Equation of Exchange

The equation of exchange is an accounting identity that links the money supply, the velocity of money, the price level, and real economic activity. It formalizes the idea that the total money changing hands in an economy equals the total nominal value of goods and services exchanged.

Key takeaways

• The basic identity is M × V = P × Q (or M × V = P × T), where P × Q is nominal spending (nominal GDP).
• If velocity and real output are stable, changes in the money supply translate into proportional changes in the price level — the core of the quantity theory of money.
• Rearranging the identity yields a simple expression for money demand: money demand is proportional to nominal income and inversely proportional to velocity.

The equation and its terms

The original form (Fisher) is:
M × V = P × T

A more commonly used form today is:
M × V = P × Q

Where:
* M = money supply (average currency units in circulation)
* V = velocity of money (average number of times a unit of currency circulates per period)
* P = average price level
* T = index of real transactions
* Q = index of real expenditures (often interpreted as real output)

P × Q is nominal GDP (total nominal spending).

Interpretation

• M × V represents the total nominal amount of money spent in a period.
• P × Q represents the total nominal value of goods and services purchased in that period.
  The identity simply states that nominal expenditures equal nominal income.

Quantity theory of money

Starting from M × V = P × Q, rearrange to solve for the price level:
P = (M × V) / Q

Expressed in growth-rate form (using hats to denote growth rates):
p̂ = m̂ + v̂ - q̂

If the velocity of money (v̂) and real output (q̂) are approximately constant (i.e., v̂ = 0 and q̂ = 0), then:
p̂ ≈ m̂
meaning the inflation rate is roughly equal to the growth rate of the money supply. This idea underpins monetarist views that emphasize the role of money supply in determining inflation.

Money demand

Solving the identity for M gives an expression for the demand for money:
M = (P × Q) / V

Interpreting this as money demand (M_D):
M_D = (P × Q) × (1 / V)

Thus:
* Money demand is proportional to nominal income (P × Q).
* The factor 1 / V can be seen as the demand to hold liquidity (the inverse of velocity).

Fisher’s version and GDP

Fisher originally used transactions (T) in the identity (M × V = P × T). When transaction data are unavailable, T is commonly replaced by Y (national income), yielding M × V = P × Y, where P × Y is nominal GDP.

The expenditure-side GDP identity is:
GDP = C + I + G + NX
(where C = consumption, I = investment, G = government spending, NX = net exports). P × Q (or P × Y) corresponds to nominal GDP in the equation of exchange.

Bottom line

The equation of exchange is a concise way to link money, prices, and economic activity. It is an identity (always true by definition) and a foundation for the quantity theory of money, which connects money growth to inflation when velocity and real output are stable. It also provides a straightforward expression for money demand as a function of nominal income and the liquidity preference captured by the inverse of velocity.