# Rothbard vs Fisher's Equation of Exchange, MV = PT

The following is abstracted from :

M.N. Rothbard, __Man, Economy and State__

This is available in both online .pdf form and as a newly released hardcover Scholar's Edition from mises.org.

It is provided here as a reference for comments that came up in response to an earlier post, Money and the Government.

CHAPTER 11?MONEY AND ITS PURCHASING POWER. . . . . . . . 755

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Money and Its Purchasing Power 831

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13. The Fallacy of the Equation of Exchange

The basis on which we have been explaining the purchasing power of money and the changes in and consequences of monetary phenomena has been an analysis of individual action. The behavior of aggregates, such as the aggregate demand for money and aggregate supply, has been constructed out of their individual components....

... Monetary theory in American economics, however ... , has been presented in entirely different terms?in the quasi-mathematical, holistic equation of exchange, derived especially from Irving Fisher. The prevalence of this fallacious approach makes a detailed critique worthwhile.

...Fisher has leaped from the real world of an array of individual prices for an innumerable list of concrete goods into the misleading fiction of a ?price level,? without discussing the grave difficulties which any such concept must face. The fallacy of the ?price level? concept will be treated further below. The ?price level? is allegedly determined by three aggregative factors: the quantity of money in circulation, its ?velocity of circulation? ? the average number of times during a period that a unit of money is exchanged for goods ? and the total volume of goods bought for money. These are related by the famous equation of exchange: MV = PT....

...This off-hand assumption of equality is not self-evident, as Fisher apparently assumes, but a tangle of fallacy and irrelevance....

...There is thus never any equality of values on the part of the two participants. The assumption that an exchange presumes some sort of equality has been a delusion of economic theory since Aristotle, and it is surprising that Fisher, an exponent of the subjective theory of value in many respects, fell into the ancient trap. There is certainly no equality of values between two goods exchanged or, as in this case, between the money and the good....

...We have seen, however, that even for the individual exchange, and setting aside the holistic problem of ?total exchanges,? there is no such ?equality? that tells us anything about the facts of economic life. There is no ?value-of-money side? equaling a ?value-of-goods side.? The equal sign is illegitimate in Fisher?s equation.

How, then, account for the general acceptance of the equal sign and the equation? The answer is that, mathematically, the equation is of course an obvious truism: 70 cents = 10 pounds of sugar x 7 cents per pound of sugar. In other words, 70 cents = 70 cents. But this truism conveys no knowledge of economic fact whatsoever....

...What we have in Fisher?s equation, in short, is two money sides, each identical with the other. In fact, it is an identity and not an equation. To say that such an equation is not very enlightening is self-evident. All that this equation tells us about economic life is that the total money received in a transaction is equal to the total money given up in a transaction ? surely an uninteresting truism....

...The only knowledge we can have of the determinants of price is the knowledge deduced logically from the axioms of praxeology. Mathematics can at best only translate our previous knowledge into relatively unintelligible form; or, usually, it will mislead the reader, as in the present case. The price in the sugar transaction may be made to equal any number of truistic equations; but it is determined by the supply and demand of the participants, and these in turn are governed by the utility of the two goods on the value scales of the participants in exchange. This is the fruitful approach in economic theory, not the sterile mathematical one....

...only individual actors can decide whether or not to buy; only their value scales determine prices. It is this profound mistake that lies at the root of the fallacies of the Fisher equation of exchange: human action is abstracted out of the picture, and things are assumed to be in control of economic life. Thus, either the equation of exchange is a trivial truism ? in which case, it is no better than a million other such truistic equations, and has no place in science, which rests on simplicity and economy of methods ? or else it is supposed to convey some important truths about economics and the determination of prices. In that case, it makes the profound error of substituting for correct logical analysis of causes based on human action, misleading assumptions based on action by things. At best, the Fisher equation is superfluous and trivial; at worst, it is wrong and misleading, although Fisher himself believed that it conveyed important causal truths....

...But Fisher is looking for an equation to explain the price level; therefore he brings in the concept of an ?average price level,? P, and a total quantity of goods sold, T, such that E is supposed to equal PT. But the transition from the trivial truism E = pQ + p 'Q ' . . . to the equation E = PT cannot be made as blithely as Fisher believes. Indeed, if we are interested in the explanation of economic life, it cannot be made at all.For example, for the two transactions (or for the four), what is T? How can 10 pounds of sugar be added to one hat or to one pound of butter, to arrive at T ? Obviously, no such addition can be performed, and therefore Fisher?s holistic T, the total physical quantity of all goods exchanged, is a meaningless concept and cannot be used in scientific analysis....

...Thus, any concept of average price level involves adding or multiplying quantities of completely different units of goods, such as butter, hats, sugar, etc., and is therefore meaningless and illegitimate. Even pounds of sugar and pounds of butter cannot be added together, because they are two different goods and their valuation is completely different. And if one is tempted to use poundage as the common unit of quantity, what is the pound weight of a concert or a medical or legal service?...

...It is evident that PT, in the total equation of exchange, is a completely fallacious concept. While the equation E = pQ for an individual transaction is at least a trivial truism, although not very enlightening, the equation E = PT for the whole society is a false one. Neither P nor T can be defined meaningfully, and this would be necessary for this equation to have any validity....

...Let us consider the other side of the equation, E = MV, the average quantity of money in circulation in the period, multiplied by the average velocity of circulation. V is an absurd concept....

...Velocity is not an independently defined variable. Fisher, in fact, can derive V only as being equal in every instance and every period to E/M....

...But it is absurd to dignify any quantity with a place in an equation unless it can be defined independently of the other terms in the equation....

...In fact, since V is not an independently defined variable, M must be eliminated from the equation as well as V, and the Fisherine (and the Cambridge) equation cannot be used to demonstrate the ?quantity theory of money.?...

...This example should reveal the fallacy of equations in economic theory. The Fisherine equation has been popular for many years because it has been thought to convey useful economic knowledge. It appears to be demonstrating the plausible (on other grounds) quantity theory of money. Actually, it has only been misleading....

...Even Fisher admitted this by conceding that a change in M would always affect V, so that the influence of M on P could not be isolated. He contended that after this ?transition? period, V would revert to a constant and the effect on P would be proportional. Yet there is no reasoning to support this assertion. At any rate, enough has been shown to warrant expunging the equation of exchange from the economic literature.

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## "Thus, any concept of

"Thus, any concept of average price level involves adding or multiplying quantities of completely different units of goods, such as butter, hats, sugar, etc., and is therefore meaningless and illegitimate."

P isn't calculated like this; it is simply the ratio of the price of a particular bundle of goods at two times. No adding or multiplying of bread and butter is involved, just adding up dollars.

There are problems with doing this, of course, but this critique of Rothbard's is nonsense, no?