Ranjit Dighe
Lecture notes to accompany Mishkin's Chapter 16 ("Determinants of the Money Supply")

* In these notes:
I.   The real-world money multiplier
II.  The real-world money multiplier and the size of the money supply


The simple money multiplier, or simple deposit multiplier, is very simple indeed.  You just calculate 1/RRR; since RRR is now 10%, then the simple money multiplier is 1/.10 = 10.  But, as noted above, it is not very realistic, since banks do hold some excess reserves (though not much) and, especially, because a sizeable fraction of money loaned out does not get redeposited into bank accounts.  (In fact, about two-thirds of the U.S. currency that's officially "in circulation" is not even held in this country; many people in other countries, especially third-world countries, hold dollars as a hedge against inflation in their own currency, since the dollar right now is a much better store of value than, say, the Russian ruble.)
-- To the extent that banks hold onto some of their excess reserves, those reserves don't get loaned out and don't get redeposited into the banking system; thus excess reserves do not increase the money supply at all.  So an excess reserves ratio (the ratio of excess reserves to checking deposits, or ER/D) above zero will cause the money multiplier to be less than 1/RRR.
-- When cash loaned out continues to be held as cash, instead of redeposited into bank accounts, those holdings of currency are still counted as part of the money supply, but there is no multiple expansion of deposits (and hence of the money supply) associated with currency. Instead, every dollar that the public holds as currency is a dollar that doesn't get redeposited in the banks and therefore is not available to be loaned out and to contribute to further expansion of the money supply.

If we dig up information on banks' actual holdings of excess reserves and the fraction of loan amounts that people hold as currency (instead of redepositing it into bank accounts), we can get a much more realistic estimate of the money multiplier. We will call that realistic estimate the real-world money multiplier, or just the money multiplier.

Why we care about the real-world money multiplier: Because the Fed seeks to control the money supply in order to influence interest rates, and yet the Fed does not control the money supply directly.  Instead, it controls the monetary base, which is the sum of bank reserves and currency in circulation.  Notationally:
                      monetary base = reserves + currency
                               MB          =      R       +     C

The money supply (Ms) is equal to the monetary base (MB) times the money multiplier (m):

Ms = m * MB

So the Fed needs to know what the money multiplier is in order to be able to control the money supply effectively.

We can derive the real-world money multiplier m with a little mathematical manipulation. Rearranging the above equation, we get
m = ----- ,

which we can calculate if we are given the Ms and the MB. For the Ms, we normally use M1, which is roughly defined as Currency in circulation (C) plus checking Deposits (D). Notationally, then, M1 = C + D.

Breaking the MB down into required reserves and excess reserves (ER) and recalling that required reserves are RRR * checking deposits, we get:

MB = RRR*D + ER + C

With just a little mathematical manipulation (done in class and in Mishkin, p. 415), we can express the real-world money multiplier (m) as a function of the required reserve ratio (RRR), the excess-reserves ratio (ER/D), and the currency-deposit ratio (C/D):

                    1 + [C/D]
m = ------------------------------
        RRR + [ER/D] + [C/D]

To find m, all we need are the values of the C/D and ER/D ratios. (We already know that RRR = .10, or 10%.) Alternatively, we could compute m with values for C, D, and ER. (The ratios are somewhat more stable than the absolute numbers of C, D, and ER, however.) Some not-completely-off-base numbers for those variables are:

C = currency held by the public = $400 billion
D = checking deposits = $800 billion
ER = excess reserves = $0.8 billion
M1 = narrowest measure of money supply = C + D = $1200 billion

Plugging those numbers into our formula for m, the money multiplier, m is:

                 1 + [400/800]                           1 + 0.5                  1.5
m = ------------------------------------- = ----------------------- = --------- = 2.496
        0.10 + [0.8/800] + [400/800]     0.1 + 0.001 + 0.5       0.601

Voila! The real-world money multiplier is roughly 2.5. [Right now, using the actual numbers for C, D, and ER in the year 2002, it was actually about 1.7, but this is not too far off.]

If the money multiplier changes, the Fed needs to make offsetting changes in the monetary base so as to keep the money supply stable.

