# Mises Wire

## The Quantity Theory of Money and the Equation of Exchange

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One of the first things that the student of economics is liable to be tortured with when he turns to monetary theory is the equation of exchange. Based on a mechanistic understanding of the quantity theory of money, the equation purports to show the relation between the supply of money and the prices of goods. It is one of the foundational fallacies of modern economics.

The quantity theory of money itself was a major landmark in the development of economic theory. It can be traced back to the Italian Bernardo Davanzati and the Pole Copernicus in the sixteenth century. But the quantity theory is better known from the classic elaborations by David Hume and David Ricardo. These theorists attempted to explain the relationship between prices and the quantity of money based on the laws of supply and demand. Their main conclusion—and the central truth established by the quantity theory—was that an increase in the quantity of money necessarily leads to an increase in prices. A corollary to this conclusion is that nothing is gained by increasing the quantity of money; any quantity is adequate to fulfill money’s social function.

Advances in monetary theory would have to look at how the supply and demand for money was determined, something that Mises investigated and discussed at length in his 1912 *Theory of Money and Credit*. Unfortunately, despite Mises’s pioneering work, a different and altogether inferior elaboration of monetary theory took hold and gained widespread popularity: a mechanistic version of the quantity theory of money summed up in the so-called equation of exchange.

**Anatomy of the Equation of Exchange**

The equation of exchange first achieved prominence with Irving Fisher’s 1911 book *The Purchasing Power of Money*, and latter-day monetarists spread its use far and wide. Milton Friedman, perhaps the best-known monetarist in the twentieth century, even had it printed on his license plates. There are different variants of the equation (*M*V=P*T*, *M*V=P*Y*, *M*V=P*Q*, to take the simplest), but none of them are substantially different from Fisher’s original formula: *M*V=P*T*. What are the components of this formula?

*M* signifies the quantity of money, or rather the average quantity in a given period.

*V* is the velocity of money and, as we shall see, is not a well-defined concept. Monetarists usually present it as money’s “turnover” or an indicator of how much “use” each monetary unit gets.

*P* is the level of prices, an average of all prices paid in a given period.

*T* is the sum of transactions over that same period.

**[RELATED: "The Fallacy of the Equation of Exchange" by Murray Rothbard]**

This equation, it is claimed, shows the relation between the money side and the “real” side of the economy. Its proponents claim that the equation clearly shows the relation between the quantity of money and the price level: if we assume that *V* and *T* are constant, then an increase in *M* necessarily leads to an increase in *P*. Now we have clear proof of the quantity theory of money. Or do we?

**A Critique of the Equation of Exchange**

Both Mises and Rothbard penned devastating critiques of the equation of exchange and we will draw freely on their works in what follows. It is worth noting at the outset that the holistic approach underlying the monetarist theory is wholly inadmissible. As Mises said in *The Theory of Money and Credit*:

For a long time it was believed that the demand for money was a quantity determined by objective factors and independently of subjective considerations. It was thought that the demand for money in an economic community was determined, on the one hand by the total quantity of commodities that had to be paid for during a given period, and on the other hand by the velocity of circulation of the money….It is inadmissible to begin with the demand for money of the community. The individualistic economic community as such, which is the only sort of community in which there is a demand for money, is not an economic agent. It demands money only insofar as its individual members demand money. The demand for money of the economic community is nothing but the sum of the demands for money of the individual economic agents composing it. But for individual economic agents it is impossible to make use of the formula: total volume of transactions ÷ velocity of circulation. If we wish to arrive at a description of the demand for money of an individual we must start with the considerations that influence such an individual in receiving and paying out money.

Although this basic error should be more than enough to disqualify the equation, it is still both worthwhile and necessary to examine it in detail. Does it work on its own terms? Let us examine its components.

*M*, the quantity of money, is unproblematic. Exactly what kind of claims and money substitutes should be counted as part of the money supply in an economy is open to discussion, but the quantity of money is a clearly defined concept.

*P* and *T* are more suspect. What is the real meaning behind these terms? In reality, they are nothing but statistical abbreviations of all the trades in an economy in a given period. *P* is the average of prices paid and *T* is the number of transactions. However, this means that *P*T* is simply what sellers received in exchange for their goods and services—their aggregate revenues or money income, traditionally symbolized by *Y*.

