San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The 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
m1=[1+(C/D)]D/[rD+(ER/D)+ (C/D]D
and hence

• m1=[1+(C/D)]/[rD+(ER/D)+ (C/D]

## The M2 Money Multiplier

The M2 money supply is defined as the M1 money supply plus time deposits T 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 given by:

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

(To be continued.)