pcecon.com Class Notes
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Mathematically speaking, the effect of the money supply on the economy can be summarized by the formula
MxV = PxY
in which
M is the money supply
V is velocity, which is the number of times each dollar is spent on GDP
P is the price level (or the ratio of the current year’s prices to the base year)
Y is real GDP
The above equation, called the “equation of exchange,” must always be true. The first part, MxV, is spending. The second part, PxQ is nominal GDP. They have to be equal because spending on GDP has to equal GDP.

In terms of changes or growth rates (also called percentage changes or %∆ for short), this formula means that
%∆M + %∆V = %∆P + %∆Y
or, the growth in the money supply plus the growth in velocity equals inflation plus the growth in real GDP.
This is the growth rate version of the equation of exchange, which also always is true.
Since velocity is an important part of the equation, and since we haven't discussed it much, here are a few details about what determines it:
Other things equal, velocity will increase (people will spend each dollar more times) if
interest rates on non-money investments increase (people hold less money)
expected inflation increases (it's more costly to hold onto money)
people decrease their demand to hold wealth as money

The usefulness of the equation of exchange formula can be expanded by realizing that the changes in the growth rates follow a pattern over time. The pattern can be seen in the aggregate demand/aggregate supply model, but the pattern also can be seen if you think of the economy as sort of a train.
In this "money train," the locomotive is the money supply. Velocity acts as a shock absorber, and the cars in the train are real GDP and the price level, in terms of which car moves first:

When the growth rate of the money supply is changed, the first effect is that velocity's growth rate changes in the opposite direction (to absorb the shock) for a few months. Then, velocity springs back, pulling real output along (in the same direction as the change in M's growth rate) about six months to a year later. Prices move very little in this time. As real output catches up to the speed of M, eventually the price level is pulled along, 18 months to three years after the change in M. In the period that prices are catching up to the rest of the train, real output may stop moving or even appear to move backwards in comparison to the rest of the train.

While we are adding up numbers involving real and nominal values, remember that we have previously done this with interest rates:
nominal interest =inflation + real interest

Real interest rates fall as the money supply increases (and more loans are made) but will rise as the price level rises (since the demand for loans will rise with higher prices).

The order of changes when monetary policy is undertaken is something like (with some overlaps)...
M growth changes
velocity changes (opposite direction from M)
real interest rates change (opposite direction from M)
velocity starts to change back (even goes in the direction of M)
real GDP changes (same direction as M)
real wages change (opposite direction from M)
price level changes (same direction as M)
price level changes (same direction as M)
real interest rates change back
real wages change back
real GDP changes back
everything real changes back, only price level has changed long run.

All of this is assuming that no other changes occur as the economy settles back to a long run equilibrium.

Here is a diagram showing the effects of a one-time increase in the money supply over time: