The Dual Nature of Money:
Insights from the Arrow-Debreu Model
by Joseph Mark Haykov
April 30, 2024
Abstract:
In the realm of mathematical economics, the Arrow-Debreu model serves as a foundational framework for comprehending economic equilibrium. This paper endeavors to dissect the intricate role of money within this model, honing in specifically on its function as a unit of account. Despite the model's abstraction from the complexities inherent in real-world monetary systems, it yields invaluable insights into the essence of money. Rooted in the classical Jevons-Menger-Walras definition, we contend that the roles of money as both a unit of account and a medium of exchange are deeply interwoven, resulting in a profound recursive entanglement. Our principal contribution lies in the introduction of the U=S+E formula, which postulates that the unit of account money supply equals the amalgamation of the store of value money supply and the medium of exchange money supply. This formula serves as the cornerstone for our discourse on the USE-value of currencies, elucidating the factors that confer relative value upon a given unit of money vis-à-vis competing alternatives. Furthermore, we refine the quantity theory of money, MV=PY, transforming it into an accounting identity EV=PY, wherein the unit of account money supply equals the sum of the store of value and medium of exchange money supplies: U=S+E. By delineating such relationships, our aim is to illuminate the multifaceted interplay between money's diverse functions within economic systems.
Introduction
Dear reader,
If the abstract seemed daunting, fear not. I understand that the intricacies of mathematical economics can be intimidating. In this paper, we acknowledge the complexity of our subject matter and aim to bridge the gap between the specialized realm of mathematical economics and a broader audience. While certain concepts may be self-evident to those formally trained in mathematical economics, we recognize that not everyone may be familiar with the intricacies of the Arrow-Debreu model. Thus, we dedicate space to elucidate these concepts, ensuring accessibility for all readers. At its core, this paper seeks to underscore the significance of the topic of money and its relevance to economic discourse. For those that are familiar with the Arrow-Debreau framework, may skip the following section, and we hope the trivial discussion below does not dissuade you from reading the rest of this paper.
The Arrow-Debreu Model: Exploring its Foundations and Implications in Economic Theory
The Arrow-Debreu model stands as a cornerstone of mainstream economic thought, permeating diverse spheres, including the policymaking realm, where institutions like the Federal Reserve rely on its principles to inform decisions such as setting interest rates. Its pervasive adoption is underpinned by its rigorous mathematical formalism, which provides compelling support for what can be construed as the Adam-Smith hypothesis. This hypothesis posits that labor specialization, operating within the confines of perfect market conditions, culminates in optimal welfare. Through independently verifiable logical deduction, the Arrow-Debreu model rigorously substantiates this assertion, rendering it a linchpin in economic theory and practice.
Much like the Pythagorean theorem, the first welfare theorem of mathematical economics, which is a fundamental component of the Arrow-Debreu model, establishes that under specific ideal market conditions such as unfettered trade and symmetric information, a Pareto efficient outcome ensues. In this paradigm, characterized by the absence of externalities or monopolies, all goods are freely available for trade. Participants in the economy, concurrently acting as consumer-producers, are driven by their inherent nature to maximize individual welfare.
Within the confines of a perfect market environment, free trade becomes mutually beneficial or Pareto-improving, propelling the economy toward a state of Pareto efficiency, where no individual can enhance their welfare without diminishing another's. This condition sets the gradient of the collective welfare function to zero, thereby indicating the achievement of a global maximum. Conditions such as diminishing marginal utility of consumption are essential to ensure that the objective function remains convex. However, a significant challenge arises in optimizing the convex sum of everyone’s collective subjective utility, given its unmeasurability by a third party and often even by the consumer themselves. How do we measure the subjective benefit of stepping into an air-conditioned room on a hot day, apart from assessing how much one would be willing to pay? This conundrum underscores the critical role of money as a unit of account in mathematical economics, where it helps to translate subjective utilities into quantifiable terms.
In the Arrow-Debreu model, the role of money as a unit of account is paramount. By quantifying subjective utility in monetary terms, money ranks subjective preferences and facilitates the optimization of collective welfare by maximizing these rankings. Consumer-producers engage in mutually beneficial trade by exchanging the fruits of their labor—measured in wages or other forms of income—for the goods and services produced by others. Here, money not only acts as a medium of exchange for purchasing goods and services but also as a unit of account, equating the subjective expected benefits (or utility) of purchases with the subjective costs associated with earning the necessary income. These costs are largely determined by the time invested in labor. In scenarios where money is not actively used for payments, it effortlessly transitions to serving as a store of value, thereby fulfilling its universally recognized roles: unit of account, medium of exchange, and store of value.
Of course, producer surplus is easy to measure. It is simply the difference between the price at which goods are sold and their total, all-encompassing or "economic" cost of production, which includes all opportunity costs. For example, the costs of advertising an apartment, the time spent by the beneficial owner(s) themselves collecting rent, effectuating repairs, and evicting non-paying tenants—all exemplify opportunity costs.
Consumer surplus, unlike producer surplus, cannot be directly measured as it hinges on both the subjective perception of benefit and the perceived cost to the consumer. Money plays a crucial role as a unit of account—or more precisely, as a measure. In mathematical economics, consumer surplus is defined as the difference between the maximum price a consumer is willing to pay for an item, such as an iPhone, and the market price at which the item is actually purchased. Money acts as a unit of account to measure the true subjective cost of an item like an iPhone—the amount of time needed to earn the income, typically in wages, required to make the purchase.
Indeed, consumer surplus, though subjective, is inherently linked to monetary units. This linkage arises because the more a consumer is willing to pay for a product, the greater their subjectively perceived benefit from the purchase. We cannot definitively say that a consumer derives twice the subjective benefit from buying Product A (Bentley) as from Product B (Ford) merely because the price of A is twice that of B. However, we can be absolutely certain that the consumer perceives a greater benefit from Product A than from Product B because they are willing to pay more for it. This willingness to pay more means that the ranking of subjective utilities is quantifiable in monetary units. Using money as a unit of account thus allows us to assert with certainty that higher dollar trading volumes in the economy correlate with increased overall welfare, as all such trade is mutually beneficial. The more we trade, the more we benefit—this is a self-evident conclusion that logically follows from the Arrow-Debreu axioms, just as the Pythagorean theorem follows from the Euclidean axioms of geometry, assuming the shortest distance between two points is a straight line.
However, as we may recall from middle school mathematics, while the accuracy of logical reasoning is guaranteed—since such proofs are independently verifiable (for instance, many of us capable of mathematical reasoning proved the Pythagorean Theorem in fifth grade)—the truths we prove are merely conditional, not absolute. If the Euclidean axioms are false, then the Pythagorean Theorem no longer applies. For example, Einstein had to employ Riemannian geometry to model curved space-time, which more accurately describes the universe as it actually exists. Similarly, the Arrow-Debreu model, with its assumptions often violated in real-world economies, faces limitations. When fundamental assumptions, like unfettered trade, are not met—as seen in cases like Haiti—inefficiencies in the real world inevitably arise.
To conclude this brief summary: Mutually beneficial trade, in a perfect market, is Pareto-improving, and represents a real-world manifestation of gradient descent optimization, where mutually beneficial (or equivalently Pareto-improving) free-trade leads to an optimality condition where the gradient of the objective function is zero, as defined by Pareto efficiency, where no one benefits without harming another.
Defining Money: Exploring its Varied Forms and Essential Functions
Defining 'money' clearly is indeed crucial at the beginning of this discussion, on account of the fact that opinions about as to what is money vary widely, fueling debates over whether assets like gold or Bitcoin qualify as money, or if the term should be reserved for fiat currencies like the dollar. These debates persist because, unlike fiat currencies, gold and Bitcoin are rarely used for everyday transactions today, except in special cases such as Bitcoin in ransomware payments. Critics, including the late Charlie Munger, have even disparagingly referred to Bitcoin as a 'turd' due to its unconventional uses. Despite gold’s historical role as a medium of exchange, its limited current use in this capacity raises questions about its status as money today.
As explained in “A Walrasian Theory of Money and Barter,”1 according to the early mathematical economists Jevons, Menger, and Walras, the key role of money in their shared hypothesis is that of a medium of exchange. This role is necessary to solve the double coincidence of wants problem in barter. As evidenced in the referenced paper, this Jevons-Menger-Walras hypothesis has indeed become an axiom in mathematical economics. Under the axiom that money is inherently a medium of exchange, Bitcoin should logically not be classified as money unless it is used as a medium of exchange - a point that is subject to debate, hence the ongoing arguments.
However, there is no debate about the fact that throughout history, money has taken various forms, including cattle, tobacco leaves, cowrie shells, and gold and silver coins. Today, diverse money units continue to facilitate global trade – as exemplified by the 30 or so currencies that are actively traded on the Forex market – each recognized as valid money within its respective country or regional economy, such as the Euro (EUR). Despite any apparent disagreements, there is a universal consensus on the three essential roles that any form of money must fulfill within the economies that issue it. This highlights the fundamental importance of money across all financial systems2.
