The Complex Money Multiplier

The simple money multiplier is 1/R, where R is the ratio of required reserves to deposits. In a more complex world, the money multiplier must allow for the possibility that individuals retain some proportion of their money in the form of cash rather than deposits. In addition, it must allow for the possibility that banks may wish to retain some reserves in excess of the required amount.

We can begin with the second possibility. Suppose that banks hold some additional proportion, say "x" for "extra reserves," in excess of the required proportion, R. For purposes of the money multiplier, we can treat this situation as if banks were required to retain both R and x. Let R* equal the sum of these two proportions: R* = R + x. Then the money multiplier is simply 1/R*, and the effect of these extra reserves is to diminish the size of the multiplier: 1/R* < 1/R. For example, suppose the reserve requirement is .20 (20%), but banks keep an additional .05 (5%) of every deposit as extra reserves. The simple multiplier is 1/.20 = 5, whereas the more complex multiplier is 1/(.20 + .05) = 1/.25 = 4.

"Cash drain," the tendency of households to hold part of any additional money as cash, is more problematic. In this case, we must compute the change in the money supply resulting from an initial change in excess reserves as the sum of new deposits into the system plus any new additions to cash balances outside of banks.

Following convention, we assume that households wish to withhold a constant proportion, c, of new deposits as cash: D C = cD D, where C is cash holdings outside of banks, and D is deposits. This assumption implies that households also hold a constant proportion of any new money as cash. Since new money is the sum of cash and deposits, D C + D D = D M, we can substitute D D = D C/c into this expression and solve for D C to obtain D C = D M. In like manner, we can substitute for D C = cD D into the expression to obtain D D = D M. For example, suppose households withhold 25% of deposits as cash balances, so that c = .25. If households were to receive $100 of new money (D M = $100), they would then retain D C = D M = $20 as cash and deposit the remainder: D D =D M = $80.

With this as background, we can now proceed to find the complex money multiplier. Suppose that new money equal to D M is created and distributed to households. (You might imagine that the Fed drops D M of newly printed money out of a helicopter, that is then grabbed up by households.) As just explained, D M will be held as cash balances, and D M will be deposited in the bank. Of this deposit, the bank will hold a constant proportion, R, in reserve and will lend out the remainder: D L = (1 – R*)D D where D L is new lending. To summarize this first round, we have:

D C₁ = D M

D D₁ = D M

D R₁ = RD D₁

Of course, this new lending becomes newly acquired money to another household. Of this amount, D L₁ is held as cash, and D L₁ is redeposited. Accordingly, this second round produces:

D C₂ = D L₁ = D M

D D₂ = D L₁ = D M

D R₂ = RD D₂

Again, this new lending becomes newly acquired money to yet another household, that divides it between cash and deposits:

D C₃ = D L₂ = D M

D D₃ = D L₂ = D M

D R₃ = RD D₃

As with the simple multiplier, this process continues indefinitely. From these three rounds, the pattern becomes clear, and the total increases in cash, deposits, and money take the form:

D C = D M x (1 + + + + - )

D D = D M x (1 + + + + - )

D C + D D = D M x (1 + + + + - ), since + = 1

From the discussion of the simple multiplier, we know that any sum of the form 1 + b + b² + b³ + - = , provided 0 < b < 1. In our example, b = < 1, so that our formula for D C + D D collapses to D C + D D = D M = D M

That is, the complex money multiplier is equal to . (Note that if there is no cash drain, c = 0 and the money multiplier is again just 1/R. To allow for the possibility that banks hold extra reserves, simply substitute R* in the formula.) Comparing this to the simple multiplier, we note that < , that is, cash drain reduces the size of the money multiplier.

Interestingly, the complex money multiplier could have been derived more easily by assuming that households hold a constant proportion of all deposits as cash, not simply new deposits. In that case, let c = C/D be that proportion. At this point, we also need a new concept-define the "monetary base" B as cash held outside banks plus total bank reserves (RR): B = C + RR. (This is sometimes called "high-powered money.") We also know that the total amount of money consists of deposits plus cash: M = D + C.

If we now divide M by B, we obtain = . Now divide each term on the right by total deposits, D, to obtain: = = . (We have made use of the fact that RR/D is the required reserve ratio, R.) Multiply by the base, and we get our final result: M = B.

As before, we see that any change in the monetary base (newly created money) will be multiplied by to obtain the final increase in the money supply.