San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 Derivation of Money Supply Multipliers

Definitions:

• C = currency in circulation
• D = demand deposits (funds in checking accounts)
• rD = required reserve ratio
• RR = required reserves = rDD
• R = actual reserves
• ER = excess reserves = R - RR
• M1 = money supply = C + D
• MB = monetary base = R + C
• m1 = M1 money multiplier = M1/MB

Derivation of m1

• MB = R + C = RR + ER + C = rDD + (ER/D)D + (C/D)D
• Therefore
MB = [rD + (ER/D)+ (C/D)]D
• But
M1 = D + C = [1 + (C/D)]D
• Therefore
m1 = M1/MB =[1+(C/D)]D/[rD+(ER/D)+ (C/D]D
• and hence

## The M2 Money Multiplier

The M2 money supply is defined as the M1 money supply

• plus time deposits
plus money market mutual fund shares
plus money market deposit accounts
plus overnight repurchase agreements
plus overnight Eurodollars.
To keep matters simple all of the above items will be grouped together as MMF. Thus

#### M2 = M1 + T + MMF.

Let rT be the required reserve ratio on time deposits. The required reserves at the Fed are then

#### RR = rDD + rTT.

Thus the monetary base and the M2 money supply are:

#### MB = rDD + rTT + ER + C = (rD + rT(T/D) + ER/D + C/D)D M2 = D + C + T + MMF = (1 + C/D + T/D + MMF/D)D

The M2 money multiplier m2 is then given by:

#### m2 = M2/MB m2 =     (1 + C/D + T/D + MMF/D)     (rD + rT(T/D) + ER/D + C/D)

For a handy calculator for computing the money multipliers and the money supplies see money.