In Ch 15, we developed a simple model of multiple deposit creation (and the SDM), where the FRS can influence the amount of checking deposits (D) and the Money Supply (M = C + D) by changing the reserve requirement (r_D), and can influence Deposits (D) and Money Supply (M) by changing the level of reserves (R) through an OMO.

However, the SDM model assumes unrealistically that: 1) the public holds no cash (C) and 2) banks hold no excess reserves (ER).
 

THE MONEY SUPPLY MODEL AND THE MONEY MULTIPLIER (m):

We now develop a more realistic money supply model that allows for the fact that the public does hold cash balances.  We link the one variable that the FRS can control directly (MB) to the money supply (M for M1), using the money multiplier (m):

m  =  M  /  MB

or equivalently:

M =  m  x  MB

dM =  m  x  dMB

change in M1(M)  =  money multiplier (m) x  the change in the monetary base (MB)

NOTE: d = delta = CHANGE, dM1 means the "Change in MI," dMB means the "Change in MB," etc.

The FRS can control the MB directly through an OMO, but is really more interested in M1 (M).  Using the money multiplier, they can determine the appropriate OMO to get the desired change in M1 (M).  Since the money multiplier (m) is always greater than 1, there is an expansion process, which is why the MB is called "high-powered money."  It expands through the deposit expansion process, after it enters the economy and the banking system through an OMO.
 

MS MODEL:

M = C + D

Money Supply (M1 = M) is equal to currency or cash (C) plus checking deposits (D)
 

MB = C + R

Monetary Base (MB) is equal to currency (C) in circulation (outside banks) plus bank reserves (R), which include a) bank deposits at FRS and b) vault cash.
 

R = RR + ER

Reserves (R) in banks is equal to required reserves (RR) plus excess reserves (ER).  We will assume for now that ER = 0, so that R = RR.
 

R = r_D x D

Therefore, bank reserves (R) are equal to the reserve requirement ratio (r_D) times checking deposits (D).
 

D = R + L/S (loans and securities)

Checking deposits (D) are equal to reserves (R) plus Loans and Securities, so a bank's balance sheet looks like this:
 
 
 
 
 
 

We assume that people hold money (M) in the form of cash (C) and checking (D), in some ratio: (C / D), which is called the currency/deposit ratio.  For example, using the money supply data on page 58, people in 1999 held $460B in cash (C) and $623B in checking (D), so the C/D ratio was $460/$623 = .7384, meaning that on average people (households, businesses, universities, etc.) held about $74 in cash (C) for every $100 held in checking (D).

Also, if we know the C/D ratio and the amount of checking deposits (D), we can calculate the predicted amount of currency (C) held by the public with the formula:

C  =  (C / D)  x  D

Point: depositors' decision to hold cash versus checking balances influences the C/D ratio, which affects the money multiplier.  FRS has to account for the publics' money demand.  The greater the MD, the lower the money multiplier (m), see below.  Also, banks do hold some excess reserves, which also lowers the money multiplier (m).
 

DERIVATION OF THE NEW DEPOSIT MULTIPLIER (DM):

MB = R + C

MB = (r_D x D) + (C/D x D) factor out D

MB = ( r_D + C/D) x D

D = MB  /  ( r_D + C/D )

D  =   MB    x              1
                             (r_D + C/D)

dD = dMB  x  DM (deposit multiplier)

The Deposit Multiplier (DM)   =           1
                                                        (r_D + C/D)
 

This equation (dD = dMB  x  DM) links changes in monetary policy, i.e. changes in the MB (dMB = OMO), to changes in bank checking deposits (dD), through the deposit multiplier (DM).  For example, suppose that the reserve requirement (r_D) is .05 and the C/D is .45, the DM = 1 / (.05 + 45)  =  1  / .5  =  2x.  Therefore, for every $1 OMO (dMB = $1), the change in checking deposits (dD) will be equal to dD = $1 x 2 = $2.

