to significant human suffering and cost. These unfortunate outcomes illustrate that deviations from the Arrow-Debreu model's assumptions, such as unrestricted free trade, do not inherently result in inefficiency. However, achieving real GDP growth without these principles requires very costly interventions to address agency costs and combat rent-seeking behaviors that inevitably result when the product of one’s labor is involuntarily expropriated by the government, providing rational utility maximizers with all the incentive in the world to steal whatever they can from the government – which is only fair – given that the government steals pretty much most of their labor from them.

For instance, under Stalin, the establishment of a surveillance network known as 'stukachi' and a punitive gulag system were drastic measures aimed at mitigating endemic rent-seeking and agency costs, compensating for the absence of free-market mechanisms like the stock market. This example underscores the complexities of applying economic theories in varied political environments and highlights the deep understanding required to navigate these challenges effectively. Failing to grasp these economic principles can lead to severe consequences, the costs of which, in terms of human suffering and economic inefficiency, continue to impact us to this day.

Measuring Pareto Efficiency

Minimizing firm profits in the absence of externalities—such as rent-seeking, barriers to entry facilitating monopolies, and negative externalities like pollution—is counterproductive. In an efficient market, the development of innovative patented technologies inevitably generates excess profits that benefit society, assuming no theft. The key is to eliminate other inefficiencies, such as rent-seeking, agency costs, theft, robbery, extortion, and asymmetric information, as these always facilitate cheating by rational cost minimizers.

How should that surplus be split between consumers and producers? Ideally, a 50-50 split is optimal. If we switch the roles of individuals as consumer-producers, under the principles of information symmetry and rational behavior, and assuming you don’t know which side you will end up on—buying or selling—how would you set the price? A 50/50 surplus split, at the mid-quote, is where the derived subjective utility is equal for both parties. Thus, the midpoint is the optimal point, where the economy operates with maximum Pareto efficiency, minimizing unearned wealth procured by non-producing arbitrageurs—or economic parasites—as described by Lenin.

The end result is the same, regardless of whether arbitrage takes place in space or time. This becomes evident by “collapsing the wave function,” comparing E to its reciprocal transpose E_T, and observing the differences. Whether unearned wealth is extracted via asymmetric information in space or time, the end result is mathematically identical: real-world inefficiency. Therefore, true economic efficiency is measured by three parameters:

1. The difference between E and E_T, multiplied by real GDP.

2. The extent to which unfettered exchange is permitted.

3. The extent to which symmetric information is available.

That’s how one could measure market efficiency, beyond simply looking at real GDP growth—as an alternative real-world measure. Oh, and by the way, Einstein was wrong. God does play dice with the universe, just loaded dice, loaded in a way such that God always wins in the end, because everything is entangled and therefore Pareto-efficient and balanced in the long run—ensuring that, in time, everyone gets their comeuppance, giving back everything they stole. In this reality, no matter what, E always equals E_T! Isn’t that what the restated E=mc² says: E⁴=E_T⋅c²?

Conclusion

In a perfect market, mutually beneficial trade not only enhances Pareto efficiency but also serves as a real-world application of gradient descent optimization. This dynamic necessitates the use of money in its multifaceted roles to maximize overall welfare without disadvantaging any participants. Therefore, it logically follows that in a competitive free market, the effectiveness of a currency in fulfilling its essential functions—unit of account (U), medium of exchange (E), and store of value (S), collectively referred to as "U=S+E"—directly contributes to maximizing economic welfare.

This concludes our discussion, whose purpose was to formally integrate the first welfare theorem with our U=S+E definition of money, highlighting the essential roles of any currency in a competitive free market. These roles are crucial for enabling consumer-producer representative agents to maximize collective welfare, making a well-functioning currency indispensable for achieving optimal economic outcomes. Although this section is comprehensive, its length is justified. Moreover, as noted at the outset, it can be skipped by those who prefer to proceed directly to the subsequent discussion, which promises to be even more engaging and enlightening.

Part III: What Makes Money 'USE'-able?

