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State-Contingent Bank Regulation With
Unobserved Actions and Unobserved
Characteristics

WP 04-02

David A. Marshall
Federal Reserve Bank of Chicago
Edward Simpson Prescott
Federal Reserve Bank of Richmond

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

State-Contingent Bank Regulation With
Unobserved Actions and Unobserved
Characteristics¤y
David A. Marshalla and Edward Simpson Prescottb
a

b

Federal Reserve Bank of Chicago

Federal Reserve Bank of Richmond

Federal Reserve Bank of Richmond Working Paper No. 04-02
February 21, 2004

Abstract
This paper studies bank regulation in the presence of deposit insur­
ance, where banks have private information on their own ability and their
investment strategy. Banks choose the mean and variance of their portfo­
lio return. Regulators wish to control banks’ risk choice, even though all
agents are risk neutral and there are no deadweight costs of bank failure,
because high risk adversely a¤ects banks’ ex ante incentives along other
dimensions. Regulatory tools studied are capital requirements and returncontingent …nes. Regulators can seek to separate bank types by o¤ering a
menu of contracts. We use numerical methods to study the prop erties of
the model with two di¤erent bank types. Without …nes, capital require­
ments only have limited ability to separate bank types. When …nes are
added, separation is much easier. Fine schedules and capital requirements
are tailored to bank type. Low quality banks are …ned when they produce
high returns in order to control risk-taking behavior. High quality banks
face …nes on lower returns to prevent low-typ e banks from pretending they
are high quality. Combining state-contingent …nes with capital regulation
signi…cantly improves upon pure capital regulation.

JEL Classification: D82, G21, and G28

Keywords: Bank regulation, moral hazard, hidden information
¤ David A. Marshall, Federal Reserve Bank of Chicago, 230 S. LaSalle St., Chicago, IL ,
60604; Tel: (312) 322-5102; Fax: (312) 322-2357; E-Mail: dmarshall@frbchi.org. Edward S.
Prescott, Federal Reserve Bank of Rcihmond, P.O. Box 27622, Richmond, VA 23261; Tel:
(804) 697-8206; Fax: (804) 697-8255; E-Mail: Edward.Prescott@rich.frb.org.
y We thank Darrin Halcomb and Scott Brave for their assistance in preparing this manu­
script. This paper was presented at the JEDC/CFS conference on Contracts and Institution
in Models with Heterogeneous Agents and at CEMFI. The views expressed in this paper are
those of the authors and are not necessarily the views of the Federal Reserve Bank of Chicago,
the Federal Reserve Bank of Richmond, or the Federal Reserve System.

1

1

Introduction

This paper studies bank regulation in a model where deposit insurance induces
a wedge between private decisions and the social optimum. In this model,
each bank has private information on its ability and chooses both the mean and
variance of its investment portfolio. The regulator can set capital requirements
and impose state-contingent …nes. Furthermore, the regulator may o¤er banks
a menu of regulatory contracts. A key feature of this model is that banks
that choose a lower portfolio variance also choose a portfolio with a higher
mean. Thus, in contrast to our usual …nance intuition, bank portfolio returns
endogenously exhibit a “reverse mean-variance trade-o¤ ”. This feature can be
exploited by the regulator to improve social welfare.
Often, the goal of regulation is described as “ensuring the safety and sound­
ness of the banking system”. That is, the regulator seeks to reduce the overall
risk of the bank sector. This goal is usually motivated by a desire to protect
taxpayer liability, reduce failure resolution costs, or prevent systemic risk. We
develop an alternative rationale for reducing bank risk that is complementary
to, but distinct from, these standard rationales. In our paper, the cost to society
of high failure risk is due to the way high risk distorts the ex-ante incentives of
banks. For example, White (1991) argues that the cost of the United States
Savings and Loans crisis was not primarily the deadweight societal cost of resolving the failed thrift institutions ex post. Rather, it was the cost of poor
investment decisions made by thrifts before the wave of thrift failures started
in the mid-1980’s.
It is well known that many savings and loan institutions were technically
insolvent in the early 1980’s because they held a large amount of low-interest
mortgages made before the in‡ation of the late 1970’s but had to pay the much
higher market rate of interest for short-term liabilities that existed in the early
1980’s. Mistaken attempts at deregulating the S&L’s without proper supervi­
sory safeguards gave these insolvent thrifts the opportunity to increase their
portfolio risk, in essence, to gamble for resurrection. White (1991) provides ev­
idence that failed thrifts were more likely to have engaged in real estate lending
and other new activities that were not in the traditional purview of thrifts.
In this paper, we argue that this sort of fall in the diligence with which
banks construct their asset portfolio is associated with an increase in bank risk.
More precisely, if banks were required (or induced) to reduce the variance of
their portfolio, they would also tend to expend more e¤ort increasing the mean
of their portfolio return. Thus, the welfare-maximizing regulator ought to be
concerned about reducing bank risk, but not necessarily because risk per se is
costly, but because reduced bank risk leads banks to make better investments,
thereby increasing the mean output of the economy and enhancing aggregate
welfare.
In our model, the regulator cannot control bank risk directly, because the
distribution of banks’ portfolio returns is private information. This di¢culty in
determining banks’ ex ante return distributions, especially the return variance, is
a fundamental practical problem for bank regulators. The Basel Committee on
2

Bank Regulation continues to struggle with a practical way of measuring bank
portfolio risk in its e¤orts to implement risk-based capital requirements. We
capture this di¢culty in an extreme way by assuming that risk is completely un­
observable to the regulator. Therefore, risk must be controlled indirectly. The
main regulatory tools available to do so in our model are state-contingent …nes.
These …nes di¤er from most current regulatory practice, which relies primarily
on non-state-contingent regulatory tools such as ex ante capital requirements.
The Basel Accord of 1988 bases capital minimums on a crude risk-weighting
of total assets held by a bank. Similarly, the Federal Deposit Insurance Cor­
poration Improvement Act of 1991 (FDICIA) imposes capital requirements for
U.S. banks. But recently, some state-contingent devices have become part of
the regulatory tool kit. The prompt corrective action provisions of FDICIA can
be viewed as state-contingent: if a bank’s capital falls below a minimal value,
sanctions can be imposed, including closure. State-contingent tools play an even
more prominent role in the “internal models approach,” a 1995 modi…cation to
the Basel accord that applies to the trading books of large money center banks.
This approach allows the bank to set its regulatory capital level using the Value
at Risk (VaR) estimate produced by the bank’s own risk model. Regulators
backtest these models to determine if the bank’s model is adequate or if it is
accurately reporting its results. If a bank’s model performs poorly, sanctions
can be imposed. As argued by Rochet (1999), these checks introduce some
state-contingency into the regulatory mechanism. 1 Our model explicitly studies
state-contingent regulation by allowing …nes to depend on the return produced
by the bank.
This paper builds on Marshall and Prescott (2001), who study state-contingent
…nes in a two-dimensional moral hazard model where both the mean and vari­
ance of the bank’s portfolio return is private information.2 They …nd that for
lognormal distributions of returns it is optimal to impose …nes on banks that
produce extremely high returns. This seemingly perverse result is driven by the
need to control risk taking. It is desirable to impose …nes on return realizations
with the highest deterrence e¤ect per dollar of …nes assessed in equilibrium. In
the absence of limited liability, the optimal …ne would be placed on the extreme
left-hand tail of the distributions. However, with limited liability …nes cannot
be assessed on the left-hand tail so they are assessed on the right-hand tail
1 The “pre-commitment approach” is another state -contingent mechanism. Under pre­
commitment, banks would be allowed to choose their own level of capital but would be subject
to a …ne if this capital did not cover ex post losses. In the proposal, …nes were to be used
as the penalty but other penalties, such as increased capital or incre ased regulatory scrutiny,
could be used as well (Kupiec and O’Brien (1995a,b)). This approach was actually put forth
by regulators for public comme nt (O¢ce of the Federal Register (1995)) but has not been
adopted.
2 We also build other papers in the literature on bank regulation in the presence of private
information. Antecedent papers include Giammarino, Lewis, and Sappington (1993), Campbell, Chan, and Marino (1992), Boyd, Chang and Smith (1998, 2002), Nagara jan and Sealy
(1998), Besanko and Kanatas (1996), and Matutes and Vives (2000). Also relevant is the
small principal-agent literature on when the agent controls risk. See, for example, Palomino
and Prat (2003).