Historical note on the Great Depression:
C/D and ER/D both tend to increase during major recessions and depressions, because the public may view banks as unsafe and banks are more likely to be pessimistic about borrowers' creditworthiness. During the Great Contraction of 1929-33, both of those ratios skyrocketed, and the money supply fell by about 25%, the most it has ever fallen. The drop was because of the increases in those ratios, which dramatically shrunk the money multiplier. A related reason was the huge number of bank failures, caused in large part by the many runs on the bank by depositors, who were trying to covert their deposits into cash (C/D was rising). Those bank failures directly and severely reduced the level of bank deposits, the key component of M1.


The money multiplier is important because the Fed needs to keep the money supply in balance with money demand -- which fluctuates a lot -- in order to keep interest rates stable, and the Fed can't control the money supply effectively unless it knows what the money multiplier (m) is.  The Fed's control of the money supply (M1) is indirect, because the Fed directly controls only the monetary base (MB, = reserves + currency in circulation), whereas the money supply (M1) is equal to the monetary base (MB) times the money multiplier (m):

M1 = m * MB

The Fed can influence the size of m by changing the RRR, but it can't control C/D or ER/D, which depend on the behavior of banks and their borrowers and depositors.  If and when m changes, the Fed needs to make offsetting changes in the monetary base so as to keep the money supply stable.

When we looked at the money market earlier in this course, in the way of an explanation of interest-rate movements, we simply assumed that the money supply was fixed by the Fed and did not depend at all on the interest rate. That assumption is not realistic. Let's look at all the players who jointly determine the size of the money supply (remember: Ms = MB * m):
Player  Variable Ms response to an increase in that variable Reason
The Fed RRR decreases less multiple-deposit expansion (m shrinks)
MB (through R) increases more reserves mean more loans (and redeposits)
discount rate decreases fewer bank reserves --> MBdecreases
Banks supply of loan monies 
(ER/D decreases if banks are looking to make more loans)
increases m increases
Depositors C/D decreases less multiple-deposit expansion (m shrinks)
Depositors and banks expected deposit outflows (ER/D increases if banks are expecting a lot of these) decreases m shrinks
Borrowers demand for loans (ER/D decreases if demand for loans increases) increases banks can now make more (safe) loans at the same or a higher interest rate, so they loan out more of their excess reserves (ER/D falls) ==> money multiplier becomes larger. If those higher interest rates allow banks to attract new deposits away from competitors like MMF's, then MB becomes larger, too.

Unlike the simple deposit multiplier (1/RRR), which is very stable because the Fed almost never changes the RRR, the real-world money multiplier m can vary quite a bit from year to year. (In the past year, for example, m fell from about 1.9 to about 1.7.) m is sensitive to changes in the business cycle and to changes in interest rates.

m is procyclical (m rises in economic upswings; m falls in recessions).
-- Why:
* the excess reserves ratio (ER/D) tends to rise during recessions, as banks tend to view loans as risky during recessions. In an economic slump, businesses and households may be more likely to default on their debts. So banks may hold a lot of excess reserves because they think holding onto their reserves is a lot safer than loaning them out. An example was the "credit crunch" during the 1990-92 recession.
* the currency-deposit ratio (C/D) tends to rise during recessions, as people may start to view bank deposits as risky (if their banks are in danger of failing. This was a bigger problem back before federal deposit insurance was established in 1933; it is not really a big deal today).
---- Since ER/D is in the denominator of that ratio formula for m, an increase in ER/D will reduce m.
---- Although C/D appears as a term in both the numerator and denominator of that formula for m, it's added to a much larger number (+1) in the numerator than in the denominator (where it's added to the RRR and the ER/D, which together add to about 0.11 or less), so an increase in C/D enlarges the denominator more than it enlarges the numerator, causing their ratio (m) to decrease.

m is also affected by changes in interest rates.
-- Why:
* higher interest rates will induce banks to loan out more of their excess reserves, causing ER/D to decrease, which causes m to become larger.
* higher interest rates will induce people to carry less cash (C) and keep more in interest-earning checking account deposits (D), so C/D will decrease, too, causing m to become larger.
---- Since both of those changes make m larger, then they'll also cause the money supply (= m * MB) to become larger.  So, when we draw the money market as a supply and demand diagram (where Qm is the quantity of money and the nominal interest rate, i, is the price of money), the supply-of-money curve should be upward-sloping, not vertical.