Fractional reserve banking demystified: how money is multiplied today

This article is part of the Money Series
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In the last article, we discovered what Bank Reserves are: in essence, they are the deposits, held at the Central Bank, that commercial banks use to make payments with each other.

Understanding the Central Reserves is fundamental if we want to understand one of the ways money is increased today inside the commercial bank system.

We have already seen how a commercial bank can expand the money in circulation by making loans and mortgages. We still lack an understanding of how this works on a systemic basis, ie taking the bank circuit as a whole.

The classic theory involves the fractional reserve banking mechanism.
The explanation sounds like this.

Mr. Adam White goes to bank A and deposit 10000€ in his bank account. If the law imposes that bank A keeps, let's say, 10% of the deposited sum in the form of bank reserves, then bank A will put 1000 € (that is 10% of the original 10000 €) in its account by the Central Bank.
There are 9000 € left for the bank to give out for a loan to another guy, let's call him Mr. Black.
So, Mr Black takes 9000 euros and pays Mr Smith, who has an account in bank B. Mr Smith deposits 9000 euros. Then, Bank B puts 900 € (10% of 9000 euros) in bank reserves, and is free to give the remaining 8100 away in the forms of loans. And the process starts over again.

So, in brief, the original 10000 have become in the overall customers' deposits, after many transactions:

10000 + 9000 + 8100 + 7290 + ... = 10000 * (1 + 0.9 + 0.9*0.9 + 0.9*09*0.9 + ..) = 10000*1/(1-0.9) = 100000

So, according to this theory, after many transactions, we have a geometric series converging to ten times the original capital, ie 10K have become 100K.
And the reserves?
Well, the principle is the same: the first bank puts 1K in reserves, the second bank puts 10% of 9K in reserves, so 100 euros, and so on.
In other terms, the total amount of reserves will be the reserve ratio times the total credit.
In this case: 10%*100000 = 100000


So, bank reserves have increased 10 times.
Total credit created: 100000 in commercial bank accounts plus 10000 bank reserves

What happens if the reserve ratio, set by the Central Bank authorities, is increased to 15%? in this case, only 85% of the deposit is available to the bank for extra loans, since 15% must be kept in the form of bank reserves.
The original 10K now becomes:

10000 + 8500 + 7225 + 6141.25 + .. = 10000 * (1 + 0.85 + 0.85*0.85 + 0.85*085*0.85 + ..) = 10000 /(1-0.85) = 66667
So, credit has increased 6.67 times.
And, for the bank reserves:

15%*66667 = 10000

the total bank reserves have not changed!
Let's make a simple table: we start with a capital of 10000 euros.

Original deposit	Reserve Ratio	Total credit in bank accounts	Total Bank reserves
10000	20%	50000	10000
10000	15%	66667	10000
10000	10%	100000	10000
10000	5%	200000	10000
10000	1%	1000000	10000

Note: I ran some mathematics, here, simple geometric series expansion, and to me the formulas are the following (feel free to correct me if I am wrong):
C is initial capital
r is reserve ratio
Total credit becomes: C*1/(1-(1-r)) = C/r
Total reserve becomes: rC*1/(1-(1-r)) = C

That's the power of mathematics.
With the same initial deposit, even if the reserve ratio decreases, the total bank reserves are the same, equal to the first deposit, while credit can explode (up to 1Million from an initial deposit of 10K).
I remind you that it's up to the Central Bank to decide the reserve ratio, that is the ratio between the loans and mortgages a commercial bank can create and the quantity of reserves the commercial bank needs to keep for customers to do bank transfers to other banks and to withdraw money from the ATMs.


Some observations:
1. This model does not explain where Mr White, the original depositor, our "Adam", the first man,  got his money from. Somebody gave him the money.
2. This model assumes that the Central Bank fixes the reserves and so the Commercial Banks fixes the maximum amount of money they can give out to people and firms in the form of mortgages and loans. In other terms, the Central Banks can decide the amount of money in the Country according to this model. The Commercial Banks can only obey. You see? you fix the reserves, column in the right-end, and by modifying the reserve ratio, you have control over the credit.
3. Due to the control of the Central Bank, each moment the amount of credit in a Country is known and can be scaled up or down, according to specific needs.


Now, we have seen that Commercial banks can increase the quantity of money in circulation by making loans basind on internal decisions (ie, is the customer able to repay the principal plus the interest?). So, the first point is easily answered if we think that the banks create "endogenously", that is from the inside, the money. They give money to our Adam, the first man, by creating for him 10000 euros out of thin air. Of this, the bank puts 10% in the form of reserves, by the Central Bank.
Then, the show can get started.
But, and this is fundamental, every bank of the chain can EXTEND credit by creating loans and mortgages. This is not considered in the fractional reserve model. So, yes, banks keep reserves in the Central Bank registers, but it is UP to the Commercial banks to decide how much money they will put there: they will put there 5%, 10% or 15% of the money they create!

So, it is not the Central Bank who can determine the amount of credit in the country. The Central Bank can fix the ratio of the reserves, but the quantity is eventually decided by the commercial banks.

That's why the fractional reserve model is inherently incorrect: it does not take account that Commercial Banks can extend credit on their own, and assumes that Central Bank has total control over the credit in a country.

Of course, if reserve ratio decreases, banks have even more stimulus to lend money, but the real driver of credit creation by commercial bank is the confidence that the depositor will be able to repay his debt.