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 objects of study for economics and choose prices (P) and output (Q)—they choose something they value more and give up something they value less: that’s the essence of economic theory—praxeology.
Thus, assuming the two sides of the EoE to be conceptually equivalent—and assuming, therefore, that changes on one side must logically imply changes on the other one—gravely misconceives the fundamentals of economic thinking (Rothbard,  2009 , pp. 833–34), because it neglects human choices. This kind of assumption, therefore, is invalid and unsuitable to build any kind of sound and consistent economic theory. Of course, M*V = P*Q will always be algebraically true, but there is no necessary causal nexus linking the two sides of the equation—variations in one variable on one side (say, M) may not necessarily cause variations in one variable on the other side (say, P).
Second: the EoE conceals the true mechanism of money propagation in the economy. According to the EoE, indeed, variations in M should mechanically cause variations in P—the “general price level.” However, this simplistic and mechanistic explanation of monetary phenomena gravely misconceives how money really enters the economy—that is, through actual exchanges of newly created money for already produced goods and services.
The process of monetary inflation (i.e., the injection of new money into the economy) is bound to alter relative prices of goods and services, thus advantaging some agents (the sellers of goods and services bought first with the new money) and disadvantaging some others (the sellers of goods and services bought last) (Rothbard,  2009, pp. 811–14). In other words, the EoE allows no conceptual room for the Cantillon effect—the alteration of relative prices.
Even worse, the idea of a “general price level” (P) has no sensible economic meaning. In fact, the right-hand side of the EoE makes sense if and only if P and Q are thought of as vectors of prices and physical quantities—i.e., [P; P’; P’’; P’’’; …] multiplied by [Q; Q’; Q’’; Q’’’; …], where the superscripts identify any individual exchange (P*Q + P’*Q’ + P’’*Q’’ + …).
Instead, the EoE aims at introducing 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) price level (P)? Provided that monetary prices are ratios and every price is defined in terms of a different good or service (prices of different goods and services never share a common unit, or denominator, because their form is “X dollars per one unit of type Y good or service”), how could you average—or even add—them?
It’s obvious, indeed, that there is no means of defining an “average” (or “general”) price level (P). It’s indeed impossible both (1) to directly sum (or average) prices of different goods and services and (2) to arrive at P as the ratio between total expenditure (E = P*Q) and total physical quantities exchanged (Q)—i.e., P*Q/Q = E/Q=P. In fact, computing this latter ratio would require a sum (or average) for the vector Q to be computable—and that cannot possibly be the case, since you cannot sum, or average, heterogeneous goods and services (Rothbard  2009, p. 839).
Third: the concept of “money velocity” is flimsy, if not nonsensical. Out of the four variables forming the equation of exchange, V is the only one that cannot conceptually stand alone—and make sense—outside the equation.
M can be conceived of as either a physical quantity (pounds of gold) or a monetary one (the total nominal value of coins and banknotes within the economy); P can be conceived of as an array of prices, which are by their very nature relative subjective values, or tradeoffs; Q is ultimately comprised of physical quantities (pounds of bread, gallons of beer, haircuts, etc.); but V, in the end, cannot be conceived of autonomously.
In fact, even if you define V as the average number of times a dollar moves from its owner to another one, you are basically question begging—because dollars change hands if and only if quantities of goods and services (Q) are exchanged for money (M) at given money prices (P). Hence, you would be trying to define a variable (V), supposed to be independent of—and to influence—a system (the EoE), in terms of the very system that the same variable is supposed to effect—a prima facie circular reasoning.
V, indeed, cannot be autonomously defined 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 moving from one owner to another one” is hence proven as inconsistent, unless you in effect assume the EoE—which should be stemming from, not underlying, the idea of money velocity (V)—to be the case. If the EoE is assumed, then dollars (M) are in effect moving from one owner to another one during exchanges of 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 it!
Concluding, the EoE proves to be unsuitable to provide a conceptual groundwork for any kind of sound and consistent economic analysis—and should be therefore expunged from economic theory.