Having established that the roles of money are intricately intertwined, as encapsulated by the U=S+E equation, it's crucial to understand what enables a currency to effectively fulfill these functions. Money serves not only as a unit of account (U) and a medium of exchange (E) but also transitions into a store of value (S) when not actively used in transactions. To function efficiently across all three roles, specific dual requirements are essential:

1. Unit of Account (U): For a currency to be an effective unit of account, it requires a stable supply—akin to the constant length of a ruler—and divisibility—similar to a ruler marked in increments. This divisibility allows prices to reflect a minimum price variation (MPV), such as one penny, which defines the precision with which values can be measured. Historically, stocks on the NYSE traded in increments of 1/8th of a dollar. However, the transition to decimalization, introducing a minimum price variation of one penny, not only streamlined trading but also enhanced the precision of price measurements. These improvements are crucial for ensuring that a currency reliably serves as a unit of account.

2. Store of Value (S): For money to effectively serve as a store of value, it must be safe to store and readily accessible when needed for exchanges. The risk of losing purchasing power through currency debasement underscores the importance of maintaining a stable money supply, highlighting the interplay between the roles of a unit of account and a store of value. Money must be difficult to steal, whether physically or virtually (via debasement), and should be immediately accessible when required. For example, gold buried on a deserted island, while secure, fails as an effective store of value due to its inaccessibility.

3. Medium of Exchange (E): To effectively serve as a medium of exchange, money must be easy to transfer and difficult to counterfeit. This requires that recipients be able to quickly verify the authenticity of incoming payments, thus necessitating the ability to exclude counterfeit transactions, including bounced checks. Cash and gold coins, for instance, are simple to use for payments and their authenticity can be relatively easily verified, minimizing the risk of fraudulent transactions.

By evaluating any currency from the comprehensive perspective of serving as a unit of account (U), a store of value (S), and a medium of exchange (E), we can assess its use-value as money. In a free, competitive market, the efficiency with which a currency fulfills all six of the dual requirements listed above will impact its exchange value relative to all other goods and services in the economy, thereby making it the preferred alternative selected to be used as money among all competing potential currencies. Next, we will explore practical examples, such as gold coins, to illustrate these principles in action.

What Makes for Good Money: Some Case Studies

As we delve into this section, let's recap our discussion thus far. Money fulfills three primary functions: it serves as a Unit of Account (U), a Medium of Exchange (E), and a Store of Value (S). To effectively serve as a unit of account—a term synonymous with 'unit of measure' in monetary contexts—a currency must be both divisible and have a stable total supply. As a store of value, it should be readily accessible to the owner, yet secure against unauthorized access. Lastly, for optimal functionality as a medium of exchange, a currency must be easy to transfer and difficult to counterfeit, traits exemplified by cash or gold coins. Furthermore, for efficient operation as a medium of exchange, the cost of processing payments must be low, and the transaction speed should be high, as is the case when handing someone a $20 bill. However, this is not always the case with bitcoin payments, international wire transfers, paper checks, and especially credit card transactions—which, while fast, are particularly expensive.

Consider the differences between making a credit card payment versus a cash payment. When you pay a merchant with cash, the $20 bill you hand over remains exactly $20 in the merchant’s possession. In contrast, when you use a credit card, the $20 you charge is subject to a fee, typically around 3%, which the merchant must pay to the credit card company. As a result, the merchant receives slightly less, losing out on about 60 cents due to processing fees. While this wealth transfer may seem minor per transaction, it accumulates significant costs for both consumers and merchants, ultimately benefiting the credit card processor the most. This scenario underscores why cash remains the most cost-effective payment option.

The Intersection of Mathematical Literacy and Cultural Icons: The Case of Mr. T

To underscore the practical implications of mathematical literacy and economic theory for our readers who have navigated through our discussions on the Arrow-Debreu model and the concept of rent-seeking, we present a compelling example involving a cultural icon. Mr. T, born Laurence Tureaud, emerged as a cultural phenomenon in the 1980s, most famously for his role as B.A. Baracus in the television series "The A-Team." Known for his rugged demeanor, mechanical prowess, and distinctive fear of flying, his character was often seen adorned with a full set of gold chains, which also became emblematic of Mr. T's persona both on and off the screen. His catchphrase, "I pity the fool!" and his unique style have cemented his place in pop culture as one of the most recognizable figures from the 1980s.

