What is Money?
By Nathan, Phillip, and Joseph Mark Haykov
May 17, 2024
Abstract
In the nineteenth century, early economists, particularly William Stanley Jevons, Carl Menger, and Léon Walras, emphasized the pivotal role of money as a medium of exchange, necessary to overcome the double coincidence of wants problem inherent in direct barter systems. The Arrow-Debreu model—the foundation of modern mathematical economics—does not directly address money, introducing it only as an afterthought as a unit of account in which prices are measured in equilibrium. The equation U=S+E integrates these perspectives by axiomatically asserting that the total spendable money supply, classified as M2 in the U.S., not only functions as a unit of account (U) but also performs dual roles as a store of value (S) and a medium of exchange (E). This framework refines the traditional quantity theory of money (MV=PY) into a more precise accounting identity (EV=PY), where E specifically denotes the portion of the M2 money supply actively used in transactions, excluding funds that remain dormant in savings accounts for extended periods. Furthermore, the equation U=S+E illuminates how the exchange value—or the purchasing power—of a currency is determined by its utility value. For instance, Bitcoin’s current market capitalization of approximately $1.3 trillion is determined by its utility, or its 'U = S + E' value, reflecting its effectiveness in fulfilling money’s three functional roles: 'E' as a medium of exchange, 'S' as a store of value, and 'U' as a unit of account. In its role as a unit of account ('U'), money not only measures the relative prices of competing products, such as cars, but also relates these prices to wages and the time required to earn them, thus representing the true opportunity cost for individuals who act as both consumers and producers within the Arrow-Debreu model.
Keywords: Money as a Medium of Exchange, Arrow-Debreu Model, Quantity Theory of Money, Utility Value of Currency, Market Capitalization, Bitcoin, Mathematical Economics
JEL Codes: E40, E41, E42, C02
Introduction
Contrary to common misconceptions, there is indeed 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 website1. 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.
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, primarily recognizes money as a unit of account. "A Walrasian Theory of Money and Barter", freely available from Harvard2, is cited for its succinct summary of the ongoing debate about the nature of money, a discussion that can be traced back to the seminal contributions of early mathematical economists, namely Jevons, Menger, and Walras. In the late 19th century, these economists proposed that the primary function of money was as a medium of exchange, a necessary solution to the double coincidence of wants problem inherent in direct barter.
Over time, the characterization of money as a medium of exchange has crystallized into a dogma—an assumption-dependent axiom within mainstream mathematical economics. However, accepting such conjectures uncritically is precarious3. All axioms, by their nature, are assumptions—essentially educated guesses that are often deemed 'self-evidently true' by those who formulated them, sometimes centuries ago. Over time, these axioms can become conflated with facts, effectively turning into dogma. This becomes particularly problematic when such assumptions are not universally applicable, as many prove to be false under specific circumstances4.
Despite ongoing debates—particularly regarding whether assets like gold, which central banks use as reserve assets but not as mediums of exchange, should be considered 'real' money—the fundamental facts remain uncontroversial: Money has assumed various forms throughout history. These have ranged from cattle and tobacco leaves to cowrie shells, and from gold and silver coins to today's diverse fiat currencies. Within the Arrow-Debreu framework, all these forms of money are categorized as units of account or simply ‘money-units,’ and have actively facilitated global trade. This diversity is exemplified by the approximately 30 different currencies actively traded on the Forex market, each recognized as a valid fiat currency within its respective country or regional economy, such as the Euro (EUR).
Drawing insights from Bertrand Russell’s 1959 interview5 on the legacy of future generations, we see a clear imperative to transcend existing dogmas and focus on factual evidence. This approach reveals that the economic theories of Jevons, Menger, Walras, Arrow, and Debreu are not in conflict with each other, nor with the empirically observed roles and functions of money as detailed by the Federal Reserve Bank of the United States. Indeed, there is a universal agreement on the three indispensable functions that any currency must serve within its 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 initiate our discussion by examining money's role as a medium of exchange ('E'), a concept extensively explored by economists such as Jevons, Menger, and Walras. The proverb 'measure twice, cut once' not only applies aptly to carpentry but also encapsulates the essence of monetary transactions. Economic agents frequently engage in comparative price evaluations—whether assessing different car models or other commodities—considering both price and its proportionality to their financial status. This dual assessment employs money as both a unit of account ('U') and a medium of exchange ('E'), enabling a thorough appraisal of purchasing options within one’s economic means.
