The Money Multiplier

The money multiplier tells us the maximum amount of new demand-deposit money that can be created by a single initial dollar of excess reserves. This multiplier, m, is the inverse of the reserve requirement, R: m = 1/R. This note will demonstrate that fact.

Suppose some initial amount, d1, is deposited into the banking system. With a reserve requirement of R, this deposit creates initial excess reserves equal to E1 = (1 - R) x d1. Assuming all of this amount is lent out and redeposited within the system, these excess reserves become new money: D M1 = E1 = (1 - R) x d1. This second deposit creates its own excess reserves equal to E2 = (1 - R) x D M1 = (1 - R) x E1. Again, this is redeposited as new money, so that D M2 = (1 - R) x E1. Continuing on like this indefinitely, we see a pattern develop:

D M1 = E1

D M2 = E2 = (1 - R) x E1

D M3 = E3 = (1 - R) x E2 = (1 - R)2 x E1

D M4 = E4 = (1 - R) x E3 = (1 - R)3 x E1

and so on ad infinitum.

The total increase in new money (call this "D") can be found by adding up all the successive changes in new money, D = D M1 + D M2 + D M3 + .... Then substituting for D Mi,
D =E1 x [1 + (1 - R) + (1 - R)2 + (1 - R)3 + ...].

Suppose we multiply D by the term (1 - R) and subtract this from D. All terms on the right-hand side with the exception of the initial "1" would cancel out; the resulting difference is D - (1 - R) x D = E1. Now collect the "D" terms on the left to obtain D x [1 - (1 - R)] = D x R = E1. Finally, divide both sides by R to obtain the desired result: D = E1 x . That is, an initial amount of excess reserves equal to E1 creates new money equal to this amount multiplied by the inverse of the reserve requirement.