According to the St. Louis Federal Reserve, money fulfills three essential functions across historical and contemporary economic contexts: it acts as a unit of account ('U'), a store of value ('S'), and a medium of exchange ('E'). These roles form a practical framework for the discussion of money in this paper. We propose the following axiom: the more effectively a currency performs these functions, the greater its utility, or use value as money, thus increasing its exchange value relative to competitors.
Defining Characteristics of Effective Units of Account
The real-world utility of a currency as a unit of account is primarily determined by the stability of its spendable money supply. Gold is renowned for its inherent scarcity and limited supply, qualities that establish it as an exceptionally precise unit of account. Its prolonged role as a form of money is bolstered by its resistance to counterfeiting, ensuring a stable and dependable money supply. Unlike bank money, which relies on trust in the issuing authority, gold's distinct physical properties—such as its unique color, density, malleability, resistance to corrosion, and inability to tarnish—render counterfeiting virtually impossible. Historically, this has been demonstrated by the alchemists' unsuccessful attempts to produce gold inexpensively, a task that remains unachievable to this day.
The enduring historical record reinforces gold’s reliability and effectiveness as a unit of account, underscoring its practical value in this role. Consequently, gold remains a critical reserve asset for central banks globally, despite its reduced role as a medium of exchange. It is noteworthy that central bank gold agreements3, once deemed crucial, are now considered redundant, with some central banks even resuming gold accumulation.
The connection between the stability of a currency's supply and the health of an economy is a principle firmly upheld by central banks, which often regard deflation as a severe threat and actively combat high inflation. This correlation finds support in compelling historical evidence. Notable economic crises, such as the Great Depression in the US during the 1930s, Zimbabwe's hyperinflation, the monetary collapse of the Weimar Republic, and the stagflation of the 1970s in the United States, vividly demonstrate the detrimental effects that fluctuations in the money supply can have on economic stability. These episodes underscore the critical importance of maintaining a stable money supply as a cornerstone of economic prosperity and growth. The lessons gleaned from these historical events emphasize the essential role of currency stability in fostering an environment conducive to economic efficiency and development.
Beyond Stability: Essential Attributes of Reliable Currency
In the context of the Arrow-Debreu model, money primarily serves as a unit of account and requires a stable supply to function effectively. However, it's crucial to recognize that while money consistently functions as a unit of account, it automatically assumes the role of a store of value when it's not actively facilitating trade as a medium of payment. Conditional on a stable money supply, we must also emphasize that this monetary unit should perform the dual functions of not only a medium of exchange but also a store of value, especially when not being used for transactions.
Medium of Exchange: A currency's effectiveness as a medium of exchange hinges primarily on the ability of recipients to reliably verify its authenticity, thereby preventing fraudulent transactions. Virtually any form of currency meets this requirement, underscoring the principle that widespread acceptance and recognizability are key factors in determining a medium of exchange. An illustrative example is the hyperinflationary period of the Weimar Republic, during which the severely devalued German Papiermark continued to be used for transactions, even necessitating the use of wheelbarrows to carry cash. This scenario highlights that a currency's function as a medium of exchange can endure even amid extreme economic distortions, provided it remains recognizable and verifiable by its users.
Store of Value: Using money as a store of value introduces additional requirements beyond easy verification of authenticity. Storing significant quantities of gold, for example, carries substantial counterparty risks, including the potential for involuntary exchanges such as theft, robbery, natural disasters, or household pilfering.
To mitigate these risks without resorting to extreme measures like burying gold on deserted islands, the concept of banks was introduced. Initially, bank money represented fractional ownership of gold securely stored in a bank’s vault. This model of fractional asset ownership is conceptually similar to owning shares in a corporation, investing in Real Estate Investment Trusts (REITs), or holding ownership in condominiums and timeshares.
The establishment of banks and the introduction of bank money provided a solution to the risks associated with directly storing physical forms of money, thus transforming the landscape of financial security and currency utilization. From this bank-money, fiat money, currently in use today, evolved, as did cryptocurrencies.
Ultimately, what makes for a good store of value is the ability of the owner to exclude all others from spending their money units without approval.
For example, the ability to independently verify gold's authenticity significantly reduces counterparty risk compared to bank-issued money. The same principle applies to cryptocurrencies like Bitcoin. While the value of bank money depends on the issuing institution's ability to verify its authenticity and maintain trust, both gold and Bitcoin allow for direct authentication by the recipient of payment. This verification process, similar to using cash, facilitates direct transactions without the need for a bank to act as an intermediary in processing fund transfers. This added layer of security and independence enhances confidence in a currency's value as both a medium of exchange and a store of value.
In summary, for a currency to effectively serve as a unit of account, it must have a stable supply. As a medium of exchange, transparent and symmetric information about the currency's authenticity is essential to mitigate the risks of fraud. Lastly, for a currency to be a reliable store of value, the owner must have exclusive control over its use, ensuring that no one but themselves—or their authorized agents—can spend or access the money. Each of these aspects is crucial in ensuring that a currency fulfills its roles effectively within the economy, encapsulating the U=S+E value of money.
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 amount represents the unit of account (U)—the total money supply in the economy that is available 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 or savings.
The M2 money supply, also referred to as the spendable money supply, is designed to reflect the concept akin to the number of minted gold coins in a bank-less monetary system. It represents the portion of the money supply that is available for use as a medium of exchange. Classified by the Federal Reserve Bank, the M2 includes all money units that are readily accessible for transactions. This encompasses cash, checking account balances, as well as savings accounts and money market funds. These 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 all equally and effectively serving as viable mediums of exchange.
People often utilize bank accounts, a component of the M2 money supply, to store funds when short-term liquidity is necessary. This could involve covering margin calls or unforeseen medical expenses, showcasing a key use of money as the most liquid asset: a store of value. In environments characterized by low interest rates and minimal inflation, savings accounts often become preferable to bonds. This preference arises from their greater liquidity and lower risk. Unlike bonds, where prices can decline if yields rise, the balance in a savings account remains stable. Consequently, using M2 components like bank accounts as savings vehicles becomes particularly appealing during periods of low interest rates, as cash assures that yields cannot turn negative and drop, though yields can always increase, as they did.
The Stock Market as a Metaphor for Money's Accounting Identity
Contrary to common perception, the quantity theory of money is actually an accounting identity and not a theory, because it can be expressed as a tautology based on arithmetic laws. This concept finds a parallel in the financial realm, specifically in Bill Sharpe's 1991 paper titled "The Arithmetic of Active Management." Sharpe shows that, when considered collectively, active investors cannot outperform the market. This is because, collectively, these investors own the market—or more precisely, the segment of the market portfolio not held by passive investors. Sharpe’s findings underscore a reality governed by accounting principles rather than theoretical speculation.
Similarly, the quantity theory of money establishing a straightforward arithmetic relationship (an accounting equality) between inflation and nominal GDP. Nominal GDP, defined as the total market value of all final goods and services produced and consumed within an economy by end users, is distinct from gross output. Gross output includes all production activities, not just the final products that contribute to GDP but also intermediate goods consumed during production, such as the lumber used in making furniture.
The quantity theory of money posits as an axiom the following equation: MV=PY, which, when each variable is precisely defined within the realm of mathematical economics, emerges as a straightforward accounting identity. Let's delve into what each variable stands for in this particular equation:
'P' represents the Price Level, a core concept in macroeconomics that serves as a formal indicator of inflation. Inflation is typically gauged by the Consumer Price Index (CPI), which tracks the general price level of a diverse basket of goods and services. This basket is carefully selected to mirror the composition of the broader GDP, acting as a barometer for average price movements over time. Thus, 'P' is crucial for understanding the cost of living, as it reflects the financial outlay required to purchase this representative basket of goods and services. The CPI is calculated by averaging the price changes of these goods and services, weighted by their significance or share in the typical spending patterns of households. For example, if households, on average, allocate forty percent of their income to housing, the change in housing prices will carry a weight of 0.4 in the CPI calculation. This method ensures that the CPI accurately reflects how price changes affect the average consumer, offering a realistic picture of inflation and its impact on daily life.
'Y' represents 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 on the physical output without the distortions due to price changes. In essence, 'Y' measures the economy's overall productive capacity and output in real terms, providing a snapshot of economic activity and health.
By combining 'P', the Price Level, with 'Y', the Volume of Final Goods and Services, we arrive at nominal GDP. This measurement, as defined in economics, encapsulates the total of all transactions within a year. It's calculated by multiplying the quantity of items purchased by their price, thus determining the overall value of trading activity during a fiscal year in an economy. A crucial detail of this calculation is the exclusion of intermediary consumption, aligning with the strict definition of nominal GDP in economics. Essentially, nominal GDP offers a thorough overview of the economy's output in dollar terms, reflecting total spending on final goods and services without adjusting for price fluctuations.
