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.
