The equation of exchange identifies the exact mathematical relationship that exists between the money supply, the price level, and the volume of economic activity. The economist Irving Fisher (1867–1947) first formulated the equation of exchange, and his version took the following form:

MV + M′V′ = PT.

Here *M* stands for the stock of currency in a given year, *V* stands for the velocity or number of times a dollar bill changes hands during a year, *M*′ measures the quantity of checkable deposits, and *V*′ the velocity of checkable deposits. *P* stands for the price involved in a typical transaction, and *T* represents the number of transactions.

Contemporary economists make use of a simplified equation of exchange that takes the following form:

MV = PY

Here *M* stands for a measure of the money stock that includes, at a minimum, currency in circulation plus checkable deposits. Time deposits and other highly liquid assets may also be included. *V* stands for the income velocity of money, defined as being equal to the money value of income and output divided by the money stock. *P* stands for the price level and *Y* stands for real output. In practice *PY* stands for Gross Domestic Product (GDP) unadjusted for inflation, called nominal GDP, and *Y* stands for GDP adjusted for inflation, called real GDP. *P* is a factor standing for the price level and is calculated by dividing nominal GDP by real GDP. Velocity is calculated by dividing nominal GDP by the money stock.

Nominal GDP divided by *M* equals *V,* which can be converted to the form *MV* = nominal GDP. Furthermore, nominal GDP divided by real GDP *(Y)* equals the price index *(P),* which is mathematically equivalent to saying that nominal GDP = *PY.* There fore *MV = PY* is what is called an identity in mathematics, true by definition.

The equation of exchange is often converted to a percentage change form, expressed as:

% change in M + % change in V = % change in P + % change in Y

A school of economists called quantity theorists assumes that velocity is relatively stable, suggesting that the percentage change in *V* is always zero. They also assume that the percentage change in *Y* is at the long-term growth rate of real GDP, approximately 3 percent. With these assumptions the inflation rate (percentage change in *P*) will always be 3 percent less than the growth rate of the money stock (percentage change in *M*). If the money stock grows at 10 percent a year, the inflation rate will be 7 percent a year. Therefore, inflation is an exact mathematical function of the money stock growth rate, and the equation of exchange furnishes us with a theory of inflation.

Empirical evidence bears out the close correspondence between money stock growth and inflation, but there is still room for some economists to argue that increases in the inflation rate force authorities to increase monetary growth, instead of the other way around. These issues still stand to benefit from further study.