While the Arrow-Debreu model primarily focuses on general equilibrium without explicitly integrating money, it implicitly necessitates money as a unit of account to articulate equilibrium prices. This requirement underscores the importance of the total spendable money supply in facilitating complex economic calculations and enhancing market operations.
In modern terms, the U.S. dollar's total spendable supply, as gauged by the M2 money supply estimated by the Federal Reserve Bank, encompasses all readily available funds—including cash, balances in checking and savings accounts, and money market funds—all denominated in U.S. dollars. This concept mirrors ancient economies such as Rome, where the total quantity of minted aureus gold coins represented a comparable measure of spendable wealth.
Defining 'U' as the available money supply for simultaneous spending illustrates its dual functionality as both a medium of exchange and a store of value. Unspent money naturally assumes a store of value role ('S'), akin to gold coins stored or funds saved rather than invested. This delineation resonates with Keynes's liquidity trap concept from his 1936 seminal work, The General Theory of Employment, Interest, and Money, depicting money also as 'U+S' (Unit of Account + Store of Value)—a metric for more abstract, generalized purchasing power.
We assert a principle of 'exclusive dual-use,' stating that money can either be saved or spent, but not both simultaneously—similar to the adage, 'you cannot have your cake and eat it too.' Thus, money functions exclusively as a store of value or a medium of exchange at any given time. By combining 'S' and 'E,' we derive the total spendable money supply, akin to the M2 money supply. This integral view, denoted as 'U', merges the roles of 'U+E' and 'U+S,' crucially linking overall purchasing power with specific and relative good and service prices. Therefore, money’s dual role, transitioning smoothly between a unit of account and a medium of exchange, and into a store of value when not used as payment, 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 delve deeper into the mathematical underpinnings of our money definition, aligning it with the first welfare theorem of mathematical economics. The upcoming technical sections aim to bolster this discourse. Readers less inclined towards mathematical detail may skip this segment and proceed to 'What Makes Money 'USE'-able?', focusing 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 stands as a foundational pillar of mainstream economic thought, significantly influencing policymaking and guiding critical decisions, such as interest rate adjustments by institutions like the Federal Reserve. Its rigorous mathematical formalism not only substantiates but also extends Adam Smith's hypothesis from 'The Wealth of Nations' (1776), which posits that labor specialization enhances productivity. Kenneth Arrow and Gérard Debreu mathematically validated this hypothesis in 1954, demonstrating that under perfect market conditions, labor specialization should theoretically lead to optimal welfare outcomes. The Arrow-Debreu equilibrium model validates this principle through logical deductions within a formal axiomatic framework, cementing its pivotal role in both theoretical economic analysis and practical policy-making.
In this model, the first welfare theorem of mathematical economics uses deductive reasoning to demonstrate that under conditions of perfect markets—characterized by free trade, symmetric information, and perfect competition—a Pareto efficient outcome is inevitable. These ideal market conditions also stipulate the absence of externalities and complete information among participants. In such environments, where all goods are freely tradable, individuals are incentivized to act both as consumers and producers in their pursuit of maximizing personal welfare. This pursuit naturally leads to an optimal distribution of resources, ensuring that no individual can improve their welfare without detracting from someone else’s. Consequently, achieving Pareto efficiency under these circumstances guarantees that resources are allocated in a manner that benefits society as a whole.
Within this idealized market framework, free trade is inherently mutually beneficial, leading to outcomes that steer the economy toward Pareto efficiency. Pareto improvements occur when the mutual benefits of trade ensure that any gains are distributed in such a way that improving the welfare of one individual does not diminish the welfare of another. Key conditions, such as the diminishing marginal utility of consumption, play a crucial role in ensuring the convexity of the objective function, which represents the collective welfare of everyone involved. This convexity is vital as it facilitates the optimization process, allowing for efficient resource allocation and maximizing overall welfare.
