# Economics – The upper limit of the money supply

By | 29/05/2011

Today I learned a simple principle in Macro-Economics, that I was surprised I was not aware of before.

As you know, Factional-Reserve banking (FRB) is a system in which commercial banks create money with every loan they give, and people destroy money when they return loans. Commercial banks are allowed to lend up to a reserve ratio dictated by the government or other authorities, depends on which country.

FRB is known to create a flexibility in the amount of money in the market, an amount that is derived from the money base (MB).

For instance, let's say person A has 100$bill. The bank reserve rate is 10%. Person A then goes to the bank and deposit the money. Up until now, money base is 100$, and M2 is 100$. Now the bank lends the money to person B up to the maximum amount allowed according to the reserve ratio of 10% – 90$.

Now we have money base 100$but M2 is 190$. Money supply has increased.

Person B deposits his 90$in the bank and this bank can now also lend it to someone else, say to person C. Again, it can lend only up to the reserve ratio – 10% or 81$.

Now we have money base 100$but M2 is 271$.

This can go on and on forever.

What I learned is that there is a cap on the maximum amount of money supply. It looks like it can grow to infinity, but actually it is a very fast converging series.

I will do a little math here, to brush off some rust from the time I learned Calculus:

First, we have a 100$bill. In the second stage we have another increment to the money supply which is the reserve ratio times the previous amount (first 100$ bill deposit).

In the third stage we have another increment to the money supply which is the reserve ratio times the previous amount, and so on.

We get:

$D_{1}=100;D_{2}=100(1-R);D_{3}=100(1-R)^2;...$

Where:

$D_{n}$ is the deposit number and $R$ is the required reserve ratio from banks.

Supply of money will be:

$S=D_{1}+D_{2}+D_{3}...$

or:

$S=\sum_{n=0}^\infty D_{n}= \sum_{n=0}^\infty 100(1-R)^n=100\sum_{n=0}^\infty (1-R)^n$

As you see above, I claim that there is a finite limit to this sum, a single number, in other words that this series converge. Lets see:

$let: (1-R)=X$

we get:

$S= 100 \sum_{n=0}^\infty X^n$

which is a known Taylor series for function $f(x)=\frac{1}{1-x}$, so using Taylor theorem:

$S= \frac{1}{1-X}=\frac{1}{1-(1-R)}=\underline{\underline{\frac{1}{R}}}$

This is a mathematical proof that the maximum supply of money is:

$S_{max} = \frac{MB}{R}$

It is interesting because using this formula regulators can control how much money there is in the market by changing the reserve requirement.

Now you know, for instance, why does the Chinese government raise the reserve requirements from banks – it immediately "deletes" money from the market, assuming MB (money base) is constant.