The quantity theory of money says quite simply that the amount of money in the economy is the amount of money spent in the economy. (When you put it that way, it sounds kind of obvious.)
M is the supply of money.
V is the velocity at which money is spent; if the average dollar circulates three times a year, V = 3/year.
P is the price level, an abstract aggregate of all prices in the economy.
Q is the quantity of goods sold, the real GDP.
MV = PQ
The money multiplier is a simple way of expressing the way that fractional-reserve banking constrains the money supply. It says that the total money supply is equal to the monetary base divided by the reserve requirement.
M1 = M0/R
This comes from the fact that as banks lend, they can lend up to 1 - R of what they have. The first bank has M0, and lends M0*(1-R), which ends up in the second bank, which lends out M0*(1-R)^2, and so on. As you add up this geometric sum, it converges to M0/R. This is actually an upper bound on the money supply---if banks don't lend out everything they can, the real money supply can be less than this.
Thus, if the Fed drops the reserve requirement from 10% to 5%, the reserve requirement has been cut in half, which means that the money supply can potentially double.
Going back to the quantity theory of money:
MV = PQ
We just made M bigger (potentially twice as big, in fact). If people's spending habits don't change, V will remain the same. If prices take awhile to adjust, in the short run P will not change much either. That means that the increase in M must cause an increase in Q in direct proportion.
In this example, if we assume that V and P really do remain fixed and banks immediately lend everything, we can double GDP overnight. The reason this doesn't happen in real life is that banks don't lend everything out immediately, and very large changes in money supply like that are transferred over to changes in V and P, particularly once Q approaches its maximum potential.
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