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. In contrast, nominal GDP parallels the dollar trading volume, reflecting the total value of these transactions.
Expanding on this comparison, think about 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, with the weights based on their market capitalization. In a similar vein, calculating CPI inflation resembles computing the return on the S&P 500, but with a twist: instead of using market capitalization as the weight, it uses the past year's dollar trading volume for each stock. This approach is analogous to how CPI inflation is calculated by weighting the price changes of goods and services according to their share of total spending. This method prioritizes the impact of price changes on the average consumer, focusing on their spending habits rather than the absolute economic size or significance of the goods and services.
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. If we take this dollar trading volume and divide it by the portion of the money supply that actively participates in these transactions (referred to as 'E' in the context of the equation 'U = S + E', rather than the entire spendable money supply, or 'M2', which is 'U'), we uncover an accounting identity. This identity illustrates how Nominal GDP, or the total economic activity in dollar terms, can be viewed through the lens of transactions facilitated by money serving as the medium of exchange.
This method enhances our understanding of economic dynamics by centering on the money that's actively facilitating transactions, rather than the entire pool of money that's potentially available for spending. It highlights the pivotal role of active, circulating money in propelling economic activities, reinforcing the idea that it's not the money lying idle but the money in motion that energizes the economy.
The 'MV' part of the equation sheds light on why some view MV=PY as theoretical rather than an accounting identity; this perspective often stems from misconceptions about what 'M' (money supply) and 'V' (velocity of money) represent. In the formula U=S+E, 'U' represents the total M2 money supply when we’re looking at the broader picture of available money.
However, in the discussion surrounding the MV=PY equation, 'M' aligns specifically with 'E', not 'U'. In this sense, the real quantity theory of money posits that EV=PY as the accounting identity, where 'E' denotes the segment of money that is actively engaging in economic activities, distinct from 'S', which indicates saved money. Importantly, 'S' includes funds in savings accounts that are counted within the M2 money supply but excludes investments like government bonds (part of M3) or any other assets not considered immediately spendable or on-demand within M2. Thus, 'S', while part of M2, is viewed as money taken out of the immediate circulation that fuels transactions and broader economic interactions.
This distinction is crucial for understanding the intricacies of money's role in the economy and its impact on overall economic activity. It underscores that the significance of money isn't solely determined 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 share trading volume velocity reflects the speed at which stocks change hands, highlights this point.
However, unlike the stock market, the volume of transactions in an economy tends to remain remarkably stable over time. Money used as a medium of exchange ('E') is primarily earned as income (typically wages) and spent on purchasing goods and services, contributing to nominal GDP. The repetitive nature of consumption spending—on essentials like clothes, food, rent, and haircuts—reflects our inherent consumer behavior, resulting in a frequency of transactions that remains relatively constant. In essence, unlike share volume in the stock market, the share volume of GDP remains largely consistent.
Therefore, inaccurately calculating 'V' by dividing Nominal GDP by the total M2 money supply (instead of 'E') leads to misconceptions. Any observed changes in 'V' are better understood as shifts in the balance between 'E' (money designated for spending) and 'S' (money saved or invested) 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' reflect changes in the distribution and utilization of money within the economy, rather than simply the rate of its movement. This approach underscores that it's the dynamics of money's distribution and its engagement in economic activities that truly shape economic conditions, rather than just the speed at which it changes hands.
This perspective serves to demystify intricate economic indicators, offering insight into how the quantity theory of money provides a framework for understanding the interplay between money supply, velocity of money, price levels, and economic output. By likening these dynamics to familiar stock market transactions, we can better comprehend how these foundational economic principles manifest in practical scenarios, rendering abstract concepts more tangible and understandable. Additionally, examining the historical correlation between nominal GDP and M2 money supply underscores the significant risks associated with relying solely on the fiat dollar money unit or dollar-denominated fixed-income securities to preserve purchasing power.
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.
1https://scholar.harvard.edu/files/maskin/files/a_walrasian_theory_of_money_and_barter.pdf
2The specific uses of money in these capacities are detailed directly by the Fed, accessible at: https://www.stlouisfed.org/education/economic-lowdown-podcast-series/episode-9-functions-of-money
3https://www.ecb.europa.eu/press/pr/date/2019/html/ecb.pr190726_1~3eaf64db9d.en.html