Today, imagine Mr. T walking into any New York airport, his neck draped with his iconic gold chains, flying off to California without a hitch. Contrast this with him carrying an equivalent value in gold coins or, especially, cash. In the latter scenario, he might face arrest under suspicion of money laundering, with his assets possibly being confiscated under current asset forfeiture laws. This stark difference in treatment illustrates not only the societal norms and legal stipulations around different forms of carrying value but also highlights the broader economic and regulatory implications.

The strict regulation of cash and the leniency towards other forms of wealth like gold jewelry can be viewed through the lens of rent-seeking. This economic concept, described by economists Gordon Tullock and James Buchanan, involves entities attempting to increase their share of existing wealth without creating new wealth, thus imposing a cost on society. Such dynamics are vividly demonstrated by the practices of payment processors that levy a 3% fee on credit card transactions, benefiting from the regulation and societal norms without contributing additional value to the economy.

This exercise in connecting theoretical economic concepts with tangible real-world scenarios, like that of Mr. T's experiences, not only enriches our understanding but also brings to light the subtleties of economic policies and their impact on everyday life. By marrying cultural references with economic theory, we aim to illustrate the pervasive influence of mathematical literacy in decoding complex, everyday phenomena.

Part I: From Cattle to Gold and Silver

Cattle have historically served as an effective unit of account, especially when their population remains stable, allowing a single cow to represent a sufficiently small unit of price variation. Indeed, cattle have functioned reasonably well as a currency in many aspects. As a store of value, cattle are notably difficult to steal due to their size and the specific conditions required for their upkeep, and they are relatively easy to transfer ownership, qualities that also make them a good medium of exchange. Furthermore, the authenticity and quality of a cow are relatively simple to verify, simplifying transactions.

However, the stability of this 'money supply' can be threatened by factors such as diseases like mad cow disease, illustrating that managing a currency system, even one based on livestock, is seldom straightforward.

Throughout history, various forms of currency have been utilized, each meeting the established criteria for effective money to varying degrees. Gold and silver, however, emerged as the dominant currencies for many centuries, especially under the bimetallic standard. Dating back to the Middle Ages, gold and silver coins were widely circulated. Notably, the exchange rate between these two metals remained remarkably stable at 15:1 for centuries, with one gold coin consistently valued at 15 silver coins. This stability persisted despite fluctuations in the available supplies of both metals. The enduring stability of this price ratio can largely be attributed to the fact that both gold and silver derived much of their 'use-value' from their application in minting circulating money.

Indeed, gold and silver coins have historically served as effective units of account due to their stable supply, which ensures a consistent measurement of relative prices. Their malleability allows for the production of coins in various denominations, facilitating transactions of differing sizes. Similar to adjustments in Minimum Price Variation (MPV) seen in stock exchanges' transitions from fractions to pennies, the metal content of minted coins can be altered to implement similar adjustments. However, historical periods like the Roman Empire witnessed adverse consequences due to increases in the money supply from lowering the gold content in coins. Despite these challenges, the adaptability of gold and silver coins as units of account is well-documented, underscoring their enduring utility in economic systems.

Part II: Why Commodity Money Leads to the Use of Bank-Money

As a medium of exchange, gold and silver coins are effective for in-person transactions but become impractical for remote payments, such as making a down payment on a Ferrari manufactured in Italy by a buyer in New York. Shipping gold to Italy presents logistical challenges and risks. While for most transactions, transferring coins directly incurs no additional cost, for high-value transactions like purchasing a luxury car, simpler methods like cash on delivery (COD) may prove inadequate. This scenario underscores the necessity for international wire transfers, which offer a more practical solution for handling significant monetary exchanges across distances.

While gold and silver coins are reliable for everyday exchanges, they are less suitable for complex international financial transactions, underscoring the necessity of banks in modern economies. Beyond facilitating long-distance payments, the role of banks is also critical due to the inherent limitations of commodity money, such as gold coins, as a store of value. These shortcomings, including vulnerability to theft, degradation, and the practical challenges of safe storage and transportation, highlight the essential functions of banks in safeguarding physical commodity money.

