The Equation of Exchange
M V = P Y
The equation of exchange comes in two forms. The early version M = k P Y with k and the newer version M V = P Y with V. The difference is of no economic consequence because k and V are inverses.
The equation of exchange can be the basis for a demand for money, an aggregate demand curve, etc.
Percentage changes. ∏ is the inflation rate (percentage rate of change in P). g is the percentage growth rate for Y. v is the percentage rate of change of V. m is the percentage rate of change of M. Then, m + v = ∏ + g.
The velocity is constant due to the technology of exchange. M = k P Y. Given a fixed money supply, the product P Y must be constant. This determines an aggregate demand curve.
Because Y is determined by real factors of production, the equation of exchange determines the price level. In particular, changes in M change only P.
The velocity of money is a stable function of the interest rate. M = P Y / V(R). This determines the demand for money.
Given a fixed money supply, the relation between P and Y determines an aggregate demand curve. That relation incorporate the simultaneous changes in R.
Y can be changed holding M and P constant because V(R) is a function of R.
Velocity is a function k(•) of multiple interest rates and asset returns. M = k(•) P Y. To a reasonable approximation the velocity is constant.
In the short run, changes in M cause changes in Y because P does not adjust.
In the long run, changes in M cause changes only in P because Y is determined by real factors of production.
An agnostic view is that the equation of exchange is simply the definition of velocity. V = P Y / M. While the velocity can be calculated given P, Y, and M, that calculation imposes no relation among those three variables.
The agnostic view does not necessarily rule out monetary policy to control R. Even if the velocity of money is an unstable function of the interest rate (and other variables), short-term adjustments in M might effectively control the interest rate.
The agnostic view is getting ever more plausible as the forms of payments multiply and evolve.
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