I have thus far found the Mystery of Banking to be an amazing book on the dynamics of the money system. But Rothbard's description of the fractional reserve system does not jive with what I have been taught and this definition becomes essential later on when he explains why fractional reserve banking could not exist without a central bank.
I have always understood fractional reserve banking to mean that the bank only keeps a fraction of the deposit and loans out the rest. So with a reserve requirement of 10% and 50k of deposits the bank would then loan out 45k which leaves the bank open to a bank run if more than 5k is redeemed. But in figure 7.2 Rothbard says that the bank would have lent out 80k, more money than had been deposited. Is that correct? The example becomes critical later on when he explains why fractional reserves would not only fail due to bank runs but also due to day to day transactions as clients shift funds and the banks don't have the cash to transfer. Under my fractional reserve they actually would have enough cash and would fail only under a bank run, but in Rothbard's example they would fail when more than 50k of bank notes get redeemed. This later sets the whole basis for why a central bank is needed to support the fractional reserve system so you can see why it is bothering me that it isn't jiving with the definition I have been taught.
Can someone explain this to me?
Start by assuming a system with one bank. The bank lends 45k to someone who spends it on something with Shopkeper A, who deposits that back at the bank. The bank now has 95k in deposits and can loan out (on the basis of this) up to 0.9 * 95 = 85.5k. They already have loans out of 45k though, so basically they can go ahead and lend out another 40.5k (which you will note is 0.9 x the most recent deposit)... and so on and so forth, with a series that tends towards 50k divided by 0.1 - i.e. 500k. So basically an initial deposit of 50k allows them to create loans of up to 500k.
In the real world, there is obviously more than one bank. But if you view the entire banking system as a whole and acknowledge that all the banks in the system have exactly the same reserve requirement, the same simple maths that you go through for an individual bank can also be applied to the banking sector as an aggregate/whole as well.
If all the banks are able to loan out as much as they possibly can then any of the banks is vulnerable if more than 10% of their depositors come in and ask for their money back (or if a few big depositors comes in and ask for their money back).
It's probably better explained with diagrams and more extensive examples though: http://en.wikipedia.org/wiki/Fractional-reserve_banking
Note: In practice, most US banks don't actually keep 10% reserves, despite that being the legal requirement. They use all sorts of complex techniques (such as sweeps) so that they end up with around about the same reserves as most English or European banks (around 3%)... so the above is a bit of an oversimplification of the US banking system, and probably of many others, but it certainly forms the basis for understanding it.