This brings us to *V*. How exactly do we establish *V*? In sharp contradistinction to the other terms of the equation, there simply isn’t a way to independently arrive at or define a magnitude for *V*. It is simply the factor necessary to make the equation *M = P*T* true.

There are two ways to find *V*. One is to simply divide *P*T* by *M*. The other—and the one Fisher used—is to start with the sum of expenditures for a given period, *E, *and the quantity of money, *M*. Then we simply define *V* as the relationship between these two magnitudes: *E/M=V*. Unfortunately, this still does not solve the problem of *V* not being independently defined. It simply introduces another variable, *E*, into our equation.

More fundamentally, a little analysis shows that the equation is truly absurd if it is meant to say anything about the role of the quantity of money in an economy:

If *V=E/M*,

then* M*V=M*E/M*,

and then* M*E/M=E*.

And since we already know that *P*T=Y*, the equation *M*V=P*T *reduces to *E=Y*.

The ground-breaking insight of the monetarist equation of exchange is therefore that the sum of money expenditures in a given period has to equal the sum of money incomes for that same period. It is certainly true that in any transaction the buyer’s expenditure is necessarily equal to the seller’s income, but one wonders why this should be elevated to a cornerstone of monetary theory.

What about inflation? A defender of Fisher and his countless monetarist epigones might claim that at least the equation can be used to show the relation between the quantity of money and the level of prices. In Friedman’s famous dictum, it shows that “inflation is always and everywhere a monetary phenomenon.” However, this says nothing more than what the primitive quantity theory had already established without the elaborate trappings of the equation of exchange. In fact, the equation depicts the relation between the quantity of money and inflation in an inferior and very misleading way, with changes in the price level simply being a function of changes in the quantity of money. And this is clearly erroneous.

**The Alternative: Misesian Monetary Theory**

Fundamentally, the equation of exchange rests on a misguided approach to economic theory. It simply postulates the existence of aggregate concepts such as velocity and the level of prices and that we can understand these without looking at what brings them about and what causes them to change. Any consideration of causality is sacrificed in favor of impressive formulae.

Mises and the other Austrians showed long ago how to think about monetary theory, based on the fundamental insights into the role of human action and subjective valuations in the economy. Rather than give a full description of Mises’s insights, let us here briefly illustrate how he conceives of an increase in the quantity of money and how it eventually affects prices.

At the outset, before the increase in the quantity of money, each individual has a certain cash holding determined by the marginal utility of money to him. Each person has a cash holding just large enough that the utility of the marginal monetary unit outweighs—is ranked higher on his value scale than—the utility he expects to gain from exchanging it for consumer or producer goods.

What happens when the quantity of money increases? This increase always means that some people gain more money than they had before. Let us assume that the close personal friends of J. Powell, money producer extraordinaire, Mr. Goldman and Mrs. Sachs suddenly find their cash holdings increased. Now their valuation of the marginal monetary unit has changed, as they have moved down their value scales, so to speak. The value of the marginal unit of money is now lower to Mrs. Sachs, and she will therefore use some of the new money on goods and services that are now ranked higher on her value scale.

In this way the new quantity of money moves through the economy: the first receivers, Mr. Goldman and Mrs. Sachs, spend the new money until their cash holdings again reflect their subjective valuation of the marginal unit. In the meantime, the added demand for goods and services leads to a rise in prices. The next receivers of the new money (those who supplied Mr. Goldman and Mrs. Sachs with goods and services) are now in the same situation as the first receivers. They too will spend their additional money, leading to increases in the prices of the goods they spend them on. And so the process continues, until the new money has been spread through the economy. Some prices rise while others remain the same. Some have gained from this process, namely, those who received the new money first, before prices had adjusted; others have lost, namely, those who only experienced an increase in their money incomes after the rise in prices, or who never saw any of the new money at all.

**Conclusion**

This brief critique of the equation of exchange and the contrast to Misesian monetary theory will hopefully have made it evident that the equation of exchange is an incoherent and mechanistic version of the quantity theory of money. Unfortunately, bad theories have a long life in the social sciences, and this has certainly been true of the equation of exchange. Still, if one wants to understand monetary phenomena, the starting point must be a complete rejection of the mechanistic quantity theory. Let it be thrown out into the outer darkness, where there shall be weeping and gnashing of teeth!

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