Rothbard’s Refutation of the Quantity Theory of Money
In chapter 11 of Man, Economy, and State  (2009), Rothbard sets out his theory of money and its influences on business fluctuations. Among the many insights Rothbard provides, we find a compelling and cogent refutation of Irving Fisher’s equation of exchange (in section 13)—which underlies the monetarist quantity theory of money.
The idea behind the equation of exchange (EoE) is trivial: given the total quantity of money (M), the alleged “general (or average) price level” (P), the total physical quantity (Q) of goods and services exchanged within the economy, and the so-called velocity (V) at which money is exchanged between agents, the relation M*V = P*Q must hold.
Rothbard’s critique of the EoE has three main pillars. First, it is conceptually wrong to think of exchange as equivalence—i.e., indifference between what you give up and what you receive. Second, the EoE does not capture heterogeneity in price inflation (Cantillon effects), and employs a variable—the general price level (P)—that cannot be sensibly conceived of in reality. Third, the very idea behind the EoE’s mechanism is nonsensical, because it is just a tautology—always true provided you conveniently define money velocity (V).
First: exchange is neither indifference nor equivalence. The first fallacy involved in Fisher’s EoE is the assumption that you can derive a conceptual equivalence from an accounting truism. It is true, indeed, that money expenditure (E = M*V) is bound to equate the prices (P) of the goods and services you are purchasing times the physical quantities of goods and services you are acquiring (Q). This holds true particularly if you consider a single transaction.
However, it is wrong to infer from this accounting truism a causal nexus, implying that one of the two sides of the equation “determines” the other one. In fact, as Rothbard points out, economics—being interested in the study of human action—is not concerned with indifference or equivalence: neither of them can be the basis for any kind of economic action (Rothbard  2009, p. 307). When human beings act—i.e., when they are obj
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