Equation of exchange
andrea.parisi | Published: 4 Sep 2023, 12:36 a.m.
The first law that I discovered in Economics is the law of prices presented here, a special case of the Equation of Exchange. This law might not appear on the first page of a book of Economics, however since it is the first that I discovered, and it is central to other economic “monetary” schools, it is a good way to start our journey. Also, it is useful to illustrate some of the problems that, in my view, underline economic theories.
The mathematical formula that describes the Equation of Exchange reads as: \begin{equation} \label{eq:exchange} P = \frac{M V}{N} \end{equation}
where $P$ is the average price, $M$ is the “monetary mass” or the amount of money in circulation, $N$ is the amount of goods, and $V$ is the speed of circulation of money. Setting $V=1$ we get the equation discussed earlier. I found the above equation presented in different ways, but almost always the definition is limited to the above lines. Often, the equation is written as: \begin{equation} \label{eq:exchange-2} N P = M V \end{equation}
by moving $N$ at the numerator, where the product $N P$ is taken as the nominal GDP of a country. To understand the equation better, we need a small excursus. Imagine that we are in the middle ages, where our only form of money available is gold. Gold exists in limited amounts, and part of it is hoarded for value storage (maybe kept hidden in a chest). Thus people have some amount of money $M$ in their pockets (which is the amount of circulating money, that is all coins, less the hoarded ones) and they use this money to buy staff. There are $K$ different types of items for sale around, and people actively buy these items. If item $j$ costs $P_j$ and $N_j$ pieces are sold, the total amount of exchanged money is $N_j P_j$. During a time interval $T$, several items will be exchanged in different numbers and each at its own price. Thus the total amount exchanged will be: \begin{equation*} M = \displaystyle \alpha \sum_{j=1}^K N_j P_j \end{equation*}
where $\alpha$ is a proportionality constant that takes into account that the same money may be used to buy multiple items within the same time period $T$. For instance, Mr A might buy bread in the morning, and the money gained by the baker is used to buy some flour from the supplier in the afternoon. Thus, the same money has been used in multiple transactions. Taking $P = \frac{1}{N} \sum_{j=1}^K N_j P_j$ as the weighted average price of exchanged goods, and $V = 1/\alpha$, we get back \eqref{eq:exchange-2}.
The product $N P$ thus represents the total value of transactions executed, and thus corresponds, in modern terms, to the nominal Gross Domestic Production (or GDP), which is exactly that: the total value of transactions executed over a specific time period, typically one solar year. Each year, thus $N$ and $P$ can be measured and they will provide a different estimate. If instead of a solar year the GDP is measured on a monthly basis, then $N$ and $P$ will provide a different estimate each month. In other words, $N$ and $P$ change with time, or, in mathematical terms, they are a function of time. The term $M V$ on the other hand, represents the total amount of circulating money, taking into account that the same money might be used for multiple transactions. It is important to point out that $M$ should represent the money being used in transaction, not the one hoarded. If a considerable fraction of coins are being hoarded, or saved for future use, then the value of the circulating coins does not match (not even approximately) the total value of all existing coins. In this case, again, $M$ and $V$ are also function of time. If we were to write this explicitly we would write $N(t) P(t) = M(t) V(t)$, but we can stick with \eqref{eq:exchange-2} as long as we understand that all terms are a function of time.
Now, fast forward to present-day. What is the current monetary mass (or the total value of circulating coins)? This value is not known. First of all, nowadays coins do not represent the full amount of circulating money, as part of the monetary mass now lives digitally through the banking system. In fact, coins represent only a small fraction of the amount of circulating currency used in everyday transactions. However, modern day economy provides a range of digital and financial assets that (according to economists) are convertible or act as money, thus the definition of monetary mass is blurred. There are various recognized definitions of monetary mass, typically classified according to increasing complexity. However, these same definitions have changed over time, and they differ also among countries. Some economists even disagree with what should be defined as monetary mass, and further unofficial definitions have also been used within certain economic schools. Modern economics has settled on a measure of monetary mass known as M2, but it is enlightening that this choice is an agreed convention, not the necessary result of economic theory. In other words, there is no compelling mathematical equation that clearly defines what the monetary mass is, but rather a definition based on logic and understanding that may or may not be agreed upon.
Let us now consider the other side of the equation. On that side, we have the total value of the transactions occurred over the time period $T$. This is typically measured by the nominal GDP; however, the GDP cannot be measured exactly. We only have access to estimates of parts of the GDP as only part of the full set of transactions is recorded. For instance, some transactions do not enter official records, typically transactions occurring between private individuals (the hidden economy), or illegal activities that, because of their nature, are typically not included in the GDP figures. Overall, the fundamental issue here is that we can only deal with estimates of the missing part of the GDP, that may or may not count correctly the unofficial transactions occurring within the economy.