Commodity money presents significant drawbacks as a store of value. It is inherently easy to lose due to its physical nature, susceptible to theft, pilferage, or confiscation. Examples range from routine theft and household burglaries to government seizures, such as the 1933 incident in the United States when President Roosevelt ordered the surrender of all gold used as money. These vulnerabilities arise from the tangible and portable nature of physical coins, making them a risky option for long-term wealth preservation.

Part III: Bank Money-Units Introduce Issuer Counterparty Risk

The advent of bank money was intended to diversify the risks of physical loss inherent in commodity money. In the United States, until 1933, bank money represented fractional ownership of gold held in bank vaults. However, the issuance of excessive fractional ownership certificates, in forms such as cash and balances in checking and savings accounts (collectively known as M2), often exceeded the actual gold reserves. This imbalance contributed significantly to the Great Depression, although a detailed discussion of this historical event is beyond the scope of our current discourse.

While bank money was initially intended to mitigate the risks associated with the physical possession of commodity money, it introduced a new set of risks associated with the issuer itself, known as counterparty risk. This risk can lead to an expansion of the money supply, resulting in currency devaluation and undermining its effectiveness as a unit of measure. Historically, fiat currencies, which were invented and utilized as early as 800 years ago in ancient China, were generally avoided precisely because of their inherent excessive issuer counterparty risk.

Today, issuer counterparty risk, stemming from asymmetric information regarding the timing and magnitude of expansions in the US dollar M2 money supply, remains a significant concern. To mitigate this risk, central banks maintain gold as a reserve asset, capitalizing on its enduring value and stability. Today, all major central banks hold substantial gold reserves as part of their strategies to stabilize their currencies. Additionally, central banks in countries like China and Russia, facing higher counterparty risks with US dollar reserves, have been actively increasing their gold holdings.

This sustained use of gold, effectively as a reserve currency, alongside its widespread role as a means to store and preserve purchasing power outside of central banks, has significantly impacted the precious metals market. Specifically, the price ratio of gold to silver has escalated to over 80:1, largely because silver ceased to be used as money after the 19th century. Thus, the market price (or exchange value) of gold is determined primarily by its use value as money, rather than as a commodity. In contrast, silver, which lacks monetary use value due to being demonetized between 1850 and 1890 by all major economies of that era, is primarily valued as a commodity.

Part IV: Mitigating Issuer Counterparty Risk

Using gold as a reserve asset serves not only as a hedge against the known risks of future currency debasement, particularly in relation to the dynamics of the M2 money supply in the US, but also as a safeguard against the uncertainties associated with relying on fiat currencies as an asset class. In reality, fiat currencies derive their value primarily from the issuing government's promise to honor its fiscal obligations. This includes paying interest on bonds and accepting the currency for tax liabilities—a foundational principle of Modern Monetary Theory (MMT). Indeed, this governmental promise, when bolstered by central bank gold holdings, provides crucial support for the value of all fiat currencies, reinforcing their stability in the global financial system. By backing fiat currencies with gold, central banks offer a tangible assurance of their currency’s value, thus enhancing investor and public confidence even during periods of economic uncertainty.

The alternative to this promise-based system involves the direct enforcement of paper money usage through government authority, compelling its acceptance in transactions, as seen in places like the Soviet Union under Stalin and North Korea. Yet, as evidenced by the devaluation of currencies like the Venezuelan bolivar and the Argentine peso, such enforcement does not in and of itself secure substantial value relative to competing monetary units. These examples illuminate the inherent weaknesses in fiat currencies and underscore the enduring importance of gold as a stable and reliable reserve asset, essential for maintaining confidence in the monetary system.

The spendable money supply of any fiat currency, such as the US dollar—which is currently considered the most stable fiat currency—is almost certain to grow at a rate significantly higher than the supply of gold. This trend has been observable since the dollar was fully removed from the gold standard in August 1971 under President Nixon, and it is especially evident when looking at the dollar gold price after 1974, once Ford again allowed people to own monetary gold, having its price de-pegged from the dollar. Why or how people look at the dollar price of gold prior to 1974, when the dollar represented fractional ownership of gold, as evidenced by De Gaulle’s repatriation of dollars into gold as per prior agreements in the late 1960s, is beyond us!