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 as the value of the next best alternative that is forgone when a decision is made6.
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 acts as both a unit of account and a medium of payment, transforming these 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 to pay.
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 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 not only captures 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 inherently 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 certainty that the consumer perceives a 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 provides a 'rank-order' measure—not of the subjective benefit in some non-existent unit of 'utils,' which does not realistically capture how subjective benefits from trade are conceptualized, but of its relative subjective utility to the consumer compared to other, lower-priced options. In essence, money does not measure the absolute subjective utility expected, but rather its rank-order value.
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 Ackerlof 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 different potential purchase options, 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.
The U=S+E Framework:Integrating Economics with Evolutionary Biology
The U=S+E framework provides an additional benefit: local nonsatiation is not merely an assumption but a logical outcome of our axiomatic framework, integrating a fundamental principle of biology—evolution through natural selection. In humans, this manifests through female choice, where females typically favor partners who exhibit traits perceived as desirable. Although these traits vary across cultures and epochs, a common constraint for most males is demonstrating purchasing power as a signal of their social ranking.
This dynamic inherently ensures local nonsatiation when measured in monetary units. Money serves multiple roles in this context: it acts as a unit of account by measuring purchasing power and as a store of wealth due to its acceptance as a medium of exchange. Individuals are inherently motivated to enhance their status to attract desirable partners. In this pursuit, the ability to store and accumulate wealth becomes a crucial factor. Consequently, money, in its role as a store of wealth, acts indirectly as a consumption constraint that individuals strive to minimize relative to their competitors, aiming to maximize their reproductive success7.
This willingness to pay more effectively translates subjective utilities into quantifiable monetary units. This translation allows us to assert with certainty that higher trading volumes, expressed in dollars, indicate an increase in overall welfare, assuming these trades are genuinely mutually beneficial. This conclusion is analogous to how the Pythagorean theorem derives from the Euclidean axioms in geometry, which assert that the shortest distance between two points is a straight line. By order-ranking these utilities, we gain valuable insights into consumer preferences and economic dynamics, thereby enhancing our understanding of market efficiency and consumer behavior.
The Arrow-Debreu Model and Its Real-World Limitations
Recalling our middle school mathematics, we appreciate that the accuracy of logical proofs is fully guaranteed because they are independently verifiable. Many individuals, including those as young as fifth graders capable of mathematical reasoning, have proven the Pythagorean Theorem. However, it's important to note that the truths established through such proofs are conditional rather than absolute.
For example, the Pythagorean theorem holds true only under the foundational Euclidean axioms. Yet, Einstein’s adoption of Riemannian geometry to describe the curved space-time provides a more accurate model of the universe, challenging traditional Euclidean perspectives. This concept has practical applications in technologies like GPS, which must account for time dilation effects due to the differing speeds of clocks on satellites compared to those on Earth. Consequently, in our objective reality, where GPS technology is indispensable, the shortest distance between two points is not necessarily a straight line—an empirical fact.
Similarly, the Arrow-Debreu model, known for its rigorous mathematical framework, encounters significant limitations due to its often unrealistic assumptions about real-world economies. The model assumes conditions such as unrestricted and symmetrically informed trade. However, when these foundational assumptions are not met, as is frequently the case, outcomes like theft, robbery, and fraud occur, facilitated by asymmetrical information. This is exemplified by the selling of a “lemon”—a defective car misrepresented as fully operational. Without robust law enforcement and free-market mechanisms to address these market failures, the model’s predictions fall short of capturing the complexity of actual market dynamics.
Limitations of the Arrow-Debreu Model:Rational Utility Maximization and Real-World Application
The limitations of the Arrow-Debreu model stem from its reliance on an axiomatic description of real-world human behavior, a method common to mathematical economics and game theory. 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 are assumed to behave in ways that maximize their welfare within the constraints of the game’s rules, employing strategies that yield the highest possible payoff for themselves.
This characterization logically implies that a representative agent in the economy aims not only to maximize utility but also to minimize 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. This discrepancy emphasizes 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 two specific Arrow-Debreu conditions—voluntary exchange and symmetric information—significantly undermine economic efficiency, both in theoretical models and real-world scenarios. 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.