3

instead.3
An obvious issue raised by this result is that if some banks are higher qual­
ity than others, and if bank quality is private information, …nes on high returns
may simply punish the high quality banks rather than deterring risk-taking by
low quality banks, and may even deter innovation. (See Boyd (2001)). By in­
corporating unobservable heterogeneity in bank types, our model can address
this trade-o¤. The model contains both low and high quality banks. Low type
banks have little incentive to expend e¤ort to increase their portfolio quality.
In contrast, high-type banks make the socially optimal e¤ort choice even in the
absence of regulation. We …nd that, as in Marshall and Prescott (2001), the
optimal contract still imposes …nes on high returns for the low-type banks. However, these …nes are not imposed on the high-type banks because the regulator
can separate types by o¤ering a menu of contracts. Low-type banks choose the
contract with high return …nes, while high-type banks choose an alternative con­
tract. However, because the regulator cannot observe bank type, the contracts
on the menu must induce self selection. Consequently, some punitive measures
need to be included in the high-type contract, even though, in the absence of
unobserved heterogeneity the high-type would be self-regulating. This must be
done to convince the low-type to truthfully report their type. Furthermore, we
…nd that the costs of private information about bank type are borne entirely
by the higher quality bank. The lower-quality bank always receives at least as
much utility as it would have under type-observability.
The problem is very di¢cult to analyze. There are two dimensions to the
moral-hazard problem and there is private information on bank type.4 Furthermore, the capital requirement, while observable, a¤ects incentives. Because of
these complications, we explore the model by using numerical methods to solve
and analyze speci…c parametric examples. We transform the problem into a
linear program as in Myerson (1982), Prescott and Townsend (1984), and oth­
ers. Even solving this linear program is not straightforward because of the large
number of o¤-equilibrium strategies that need to be checked to preserve incen­
tive compatibility. We use a method developed in Prescott (2003) that e¢ciently
checks these strategies. The numerical methods are described in Appendix B.
The remainder of the paper is organized as follows: Section 2 develops the
model. Section 3 derives some comparative static results on the connection between bank behavior and capital requirements. It also provides some conditions
under which a decreased return variance induces a bank to increase its portfolio
mean. Section 4 reports in detail the optimal contracts for a variety of parametric examples. The …nal section o¤ers some concluding comments. Technical
3 The result is similar to that of Green (1984) who studied risk control in a non-banking
environment.
4 There is a small literature on two-stage problems that start with hidden information and
then follow with moral hazard. Papers on this problem include Christenson (1981), Baiman
and Evans (1983), Penno (1984), La¤ont and Tirole (1986), McAfee and McMillan (1987),
Demougin (1989), Melamud and Reichelstein (1989), and Prescott (2003). These papers study
a substantially simpler moral hazard problem than we do, and do not incorporate elements like
capital requirements. Furthermore, this literature has been primarily focused on conditions
under which a report on private information is valuable.

4

details can be found in the two appendices.

2
2.1

The Model
Households

There are two periods and a single consumption goo d. There is a continuum of
risk-neutral households of measure one who consume in the second period only.
The households own all the assets in the economy, consume all the output, and
operate all the banks. Each household includes one “banker”, who is one of
two types, low and high. Low-type bankers are bad at operating a bank while
high-type bankers are good at bank operation. (We discuss the consequences of
bank type more formally below.) Let h i denote the fraction of households with
a type i banker, for i 2 flow; highg.
In the …rst period, a household receives an endowment of one unit of the
consumption good. Each type of household must split this endowment between
capital to use in its own bank and funds to deposit in other banks. Demand
deposits pay o¤ one unit of the consumption good in the second period for each
unit invested in the …rst period. In addition to its pecuniary payo¤, a unit of
bank deposits provides liquidity services with utility value ½ > 0. All demand
deposits are government insured, so the household is indi¤erent about which
bank holds its deposits.
For simplicity, we assume that each bank can only be of size one.5 A typei bank funds its investments with deposits from outsiders Di 2 [0; 1]. The
remaining portion of the investment, 1 ¡ Di , is funded by the banker’s own
funds. These own funds will be called capital. Since every bank must be the
same size, the deposits made by the household must equal Di, the deposits its
bank takes from other households. We make the assumption that a household
may not place deposits in its own bank. This assumption captures the idea that
depositors do not monitor their bank because of deposit insurance.
We do not model the individual assets of a bank’s investment portfolio.
Instead, we assume that the bank chooses the distribution of its portfolio return.
Let r denote the gross return accruing to a bank. This portfolio return has a
cdf F (¢j¹; ¾ ), where F is a two-parameter family of probability distributions
For simplicity of
completely characterized by its mean ¹ and variance ¾ 2 .
exposition, we assume that there exists a pdf corresponding to F (¢j¹; ¾). This
pdf is denoted f (¢j¹; ¾). For most commonly used two-parameter distributions,
the value of f (rj¹; ¾) is decreasing in ¹ for r su¢ciently small. Accordingly,
we assume that there exists an r¤ (¹; ¾) such that
@f (rj¹; ¾)
< 0; 8r < r¤ (¹; ¾) :
@¹

(1)

5 This extreme span of control assumption is often made in the bank regulation literature .
For example, see Boyd, Chang, and Smith (2002).

5

For example, equation (1) holds for the normal distribution with r¤ (¹; ¾) = ¹;
it holds for the log normal distribution for an r ¤ (¹; ¾) > ¹:
The bank chooses two characteristics of the portfolio. The …rst is the portfolio standard deviation ¾, which measures the bank’s risk choice. The second
is the bank’s level of screening e¤ort, s ¸ 0. We think of screening e¤ort as
the amount of diligence applied in evaluating loans and other assets. Screening
positively a¤ects the mean of the distribution, denoted ¹i (s), where
¹0i (s) > 0

(2)

¹0i0 (s) < 0:

(3)

The only di¤erence between the two bank types is that a given amount of
screening e¤ort results in a higher mean return for the high types than for the
low types. That is,
¹h ig h (s) > ¹low (s) ; 8s:
The cdf of the return of bank of type i that chooses screening s and risk ¾
will be denoted Fi (¢js; ¾) where
Fi (rjs; ¾) ´ F (rj¹i (s) ; ¾) :

(4)

The pdf corresponding to this cdf will be denoted f i(¢js; ¾). Screening s also
has a utility cost °s, with ° > 0.
In addition to the banks and households, there is a regulator who seeks to
maximize social welfare. (We characterize explicitly the regulator’s objective
in Section 2.2, below.) The regulator may impose …nes, g i(r ), as a function
of the bank’s type and ex post return. These …nes are non-negative, and are
constrained by limited liability of the bank. Consequently,
0 · gi (r) · maxf0; r ¡ Dig:
The expected payo¤ of the i th bank, denoted vi, is
Z 1
vi ´
[r ¡ Di ¡ gi (r)] f i (rjsi; ¾ i ) dr:

(5)

(6)

Di

The household purchases consumption in the second period using its bank
deposits (which are distinct from but equal in amount to the deposits received
by its banker), plus the pro…ts from its banker’s activities, less a lump sum tax
(common across types) with expected value T that is used by the deposit insurer
to pay o¤ the depositors of failed banks. Therefore, the expected consumption
for a household with bank type i, denoted Ci , is subject to the constraint
C i · Di + v i ¡ T :