While the broader implications of the disparity between the growth rate of fiat currency money supply and the supply of gold warrant further exploration, one inevitable result is the instability of the money supply, supported by a consensus from academic sources. This includes insights from the referenced Walrasian theory paper, which also asserts that instability in the fiat money supply is inevitable. This perspective highlights the inherent challenges in managing fiat currencies within modern economic frameworks.

This instability is a product of the rational utility maximization axiom, as no politician acting rationally within this framework will opt to raise taxes when they can simply increase the money supply by default. Thus, to introduce a rare biblical reference in an academic discussion, even if King Solomon himself, endowed with divine wisdom, were appointed head of the Federal Reserve, he would likely decline the position. However, hypothetically assuming he accepted, the instability in the M2 money supply would still inevitably ensue as a simple consequence of unfunded spending.

It is precisely for these reasons that cryptocurrencies have gained substantial value in the marketplace. They epitomize the ideal of what money could be: decentralized, stable, and unaffected by the whims of political changes. Interested in learning more about how cryptocurrencies achieve these attributes? Explore their potential and compare them to traditional forms of money by visiting us at tnt.money!

While this article has extended beyond its intended length, we fulfill our promise by demonstrating how replacing M⋅V (Money Velocity) with E⋅V (Medium of Exchange Velocity) transforms MV=PY into a comprehensive accounting identity: EV=PY. This adaptation provides a clearer understanding of how money circulates within an economy, reflecting the direct impact of the medium of exchange on overall economic activity.

Practical Example of the U=S+E Formula

Let's explore the quantity theory of money and the concept of the money supply using the U=S+E formula. Consider the M2 money supply in the US, which currently stands at approximately $21 trillion. This figure represents the unit of account (U)—the total money available in the economy for spending or saving. In this example, we assume that $15 trillion of the total money supply is actively used to facilitate transactions, denoted as E (Exchange). The remaining $6 trillion, represented as S (Savings), indicates the portion of the M2 money supply that is held as a store of value.

The M2 money supply, also referred to as the spendable money supply, mirrors the number of minted gold coins in a bank-less monetary system. It includes all money units readily accessible for transactions, encompassing cash, checking account balances, savings accounts, and money market funds. Classified by the Federal Reserve Bank, these components are considered liquid assets due to their ability to be quickly mobilized for transactions through various methods such as wire transfers, check writing, or electronic transfers, thus serving effectively as mediums of exchange.

Bank accounts, a key component of the M2 money supply, are often used to store funds when short-term liquidity is necessary, much like gold coins used to be stored in a vault in case purchasing power was needed—gold (and money in general) being the most widely accepted payment medium. This could be for covering margin calls, unforeseen expenses, or temporary job losses, showcasing money’s role as the most liquid asset—a store of value characterized by purchasing power.

Particularly in environments marked by low interest rates and minimal inflation, savings accounts often become preferable to bonds due to their greater liquidity and lower risk. Unlike bonds, where prices can decline if yields rise, the balance in a savings account remains stable. This makes using components like bank accounts as savings vehicles particularly appealing during periods of low interest rates, as cash assures stability of value. While there is no upside to holding bonds given that yields cannot turn negative and drop because cash exists, yields can always increase, as they have historically. This introduces an additional price risk to holding bonds, absent in bank accounts classified as M2, yet both having roughly comparable yields. This exemplifies how a part of the M2 money supply can be used as S (Store of Value).

Understanding the Quantity Theory of Money

Contrary to common perception, the quantity theory of money is not merely a theoretical construct but an accounting identity, expressed as a tautology based on arithmetic laws. This principle is paralleled in the financial realm, as demonstrated in Bill Sharpe's 1991 paper, "The Arithmetic of Active Management." Sharpe illustrates that active investors, when considered collectively, cannot outperform the market because they collectively own the market—or more precisely, the segment of the market portfolio not held by passive investors. Sharpe’s findings emphasize a reality governed by accounting principles rather than theoretical speculation.

