Michael Foster wrote:
> Stephen A. Lawrence wrote:
>
>> Your earlier claim, quoted below, was that an individual bank could
>> lend out 10 times the amount of that bank's total deposits. The
>> above quote from Wiki does not support that claim.
>
> OK then, banks acting in concert lend out 10 times as much money as
> is deposited in individual banks. What difference does it make?
None; it's false either way.
> The
> fact is that given the current reserve requirement of 10%, the effect
> is that for every dollar deposited in a bank, ten dollars can and
> usually will be lent out by whatever convoluted mechanism. Quoting
> again from that section of the Wikipedia article:
>
> "As the process continues, the banking SYSTEM (emphasis mine) can
> expand the change in excess reserves of $90 into a maximum of $1,000
> of money ($100+$90+81+$72.90+...=$1,000), e.g.$100/0.10=$1,000."
>
> I don't see any room for equivocation there. If there is, explain it
> to me.
OK, here's an explanation.
Let's suppose banks lend out some fraction, call it "a", of their
deposits. If the reserve requirement is 10% then a=0.9 .
As an aside, "a" actually represents the *effective* rate at which
deposits are lent out. In principle, most (non-demand) deposits in the
United States have a 0% reserve requirement, and in a lot of Europe the
reserve requirement is enforced "voluntarily" which means the mandated
reserve requirement is also 0%. However, it turns out bankers are not
total idiots and in fact they don't lend out everything "down to the
pictures on the walls". From what I've read, in Europe the *actual*
reserve level runs around 3 or 4 percent, which would make "a" about
0.96 or 0.97. I don't know what the 'effective' reservation rate is in
the United States. The actual level doesn't affect the following
derivation, except for the basic assumption that 0 < a < 1. (If a=1
then things go badly awry, not just with the derivation but with the
real-world money supply.)
Back to the derivation.
We will assume there is some hypothetical primordial deposit (either due
to issue of goldbacks, or gold coins, or cowry shells, or "high powered"
paper money) which we will call D. Then the initial bank deposits total:
D = initial state
However, the banks are assumed to lend out the fraction "a" of their
deposits. Furthermore, we make an additional assumption: The loan
recipients place 100% of the borrowed money back into a bank. In
practice the fraction will be slightly less than 100%, and the time it
takes the money to get back into another bank will be nonzero as well,
which will slow down the rate at which things happen in the real world;
in the model we're assuming everything happens at once.
So, an amount a*D will be lent out, and *RE-DEPOSITED*. Consequently
total deposits at step 2 will be
D + a*D = deposits at step 2
Note that we're concerned with DEPOSITS here, not the amount out on
loan! Note also that we already have more money *on* *deposit* than we
started with, because multiple people think they "own" the same money.
Again, a fraction "a" of the "stage 2" deposits will be lent out, and in
turn redeposited, which will inflate deposits yet further, to:
D + a*D + a*a*D = deposits at step 3
It's easy to see that the final result of this, if allowed to continue
to its logical conclusion, will be
total deposits = T = D + a*D + a^2 * D + a^3 * D + ....
or
(1) T = D * sum{n = 0 to infinity} a^n
We can evaluate this by using a trick. We multiply the sum by "a" to obtain
a * sum{n = 0 to infinity} a^n
= sum{n = 0 to infinity} a^(n+1)
= sum{n = 1 to infinity} a^n
You may have to think for a moment to see why this is true, but all
we've done is multiply the sum term-by-term by "a". Next we rewrite the
right hand side as sum{0->infty} minus the first term:
a * sum{n = 0->infty} a^n = sum{n = 0->infty} a^n - 1
We then collect terms and divide through by (1-a), which I'll show in
two steps, to get the final value for the sum:
1 = (1 - a) * sum{n = 0->infty} a^n
(2) sum{n = 0->infty} a^n = 1 / (1 - a)
Now plugging equation (2) back into (1) we obtain
(3) T = D / (1 - a)
and the factor "1/(1-a)" is the MONEY MULTIPLIER which I have mentioned
a number of times, and which Michael Foster was talking about in his post.
