Economics / Finance - Banks - Central & Commercial, and Fractional Reserve banking

mods. i also posted this in the general forum but there is very little action there, in comparison to here, and this was the next closest thing. I hope you understand why i have posted it here an there. Just trying to get as quick and detailed an answer as possible. thanks.

Hi. To begin with i will quickly explain my position. I know relatively little about economics and finance, yet, i know enough to have driven myself into a ditch. Will somebody help me get out?
I asked this in a seminar just passed and nobody could get their head around it, never mind explain it. Then again, that was probably because of the awful way i explained it.

Ok. Here is where i sit.

I know the standard story. I think. That is....

*Central banks, independent from government control yet still a permanent institution, set an interest rate - particularly chosen to control inflation.
*Commercial banks borrow money from the central bank, at the set interest rate, and then, in turn, borrow that money, along with money taken from depositors savings, to business and individuals.
*Using fractional reserve banking - the ability to lend by a factor of more than ten the amount of reserves held - the commercial bank enables the economy to grow at a much faster rate than it should otherwise be able to. What the bank holds in savings reserves and money borrowed from the central bank it lends out with a generous margin, for itself, between what it is paying savers in interest and what it is paying the central bank, in interest, for borrowing the money.

So, put bluntly, in this story, once fractional reserve banking is factored in, the commercial banks have a license to print money that doesn't exist, that they create, and to expand the amount of money in the system. Inflate the economy.

Now, there is an obvious paradox here.
The only new money that comes into the system is from a bank. There is a set amount of money in the system and the only way new money enters into it, and expands the system, is when it is borrowed. There is [/i]x[/i] amount of money in the country / system. Much of that money is owed to banks, with interest i. The only way that debt can be repaid is by collecting the principal, money borrowed, p, plus the interest i, and giving it back. This is a detraction of the total money supply. x - p - i So, if all debts are paid back the money system slowly shrinks.
So, new money comes into the system, in the shape of new borrowing. x + p. This p is accruing interest, i. If the debt principle p is paid back immediately with even the minimal amount of interest i there will still be a total detraction in the amount of x, namely, the money system ends up like this at the time of the loan being made x + p and then ends up like this once the loan is paid back x - p - i.

Every time a loan is made, which is necessary for growth, it means the amount of total money in the system increases.
And every time a loan is repaid the amount of total money in the system decreases.

Then, some loans are called bad and cancelled. The principle exists somewhere in the system and the interest is forgotten about. In this instance the total money supply has grown.

In spite of this more loans are still repaid then called bad. The money supply still grows exponentially. It is a debt reliant system. The economy can only grow with the issuing of new debt.
If new debt is not issued, there is no growth.

Now, this observation is not revelatory. It is quite obvious.

But my confusion now, is this.
What actual role do central banks play in this relationship between commercial banks and the economic system?
They are supposed to control the levers on inflation in the economy but, considering fractional reserve banking and the benefits it offers the commercial banks, the commercial banks can still grant loans by a factor of ten, so even nominal increases in the base rate of a central bank will make little difference. In fact, it isn't even a factor of ten, it is a multiple of that. A business deposits 10 pounds with the bank. The bank has 10 pounds and lends out 90 pounds to 4 different businesses, 22.50 pounds to each. It keeps 10 pounds in deposits and lends out 90 pounds from the original 10 pounds. That money lent out now goes to other banks, or even in an account of the same bank from whom it was lent, where it is now deposited, and this money is now in turn also lent out according to the fractional reserve banking system. One deposit of 10 pounds allows loans to be made out by a seemingly infinite factor. Of course each bank is just dependent on their being no bank run and that a proportion on the loans come good. It doesn't even need to be a high percentage.

So, if the commercial banks have this ability to turn one small deposit into a multitude of loans, well, what restraint and power does the central bank have?
It's pretty much a lender of last resort, and that's all it is, no?
That is all i can make it out to be.

And then, in one final confusion, i understand the central bank itself doesn't even create it's own money, per se. Rather, the central bank, in it's standardised form, actually borrows money from commercial banks. Now that confused the hell out of me.
If i understand that correctly, the central bank is the lender of last resort to the commercial banks, and these commercial banks enable the central bank to be that lender of last resort by lending to it....?

And then, i also understand, that in some instances commercial banks are required to entrust their deposits to central banks, and these central banks then pay an interest to commercial banks for those deposits, because the central banks in turn lend those deposits out....

so, with a debt necessary economic system, where debt is an inevitable and never ceasing but always growing part of the system, and fractional reserve banking, and next to no understandable constraint on inflation, what exactly is the relationship between commercial and central banks?

how does this whole thing work?

Please explain in as straight forward a manner as possible.
In my class, when i raised this issue, nobody seemed to understand the paradox very well, and as for the rest of it, well, it felt like i was barking up the wrong tree. Sorry, when i say my class i meant my philosophy seminar Undergraduate here.

Thanks. Can't wait to read some answers.