Similarly, the quantity theory of money establishes a straightforward accounting equality between inflation and nominal GDP. Nominal GDP is defined as the total market value of all final goods and services produced and consumed within an economy by end users. This contrasts with gross output, which includes all production activities, accounting not only for final products that contribute to GDP but also for intermediate goods consumed during production, such as the sand used in making semiconductors.

The quantity theory, often described as an identity or equality in monetary economics, asserts that MV=PY. This equation becomes a straightforward accounting identity when each variable is precisely defined within the realm of mathematical economics. Let's define what we mean by variables P and Y.

• 'P' - Price Level: P represents the price level, a core concept in macroeconomics and a formal indicator of inflation. Inflation is typically measured by the Consumer Price Index (CPI), which tracks the general price level of a diverse basket of goods and services. This basket is selected to reflect the composition of the broader GDP, serving as a barometer for average price movements over time. The CPI is calculated by averaging the price changes of these goods and services, weighted by their significance or share in typical household spending patterns.

• 'Y' - Volume of Final Goods and Services (Real GDP): Y denotes the volume of final goods and services, or real Gross Domestic Product (GDP). This metric quantifies the total amount of goods and services produced and consumed within an economy, focusing exclusively on physical output and thus eliminating distortions due to price changes. Y measures the economy's overall productive capacity and output in real terms, offering a snapshot of economic activity and health.

By combining P, the price level, with Y, the volume of final goods and services, we calculate nominal GDP. This metric encapsulates the total value of all transactions within a year, obtained by multiplying the quantity of items purchased by their prices. Nominal GDP reflects the total spending on final goods and services, excluding intermediary consumption like the lumber used in furniture making, without any adjustment for price fluctuations. It thus provides a comprehensive overview of the economy's output in dollar terms, summing up the market value of all final goods and services consumers paid for in a year.

Exploring Economic Concepts Through the Stock Market Analogy

Using the stock market as an analogy can help illuminate the concepts of real and nominal GDP. Real GDP is akin to the share trading volume of the S&P 500 index, representing the quantity of transactions. If we extend this analogy, the S&P 500 would be analogous to the final goods and services, while non-S&P stocks would represent intermediate consumption. In this context, nominal GDP is similar to the dollar trading volume, reflecting the total value of these transactions. This comparison helps distinguish between the physical volume of economic activities (real GDP = share trading volume of the S&P 500 index) and their total monetary value (nominal GDP = dollar trading volume of the S&P 500 index).

Expanding on this comparison, consider how the return on a market index, like the S&P 500, is determined. It's typically calculated as the weighted average of the returns on individual stocks, where the weights are based on their market capitalization. Similarly, calculating CPI inflation mirrors this approach but with a crucial modification: instead of using market capitalization as the weight, it uses the past year’s dollar trading volume for each stock. This method is analogous to how CPI inflation is calculated by weighting the price changes of goods and services according to their share of total consumer spending. By adopting this approach, the calculation emphasizes the impact of price changes on the average consumer, focusing on their spending habits.

Using the U=S+E Formula to Clarify that 'M=E' in MV=PY

By understanding 'PY' as Nominal GDP—or in our stock market analogy, the dollar trading volume of the S&P 500—we can delve deeper into the mechanics of the economy. When this dollar trading volume is divided by the portion of the money supply actively participating in transactions (referred to as 'E' in the equation U=S+E, rather than the entire spendable money supply or 'M2', which is 'U'), we reveal an accounting identity. This identity illustrates how Nominal GDP, or the total economic activity in dollar terms, is facilitated by transactions using money as a medium of exchange.

The 'MV' part of the equation often leads to discussions about whether MV=PY is theoretical rather than an accounting identity. This perspective typically stems from misconceptions about what 'M' (money supply) and 'V' (velocity of money) represent. In the U=S+E formula, 'U' represents the total M2 money supply when considering the broader spectrum of available money.