Please note, 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 and actual use value frequently 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, 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.
Upon reflecting on the mechanics of arbitrage, it becomes evident that arbitrage is untenable if a uniform price exists for any given asset across different markets. In the foreign exchange (FX) market, this principle implies that for any two currencies, A and B, the exchange rate from A to B should be the reciprocal of the exchange rate from B to A. This relationship ensures that no arbitrage opportunities arise purely from differences in these exchange rates.
In matrix form, if E is the exchange rate matrix, then the no-arbitrage condition imposes a constraint on E. The Hadamard inverse of E is defined by simply replacing each individual element of a matrix with its reciprocal: . We use the notation ET to refer to the transpose of the Hadamard inverse of a matrix E, . Using this notation, the no-arbitrage condition can be restated as E=ET. Here, E becomes the reciprocal of its own transpose, ensuring for all j and i.
This constraint is somewhat similar to the property of a matrix being involutory, where an involutory matrix is its own inverse, E=E-1. However, while E⋅E−1=I (the identity matrix), E⋅ET=E⋅E=n⋅E, where ET refers to the transpose of the Hadamard inverse of matrix E. As we can see, the resulting matrix is not the identity matrix but rather a scalar multiple of E scaled by its row count, n. This occurs because this matrix has a single eigenvalue, which is also its trace, and is equal to n, due to the fact that the exchange rate of a currency with itself is, by definition, 1. We note, just to be fully thorough here, that while E·ET≠ET·E, (E·ET)2=E4, because (E·ET)·(ET·E)=(E·ET)·(ET·E).
By imposing the E=ET 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.
We believe it is worthwhile to highlight what might be an obvious connection. In the context of Einstein's equation E=mc2, we can substitute (E·ET)2 for E, and (E·ET)2 for m, giving us the equation (E·ET)2 = (E·ET)2 c2. If we take the square root of both sides, the equation simplifies to:
By taking the square root of both sides, we see not only that mass is the reciprocal of energy but also that, unlike mass, energy has two square roots – dually computed by not only post-multiplying but also pre-multiplying the E matrix by its actual transpose, not the transpose of the Hadamard inverse. (E⋅ET)2 has just the one root, namely E⋅ET, because E=ET. Also, it facilitates modeling quantum entanglement1
While this is all purely conjectural, it aligns not only with the supersymmetry principle in theoretical physics, but also—shockingly—with the ancient Hermetic principle of "as above, so below." However, we are not theoretical physicists, and you should take everything we say about physics with a big grain of salt. We only know a wee bit of theoretical physics, having interacted with physicist colleagues during our time on Wall Street. For this reason, and also because the concept of money does not yet have an equivalent in physics, let us return to the main topic of this paper.
As mathematical economists, we find that this linear algebra formulation captures the essential idea that in an arbitrage-free market, the reciprocal relationships between exchange rates across different currencies, as well as all goods and services, must be consistent. This consistency prevents opportunities for arbitrage by merely transposing and reciprocating the matrix of exchange rates. Within this framework, prices effectively represent the exchange rates of all goods and services relative to a single specific row or column in the full exchange rate matrix E, selected as the optimal unit of account. Remarkably, this framework supports the theories of Arrow and Debreu and even, astonishingly, aligns with the ideas posited by Marx. Indeed, the fundamental role of money is to regulate markets by preventing arbitrage.
In the real world, the practice of quoting all currencies in the foreign exchange (FX) market against a single standard currency, such as the U.S. dollar, plays a pivotal role in reducing the scope for arbitrage, thereby nudging the market towards 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.
Measuring Pareto-efficiency
Introducing the ET notation (or simply ET for ease of typing, as constant subscripts can be cumbersome) offers significant benefits. Primarily, it facilitates the computation of Pareto efficiency in an economy by examining the prices at which trades for specific goods or services occur. It becomes evident that the more “inefficient” the market is, the more E will diverge from ET. This concept is mathematically analogous to simple arbitrage, where an arbitrageur, X, buys a stock from counterparty A in market 1 and sells it to counterparty B in market 2, generating a profit from asymmetric information. This profit, known as “economic rents,” diminishes productivity because it allows a non-producing arbitrageur, X, to consume goods and services produced by others without contributing to their production in any way.
