Fractional Recursion – Theory
At the core of the banking system, is an interesting, and oft misunderstood recursive function. The next couple of essays will attempt to explain this recursive process, and analyze how it might in fact behave within the economy.
Banks are allowed to create loans from a fraction of their deposits, whilst the owners of the deposits retain a right to withdraw those funds. As the loans are re-deposited at other banks within the banking system, money is created “out of thin air”, as many like to put it with a certain degree of apoplexy. It’s quite true that money is created initially by this process, but at least in theory, the money creation is limited, or bounded.
Consider two banks, and an initial deposit of $1000 at Bank Alpha. For now, we will ignore the role of the central banks, and concentrate on the theoretical model, described in undergraduate textbooks such as Mankiw.
| Bank | Deposits | Loans | Reserve |
|---|---|---|---|
| Alpha | 1000 |
Bank Alpha is founded with an initial deposit of 1000 monetary units. If the fractional reserve requirement is 10%, then it is required to keep10% of all deposits in reserve, and can make loans totalling no more than 90% of deposits. Assume that Bank Alpha creates a loan of 900.
In making that loan, Bank Alpha effectively creates money, as soon as the recipient of the loan deposits that money back into the banking system, say at Bank Beta.
| Bank | Deposits | Loans | Reserve |
|---|---|---|---|
| Alpha | 1000 | 900 | 100 |
| Beta | 900 |
Understanding the loan/deposit duality that fractional reserve banking creates within the commercial bank’s part of the monetary system is very important, but the concentration on the process of money expansion in most textbooks can be misleading. Deposits cost banks money, interest may have to be paid, checking and ATM facilities have to be maintained, etc. Loans on the other hand, bring income. So a bank in bank Beta’s position, with deposit liability, and no assets (loans) to bring in income, isn’t going to be very happy.
We would expect each individual bank then, to attempt to maximise its lending, and as we will see later, this does appear to be what happens in practice. So Bank Beta in turn, lends the maximum that its reserve ratio permits, and more money is created as this loan is redeposited.
This sets up the recursive process, of lending and redepositing, that can be expressed as a geometric series:
where x is the original deposit. In general the series converges to 
.
So that for example F = 20 yields the limit 
[Credit: Mathias, Talk:Fractional Reserve Banking 2/11/2008]
This is however just the initial starting conditions for the banking system. Once the deposit and loan process has run for a few iterations, the money and loan supply expansion, as described in the theoretical model, should stabilize. If it is assumed that banks will tend to maximise their lending, in order to maximise their profitability, then the money and loan supplies, the sum of the deposits and loans at all the banks, would be expected to increase to the maximum allowed and stay there.
Once at the maximum allowed by the reserve requirement, and assuming no loan defaults, then the banks can only create new loans as old loans are repaid. Since loans are repaid from money from the deposit side, money is removed from thin air when that occurs. So in continuous, day to day operation, there should be no money or loan creation per se, assuming no additional money is introduced into the system, and assuming there are no loan defaults.
So why then is there a widespread belief across the Net, that the fractional reserve banking system is creating money?
Because it is. Any examination of the macro-economic statistics for the USA and other major industrial economies today shows that quite clearly. This is why the Wikipedia Fractional Reserve Banking page has some quite gratuitous graphs of several different currency’s money supplies, helpfully showing an exponential curve.
Or at least, something about the way fractional reserve banking system has been implemented is causing money creation, because the textbook description described above doesn’t do this. But the textbook description leaves out a lot of details, that are from a distributed systems perspective, extremely important.
For example, the textbook description shows a clear and presumably distinguishable difference between loans and money. In practice, this is not necessarily the case, some financial instruments are treated as money, but are in fact loans. Basel 2, the banking regulatory framework, allows some of those loan based instruments to be used for reserves. The textbook says very little about what happens when loans default, that will be examined next time, and nothing about the effect of sale of loans outside of the commercial banking system.
Critical commentary on the fractional reserve system is fixated upon money, on the deposits, on the pulling of money out of thin air, and the lack of a solid gold basis for that money. But at least in the opinion of this author, that’s not where the actual problems are. It’s the loan supply, loan regulation, and the masquerading of loans for money, that have caused systemic instability problems, quite possibly for several centuries.
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