What happens to the money supply when a bank is allowed to go bust
This depends on definition of money supply. I will use M2 Fed definition because it is very common.
Scenario 1. There is no deposit insurance and the central bank take no special action in response to the bank failure.
M2 unambiguously declines because some portion of deposits will be lost during bankruptcy, and by definition of M2 that means there is smaller money supply.
Scenario 2. In the real world, i.e. there is some deposit insurance and a central bank may take some kind of action.
The effect will be indeterminate because of lack of information.
- Deposit insurance is limited, so without knowing if any accounts exceed it
we can't say what happens.
- monetary policy has an effect on M2, so regardless what happens in one bank failure overall M2 can increase, decrease or stay constant depending on various policies central bank decides to pursue. Since what exact policy CB pursues is not specified nothing more can be said about this.
The reason I ask, is because I have noticed that governments will rarely rescue companies if they are not banks however large they are whereas when a bank is in trouble there appears to be a sense in which it is almost imperative that they must be rescued. My suspicion is that at significant part of this imperative is due to the potential shrinkage of the money supply but this appears rarely mentioned. Basically, in order to understand the panic measures, I'd like to fully understand what it is that is being averted.
Central banks do this to prevent collapse of money supply, since Friedman and Schwartz (1963) showed that Great Depression was precipitated by collapse of money supply in US, caused by bank runs/failures and by Fed further reducing money supply through tight monetary policy.
However, you are incorrect in stating that government officials are not open about this. Bernanke literally openly talks about this in his speech that became basis for so called 'Bernanke doctrine'. In his 2002 speech Bernanke claims:
- no 1. job of central bank is to keep prices stable which means not just fighting inflation but also preventing deflation
- the way how you prevent deflation is by A) expansion of money supply (hence not only collapse of money supply should be prevented, money supply should be explicitly expanded). B) provision of sufficient liquidity. C) by keeping interest rate low.
Bernanke doctrine is not some secret and central bankers talk about it openly. You might never hear it in news because it might not be sexy for journalists to report on such matters, but it is simply not true that is is not explicitly discussed.
In addition, the reason why central banks rush to help of banks is that some banks are systemically important. Collapse of a single interconnected bank my cause financial meltdown and hurt even 'healthy' banks. So preventing bank failure is considered prudential, even if overall money supply can be expanded in spite of allowing single bank to collapse.