(7)

Household utility is a linear function of consumption, liquidity services pro­
vided by deposits and the disutility of screening e¤ort:
C i + ½Di ¡ °s i:
6

(8)

Using equations (6) and (7) we can write the bank’s utility function (8) in
an alternative form:
Z

Di
0

(Di ¡ r ) fi (r js i; ¾ i) dr ¡

Z

Di
0

g(ri )f i (r js i; ¾ i) dr + ½Di + ¹i (si ) ¡ °si ¡ T :

(9)
The …rst term in equation (9) has the form of the payo¤ to a put option with

strike price Di. This term captures what is commonly referred to as the deposit
insurance put option. In e¤ect, deposit insurance gives the bank an option to
put the bank to the deposit insurer in exchange for the insurer taking over the
liability Di owed to the depositors. The second term is the income lost from
…nes. The third term is the value of liquidity services received from deposits.
The fourth term is the mean return to the bank’s portfolio. The …fth term is
the lost utility due to screening. Notice that the household/banker takes taxes
as given.6

2.2

Regulator

To cover the cost of bank failure, the regulator can use lump sum taxes and the
…nes it collects. We assume that …nes are costly to collect, due to their punitive
nature. In particular, there is a deadweight cost of ¿ ¸ 0 per unit …ne collected.
Therefore, taxes must satisfy
"Z
#
Z 1
Di
X
T =
hi
(Di ¡ r)f i (rjsi; ¾ i ) dr ¡
(1 ¡ ¿ )gi(r )f i (r js i; ¾ i) dr :
i

0

0

(10)
The regulator maximizes the share-weighted average of the ex ante utilities of

the two types of households, as given in equation (9), subject to equations (5),
(6), (7), and (10).7 However, the regulator takes into account the e¤ect on utility
of taxes T in equation (9) while the bank takes them as exogenous. These taxes
lay at the heart of the distortion caused by deposit insurance.
Suppose the regulator could observe banks’ types and control their choices
of fDi ; si; ¾ i g. Substituting equations (6), (7), and (10) into the utility function
of each bank (equation (8)) and then weighing each type by their fraction of the
population, one obtains the following expression for the regulator’s objective
function
6 Static models of bank regulation often include a franchise value or charter value term in
the bank’s objective function. (See, e.g., Keeley (1990), Marshall and Venkataraman (1999).)
This term is a stand-in for the present value of the bank’s future operations, which is lost
to the bank owners in the event of bankruptcy. Concern for lost franchise value acts as
a disincentive to risk taking, and thus can o¤set the risk-encouraging e¤ects of the deposit
insurance put option. Franchise value could easily be incorporated into this model. We
refrain from doing so in order to focus attention speci…cally on the way the deposit insurance
put option distorts bank incentives.
7 We do not address potential incentive problems with the regulator’s behavior.

7

X
i

·
Z
h i ½Di + ¹i (s i) ¡ °si ¡

0

1

¸
¿ gi (r)fi (rjsi ; ¾ i) dr :

(11)

The object in square brackets in equation (11) is the utility of the type-i household from the perspective of the regulator. Ignoring …nes and taxes (the latter
of which does not a¤ect the bank’s decisions), the only di¤erence between this
expression and equation (9) is that equation (9) includes the payo¤ to the deposit insurance put option. Because of this di¤erence in objective functions, an
unregulated banks’ decisions would generally be socially suboptimal. As we
shall see, banks have an incentive to take on too much leverage, too much risk,
and apply insu¢cient screening e¤ort.

2.3

Formal Statement of the Regulator’s Problem

In the most general speci…cation of the model, the regulator observes deposits
D and the bank’s ex post return r , but the bank’s type i and action pair (s; ¾)
is private information. Thus, the regulator’s problem is to use instruments that
condition on D and r to elicit information about bank type and to in‡uence the
bank’s action choices.
Formally, the problem takes the following steps: First, banks send reports
to the regulator on their type i. The content of these reports cannot be veri…ed
by the regulator so the bank can say anything. However, we know by the
Revelation Principle that as long as we impose the right incentive constraints,
we can restrict ourselves to a direct mechanism where a bank directly reports
its type. Second, based on this report of i, the regulator sets a deposit level
Di , recommends a screening-risk pair fsi; ¾ i g, and sets a schedule of …nes gi (r)
that depends on the portfolio return r. The deposit level is interpreted as
a capital requirement, since capital equals 1 ¡ Di. We refer to a triplet of the
deposit level, screening, and risk, (D; s; ¾); as an assignment. Third, in response
to the assignment and …ne schedule, the bank chooses its screening and risk
levels (which need not equal the (s; ¾) pair recommended in the regulator’s
assignment). Fourth and …nally, the return is realized, …nes and taxes are
assessed, the depositors are paid o¤ (either by the bank or the deposit insurer),
and each household consumes.
Using the Revelation Principle, the regulator’s problem can be summarized
as follows:
Regulator’s Problem

max

D i;si;¾ i;gi (r)

X
i

hi

µ

½Di + ¹i (si ) ¡ °si ¡

8

Z

0

1

¿ gi (r)f i (rjsi; ¾ i ) dr

¶

subject to equation (5), the following truth-telling incentive constraints
Z

Di

(Di ¡ r ) f i (r js i; ¾ i) dr + ½Di + ¹i (si)
Z 1
¡°s i ¡
gi (r)f i (rjsi; ¾ i ) dr ¡ T
¸ max
s; ¾

Z

0
Dj

0

¡°s ¡

Z

0

(Dj ¡ r ) fi (r js; ¾) dr + ½Dj + ¹i (s)
1

gj (r)f i (rjs; ¾) dr ¡ T ; 8i 2 flow; highg ; j 6= i;

the following moral-hazard incentive constraints that for all i
"Z
Di

(s i; ¾ i)

(13)

0

=

arg max
s; ¾

¡°s ¡

Z

0

0

1

(Di ¡ r) fi (rjs; ¾) dr + ½Di + ¹i (s)

(14)

¸
g i(r )fi (rjs; ¾) dr ¡ T ;

and the constraint in equation (10) that lump-sum taxes cover the costs of bank
failure resolution net of …nes collected.
The moral hazard incentive constraints (14) require that, for the deposit
level and …ne schedule speci…ed for each bank type, the recommended values
of screening e¤ort and risk are those that would be chosen by that type. The
truth telling constraints (13) guarantee that banks truthfully report their type.
The max operator on the right-hand side of equation (13) is needed because the
optimal contract does not specify the o¤-equilibrium strategy to be used by a
bank that lies. The utility from this strategy needs to be calculated to properly
assess the value to a bank of misrepresenting its type. Finally, note that while
T is in both sets of incentive constraints it enters as a constant and has no e¤ect
on sets of feasible allocations that satisfy either constraint.

3

Some Useful Comparative Static Results

Before analyzing the complete Regulator’s Problem it is useful to look at the
bank’s incentives in the absence of …nes. These comparative static results in­
dicate the direction in which the bank would change its actions in response to
possible regulatory policies.

