One of the main models of the role of banks is Diamond-Dybvig (DD). This seeks to explain how banks might be able to provide valuable intermediation between:
a) savers who are uncertain when they might want their money back, and
b) entrepreneurs who need funds for a minimum period of time in order to generate returns.
In this model, entrepreneurs can only pay an interest return if they can borrow for the time required. If required, they can repay early, but with zero interest. Savers can lend to entrepreneurs directly, but they face an uncertain return, because they don't know when they might want to realise their investment.
Instead, savers deposit with the bank, which lends to entrepreneurs. The bank deposit contract pays interest, even on an early withdrawal. The bank can do this because the deposit contract also provides that, in the event the deposit is held for the full term, it pays less than the saver would get from lending directly. The bank subsidises early withdrawers at the expense of late withdrawers, and can do this because it knows that not everyone will be an early withdrawer. In the model, savers prefer the deposit contract to direct lending, because they are risk averse and it offers less uncertainty in returns.
This is a model of banking where deposits seem to precede loans. Savers place deposits with the bank, which lends them out. In fact, there isn't even any money in the model, just goods. Loans and deposits are loans of goods and deposits of goods. So it is physically impossible for the bank to make a loan, before it has taken a deposit.
This is not necessarily a problem for the model, but it seems at odds with the story where "loans create deposits". So I thought it would be interesting to construct a version of DD that looked a little like something a circuit theorist might identify with. There's no great message here - I just like thinking about the connection between different ideas.
So, as with DD we imagine there is a single good that can be consumed or used in production. Entrepreneurs have productive projects that take two periods to generate a non-zero return.
Both savers and entrepreneurs open accounts with the bank. The account balances all start at zero. The bank agrees to provide the entrepreneurs with overdraft credit. Overdrafts must be repaid after two periods with interest at 6.25% per period (simple interest, so 12.5% in total). The bank may require early repayment, but no interest is then payable.
The bank pays interest on positive balances at 4% per period (again simple interest, with no compounding). All interest is paid simply by debiting and crediting the appropriate accounts.
There is no provision for positive balances to be "withdrawn". They can only be used for payments. However, the bank undertakes to try and maintain the real value of such deposit balances, by targeting the dollar price of goods. (We might then think of this bank as a whole banking system, with the central bank trying to maintain price stability by influencing the volume of lending).
Having got their credit agreed, the entrepreneurs then buy goods from the savers for $100. This involves the entrepreneurs instructing the bank to debit $100 from their accounts and credit it to the accounts of the savers. The entrepreneurs, as a group, then has a negative balance of $100 on their accounts (they are in overdraft) and the savers, as a group, have positive balances of $100.
In period 1, the bank credits the accounts of all savers at 4%, so total deposit balances are $104.
Now, savers want to realise half their savings ($52 in total). They can only do this through buying goods from entrepreneurs. To maintain the value of the dollar, the bank must create demand for dollars (and prompt supply of goods). It does this by calling in $52 of the entrepreneurs' overdraft. Entrepreneurs terminate 52% of their projects and sell the goods they were using to savers. Appropriate debits and credits reduce both savers' and entrepreneurs' balances by $52. Deposits are now $52 and loans (overdrafts) are $48.
In period 2, entrepreneurs' investments are realised. The bank credits a further 4% interest to the remaining balance of savers accounts taking them to $54. It debits 12.5% interest from the remaining overdraft balances of $48, taking these also to $54.
Entrepreneurs sell goods to savers for $54. The bank makes the appropriate debits and credits, leaving all balances at zero.
By arranging things this way, the bank has provided savers with a certain return of 4%, regardless of when they want their money, rather than an uncertain return of either 0% or 6.25%.
DD use their model to consider bank runs. We can do the same here. The more savers try to realise their savings early, the more the bank has to call loans to maintain the value of the deposits. This reduces returns to those who are saving for the full term, eventually to the point that they lose their entire investment. There is therefore an incentive not to be the last one holding deposits.
However, in this model this run is more in the nature of a run on the currency, not quite what DD is concerned with. To look at a run on an individual bank here, we would need to consider a model with more than one discrete banking entity.