In contrast
to the **aggregate
liquidity model**, another way to model interest rates is to take the individual
bank’s perspective. This model may be more appropriate if reserve requirements
are very low and/or if banks cannot average reserve requirements across the maintenance
period. (Ultimately, a central bank may use both models as inputs to its
decision-making.)

First,
consider the idea that the demand for reserve balances depends on uncertainties
relating to payment flows, which vary all the time. For instance, if a bank experiences
a liquidity shock after the interbank market has closed and does not have
enough reserves on hand, it will be forced to go into a costly overdraft with
the central bank. Such uncertainty creates a demand for reserves, since
reserves aren’t always acquirable in the interbank market or through cheap,
daylight overdrafts.

To
further develop this idea, let’s introduce the variable *q*, which is the
amount of reserves each bank holds before the liquidity shock. Also recall the simplified form of Equation 6 discussed **here**: *B - D = M - A*. Assuming this
amount of reserves is spread equally across all banks *N, q = (M – A) / N.*
If we assume the liquidity shocks, represented by the variable *x*, are
normally distributed (with *φ *representing the probability density
function (pdf) of the standard normal distribution – see this **Appendix **for an description of pdfs), then the expected value of the
liquidity shocks is zero across all banks. EDIT: Also assume no deposit facility.

Let’s
now consider the expected cost of going into overdraft, *C(q)*. (Bindseil
uses *C(q)*. I would have preferred something more of the form *E[f(x)]*.)
A bank will go into overdraft if it has negative reserves at the end of the
day: *x+q < 0*. The cost of borrowing in this case will be *i*_{b}*(q+x).
If* (q+x) >0*, then there is no need for the bank to borrow from
the central bank, so there are no borrowing costs. In summary, the cost of borrowing, call it f(q), is:

*f(q) = i*_{B}*(q+x) , if x < -q

*f(q) = 0 , if x *__>__ -q

We can now calculate the expected cost, *C(q)*, of going into overdraft:

See the **A****ppendix** for an explanation of this
equation if its form is not familiar to you.

From
this, one can understand that if the bank were to increase the number of
reserves *q* that it holds, then the probability of it going into
overdraft should decline, and with that, the expected cost of going into
overdraft. But holding more reserves *q* entails a cost as well, which is the
interbank interest rate. Thus, a bank will only acquire more *q* if the cost
of doing so, *i*, is less than or equal to the benefit of doing so, which
is the corresponding decline in the expected cost of going into overdraft *C(q)*. Or, another way of viewing it is that the interbank rate, which is the opportunity cost of holding reserves, should settle at a level where banks are indifferent between holding one more unit of reserves and not. This is econ 101 marginal analysis, where we determine the equilibrium market
interest rate by setting marginal cost equal to marginal benefit. We can kick
the complexity up a notch by representing the equilibrium market rate using
partial derivatives:

*Equation B**: - ∂(C(q))
/ ∂(q) = i*

Since *q
= (M – A)/N*, the central bank can clearly control *i *by controlling [*i*_{D}, i_{B}] and/or by altering *M* to an appropriate
level, which is in a simple linear relationship with *q* (in this
simplified model). The appropriate level will be determined by the shape of the
probability density function perceived by the market, which the central bank
needs to estimate. Of course, in the real world, the “market” may not
transparently be using such a clean and simple model of probability. The model
is a proxy for a more complicated process in the real world, and so certainly
central banking is part art-form as well. Nonetheless, the model is helpful, as
it helps the central bank estimate the appropriate amount of open market
operations and grapple with the expectations of the market. All of that,
combined with building a credible reputation, helps the central bank achieve
its interest rate target with precision.

For a more complete version of the individual shock model, see this post on **Michael Woodford's model**.