3.1

Unregulated banks choose suboptimally low screening

Let us consider …rst the incentives of unregulated banks. In particular, we set
g i (r ) = 0; 8r. According to equation (11), the choice of ¾ does not directly
a¤ect the value of the regulator’s ob jective in the absence of …nes. Thus, the
key concern of the regulator is to move the banks’ screening e¤ort toward the
9

social optimum. According to equation (11), the socially optimal screening
level for bank i is characterized by
¹0i (si ) = °;

(15)

that is, the marginal increase in the mean return must equal ° , the marginal
cost of screening. However, according to equation (9), the privately optimal
screening level is characterized by
Z Di
@f i (rjsi; ¾ i )
¹0i (si ) = ° ¡
(Di ¡ r )
dr:
(16)
@s
0
We now show that if D is not too big relative to the mean of the distribution
then the bank’s choice of screening is strictly lower than the social optimum.
Equations (15) and (16) di¤er by a term that captures the way the deposit
insurance put option varies with s. Using equations (1), (2), and (4), we can
sign this term, as follows:
Z Di
Z Di
@fi (rjsi ; ¾ i)
@f (r j¹i (si ); ¾ i)
(Di ¡ r)
dr = ¹0i (si )
(Di ¡ r)
dr
@s
@¹
0
0
< 0 if D < r ¤ (¹; ¾) :
(17)
According to equation (17), the left-hand side of (16) exceeds the left-hand side
of (15) as long as the deposit level D is not too big.8 If this condition holds then
equation (3) implies that the value of s i ensuring equation (16) is strictly lower
than the value of si implied by equation (15). In other words, the unregulated
bank’s screening choice is strictly less than the socially optimal screening level.
The condition that D is not too big is not too restrictive. For example, in the
case of the normal or log normal family of distributions, inequality (1) holds
for all r < ¹. Since D · 1 and the mean of the gross return exceeds unity
(assuming that the bank expects a positive net return to its investments), then
the condition that D not be too big holds automatically.
If bank screening e¤ort were observable, the regulator could presumably
mandate the optimal screening level directly. However, throughout this paper
we assume that screening e¤ort is unobservable to the regulator. So how might
the regulator induce the bank to increase its screening level? The conventional
regulatory tool is increased capital. It turns out that mandating higher capital
does indeed induce higher screening e¤ort. A second, less obvious tool, which
will be an important focus later in this paper, is to induce the bank to reduce
its risk choice. In the following, we explore each of these approaches in turn.

3.2

Inducing higher screening e¤ort via capital regulation

Suppose …nes are excluded from the available regulatory instruments, so the
only regulatory tool is capital requirements. We show here that higher capital
8 In fact, the condition D < r¤ (¹; ¾) is su¢cient, but not necessary for the left-hand side
of equation (17) to be negative. All that is needed is for D to be su¢ciently small that the
set of r < r ¤ (¹; ¾) dominates the sign of the weighted integral in equation (17).

10

tends to induce higher screening. Recall that capital simply equals 1 ¡ D,
so mandating increased capital is equivalent to mandating a lower value for
D . Suppose the regulator sets D and the bank then chooses s. The following
comparative static result holds:
Proposition 1: Suppose s > 0; D < r¤ (¹; ¾) : Then, holding ¾ constant,
the bank’s choice of s is decreasing in D .
(Proof See Appendix A.)
According to Proposition 1, the regulator can induce banks to increase s by
requiring them to reduce D , in other words, by increasing capital.

3.3

How does the risk level a¤ect screening e¤ort?

For most of this paper we assume that the bank’s choice of risk, ¾, is private
information. But suppose for a moment that ¾ were observable to the regulator,
and that the regulator could mandate a ¾ level for the bank. How would the
regulator’s choice of ¾ a¤ect the bank’s choice of s? In this section we show
that, for a wide range of speci…cations, s would be decreasing in ¾. That is,
the regulator could induce higher screening (thereby o¤setting the distortions
induced by the deposit insurance put option) by reducing the bank’s portfolio
risk.
To demonstrate this assertion, let us totally di¤erentiate the bank’s …rst
order condition (16) with respect to ¾ and rearrange to get
hR
i
D
¡ 0 (D ¡ r) f s¾ (r js; ¾) dr
@s
= RD
(18)
@¾
(D ¡ r ) f ss(r js; ¾)dr + ¹00 (s)
0

According to equation (4),

f s¾ (r js; ¾ ) = f ¹¾ (rj¹; ¾) ¹0 (s )
so equation (18) can be written
hR
i
D
¡ 0 (D ¡ r ) f ¹¾ (rjs; ¾) dr ¹0 (s)
@s
= RD
@¾
(D ¡ r) fss (r js; ¾) dr + ¹00 (s)

(19)

0

According to second-order condition (25) in the Appendix A, the denominator
@s
of (19) is negative. So to sign @¾
we must determine the sign of the numerator
of equation (19). This object cannot be signed unambiguously. ¹0 (s) > 0 by
construction, so

Z

0

@s
< 0 if and only if
@¾

(20)

(D ¡ r ) f ¹¾ (rj¹; ¾) dr < 0:

(21)

D

11

The integral in equation (21) is the second (cross) derivative of the deposit insur­
ance put option with respect to the mean and standard deviation of the return
distribution. So equations (20) - (21) say that increased portfolio risk induces
a reduction in screening e¤ort if and only if the sign of this cross derivative is
negative.
The sign of this cross derivative depends on the sign of f ¹¾ (r ), which in turn
depends on where r is located relative to the mean and standard deviation of the
distribution, as well as on the shape of the distribution itself. It is a property
of most commonly used two-parameter distributions that f¹¾ (r) < 0 for all r
su¢ciently small. In particular, if f is normal, then
p
f¹¾ (r j¹; ¾) < 0 if r < ¹ ¡ ¾ 3:
(22)
We were unable to obtain a similar analytic result for the log normal distribution,
but a grid search over a wide range of ¹’s and ¾’s reveals that, for f lognormal,
a su¢cient condition for f¹¾ (r j¹; ¾) < 0 is9
p

r < e E log( r)¡¾ (log(r))

3

(23)

:

where E log (r) and ¾ (log(r)) denote the mean and standard deviation, respec­
tively, of log (r). The analogy between equations (22) and (23) is obvious.
These results suggest that, unless ¹ is very small or ¾ is very big, inequality
(21) should hold. To check this conjecture, we numerically evaluate the lefthand side of inequality (21) on a grid of f¹; ¾ :D g combinations with support
¹
¾
D

2

[0:5; 2:5]

2

[0:1; 1:0]

2

[0:05; 1:0]

If we think of the return horizon as one year, these grids encompass the realistic
cases. For the log normal distribution, inequality (21) holds for all values of
f¾; Dg as long as ¹ ¸ 1:2 (that is, a mean net return of at least 20%). If ¹ ¸ 1:0
(positive mean net return), inequality (21) holds except for very high risk levels
( ¾ ¸ :7), and is there only when D = 1 (i.e., zero capital). We perform the
same experiment with the beta distribution with support (0; 3) that is used in
Section 4.2, below.10 With this distribution, inequality (21) holds for all f¾; Dg
in this grid as long as 1:0 · ¹ · 2:0 (that is, the mean net return is between
9 More

s .t.

precisely, there exists an r¤¤ (¹; ¾) satisfying

log [r ¤¤ (¹; ¾)] > E log (r) ¡ ¾ (log(r ))
f¹¾ (rj¹; ¾) < 0 if r < r¤¤ (¹; ¾) :

p

3

p
In practice, we …nd that for most values of f¹; ¾g ; log [r¤¤ (¹; ¾)] ¼ E log (r) ¡ ¾ (log(r)) 3:
1 0 The standard beta distribution has support (0; 1) : As in our baseline numerical example
of section 4.2, we assume in these experiments that the support on r is (0; 3), so the relevant
density function f (rj¹; ¾) is the standard beta evaluated at fr=3; ¹=3; ¾=3g.

12

0 and 100%). Thus, @@s¾ < 0 for these two distributions when the mean and
variance are in the empirically plausible region.
This section showed that, for a wide variety of plausible distributions, reduc­
ing the risk of a bank’s portfolio tends to increase its level of screening e¤ort.
But, for most of this paper we assume that the regulator cannot observe the
bank’s portfolio risk. How then are the results of this section useful? While the
regulator cannot mandate a risk level, the regulator may be able to convince the
bank to choose lower risk by indirect means, such as ex post return-contingent
…nes. According to the results of this section, doing so will result in the bank
choosing a higher screening level. This is what we …nd in our numerical simu­
lations, described in Sections 4.3 and 4.4.2, below.

