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**.

Classical View

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**.

Keynesian View

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**.

Monetarist View

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.

Agnostic View

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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