If the banks hold 10% of their deposits in reserve, then a=0.9 and the
multiplier is exactly 10. That means that, if we are given enough time
for all the money to circulate as far as it's going to go, then for
every dollar of "primary" money injected into the system (where primary
money may be gold coins, or cowry shells, or Fed-produced high powered
money, or whatever you like) we will end up with $10 in bank deposits.
If the reserve requirement is smaller, the multiplier will be larger.
In fact, for time deposits, CDs, money market funds, and other
non-demand accounts, the requirement in the United States is currently
0%. It's a safe bet that banks are maintaining *some* reserve even when
they are not required to (else they can't deal with withdrawals at all).
Consequently, the value of "a" will still be somewhat less than 1.
Furthermore, it takes finite time for deposits to go "around the loop"
each time; this relates to the "velocity of money". If this were not
true, if the real world behaved exactly like the model world, then a
reserve requirement of 0% (or simple absence of any law pertaining to
required reserves, which is presumably the situation the world was in
300 years ago) would result in the money supply going instantly to
infinity. That has never happened, even approximately, save with
enormous help by the government issuing huge amounts of paper money
(e.g., Zimbabwe, or the Weimar republic).
Note well: This multiplier affects the total *DEPOSITS*. We have
ignored the total amount of money on loan up to this point. Let's look
at it now. The total on loan is:
L = a*D + a^2 * D + a^3 * D + ....
Factoring out the "D" and one "a", we see
L = D * a * sum{n = 0->infty} a^n
and once again using equation (2), we see:
(4) L = D * a / (1 - a)
Comparing (4) with (3) we see that the amount on loan, L, related to the
total amount deposited, T, is
(5) L = a * T
Thus, the amount of money on loan will always be less than the total of
all deposits, as long as a<1.
As I've been saying all along, banks, individually or in the aggregate,
lend out no more than the total amount they have on deposit.
Finally, what Michael seems to have had in mind is the amount of money
in circulation versus the amount of the hypothetical "original deposit".
That is, of course, just the total of all deposits divided by the
multiplier, or
(6) D = T * (1 - a)
Note that I called it a "hypothetical" original deposit. This is
because money is *fungible* -- there is absolutely no way to tell which
money is "real" and which is the "imaginary" stuff which resulted from
the multiplier effect. In the parable I posted earlier, it happens to
be easy to trace the path of the gold coins around the loop, but none
the less there is no way to say, at the end of the story, "Omar's
deposit of $171 was not valid!". It was valid -- Omar deposited
perfectly good gold coins.
I've said banks are not magic. Now let's consider a case which doesn't
involve a bank. Let's suppose you take out a mortgage on your house,
for, say, $10,000. Someone, somewhere, coughed up that money, and gave
it to you in exchange for an IOU. They count the IOU (the signed
mortgage paper) among their assets, and the fact that they have that
asset is going to affect their decision to buy a boat, take a vacation,
or just spend more on Christmas presents -- it's a real asset. But
suppose you take the money from the mortgage and buy a $10,000 corporate
bond. The company you bought it from is going to give your $10,000 to
their employees as wages. Now, those employees *think* they have the
$10,000 and will act accordingly. But you *think* you have it, too, in
the form of a bond which you can sell if you need the capital. And the
note holder for your mortgage *thinks* they have the $10,000, too. So,
you've magically transformed $10,000 into $30,000 ! And you're not even
a bank.
To stop this from happening you would need to ban the lending of money.
Finally, note that *noplace* in the derivation did we assume banks
could, themselves, borrow money, nor did we make any assumptions about
the behavior of the Fed, nor about the existence of "paper money". This
discussion would have been every bit as valid in the year 1200 as it is
today.