4

An Example

It is very di¢cult to characterize the Regulator’s Problem analytically. First, it
contains a moral hazard problem with two dimensions (screening and risk) on
which the bank chooses its hidden action. Second, the moral hazard is preceded
by hidden information on bank quality. Third, the regulator also chooses a de­
posit level, which, while observable, complicates the problem because it directly
a¤ects incentive constraints.
Because of these di¢culties, we adopt the strategy of solving numerical ex­
amples to learn about the properties of this model. In this section we …rst
describe our approach to solving the model. We then use this approach to
solve a set of examples that illustrate how …nes are an e¤ective way to control
risk, and by extension, control screening. In particular, our examples illustrate
that high-return …nes can be used to deter risk-taking by the low-type banks
whether or not bank type is public information.

4.1

Solving the model numerically

We solve the Regulator’s Problem numerically by formulating it as a linear program. Linear programs are an e¤ective tool for computing solutions to mech­
anism design problems.1 1 They can be used to solve problems with arbitrary
speci…cations of preferences and technologies. Furthermore, linear programs can
be e¢ciently solved using widely available, high quality software.
To formulate the problem as a linear program, one …rst must discretize all
underlying variables. In our model, we discretize returns r, and …nes g, screening
levels s, risk choices ¾, and deposit levels D. The regulator then chooses a
joint probability distribution over the possible combinations of these variables.
Embedded in this joint probability distribution is the terms of the regulatory
contract. Thus, this solution method allows for the possibility of randomized
contracts. Indeed, allowing for randomization is the key step in transforming
1 1 Myerson (1982) and Pre scott and Townsend (1984) are early pape rs that set these prob­
le ms up as linear programs. For a survey on the use of linear programming methods to solve
mechanism design problems see Prescott (1999).

13

the problem into a linear program. However, we will focus primarily on cases
where the contracts are deterministic (that is, where the probability distribution
associated with the optimal contract is degenerate). 12
The major complication with this approach is checking the truth-telling in­
centive constraints. As mentioned above, the truth-telling constraints guarantee
that the utility from correctly reporting type dominates the utility from lying
and then taking some feasible o¤-equilibrium strategy. One way to guarantee
truth-telling is to check each possible o¤-equilibrium strategy. This is the stan­
dard method, as in Myerson (1982). Unfortunately, as we describe in Appendix
B, our problem involves an astronomical number of such constraints. Instead,
we use a method developed in Prescott (2003) that takes advantage of the max
operator in equation (13) to more e¢ciently check the truth-telling constraints.
A detailed, self-contained, description of the numerical solution procedure can
be found in Appendix B.

4.2

Parameterization

We assume that there are four possible levels of screening e¤ort,
s 2 f0:2; 0:3; 0:4; 0:5g
and two possible standard deviations for the bank portfolios:1 3
¾ 2 f0:6; 1:0g:
We assume that the increasing, concave function ¹i mapping bank type i’s
screening e¤ort into its portfolio mean takes the negative exponential form:
¡ ax
¢ ¡a s
ax
in
¹i (s) = ¹m
¡ ¹m
¡ ¹m
e i
i
i
i
© ax m in ª
where, for i = low; high, ¹m
; ¹i ; ai are non-negative type-speci…c parai
in
ax
in
ax
< ¹m
. Note that ¹i (0) = ¹m
and ¹i (1) = ¹m
. In the
meters with ¹m
i
i
i
i
parametric examples we present below, we assume that
¹min
low = 0;

¹min
high = 0:6;

¹max
low = 1:7;

¹max
high = 2:3;

alow = ahigh = 5:
12

In the case where bank type is observable, the optimal contract can always be achieved
with deterministic contracts. Intuitively, the only way randomization could be useful is if
the bank makes a decision prior to the realization of the random assignment. When type is
observable, the only choices that the banks make are their screening and risk choices. These
are made after the assignment is made, so the regulator need never randomize the assignment.
1 3 The use of only two risk levels can be motivated by the result in Marshall and Prescott
(2001), that if banks’ portfolio returns are log normal, and if banks can choose from a closed
interval of risk levels, their optimal choice would always be one of the two endpoints. That
is, for any give combination fD; sg, an interior solution for ¾ is never chosen by the bank.

14

Note that the ¹h ig h (¢) schedule represents a parallel upward shift of the ¹low (¢)
schedule.
Our solution method requires discretization of the distributions. Since dis­
cretization e¤ectively imposes upper and lower bounds on the distribution, it is
natural to start with distributions that have bounded support. In particular,
we assume that all return distributions are from the beta family of distributions.
Since the beta is a two-parameter distribution, it is completely described by its
mean and variance (along with its upper and lower bounds). We assume the
support of all distributions is the open interval (0; 3:0). We thus must construct
a beta distribution with this support for each combination of ftype; s; ¾g : We
then discretize these distributions on the following seven-point grid:14
r 2 f0:04; 0:5; 1:0; 1:5; 2:0; 2:5; 2:96g:

Finally, we set the cost of capital ½ = 0:05, the social cost of …nes ¿ = 0:01, and
the cost of screening e¤ort ° = 1:0.
Before turning to the optimal regulation in this example, let us look at the
optimal bank choices in the unregulated case where no …nes are imposed and
D = 1 (zero capital). Table 1 gives the value of the bank’s objective function
(equation (9)) and the regulator’s objective function (equation (11)) for each
type and each choice of fs; ¾g. (The value of the bank’s objective excludes
the lump sum tax, which does not a¤ect bank incentives.) Note …rst that the
regulator wants both banks to choose s = 0:4. Lower values of screening e¤ort
are insu¢ciently productive, but the highest value s = 0:5 incurs a suboptimally
high disutility of e¤ort. Note also that, with no deadweight cost of bankruptcy,
the regulator is indi¤erent between high and low risk.
Turning to the values of the banks’ objectives, the high type’s preferred
action is s = 0:4 and ¾ = 1:0. Since this maximizes the regulator’s objective,
the high-type bank is self-regulating in this example. In contrast, the low-type
bank prefers the highest risk level with s = 0:3. So the task facing the regulator
is to induce the low-type bank to increase its screening to s = 0:4: Note also that
if the low type were forced to choose the lower risk level, the optimal choice of
screening would coincide with the regulatory optimum s = 0:4. However, if the
high type were forced to choose low risk, its ob jective would be maximized at the
suboptimally high level s = 0:5. These e¤ects of risk reduction are examples of
the result in Section 3.3 that for a large set of plausible portfolio distributions,
screening is decreasing in risk. This will be an important consideration in the
optimal regulatory design. Inducing the low type directly to choose the socially
1 4 For each combination of screening e¤ort and risk, there is a unique beta distribution
on support (0; 3). We discretize each distribution by evaluating the be ta density at each of
the seven grid points, and then adding " i to the ith grid point’s probability, where f"i g7i=1
P7
2
are chosen to minimize
i=1 "i subject to the constraints that the resulting probabilities
sum to unity and that the mean and variance of the discrete distribution exactly match
the mean and variance of the original beta distribution. Since very small probabilities often
introduce numerical instability in our solution algorithm, we then do a second round of ad
hoc adjustments to ensure that no single probability is less than 0.001, that the means hold
exactly, and that the variances are close to the target variances.

15

optimal screening level is di¢cult. However, if the regulator can induce the low
type to choose low risk, it will be optimal for the bank to then select the optimal
screening level, which is the prime concern of the regulator.
At …rst glance, the results in Table 1 suggest that the screening level s = 0:2
is extraneous since it appears to be neither privately nor socially optimal: No
unregulated bank would ever choose this screening level, nor would the regulator
ever mandate it. However, as we shall discuss below, once …nes are imposed
screening may be so unappealing to the bank that a s = 0:2 deviation is a
possibility that needs to be prevented.

