P. Honohan International Review of Economics and Finance 8 1999 147–163 151
optimism by one bank can result in too much risky lending throughout the system. That certainly rings true for the kinds of bandwagon effect or herd behavior observed
in credit-sustained property booms. The effects of entry can also be modeled as a disturbance to the Cournot equilibrium.
Here, the interesting point is that the arrival of a new bank can move the equilibrium out of the no-failure zone, leading not only to a risk of failure, but to the exploitation
of the deposit-put and increasingly risky behavior. The problem is that, having inherited staff levels that justify a certain scale of lending, incumbents will treat these as sunk
costs and continue to lend at a level which is beyond what is now prudent, given the lower returns and higher risk that is emerging in the market as a result of the arrival
of the new entrant.
In the long-run, when the fixed costs incurred by the old banks are no longer relevant, it is possible that the system could return to the no-failure zone. To that
extent, the disturbing effect of entry may vanish over time, and this disturbance may well be an acceptable cost to pay for a more efficient long-term equilibrium.
3. The Basic Model
To begin, we describe those features of the set-up that are common to both variants of the model. There are n banks in the market. Each bank i can access deposits at
an exogenously fixed return r interest rate r
2 1. It can supply loans at a per- dollar unit administrative cost w
i
. There are two types of loan: low- and high-risk. The low-risk loans risk-free loans yield a certain return r
L
. Each bank’s portfolio of high risk loans yields r
H
if repaid, nothing otherwise. The probability of repayment is denoted u. We assume this risk is non-diversifiable; the entire risky portfolio of a
given bank either repays or does not.
8
The opportunity cost of capital is exogenously fixed at r
C
. It is assumed that the maximum yield on a high-risk loan r
H
is exogenously fixed, but the low interest yield r
L
and the probability of repayment u depend on the aggregate volume of loans of each type. We write x
i
and z
i
, X and Z, as the quantity of low-risk and high risk loans of bank i and the aggregate of the banks respectively.
Thus the dependence of the returns on volume is: r
L
5 f X and u 5 g Z with elasticities: e 5 Xf9Xr
L
and h 5 Zg
1
Zu. Banks are required to hold a minimum capital-to-assets ratio of g, and we will
assume that the ex ante franchise value of banking is sufficient to remunerate this capital, but that the opportunity cost of capital r
C
is higher than r so that the minimum
capital ratio is binding. The weighted average cost of funds is thus [Eq. 1]: r
5 1 2 gr 1 g r
C
. 1
3.1. Equilibrium in the no-failure zone If the capital requirement is set sufficiently high so that the probability of failure
is zero in the relevant range—in other words if we are in the “no-failure zone”—then the appropriate maximand is:
p
i
5 u r
H
z
i
1 r
L
x
i
2 r 1 w
i
z
i
1 x
i
2
152 P. Honohan International Review of Economics and Finance 8 1999 147–163
Maximization of the expected value of Eq. 2 by each bank under the Cournot assumption that the volume decisions of the others are not sensitive to one’s own,
leads to the familiar Lerner price-cost ratio conditions for each bank in the two markets:
1 2 r
1 w
i
r
L
5 2 x
i
X e
3 1 2
r 1 w
i
ur
H
5 2 z
i
Z h
4 The intuition here is that each bank takes into account the impact of its own lending
on the rate of return; thus it will expand its lending only to the point where, conditional on the lending of all other banks, a further increase would lower the gross margin
on other loans by more than it would generate in new net revenue. These conditions Eqs. 3 and 4 can thus be seen as reaction functions
9
determining each bank’s loan supply x
i
and z
i
as a function of the lending by others: X 2 x
i
and Z 2 z
i
. Aggregating these over all of the banks provides the market equilibrium conditions:
1 2 r
X 1 S w
i
x
i
r
L
X 5 2
o
x
2 i
X
2
e 5
and 1 2
r Z 1 S w
i
Z
i
ur
H
Z 5 2
o
z
2 i
Z
2
h 6
The market equilibrium conditions Eqs. 5 and 6 thus determine the risk-free low-risk lending rate r
L
and the risk parameter u. In the case where all banks are identical, the sum of squared market shares Herfindahl index is simply the reciprocal of
the number of banks, so the market equilibrium conditions reduce to [Eqs. 7 and 8]: 1 2
r 1 w
r
L
5 2 e
N 7
1 2 r
1 w ur
H
5 2 h
N 8
In words, oligopolistic competition between the banks will drive loan supply up to the point where the expected profit margin, as a percentage of gross return on each
sub-portfolio, is inversely proportional to the price elasticity of the loan supply. These conditions are, of course, directly analogous to classic IO results defining the profit
share as directly proportional to the absolute value of the elasticity of demand and inversely proportional to the number of firms.
A small number of banks and a high capital requirement means that the gross return on low-risk assets is high: writing the interest and administrative costs devoted
per dollar of lending as: b
i
5 1 2 gr 1 w
i
P. Honohan International Review of Economics and Finance 8 1999 147–163 153
we may regard the gross profit per dollar lent r
L
2 b
i
as a measure of the franchise value of low-risk banking.
