Then, assuming [FOSD] or [SOSD], at the solution to the regulator’s problem,
∀qo ∈ [q,q]:
r:
{δ[rL-γ(L) - (1
k*(qo))L]}dGq (r∣q*(qo))
- C0(q*(qo) - qo) = 0
(10)
and
(1 + b)G(rb*∣q*(qo)) + (1 - G(rb*∣q*(qo))) — re = 0 (11)
where rb*L*—γ(L*)-ρ(q*)(1-k*)L* = 0 and asterisks indicate optimal levels.
Equations (10) and (11) are obtained by differentiating (9) with respect to q
and k respectively.8
Equation (10) identifies the first-best level of quality for the bank’s loan
portfolio. Increases in quality increase the expected cash flows of the bank
and reduce the probability of failure. At the optimal level of quality, these
marginal gains are equal to the marginal costs of additional quality C0(∙).
The optimal capital structure of the bank is set out in (11) which can be
further simplified to
bG(rb*∣q*(qo)) = re — 1. (12)
The first term in (12) measures the expected savings in bankruptcy cost from
increasing the capital-asset ratio. The second term represents the marginal
cost of financing an increase in the capital asset ratio. The optimal capital-
asset ratio is given where these two quantities are equated. Apart from
differences in the costs of equity, debt, and financial distress, the bank’s
induced capital structure is, hence, similar to that of any other type of firm.
(12) implicitly determines the equilibrium probability of bankruptcy:
re 1
G(rb*∣q*(qo)) = — (13)
From (13), corollary 1 is obvious:
Corollary 1 Given [FOSD] or [SOSD], the equilibrium probability of bank
failure is independent of the level of realized quality.
This may seem surprising as one might expect high-quality banks to ex-
perience bankruptcy less often. However, equity is a substitute for quality in
reducing bankruptcy (see corollary 4). So, the regulator forces low-quality
8AsI assume that the lowest quality q is sufficiently large that it is socially optimal to
have all banks to make loans, I can concentrate on interior solutions.
11