Please note: X is able to earn money by facilitating a trade between A and B that should have occurred directly between them in a more efficient market if both had symmetric information. The fact that both parties had 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 99.99999% of the public.
Consider two scenarios, with the only difference between them being the presence of arbitrageur X, who acts as an unwanted intermediary preventing A and B from trading directly with each other. In the absence of X, all trades will occur at the mid-quote. However, if X is present, some trades will occur at the bid and others at the offer. In other words, the difference between E and ET becomes greater in the presence of X—by definition—as this bid-ask bounce volatility represents the “alpha” that X earns.
What we are highlighting here is that the root cause of market inefficiency, as 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 ET (mid-quote) multiplied by the trading volume. In this specific scenario, the calculation equals the profits earned by the arbitrageur (half bid-ask spread), down to the penny.
Of course, not every trade in reality is facilitated by an arbitrageur. However, we can approximate ET by the Volume Weighted Average Price (VWAP) for the time 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 effectively doing in this case is “collapsing the wave function”—taking all the different E matrices and averaging them out. When we measure Pareto-efficiency this way, it becomes clear that the more volatile the prices, the greater the difference between E and ET, and the less Pareto-efficient the market.
Also, if you just think about it for a second, it becomes clear even without any math. Why do you think we still use the imperial system of units in the U.S., even though the metric system is much easier to use, given that we use a base 10 system for math and the scales between the two are much 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. Can you imagine how inconvenient and difficult it becomes to measure relative prices with a ruler whose length is effectively constantly changing, as the spendable money supply, like M2, keeps going up (or down, as is the case currently), as if you were switching from metric to imperial, and back to metric every other day? Not to mention the fact that some prices, like wages, tend to be stickier than others, like prices of gasoline at the pump, further destabilizing relative, as opposed to absolute, prices during inflationary periods, for example. 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, by the way, shows exactly why price volatility, as measured by inflation and deflation, is bad for the economy—an obvious fact that is now mathematically proven. This is also why Bitcoin is worth over a trillion dollars, another fact.
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 for an optimal result.
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 individual 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 illiterate governments.
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 significant interventions to address agency costs and combat rent-seeking behaviors.
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.
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 detailed section integrates the Arrow-Debreu model 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; we previously noted that it could be skipped for those who prefer to proceed directly to the subsequent discussion, which promises to be even more engaging and enlightening.
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:
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.
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.
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, aside from TNT-bank funds, which offer an innovative alternative. TNT-bank has addressed the inherent challenges of both traditional and digital payment systems with a robust, patent-pending, open-source solution. Our system replaces the traditional proof-of-work with a superior consensus algorithm, allowing anyone to monitor TNT-bank money for suspicious activities and prevent illegal transfers by refusing credits. This capability is enabled by our dual-approval private key system, specifically designed to authorize credits, not debits. In TNT, transactions must be dually digitally signed by both the spender and the recipient of funds in order to be deemed valid and recognized, and any transaction without both signatures is deemed invalid and does not impact account balances of either wallet.
Soon, this innovative feature will be available in our patented open-source, peer-to-peer distributed software, similar in deployment to Bitcoin software where all nodes run it. This enables any government-licensed bank, regardless of size, to adopt TNT-bank money as their general ledger. It ensures full compliance with all relevant financial regulations, offering a versatile and secure option for financial institutions eager to innovate and enhance their payment systems.
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 a 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
The Role of Gold in Stabilizing Monetary Systems as a Reserve Asset
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. However, as evidenced by the devaluation of currencies like the Venezuelan bolivar and the Argentine peso, such enforcement does not necessarily 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.
Implications of Fiat Currency Expansion and the Rise of Cryptocurrencies
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.
