The gold in a FR bank does not have multiple owners. The depositor is really a creditor, and once he “deposits” gold in a fractional reserve bank, he is no longer the owner. The FR bank owns the gold and is indebted to the depositor/creditor for the same amount plus interest. (The “plus interest” is the motivation for the depositor/creditor to deal with an FR bank in the first place.) It seems to me that most people understand this, albeit imperfectly, and that is why bank runs can happen, (i.e. because people know the bank doesn’t really have all deposits stored in a vault).
Question: In your conception as you outline it above, how do you factor in the creation of new claims (let’s say, additional banknotes) in relation to an assumed fixed amount of gold by the FR bank?
Where does that fit in, in your conception or example?
When a depositor/creditor spends their banknotes, they are calling in a debt with the bank – “hey, Mr. Bank, I want that gold you owe me.” All important to the bank is the rate at which their depositors/creditors call in the debt. When the depositors/creditors reduce their spending, they call in their debt with the bank less frequently. The bank can now afford to lend more without more risk, and thus an increase in saving is converted into an increase in investment.
Banknotes are not “claims” to some quantity of gold, but bank liabilities. It is not necessary that a bank be capable of satisfying all its liabilities at once, but merely that it can when the liability is called upon. This intertemporal coordination is key to understanding how the system can work.