In chapter 11 of Man, economy and state  (2009), Rothbard presents his theory of money and its influences on corporate fluctuations.
Among the many ideas provided by Rothbard, we find a compelling and compelling rebuttal of the Irving Fisher trading equation (in Section 13) – which underlies the monetarist theory of the quantity of money.
The idea of the exchange equation (EoE) is trivial: given the total quantity of money (M), the so-called “general (or average) price level” (P), the total physical quantity (Q) of goods and services exchanged within the economy, and the so-called speed (V) at which money is exchanged between agents, the relation M * V = P * Q must hold.
Rothbard’s critique of EoE rests on three main pillars. First, it is conceptually wrong to think of the exchange as an equivalence, that is to say an indifference between what you give up and what you receive. Second, the EoE does not take into account the heterogeneity of price inflation (Cantillon effects) and uses a variable – the general price level (P) – which cannot be reasonably conceived in reality. Third, the whole idea behind the EoE mechanism is absurd, because it’s just a tautology – always true provided you set the speed of money (V) appropriately.
First: the exchange is neither indifference nor equivalence. The first error involved in Fisher’s EoE is the assumption that you can derive a conceptual equivalence from an accounting truism. It is true, in fact, that monetary expenditure (E = M * V) is related to equating the prices (P) of the goods and services that you buy to the physical quantities of goods and services that you buy (Q). This is especially true if you are considering a single transaction.
However, it is wrong to infer a causal link from this accounting truism, implying that one side of the equation “determines” the other. In fact, as Rothbard points out, economics – interested in the study of human action – is not interested in indifference or equivalence: neither can be the basis of any economic action (Rothbard  2009, p. 307). When human beings act, that is, when they are objects of study for the economy and to choose prices (P) and output (Q) – they choose something they value more and give up what they value less: this is the essence of economic theory – praxeology.
Thus, assuming that the two sides of the EoE are conceptually equivalent – and assuming, therefore, that changes on one side must logically imply changes on the other – seriously misunderstands the fundamentals of thought. economic (Rothbard,  2009, pp. 833–34), because it neglects human choice. This type of assumption is therefore invalid and unsuitable for the development of any sort of solid and coherent economic theory. Of course, M * V = P * Q will always be algebraically true, but it is not necessary causal link connecting the two sides of the equation – variations in one variable on one side (eg, M) may not necessarily cause variations in a variable on the other side (eg, P).
Second: EoE hides the real mechanism of currency propagation in the economy. According to the EoE, indeed, changes in M should automatically lead to changes in P – the “general price level”. However, this simplistic and mechanistic explanation of monetary phenomena seriously misunderstands the way in which money actually enters the economy, that is, through real exchanges of newly created money for goods and services already produced.
The process of monetary inflation (i.e. the injection of new money into the economy) is called upon to modify the relative prices of goods and services, thus favoring certain agents (sellers of goods and services purchased in first with the new currency) and disadvantage others. (sellers of goods and services purchased last) (Rothbard,  2009, p. 811–14). In other words, EoE leaves no conceptual room for the Cantillon effect – the alteration of relative prices.
Worse still, the idea of a “general price level” (P) has no significant economic significance. In fact, the right side of EoE makes sense if and only if P and Q are considered to be vectors of price and physical quantity, i.e. [P; P’; P’’; P’’’; …] multiplied by [Q; Q’; Q’’; Q’’’; …], where the exhibitors identify any individual exchange (P * Q + P ‘* Q’ + P ” * Q ” +…).
Instead, the EoE aims to introduce averages for both Q and P, but how could you average (say) pounds of butter, gallons of beer, haircuts, etc. – heterogeneous goods and services? And even if you could, how would you define their “general” (or average) (P) price point? Provided that the money prices are ratios and that each price is defined according to a different good or service (the prices of different goods and services never share a common unit or denominator, because their form is ” X dollars per unit of Y-type good or service ”), how could you value or even add them up?
It is obvious, in fact, that there is no way to define an “average” (or “general”) price level (P). It is in fact impossible both (1) to directly sum (or average) the prices of different goods and services and (2) to arrive at P as the ratio between total expenditure (E = P * Q ) and the total physical quantities exchanged (Q) – that is, P * Q / Q = E / Q = P. In fact, calculating the latter ratio would require a sum (or average) for the vector Q is computable – and that cannot be the case, because you cannot add or average heterogeneous goods and services (Rothbard  2009, p. 839).
Third: the concept of “speed of money” is fragile, if not absurd. Of the four variables forming the exchange equation, V is the only one that cannot, conceptually, be isolated – and have a meaning – outside the equation.
M can be thought of either as a physical quantity (pounds of gold) or as a monetary quantity (the total face value of coins and banknotes in the economy); P can be thought of as a table of prices, which are by their nature relative subjective values or tradeoffs; Q is ultimately made up of physical quantities (pounds of bread, gallons of beer, haircuts, etc.); but V, in the end, cannot be conceived in a stand-alone way.
In fact, even if you define V as the average number of times a dollar passes from its owner to another, you are basically wondering about begging – because dollars change hands if and only if quantities of goods and services (Q) are exchanged for money (M) at given monetary prices (P). Therefore, you try to define a variable (V), which is supposed to be independent – and to influence – a system (the EoE), depending on the very system that the same variable is supposed to perform – a At first glance circular reasoning.
V, indeed, cannot be defined autonomously as a meaningful concept, but only as a ratio – the nominal value of transactions (P * Q) within the economy divided by the total quantity of money (M) (Rothbard  2009, p. 841). The vague idea of ”dollars passing from one owner to another” is therefore proven to be inconsistent, unless you actually assume that EoE – which should stem from, and not underlying, the idea of speed of money (V) – to be the case. If EoE is assumed, then dollars (M) actually move from owner to owner when exchanging dollars for goods and services (Q) at given monetary prices (P), and V can be (sort of) thought of in terms of the other three variables. But V should serve as an explanation for the EoE. It should not be postulated or defined by him!
Final, EoE is proving to be unsuitable for providing a conceptual basis for any kind of sound and coherent economic analysis – and should therefore be removed from economic theory.
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