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).