3.2. Equilibrium in the failure zone If the capital adequacy requirements are not sufficient to rule out failure, then the
optimization criteria change. This is because, by increasing the share of high-risk loans in its portfolio, each bank can influence the value of the deposit-put, the possibility
that losses by the bank in excess of its capital thereby threatening the depositors will be met by the government.
If the high-risk loan is not repaid, then the government may have to pay out to depositors. The government is at risk of having to make such a pay-out if the sum
due to the depositors and factors of production the latter assumed already paid out of depositor funds exceeds the amount received by the bank from the low risk
borrowers, i.e. the government is at risk if Condition F holds, defined by the inequality
b
i
x
i
1 z
i
2 r
L
x
i
; F
i
. 0. As the deposit-put will only have value if this inequality is satisfied, we say that a
bank is ex ante in the failure zone whenever Condition F holds. F
i
is effectively the value of the deposit put if exercised. If Condition F does not hold, then the bank will
still be able to pay depositors even if the high-risk loan is not repaid; it is not in the failure zone, and the deposit put has no value.
Note that Condition F will not hold if what we have termed above the franchise value of low-risk banking r
L
2 b
i
is sufficiently high. The bank maximizing expected profits in the failure zone will thus maximize [Eq. 9]
p
i
1 1 2 uF
i
5 u r
H
z
i
1 r
L
x
i
2 r 1 w
i
x
i
1 z
i
1 1 2 u[b
i
x
i
1 z
i
2 r
L
x
i
] 5 u r
H
z
i
1 r
L
x
i
2 u b
i
1 gr
C
x
i
1 z
i
. 5 u r
H
z
i
1 r
L
x
i
2 ur 1 w
i
1 a x
i
1 z
i
, 9
where, a 5 gr
C
1 2 uu In examining this expression it is worth noting that, in a failure zone, the shareholders
of a failing bank can lose their entire capital, including the profit from the low-risk loans. On the other hand, from the bank’s shareholder’s viewpoint the costs, other
than opportunity cost of capital, also become contingent on not failing. Although the precise formulation is an implication of the very simple dichotomous risk specification,
the insight that the risk differential between the two loan categories is reduced seems robust. An increase in u therefore affects the expected return not only on the high-
risk asset, but also on the low-risk asset and the effective cost of lending.
With this maximand and Cournot behaviour, conditions Eqs. 3 and 4 above for individual bank asset choices in the no-failure zone are replaced by failure zone
conditions Eqs. 10 and 11:
154 P. Honohan International Review of Economics and Finance 8 1999 147–163
1 2 r
1 w
i
1 a r
L
5 2 x
i
X e
10 and
1 2 B
i
r 1 w
i
ur
H
5 z
i
Z h ,
11 where
B
i
5 1
1
1 1 b
i
gr
C
2 1
1 2 b
i
r
H
2
. modifies the mark-up ratio, and is a grossed-up risk asset quantity, effectively taking
account of the increased risk of failure, and
z
i
5 z
i
1 r
L
2 b
i
r
H
2 b
i
x
i
hence of losing the profits from the risk-free asset, imparted by additional purchases of the high-risk asset.
The only difference between Eq. 3 and Eq. 10 is that the cost term in Eq. 10, i.e. r 1 w
i
1 a, is evidently larger than that in Eq. 3. Thus the existence of the deposit-put acts just like an increase in the cost of capital.
The relation between Eq. 4 and Eq. 11 is more complex. The factor B
i
by which the LHS of Eq. 11 differs from that of Eq. 4, tends to zero for small capital
requirements g → 0 making the LHS of Eq. 11 larger than that of Eq. 4. Accordingly, the grossed-up risk assets total is higher than the total satisfied:
z
i
, z
i
, z
i
1 x
i
. Comparing the incentives of an individual bank in the failure zone with that of the
same bank not in the failure zone, conditional on market prices r
L
and u, we see that the failure zone bank will place slightly less in low-risk assets. Furthermore, its grossed-
up risk position, z
i
, will be higher provided that B
i
, 1; and the net amount placed in high-risk assets, z
i
, will also be larger, unless what we have termed the franchise value of low-risk banking is sufficiently high. The condition
10
under which the bank will place more in high-risk assets will be referred to as Condition R.
The market equilibrium condition for identical banks in the failure zone is a straight- forward aggregation of Eq. 10 and Eq. 11. Therefore, for the same elasticity e,
the equilibrium low-risk rate r
L
is unambiguously higher, though by a small amount if capital requirements g are small. Equilibrium in the high-risk asset market will
involve if elasticity h is little changed and under condition R increased lending and lower probability of success u.
One way of summarizing these findings is to say that, comparing the equilibrium of a banking system with unlimited liability with a system such as we have described
in the failure zone, there is:
P. Honohan International Review of Economics and Finance 8 1999 147–163 155
i. an increase in the low-risk lending rate sufficient to offset the relatively higher burden being borne by the capital committed to “low-risk” lending, but which
will be lost if the bank fails; ii. an expansion of high risk loans, to the point where the value of u has fallen
sufficiently to ensure that ur
H
gives the same margin over the lower effective marginal cost of lending.
Note that we have not yet discussed the question of contagion, to which we now turn.
4. Informational Disturbances
