In monetary economics, the equation of exchange is the relation: where, for a given period, Thus PQ is the level of nominal expenditures. This equation is a rearrangement of the definition of velocity:. As such, without the introduction of any assumptions, it is a tautology. The quantity theory of money adds assumptions about the money supply, the price level, and the effect of interest rates on velocity to create a theory about the causes of inflation and the effects of monetary policy. In earlier analysis before the wide availability of the national income and product accounts, the equation of exchange was more frequently expressed in transactions form: where

Foundation

The foundation of the equation of exchange is the more complex relation: where: The equation: is based uon the resumtion of the — that there is a relatively clean distinction between overall increases or decreases in rices and underlying, “real” economic variables — and that this distinction may be catured in terms of rice indices, so that or comonents of may be extracted as the multilier P, which is the aggregate rice level: where is a row vector of ; and likewise for In 2008 economist proposed an integral form of the equation of exchange, where on the left side of the equation is M(V)dV under the integral sign, and on the right side is a sum PiQi from i=1 to N. Generally, N could be infinite. There are two variants of this formula:

and The simplest cases for the dissipative scaling factors and are: k_i = \pm 1,. Also, k_i can be determined by the methods of the . If liquidity function, then, by the :

Naganoff's formula is used to describe in details the processes of inflation and deflation, Internet trading and cryptocurrencies.

Applications

Quantity theory of money

The quantity theory of money is most often expressed and explained in mainstream economics by reference to the equation of exchange. For example, a rudimentary theory could begin with the rearrangement If V and Q were constant or growing at the same fixed rate as each other, then: and thus where That is to say that, if V and Q were constant or growing at equal fixed rates, then the inflation rate would exactly equal the growth rate of the money supply. An opponent of the quantity theory would not be bound to reject the equation of exchange, but could instead postulate offsetting responses (direct or indirect) of Q or of V to.

Money demand

Economists Alfred Marshall, A.C. Pigou, and John Maynard Keynes, associated with Cambridge University, focusing on money demand instead of money supply, argued that a certain portion of the money supply will not be used for transactions, but instead it will be held for the convenience and security of having cash on hand. This proportion of cash is commonly represented as k, a portion of nominal income (nY). (The Cambridge economists also thought wealth would play a role, but wealth is often omitted for simplicity.) The Cambridge equation for demand for cash balances is thus: which, given the classical dichotomy and that real income must equal expenditures Q, is equivalent to Assuming that the economy is at equilibrium (M_{D} = M), that real income is exogenous, and that k is fixed in the short run, the Cambridge equation is equivalent to the equation of exchange with velocity equal to the inverse of k: The money demand function is often conceptualized in terms of a liquidity function, L(r,Y), where Y is real income and r is the real rate of interest. If V is taken to be a function of r, then in equilibrium

History

The equation of exchange was stated by John Stuart Mill who expanded on the ideas of David Hume. The algebraic formulation comes from Irving Fisher, 1911.