6.1 The Case Without Deposit Insurance
Up to now we have considered only the case where deposits are fully insured, so that the
interest rate paid on deposits is determined entirely by depositors’ opportunity cost, given
by rD . A concern, however, is that banks’ incentives to economize on the use of capital may
be driven by the fixed-cost nature of deposits, which are not sensitive to risk when they are
fully insured. In this section we analyze the case where deposits are not insured, so that the
promised repayment must compensate depositors for the risk they face when placing their
money in banks that may not repay. This introduces a liability-side disciplining force on
bank behavior and capital holdings since banks will have to bear the cost of their risk-taking
through a higher deposit rate.
Consider the following slight change to the model. The timing is modified as follows.
First, banks choose how much to raise in deposits (1 - k) and capital (k); the promised
repayment on deposits (i.e., the deposit rate) cD is then also set. Second, the loan rate rL
is determined. Third, borrowers choose the loan that is most attractive to them. Fourth,
banks choose their monitoring effort q once the terms of the loan have been set. Note that
the only change is the introduction of the setting of the deposit rate cD in stage 1.
Deposits are uninsured, so that the expected value of their promised payment cD must be
equal to depositors’ opportunity cost rD .Givenk , depositors conjecture a level of monitoring
for the bank, qc , and set the deposit rate to meet their reservation return, which is given by
rD. This implies that qccD = rD, or that cD = rD.
We now solve the model by backward induction. For a given cD , banks choose monitoring
to maximize
max Π = q(rL - (1 - k)cD) - krE - cq2. (10)
q
For an interior solution, this problem yields q* = rL-(2-k)cD. In equilibrium, depositors’ con-
jecture about monitoring must be correct, so that qc = q*. We can therefore substitute cD =
rDD into the solution above for q and solve for the equilibrium value of monitoring. There are
19