4.3

Optimal Regulation when bank type is observed

Marshall and Prescott (2001) studied optimal regulation in this model when
bank type is observable. It is useful to revisit this simpler case in the context of
our baseline parameterization to provide a benchmark against which the results
with unobserved heterogeneity can be compared. Several of the forces opera­
tional in that model are operational in the heterogenous agent model as well.
Furthermore, the implications of the model for this case raise some interesting
issues that the analysis with unobserved heterogeneity may be able to clarify.
We determine the optimal contract for each of the two types, both for pure
capital requirements (no …nes permitted) and for the case where both capital
requirements and ex post …nes are available. The results in the case of pure
capital regulation for the two bank types are displayed in the …rst column of
Table 2. The optimal contracts induce high screening and high variance for
both types. The high type is fully leveraged, while the low type has a capital
level of 14%. The low type needs this capital requirement to induce socially
optimal screening. This is an example of Proposition 1 in Section 3.2, above.
In contrast, as we discussed in Section 4.2, above, the high type chooses this
optimal screening level even in the absence of a capital requirement.
We now introduce …nes as a second regulatory instrument in addition to
capital. The results for this case are displayed in the …rst column of Table
3. Note that in this case …nes completely displace capital, since capital is a
more costly way of in‡uencing bank incentives than state-contingent …nes. In
particular, the optimal contract assigns the low type 0% capital (D = 1), with
the low risk and s = 0:4. The low type is induced to take this strategy by the use
of a …ne of 1.8910 that is assessed if the highest return is produced. There are
no …nes imposed on the other returns. Fining a bank for doing well may seem
counterintuitive but it is actually a simple application of the likelihood ratio
principle; which is relevant for optimal incentive contracts. This principle can
be understood as follows: Fines have two e¤ects on the moral-hazard incentive
constraints: They hurt banks who take the recommended strategy, and they
also hurt those that deviate. The relative size of these e¤ects at a particular

16

return level r , is determined by the likelihood ratio, LR (r ), de…ned as 15
LR (r ) ´

prob (rj deviating strategy followed)
:
prob (rj recommended strategy followed)

This ratio can be interpreted as the prevention per unit of …ne assessed in
equilibrium. It is generally desirable to impose the …ne on the return with the
highest likelihood ratio.
For this example, the investment strategy of most concern to the regulator
is where s = 0:2 and ¾ = 1:0:16 The likelihood ratio for this deviation is
highest at r = 0:04, the return in the extreme left-hand tail of the distribution:
The regulator would like to impose the …ne on this return but cannot because
of limited liability. The return with the next highest likelihood ratio for this
deviation is r = 2:96, the return in the extreme right-hand tail. Fines can be
assessed for this return without violating limited liability, so this is the return
level that receives the …ne.
Inspection of this contract reveals why capital is not used. The …ne on the
highest return is strictly below the maximum feasible …ne, given limited liability,
of 1.96. If this …ne were insu¢cient to induce optimal screening, it could be
increased at the margin, thereby strengthening incentives without using costly
capital. In our numerical simulations, we …nd that as long as …nes have a
low regulatory cost compared to capital, capital and …nes coexist in the same
contract only if at least one …ne is at the bound imposed by the limited liability
constraint.
Another important point that is illustrated by this example is that the bind­
ing incentive constraint need not be near the equilibrium choice. That is, it
need not be a local incentive constraint.
In particular, while the lowest screening level s = 0:2 is never chosen by an
unregulated bank, the binding incentive constraint in this example is to prevent
deviation from fs = 0:4; ¾ = 0:6g to fs = 0:2; ¾ = 1:0g. The reason is that the
high …ne on the right-hand tail of the distribution induces the bank to reduce
the probability of the highest return. But there are two ways it can do so: by
reducing ¾ or by reducing ¹ via a reduction in screening e¤ort all the way to
s = 0:2. The former is the e¤ect desired by the regulator; the latter needs to be
avoided as represented by the binding incentive constraint.
This example illustrates why the …rst-order approach to incentive constraints
does not work in this model. In the …rst-order approach (see Hart and Holm­
strom (1987)) the incentive constraints are replaced with the …rst-order condi­
tions to the agents subproblem; but these are necessary rather than su¢cient
conditions. Without strong assumptions, there is no guarantee that a solution
to the program with the …rst-order conditions is the same as the solution to
the global program. In particular, the program utilizing …rst-order conditions
would treat certain globally infeasible allocations as if they were feasible. In our
example, since local incentive constraints do not bind, it would actually select
1 5 For more on likelihood ratios in moral hazard models see Hart and Holmstrom (1987).
1 6 This is the deviating strategy with respect to which the incentive constraint binds.

17

one of these (infeasible) allocations as the solution to the regulatory problem.
Our linear programming method is a global method so we do not have to worry
about this possibility.

4.4

Optimal regulation with unobserved heterogeneity

According to the results in Section 4.3, the optimal structure of ex post …nes
for the low type is to impose large …nes on the highest return level. In other
words, banks are penalized for doing extremely well. This seemingly perverse
result is actually quite intuitive, since an extremely high return to a low-type
bank is a reasonably good signal of excessive risk-taking. The penalty structure
could be duplicated by requiring the bank to issue warrants or convertible debt
with a high strike price. However, one important concern with such a regulatory
…ne schedule is that some banks may have a high return not because they took
excessive risks, but simply because they are good banks. They may have better
management or they may have a more favorable investment opportunity set. In
this section, we address this concern by solving the Regulator’s Problem de…ned
earlier, in which bank type is unobservable. With unobserved bank quality, we
can investigate the trade-o¤s between the good and bad incentive e¤ects of
high-return …nes.
4.4.1

Optimal capital requirements without …nes in the presence of
unobserved heterogeneity

When …nes are not used, the optimal capital requirements when bank type is
unobserved in the baseline case are displayed in the last three columns of Table
2. The table gives the optimal regulatory contracts for the two types as the
fraction of high types, h high , varies from .01 to .995.
When h high is not too big (less than 86%), the optimal regulatory strategy is
to assign both types 14% capital, which is the optimal type-observable contract
for the low type. That is, no e¤ort is made to separate types. The reason is that
this contract is the least expensive contract (from the regulator’s standpoint)
that induces the low-type bank to choose the highest screening level. While
this contract is decidedly inferior from the perspective of the high-type bank,
the only alternative from the regulator’s perspective is to let the low-type bank
choose a lower screening level. The social cost to this alternative exceeds the
social cost of the capital requirements imposed on the high-type bank.
When h high is very high (87% or higher), the regulator again makes no
e¤ort to separate type. For these values of hhig h , the regulator simply assigns
both types of banks zero capital, which of course is the optimal type-observable
contract for the high type. The low type bank responds by setting s = 0:3, a
socially suboptimal screening level. There are so few low-type banks that the
regulator is willing to tolerate this suboptimal screening by low-type banks in
order to save the capital costs incurred by the high-type banks.
Finally, for hh ig h in a narrow range near 0:86; the optimal regulation is to
separate types by o¤ering di¤erent contracts to each type. While this would be
18

impossible with non-random contracts (since both bank types would choose the
lower-capital alternative), it is possible in theory to separate types by o¤ering
the low-type bank a random contract while o¤ering the high-type bank a nonrandom contract. The reason this is possible is that the bank’s objective function
is nonlinear in D because it incorporates the payo¤ to the deposit insurance put
option. The expected payo¤ of a put option is strictly convex in its strike
price. In our example, the put option term has strictly greater curvature
for the low type bank than for the high type bank, so, as an implication of
Jensen’s inequality, the low-type bank has a greater preference for randomness.
To separate types, one simply o¤ers the high type bank a nonrandom capital
level while o¤ering the low type bank a random capital level with a somewhat
lower mean. We see this in the second-to-last column of Table 2 for h hig h =
86%. The optimal contract exploits this possibility by giving the high type
a deterministic contract with 1% capital, while giving the low type a mixed
contract that imposes 14% capital with 7% probability and zero capital with
93% probability. We …nd it interesting that in the numerical simulations we
have studied, this type of randomized contract is optimal only for a very narrow
set of parameters. For the vast majority of examples, separation of types with
pure capital regulation is suboptimal.
4.4.2

