Fractional reserve is a system under which banks hold a stock of monetary base or reserves equal to some fraction of their total assets or liabilities.
In contrast, under full reserve, 100% of sums deposited at a bank are kept in the form of monetary base, or “reserves”. Thus a full reserve bank cannot lend. Instead, lending is done by institutions much like unit trusts (mutual funds in the US) or corporations. That is, those latter entities’ principle asset would be loans, and their main liability (if you can call it that) is shares.
Now let’s revert to a fractional reserve bank which is a compromise between the latter two types of entity (a full reserve bank and a lending entity funded just by shareholders).
The liabilities of a fractional reserve bank consists first of depositors (I’ll ignore bondholders for the sake of simplicity). And second there are shareholders.
Now what happens when the bank is in trouble? Depositors who don’t get their money out before the bank collapses make a loss. But losing part or all of an investment makes one a shareholder!!! That is, a shareholder by definition is one who “shares” in the profits and losses made by the entity in which they hold shares. Whether those “loss absorbing” depositors are ACTUALLY CALLED shareholders is immaterial. The indisputable point is that in reality they ARE SHAREHOLDERS. Though obviously they are a different TYPE OF shareholder to your typical ordinary shareholder in a corporation. For example, so called depositors are senior to shareholders when administrators pay off creditors in the event of insolvency.
But then there are different types of share ANYWAY (e.g. ordinary shares, preference shares, etc).
Ergo there is no such thing as a fractional reserve bank. A fractional reserve bank is an imaginary entity that exists only in the minds of deluded individuals who make dishonest use of the words “depositor” and “shareholder”.
You could of course argue that fractional reserve banking becomes a reality if the taxpayer backs such banks, i.e. guarantees deposits. But that’s a subsidy of banks. And subsidies do not make economic sense.
Ergo, in a GDP maximising system, that is system which is free of unjustified subsidies, fractional reserve banking is a mirage.