Specifically, δi is uniformly distributed between 0 and 1, and it is i.i.d. across banks.4
This introduces uncertainty at the level of each individual bank and in the aggregate. All
uncertainty is resolved at date 1, when liquidity shocks materialize.
Given the structure of liquidity shocks, each bank faces a demand for liquidity xi = δi Di
at date 1 and uses its reserves Ri to satisfy it. Reserves represent a storage technology that
transfers the value of investment from one period to the next. We may think of cash, reserve
holdings at the central bank, or even short-term government securities and other safe and
low yielding assets. (The interest rate on reserves need not be zero.) The stochastic nature
of δi means that the realized demand for liquidity xi may exceed or fall short of Ri , thus
introducing the need for a market where liquidity can be traded at date 1, as described more
below. Denoting as f (xi) the density function of xi, at date 0 each bank faces a liquidity
risk -the probability to experience a liquidity shortage at date 1- given by
φi = prob(xi >Ri
Di
)= Ri
f(xi)dxi,
(3)
and has an expected liquidity need -the expected size of liquidity shortage that needs to be
refinanced at date 1- equal to
ωi
Di
= Ri (
xi
- Ri)f(xi)dxi.
(4)
Interbank refinancing and aggregate liquidity
At date 1 an interbank market opens where banks can either borrow or lend depending
on whether they have shortages (xi <Ri) or excesses (xi >Ri) of reserves. We focus
on the ultra-short interbank or money market, such as the unsecured market for wholesale
deposits, where both banks and the central bank operate.5 Since in this market rates
are always in between the policy rates at which sound individual banks may receive(give)
overnight deposits from(to) the central bank (e.g., the marginal lending and the deposit
4We assume for simplicity that liquidity shocks are independent across banks, but all our results remain
valid as long as liquidity shocks are not perfectly correlated.
5 The most relevant and largest ultra-short market is the overnight market, in which banks exchange
liquidity at the so-called ‘overnight’ or ‘Fed funds’ rates (e.g., bid and ask rates). Most central banks
stabilize those market rates around an ‘official rate’ (e.g., the Fed Fund target rate in the US, and the
minimum bid rate in the euro area) by adjusting the supply of liquidity to changes in the aggregate demand.
Recent evidence indicates that central banks control overnight rates quite successfully (e.g., Carpenter and
Demiralp, 2005; Perez Quiros and Rodriguez Mendizabal, forthcoming).