Optimal regulation using both capital and …nes in the presence
of unobserved heterogeneity

The optimal contracts for the baseline parameterization when both …nes and
capital are used are displayed in Table 3. The …rst two rows of Table 3 give
details of the contract for the low type. The second row is only used when the
optimal contract is a mixture of two contract elements. The third row gives the
optimal contract for the high type. (In the baseline parameterization mixed
contracts are never used for the high-type bank.) As described in Section 4.3,
the …rst column gives the optimal type-observable contracts. The remaining
three columns give the contracts as the fraction of high types hh ig h ranges from
1% to 99.5%.
When bank type is unobservable, the optimal type-observable contracts for
the two bank types cannot be implemented simultaneously. The reason is that
the contract for the high type (zero capital, zero …nes) is more attractive to the
low type than its own contract (zero capital, high …ne on the highest output
level). So, the low type would invariably misrepresent its type, receive the
high-type contract, and then use a suboptimally low screening level.
To remedy this problem the regulator can choose one of three alternatives:
1. Assign the low type its optimal observable-type contract, and impose suf­
…cient penalties on the high type (either in the form of …nes or capital
requirements) so that the low type has no incentive to misrepresent type.
2. Assign the high type its optimal observable-type contract, and give the
low type a (socially suboptimal) contract that is su¢ciently attractive so
that low-type banks have no incentive to misrepresent type.
19

3. Assign neither type its optimal observable-type contract; and craft a set
of contracts such that neither bank type has an incentive to misrepresent
type.
Not surprisingly, we …nd that the optimal choice among these three alterna­
tives depends on the fraction h high of high-type banks. In particular, alternative
(1) is used when h hig h is relatively low, alternative (2) is used when h high is
extremely high, and alternative (3) is used for intermediate values of h high .
In the baseline case, the low-type contract is the same as the optimal contract
under type-observability as long as the fraction of high types is not too high
(hhig h below 88%). This is clearly evident in Table 3. For these values of h high ,
however, the contract for the high type imposes …nes at return levels 2:0 and
2:5. These …nes are substantial. At return 2:0, over 89% of the bank’s pro…t
(after paying o¤ depositors) is …ned away; at return 2:5 approximately 29% of
the pro…t is …ned. So, while the low type receives the same contract as she
would if type could be observed, the high type must pay heavy …nes in order to
dissuade the low types from masquerading as high types.
Neither …ne in the high-type contract is constrained by limited liability, so
one might think that these two …nes play distinct roles. This is partially correct.
Two constraints bind in this contract. The …rst binding constraint is the truth­
telling constraint preventing low types from pro…ting by misrepresenting their
type and choosing fs; ¾g = f0:3; 1:0g. The second binding constraint is the
incentive constraint preventing the high type from choosing the suboptimally
high screening e¤ort of s = 0:5. Consider Table 4 , which gives the values of
bank and regulatory objectives for fs; ¾g pair under both the optimal low-type
contract and the optimal high-type contract. Note that the highest value of the
low type’s objective from truth-telling is 1.7924, attained at the socially optimal
action pair fs; ¾g = f0:4; 0:6g. But the low type can attain this precise value
by misrepresenting herself as a high type and choosing the socially suboptimal
action pair fs; ¾g = f0:3; 1:0g. Thus, the truth-telling constraint binds with
respect to this deviating action. If the …ne on either return 2:0 or 2:5 were
reduced, this constraint would be violated and the low-type bank would pro…t
from lying and misbehaving.
Note also that the high-type’s objective attains its maximum of 1.69761 at
either the socially optimal action pair fs; ¾g = f0:4; 0:1g or at the socially
suboptimal pair fs; ¾g = f0:5; 0:1g. Thus, the high-type’s incentive constraint
binds with respect to this latter action. If the …ne on r = 2:5 were reduced
(without a concomitant reduction in the …ne on r = 2:0), this constraint would
be violated. 17 In other words, if only a …ne at r = 2:0 were used to dissuade
low-type banks from lying, the …ne itself would induce high-type banks to screen
excessively, an action that bank would never take in the unregulated case. An
additional …ne at r = 2:5 is needed to correct this perverse incentive. This is
an example of a possibility that can occur in optimal mechanism design when
both moral hazard and adverse selection are present: A contractual provision
1 7 If both …nes are reduced at an appropriate rate, the incentive constraint need not be
violated.

20

designed to alleviate the adverse selection problem can itself exacerbate the
moral hazard problem, requiring additional regulatory correction.
This contract on the high types is inferior to their optimal type-observable
contract, both from the perspective of social utility and private bank utility.
Table 5 reports the social value of each contract, the private value of each
contract before taxes (which is what matters for incentives) and the private
value of the contract after taxes. For purposes of utility comparison, we treat
the type-observable case as if the tax levies on the two types acted as actuarially
fair deposit insurance premiums. That is, the costs of failure resolution for a
given bank type are assessed only on the banks of that type. 18
Table 5 shows that the social utility of the optimal observable-type contract
for high banks is 1.71993. In contrast, the social utility of the equilibrium
high-type contract when bank type is unobservable is lower. For h hig h below
0.88, this value is 1.71867. This small di¤erence is the social cost of private
information about bank characteristics. The reason this cost is so small is that
in both contracts the high bank makes the optimal screening choice (s = 0:4), so
the only di¤erence from a societal standpoint is the deadweight cost of the …nes.
When ¿ is only 1%, this deadweight cost is small; the regulator is willing to pay
this cost when there are not too many high banks in the economy. However,
the private cost to the high banks of type-unobservability is much larger. The
private after-tax value to the high bank of its optimal observable-type contract
equals the social value of 1.71993, 19 while the corresponding private value of the
equilibrium contract under type-unobservability varies from 1.63905 to 1.70902
as h high varies from 0.01 to 0.88. All these values are less than the private value
accruing to the high type when bank type is observable. In contrast, Table
5 shows that, when bank type is unobservable, the low-type banks receive a
private after-tax value that in all cases exceeds what they receive under type­
observability. In particular, this value when type is observed is 1.11990, whereas
when type is private information these values range between 1.12067 and 1.19065
(for h hig h between 0.01 to 0.88).
Thus, the optimal contracts under type-unobservability in e¤ect transfer
value from high to low types. In this sense, the more productive banks bear the
full costs of type-unobservability, even though these high-type banks are com­
pletely self-regulating when type is public knowledge. This is relevant for regu­
latory practice. It is often argued that the vast majority of banks have strong
incentives to behave in a prudent and value-maximizing manner. These banks
are essentially self-regulating, since their private incentives are well aligned with
social imperatives. However, if there are enough po or quality banks, and if the
regulators have imperfect information about bank quality, then it may be nec­
essary to impose a heavy superstructure of regulation on the high banks just in
order to a¤ect the incentives of the low banks.
1 8 Note

that this sort of type-dependent tax levy is infeasible when type is unobservable.
these two values are identic al follows from our assumption that, when type is ob­
servable, the costs of bank failure resolution for type i banks are assessed only on banks of
that type, along with the fact that the value of the deposit insurance put option exactly equals
the cost of paying o¤ the depositors in failed banks.
1 9 That