In the context of the MV=PY equation, it is crucial to recognize that 'M' specifically aligns with 'E', not 'U'. This distinction redefines the traditional interpretation of the quantity theory of money, positing E⋅V=PY as the accounting identity. Here, 'E' represents the portion of the money supply that is actively engaged in economic activities, distinct from 'S', which denotes saved or non-circulating money.

Importantly, 'S' includes funds held in savings accounts, which are part of the M2 money supply, but it excludes investments like government bonds (typically categorized under M3) or other assets that are not considered immediately spendable or on-demand within M2. Therefore, while 'S' is included in M2, it is viewed as money that is removed from immediate circulation, not contributing directly to the transactions and broader economic interactions that drive nominal GDP.

This clarification allows for a deeper understanding of how money circulates within the economy and the dynamics between circulating and non-circulating funds, emphasizing the active role of 'E' in influencing economic output as measured by nominal GDP.

Exploring the Dynamics of Money: Is It V or the E/S Balance in the U=S+E Equality?

The distinction between 'E' (money designated for spending) and 'S' (money saved or invested) in the use of the M2 US dollar money supply is key for understanding the intricacies of money's role in the economy and its impact on overall economic activity. This distinction underscores that the significance of money is not determined solely by its quantity but also by its velocity—the rate at which it circulates and fuels economic transactions. Drawing an analogy to the S&P 500, where the velocity of share trading volume reflects the speed at which stocks change hands, further emphasizes this point.

However, unlike the stock market, the volume of transactions in an economy tends to be remarkably stable over time. Money used as a medium of exchange ("M", properly redefined as 'E' in the U=S+E equation) is primarily earned as income (typically wages) and spent to purchase goods and services, contributing to nominal GDP. The repetitive nature of consumer spending—on necessities such as clothing, food, rent, and haircuts—reflects our inherent consumer behavior, resulting in a frequency of transactions that remains relatively constant. Unlike the volume of shares in the stock market, the transaction volume of GDP remains super-consistent.

Therefore, when we estimate 'V' by dividing Nominal GDP by the total M2 money supply (instead of by 'E'), the result can lead to misconceptions. Any observed changes in 'V' are better understood as shifts in the balance between 'E' (money actively participating in economic transactions) and 'S' (money held out of the active circulation) within the overall money supply ('U'). This adjustment in perspective shifts the focus from the speed of money's circulation to how the balance between its active and inactive segments evolves over time. It suggests that variations in 'V', when computed using M2 instead of 'E', reflect changes in the distribution and utilization of money within the economy, rather than merely the rate at which it moves. This approach underscores that it is the dynamics of money's distribution and its engagement in economic activities that truly shape economic conditions, not just the speed at which it changes hands.

Understanding Monetary Dynamics with the U=S+E Framework

When we examine the velocity of money, V, the picture isn't very clear from M2 Velocity⁶ alone. However, analyzing the relationship between the M2 Money Stock⁷ and Nominal GDP⁸, both indexed⁹, significantly clarifies the situation. The implementation of Quantitative Easing (QE), which pushed bond yields and inflation rates down to nearly zero, led to a noticeable shift in the utilization of M2 assets from E (exchange) to S (store of value), as evidenced in the M2-GDP indexed graph post-2010. This trend is now reversing due to visible monetary inflation. Moreover, the precariousness of the current situation is significantly exacerbated by the fact that the $27 trillion in publicly held Federal debt¹⁰—interest payments on which are added to M2—exceeds the $21 trillion M2 money supply. This ensures that, regardless of the interest rate level, real interest rates remain negative.

At TNT-bank, we believe this perspective provides a clear understanding of complex economic indicators by elucidating how the quantity theory of money serves as a framework for comprehending the dynamics between money supply, velocity of money, price levels, and economic output. Drawing parallels to familiar stock market transactions helps make these concepts more tangible and accessible, demonstrating how foundational economic principles manifest in practical scenarios.

Moreover, examining the relationship between nominal GDP and the M2 money supply, given the reality of the existing welfare system and significant personal savings, the actual count of real-world transactions facilitated using money as payment does not change very much. This highlights the risks associated with relying on fiat currency or dollar-denominated fixed-income securities to preserve purchasing power. This distinction underscores the importance of understanding not just the quantity of currency in circulation but also its velocity and the broader economic context in which it operates.