While the broader implications of the disparity between the growth of fiat currency and the supply of gold warrant further exploration in a separate discussion, they are supported by a consensus from academic sources. This includes insights from the referenced Walrasian theory paper, which 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 the traditional equation MV (Money Velocity) with EM·V (Exchange Medium Money-Units times Velocity) transforms the theoretical model MV = PY into a comprehensive accounting identity. We simplify this to EV = PY for short, preferring not to use two-letter names. This adaptation provides a clearer understanding of how money circulates within an economy, reflecting the direct impact of the medium of exchange on the 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. 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.
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.
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 arithmetic relationship—an 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 lumber used in making furniture.
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=M Formula to Clarify the 'M' 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 EM·V = PY (or EV = PY for simplicity) 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 crucial 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 Velocity8 alone. However, analyzing the relationship between the M2 Money Stock9 and Nominal GDP10, both indexed11, 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 debt12 — 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
Explore further insights at tnt.money; you're sure to find valuable information—or your money back. At TNT-bank, we embody the True-NO-Trust principle, represented by our commitment to Transparent Network Technologies, ensuring free and open access to our website. Our innovative approach to TNT-bank money, which can represent fractional ownership backed by real or intellectual property assets, 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’!
And finally, for a novel twist on bolstering economic systems, why not 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. Could this straightforward yet revolutionary idea be the key to a more dynamic and sustainable economic future?
References
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1https://www.stlouisfed.org/education/economic-lowdown-podcast-series/episode-9-functions-of-money
2https://scholar.harvard.edu/files/maskin/files/a_walrasian_theory_of_money_and_barter.pdf
3Any axiom, even the Euclidean postulate that the shortest distance between two points is a straight line, is by definition merely a hypothesis introduced without empirical proof. It is important to recognize that such axioms can, and sometimes do, prove to be erroneous. For instance, the 'straight line' axiom does not hold in non-Euclidean geometries, such as those employed by GPS systems that adjust for the curvature of space-time. These systems utilize Riemannian geometry, which is based on axioms that more accurately represent the curvature of space as described by Einstein.
4For example, the axiom of pairing in Zermelo-Fraenkel (ZF) set theory posits that any set consisting of two elements can be separated into two subsets, each containing one of the elements in the original set. This axiom seems intuitively self-evident. However, it is self-evidently logically false when applied to entangled elementary particles such as electrons or photons, revealing the limitations of traditional axiomatic approaches in the face of quantum mechanics.
The application of this axiom becomes problematic in real-world scenarios, as evidenced by the empirical falsification of Bell’s Inequality, which is derived using deductive logic from the axioms of Zermelo-Fraenkel (ZF) set theory, specifically the axiom of pairing. The 2022 Nobel Prize in Physics underscored issues related to the behavior of entangled photons, which do not conform to traditional set theory predictions. This inconsistency suggests that theoretical physicists may need to develop a more robust set theory that better accommodates the complexities of multiple simultaneous quantum states, moving beyond speculative theories such as dark matter or the holographic universe.
However, the direct economic implications of these physical theories are limited, as the field of physics, unlike mathematical economics, does not directly generate financial insights—or income for the reader. This highlights the importance of reevaluating axioms within mathematical economics, particularly as they pertain to the utility and function of money.
6For 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.
7To clarify, even in non-traditional relationships, such as those involving two or more biological males and/or females of the same species, regardless of subsequent medical interventions or alterations, one partner typically assumes the dominant role of a money provider, often perceived as the de-facto ‘provider’ role traditionally associated with males.
8https://fred.stlouisfed.org/series/M2V
9https://fred.stlouisfed.org/series/M2SL
10https://fred.stlouisfed.org/series/GDP
11https://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
12https://fiscaldata.treasury.gov/datasets/debt-to-the-penny/debt-to-the-penny
Condensation of Information
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:
Reduced Dimensionality: Instead of needing to know all elements of a matrix (which in a full matrix would be n×mn×m values), you only need to know the elements of a single vector (either nn or mm values depending on whether it's a row or a column vector). This drastically reduces the dimensionality of the information required.
Data Compression: 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 EE.
Basis for a New Set Theory
Extending this idea to a theoretical framework, especially in contexts like quantum mechanics, can lead to intriguing possibilities:
Modeling Quantum States: 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.
Set Theory and Quantum States: 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.
Applications: 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 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.