The third and most crucial point is that the equation is unverifiable: even if we had a fully reliable estimate of the monetary mass $M$ and of the nominal GDP, it is inconceivable to be able to measure the speed $V$ in a national economy, thus it is not possible to verify the validity of \eqref{eq:exchange-2}. Recall that $M$, $N$, $V$ and $P$ are all function of time despite time not being explicit in \eqref{eq:exchange-2}: thus the speed $V$ acts as a time-dependent factor that makes the equality always true by definition. In other words, the equation is valid because it was constructed to be always valid, not because it is a result of a proven mathematical economic theory. The term $V$ has the role of adjusting any mistake we may make in the estimates of $N$, $M$ or $P$. It does not matter what monetary mass we choose or whether our estimate of the GDP is reliable: an appropriate value of $V$ will render the equation still valid. The exercise of choosing the most appropriate measure of monetary mass (currently M2), is performed by economists on the basis on what makes sense, not on the basis of a proven scientific approach. In many cases, economists assume that the velocity $V$ is a constant over a short period of time, and relate the increase in prices to the increase in monetary mass, as I did earlier in one of my previous posts. However, there is nothing that may justify this approximation except the feeling that the speed of money exchange is not likely to change so quickly. The problem is of course that $V$ is not acting as an independent quantity, but simply as a factor fixed by the equality \eqref{eq:exchange-2}. This means that we have no idea of what this factor truly means, whether it does represents a velocity or some convoluted quantity which we cannot precisely identify, what its expected value should be, or what is the typical timescale of its fluctuations (if it exists at all). Thus any approximation we might apply has no formal justification.
Now, let us consider an aspect of equations in science that is completely ignored by economic theory: dimensional analysis. As explained here, an equation relates measurable quantities, each having some physical dimension. For instance, $x = v\cdot t$ relates the distance $x$ to the time interval $t$ through the velocity $v$. The physical dimensions of these quantities are length (for the distance), time (for the time interval) and a length over time (for the velocity $v$). If the left hand side of the equation has a physical dimension of a length, the same must occur for the right hand side: since length/time x time = length, the equality is dimensionally valid. This kind of fundamental analysis is not part of economic theory: in no book of economics I found a discussion of the dimensionality of the terms appearing in the equations of the economic theory, and that is a problem as I will try to illustrate. Let us consider thus equation \eqref{eq:exchange-2}. On the right-hand side we have $M$ and $V$, and on the left-hand side we have $N$ and $P$. The monetary mass $M$ represents the amount of circulating money. How do we measure it? Presumably we need to count banknotes and coins (and bank deposits, etc). We do not measure it with a scale, nor with a distance metre. Given we are counting units, we may assume that money is a pure, dimensionless number: it does not fall into any of the known “physical” quantities. Presumably, the number of goods $N$ also is a dimensionless pure number since we are counting goods. Then price $P$ is the amount of money that must be used to exchange a good. Because we assumed that $M$ (the amount of circulating money) is a pure number, then $P$ should also be a pure number, thus dimensionless. It follows that the velocity $V$ also must be dimensionless. So one possibility could be that equation \eqref{eq:exchange-2} is an equation between dimensionless quantities, but that would be weird. Dimensionless equations can be modified without much constraints, which is not a good thing. Multiplying two dimensionless quantities would still result in a dimensionless quantity, which means that the equations is potentially modifiable at will without breaking any fundamental principle.
There are some hints that the terms of the equations should have some physical dimension of some kind. For instance, $N$ is the amount of goods exchanged during a fixed interval of time, so at least $N$ should have the dimensions of an inverse of time. This should be matched by an inverse of time on the other side, possibly $V$. What about prices? If price is dimensionless, multiplying two prices would be a dimensionally valid operation. But what meaning does the product of two prices have? Indeed, a price is expressed in terms of a currency, which has a role similar to a unit of measurement: multiplying two prices produces something that we do not know how to handle. In this case, what are the dimensions of price? What instrument allows us to measure and quantify prices unequivocally? And what exactly are we measuring?
This discussion is crucial in identifying the lack of proper definition of money. In economic theory, money is ill defined: there is no definition of how one can operationally ‘measure’ money. Yes, you count banknotes, but that is not enough: that does not define the intrinsic quantity that the number on the dollar/euro/pound/whatever bill is measuring. This lack of proper definition, as well as a lack of universally defined unit of measure, makes any equation that deals with money a mathematical formula, but not a law amenable to be used for predictions. That is owing to the fact that if a unit of money (for instance the unit of a currency) modifies its meaning in time, the law will change its meaning accordingly. To go back to the example of the equation of motion in physics, if time were not universally defined, different people would provide different estimates of where the driver would be in two hours. And if a forecast is made at a certain point in time, when you actually check where the car is, you might find something different as the underlying time unit or distance unit has changed.