This understanding is pivotal for appreciating the multifaceted roles of money within the economy, highlighting the strategic utilization of money to align with financial objectives and prevailing economic conditions. Furthermore, it elucidates how the inherent instability of the fiat money system prompts broader adoption and subsequent price increases of alternative monetary units to fiat currencies, such as gold and cryptocurrencies.

Conclusion

We invite you to explore further insights at tnt.money, where you will find valuable information at absolutely no obligation or cost. Our goal is to ensure symmetric information about the products we offer. We want you to know what you buy as a client, which is why everything we sell is fully open source. At TNT-bank, we embody the True-NO-Trust principle, represented by our commitment to Transparent Network Technologies, ensuring free and open access to all information on our site. Our innovative approach to TNT-bank money, which can represent fractional ownership backed by real or intellectual property assets, including copyrights and patents, caters to a growing preference for alternatives that may mitigate the risks of volatility and devaluation inherent in traditional fiat currencies.

Our discussion does not conclude here. Additional essays that delve into the intricate realms of 'mathecon game theory black papers' and subjective logical claim rings—an abstract algebraic framework designed to mathematically model cognitive biases—are available at tnt.money. We are committed to avoiding derogatory labels and unjust accusations of irrationality simply because someone is influenced by prevailing dogmas or affected by cognitive biases such as theory-induced blindness. Attentive readers will recognize that such blindness often stems from a dogmatic belief in the implicit assumptions underlying faulty, assumption-dependent axioms, rather than from the logical conclusions of the theories themselves. This phenomenon is akin to how the Pythagorean theorem inevitably results from the Euclidean axioms, a concept understandable even to a diligent fifth grader. Remember, 'garbage in, garbage out’!

In closing, we propose a novel approach to bolstering economic systems: consider backing money with patents. This approach could synchronize currency with innovation, fostering a culture of continuous technological advancement and replacing the negative externalities of traditional mining with the positive externalities of mining intellectual capital. Imagine a world where every new invention not only advances society but also strengthens the economy. This straightforward yet revolutionary idea could be the key to a more dynamic and sustainable economic future.

We urge the academic community, policymakers, and innovators to explore and adopt these ideas, advancing our scientific and technological capabilities to create a more equitable and prosperous world. Let us work together to harness the power of intellectual capital, transforming it into a cornerstone of our economic systems, and thereby driving sustainable growth and development.

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1https://en.wikipedia.org/wiki/Schools_of_economic_thought

2https://www.stlouisfed.org/education/economic-lowdown-podcast-series/episode-9-functions-of-money

3https://www.quantamagazine.org/pioneering-quantum-physicists-win-nobel-prize-in-physics-20221004/

4

5

6https://fred.stlouisfed.org/series/M2V

7https://fred.stlouisfed.org/series/M2SL

8https://fred.stlouisfed.org/series/GDP

9https://fred.stlouisfed.org/graph/fredgraph.pdf?hires=1&type=application/pdf&bgcolor=%23e1e9f0&chart_type=line&drp=0&fo=open%20sans&graph_bgcolor=%23ffffff&height=450&mode=fred&recession_bars=on&txtcolor=%23444444&ts=12&tts=12&width=1318&nt=0&thu=0&trc=0&show_legend=yes&show_axis_titles=yes&show_tooltip=yes&id=M2SL,GDP&scale=left,left&cosd=1959-01-01,1947-01-01&coed=2024-03-01,2024-01-01&line_color=%234572a7,%23aa4643&link_values=false,false&line_style=solid,solid&mark_type=none,none&mw=3,3&lw=2,2&ost=-99999,-99999&oet=99999,99999&mma=0,0&fml=a,a&fq=Quarterly,Quarterly&fam=eop,avg&fgst=lin,lin&fgsnd=2020-02-01,2020-02-01&line_index=1,2&transformation=nbd,nbd&vintage_date=2024-05-06,2024-05-06&revision_date=2024-05-06,2024-05-06&nd=1959-01-01,1959-01-01

10https://fiscaldata.treasury.gov/datasets/debt-to-the-penny/debt-to-the-penny