Banks offer differentiated loans and compete on price. The differentiation of loans may
emerge from long-term lending relationships (see, e.g., Sharpe, 1990; Rajan, 1992), special-
ization in certain types of lending (e.g., to small/large firms or to different sectors) or in
certain geographical areas. Following Shubik and Levitan (1980), we assume that each bank
i faces a linear demand for loans given by
Li =l - γ ( rL - NN
N ∖
∑rL), (2)
j=1
where riL and rjL are the loan rates charged by banks i and j (with j =1,...,i,...,N), and
the parameter γ ≥ 0 represents the degree of substitutability of loans. The larger γ the more
substitutable are the loans. Note that expression (2) implies a constant aggregate demand
for loans PN=1 Li = Nl, as in Salop (1979).
Processing loans involves a per-unit lending cost c, which can be thought of as a set up
cost or a monitoring cost. Loans mature at date 2 and yield nothing if liquidated before
maturity.
Deposits, individual liquidity shocks and reserve holdings
Banks raise deposits in N distinct ‘regions’. A region can be interpreted as a geographical
area, a specific segment of the population, or an industry sector in which a bank specializes
for its deposit business. There is a large number of potential depositors in every region,
each endowed with one unit of funds at date 0. Depositors are offered demandable contracts,
which pay just the initial investment in case of withdrawal at date 1 and a (net) rate rD at
date 2. The deposit rate rD can be thought of as the reservation value of depositors (the
return of another investment opportunity), or, alternatively, as the equilibrium rate in a
competition game between banks and other deposit-taking financial institutions.
As in Klein (1971) and Diamond and Dybvig (1983), deposits are subject to liquidity
shocks. A fraction δi of depositors at each bank develops a preference for early consumption,
and withdraws at date 1. The remaining 1 - δi depositors value consumption only at date 2,
and leave their funds at the bank until then.3 The fraction δi is assumed to be stochastic.
3 The fraction δi can also be interpreted as a regional macro shock. For example, weather conditions may
change the general consumption needs in a region, so that each depositor withdraws a fraction δi of his initial
investment.