(Note: Remember that under the assumption that the public holds no additional currency, the simple deposit multiplier (SDM) was equal to 1 / r_D.  The deposit multiplier (DM) adds the currency/deposit ratio (C/D) to the denominator of the equation.)
 

DERIVATION OF THE MONEY MULTIPLIER (m):

M  =  D + C  (M1 (M) is equal to currency (C) + checking deposits (D))

M  =  D  +  (C/D) *  D

M  =  D ( 1 + C/D)

M   =  ( MB  / r_D x C/D)  x  (1 + C/D)

M    =    MB    x       (1 + C/D)
                               (r_D + C/D)

dM  =   dMB   x        (1 + C/D)
                               (r_D + C/D)

The Money Multiplier (m) =      (1 + C/D)
                                                  (r_D + C/D)

The equation (dM = dMB x  m) links changes in monetary policy, i.e. changes in the MB (dMB = OMO), to changes in the money supply (M), through the money multiplier (m).  For example, suppose that the reserve requirement (r_D) is .05 and the C/D is .45, the money multiplier (m) would be:
m = (1 + .45) / (.05 + .45)    =     ( 1.45 / .5 )  =    2.9x.   Therefore, for every $1 OMO (dMB), the change in the money supply (M) would be:

dM = dMB x m =  $1 x 2.9x = $2.9.
 

SUMMARY OF MULTIPLIERS:

1. Simple Deposit Multiplier (SDM):   1 / r_D
dD = dR x SDM, assuming no additional cash balances (dC = 0), and no excess reserves (ER = 0).  For a change in RESERVES, what is the potential MAXIMUM amount that DEPOSITS can increase?  Answer: SDM times dR.

2. Deposit Multiplier (DM):   1  / (r_D + C/D)
dD = dMB x DM, assuming additional cash balances are held, and no excess reserves.  For an open market operation (OMO = dMB) which affects the monetary base (MB) one-to-one, how much do CHECKING DEPOSITS (D) increase?  Answer: DM times the OMO.

3. Money Multiplier (m):  (1 + C/D)  /  (r_D + C/D)
dM = dMB x m.   For an open market operation, which affects the monetary base (MB) one-to-one, how much does M1 (M) increase?  Answer: m times the OMO.

Once we know the reserve requirement (r_D) and the currency/deposit ratio (C/D), we can calculate all the multipliers.

FUNNEL DIAGRAM:
 
 
 
 
 
 

SAMPLE PROBLEM

Assume:
r_D = .05
C = $150B
D = $750B

What is the effect of an expansionary open market operation of $10B on currency (C), reserves (R), monetary base (MB), M, deposits (D) and L/S?  What is the new amount of M and what was the percentage change in the money supply?

C/D =
 

DM =
 

m =
 

SDM =
 

dD = (change in D) =
 

dM = (change in M) =
 

dC = (change in C) =
 

or $48 = C + $40
 

R =
 

MB =
 

L/S =
 

New M1 =
 

% M1 =
 
 

CONCLUSIONS:

1. The effect of an OMO on the MB is always the same ($1 OMO = $1 dMB)

2. The effect of an OMO on reserves (R) depends on whether the seller of the bond deposits the FRS check in a bank or cashes the check for currency.

3. The effect of OMO on MB is certain, the effect on reserves (R) is uncertain.

4. A $1 increase in the MB that goes into currency (C) is not multiplied, but M goes up by $1.  An increase in the MB that goes into reserves (R) is multiplied by the SDM and Deposits (D) and Money (M) increase by that amount. There is multiple expansion of checking deposits (D), but there is no expansion for currency (1 to 1).

5. The money multiplier (m) and MS (M) inversely (negatively) related to the required reserve ratio.

6. The money multiplier (m) and the MS (M) are negatively related to the C/D ratio.

7. The money multiplier (m) and the MS (M1) are negatively related to the excess reserve ratio - ER/D.
 

SUMMARY:

VARIABLE   CHANGE          REPSONSE IN M1 (M)

 MB               Increase                 Increase

 r_D                Increase                 Decrease

 C/D              Increase                 Decrease

 ER               Increase                 Decrease