21

Contrast the optimal contracts in the case described above to the case where
the fraction of high types reaches 99.5% or higher. This case is displayed in the
last column of Table 3. At this point the regulator simply assigns the high type
its optimal contract under type-observability. With our parameterization, this
contract is completely unregulated (zero capital, zero …nes). Since we require
both …nes and capital to be non-negative, there is no way to dissuade the low­
type bank from misrepresenting type. (No other contract can dominate the zero
capital, zero …ne contract in the low type’s private valuation.) As a result, the
two types must be assigned the same contract and the truth-telling constraints
hold trivially. The low-type bank then chooses a suboptimally low screening
level of 0.3. However, the fraction of low-type banks is so small that the
regulator simply does not care about the suboptimally low mean output from
these banks. (For higher values of ¿ , this threshold is reached at lower values
of h hig h .)
Finally, let us brie‡y consider the intermediate case where h high is between
0.89 and 0.99: When the fraction of high types is in this range, the social cost of
…nes on the high-type banks is su¢ciently onerous that the regulator wishes to
reduce these …nes. But to maintain truth-telling, the regulator must simulta­
neously increase the value of the low-type contracts. If it did not do so, the low
type banks would misrepresent themselves as high types. In all the examples
we have computed, the regulator gives additional value to the low-type banks
by assigning them a mixed contract that randomizes between a high-…ne, low
(or zero) capital contract and a low-(or zero-) …ne, high capital contract. As
an example, consider the second-to-last column in Table 3, which exhibits the
optimal contract in our baseline model with h high = 0:89. As compared to
the contract for h high = 0:88, the regulator reduces the …nes on the high type,
imposing a zero …ne on r = 2:5 and a lower …ne on r = 2:0. In order to
increase the utility of the low type, the regulator randomizes between the opti­
mal type-observable contract (zero capital, …ne of 1.8910 on the highest return)
and the optimal type-observable contract with capital only (14% capital, zero
…nes). This random contract gives the low-type bank higher private value, …rst
because it imposes a lower expected …ne, and second because (as discussed in
Section 4.4.1) it exploits the di¤erences in the curvature of the two types’ utility
functions. As shown in the second-to-last column of Table 5, the social value
of this mixed contract is lower than the non-random contract, but the regulator
is willing to forego this value in order to reduce the …ne on the high-type banks.
Note that both the private and social values of the high-type contract are higher
in the case than when h hig h = 0:88.

5

Conclusion

This paper studies a bank capital regulation model in which deposit insurance
causes a potentially lower level of expected output because it creates a taste for
risk that reduces marginal incentives to exert screening e¤ort. Capital regulation
of the sort commonly seen in regulatory practice is fairly e¤ective at o¤setting
22

this distortion. However, state-contingent tools are shown to be more powerful.
Fines can induce optimal screening e¤ort while economizing on (or eliminating
entirely) the use of costly capital.
We learned a number of lessons from this exercise. First, a powerful reason
the regulator might seek to deter risk is to induce banks to choose a higher mean
portfolio. Second, unobserved heterogeneity does not eliminate the usefulness
of state-contingent …nes as a regulatory tool. On the contrary, …nes are still
useful, both to deter misrepresentation of type and to deter suboptimal choices
after type has been truthfully revealed. Third, the likelihood ratio principle
guides the choice of return levels on which to impose the …nes. In particular,
…nes at the highest return level tend to be used whenever the objective is to deter
high risk-taking. Fourth, for most distribution of types, the low type receives
its optimal type-observable contract. The utility given to the low type by these
contracts is at least as great as that under type-observability. In contrast, the
high type only receives its type-observable contract when there are so few low
types that the regulator is unconcerned with separating types. Otherwise, the
high type is given a contract that provides a strictly lower utility level than
he would receive under type-observability. Finally, …nes are potentially a less
costly way of separating types than the pure capital requirements that are the
focus of much regulatory practice. When …nes are precluded, the regulator
generally gives up any attempt to separate types, even though this is feasible in
principle via randomization. However, when …nes are included in the regulatory
tool kit (with a relatively low cost ¿ of 0.01), the regulator almost always chooses
to separate types.
The model justi…es the regulatory focus on capital adequacy and safety-and­
soundness (interpreted as risk reduction), since both of these approaches can
potentially o¤set the distortions induced by the deposit insurance put option.
However, a reservation one might raise with the results of this paper is that
state-contingent …nes per se are not typically observed in regulatory practice.
Furthermore, the equilibrium contracts in this paper often require …nes on high
returns, an approach that could encounter political and even legal obstacles. In
future research, we are considering other regulatory instruments, such as costly
risk audits, that have the potential of delivering similar results as those found
in this paper while conforming more closely to observed practice.

23

A

Appendix A: Proof of Proposition 1

When …nes are set to zero, the bank’s objective in equation (9) becomes
Z

D
0

(D ¡ r ) f (r js) dr + ½D + ¹ (s) ¡ °s ¡ T :

If s > 0, then the …rst- and second-order necessary conditions for optimal choice
of s are:
Z
D

0

and

Z

0

(D ¡ r) fs (r js) dr + ¹0 (s) ¡ ° = 0

D

(24)

(D ¡ r) f ss (rjs) dr + ¹00 (s) < 0

(25)

To determine s0 (D ), the response of s to a change in D, totally di¤erentiate
equation (24) with respect to D :
"Z
#
D

Fs (Djs) + s0 (D)

0

which implies
s0 (D ) = R D
0

(D ¡ r) fss (r js ) dr + ¹00 (s) = 0
¡Fs (D js)

(D ¡ r) f ss (rjs) dr + ¹00 (s)

:

(26)

(27)

Second order condition (25) implies that
sign [s 0 (D)]

=

sign [F s (Djs)]

(28)

=

sign (¹0 (s) [F ¹ (D j¹; ¾)])

(29)

According to equations, (1), (2), and (29), s0 (D ) < 0, 8D < r ¤ (¹; ¾) as in the
statement of the proposition.

B

Appendix B: Solving the Regulator’s Prob­
lem with Heterogeneous Agents

We solved our numerical examples by formulating the Regulator’s Problem as a
linear program and then solving the problem using standard linear programming
code. There are two steps to the linearization. The …rst step is to allow random­
ization in the contractual terms. This means that the regulator may randomly
recommend (D; s; ¾) combinations to each bank type. We write this probabil­
ity distribution as ! i(s; ¾; D). Fines now need to depend on the realization of
! i (s; ¾; D). Fines could also be random but because of the linear preferences
and objective function we can write them as gi (r; s; ¾ ; D ). The second step of
the linearization is to discretize the sets of variables, that is, the g, r , s, ¾, and
D . These grids are straightforward except for the …ne grid. The upper bound
24

on …nes depends on D and r because of limited liability. For this reason we use
as our …ne grid f0; maxf0; r ¡ D gg. Because of the linear preferences, a two
point grid is all we need to capture all relevant …nes. (Lotteries over the two
points capture everything in between.)
To formulate the problem as a linear program we solve for the joint distri­
bution over the grid of variables for each type. Let ¼ i (g; r; s; ¾; D ) denote the
conditional joint probability of a type-i bank receiving assignment (D; s; ¾); re­
alizing return r (if the recommended (s; ¾) are taken) and being assessed …ne g:
(To keep the notation simple, we do not write out the explicit dependence of g
on the realization of r and D.) Embedded in this ob ject are the two choice vari­
ables of the regulator, the mixing probabilities !i (s; ¾; D) and the …ne schedule
g i(r; s; ¾; D). They are related to the joint distribution as follows:
X
! i(s; ¾; D) =
¼ i (g; r; s; ¾; D )
(30)
g;r

and
g i(r; s; ¾; D) =

X

¼ i(gjr; s; ¾ ; D )g:

(31)

g

Equation (31) gives the expected level of the …ne given the return, assignment,
and reported type, which is all we need for utility and welfare purposes.
Our strategy is to let the regulator directly choose the joint probability
distribution ¼ i (g; r; s; ¾; D). To guarantee that this object is a probability dis­
tribution, we restrict its elements to be non-negative and we require that
X
8i;
¼ i(g; r; s; ¾; D) = 1:
(32)
g ;s;¾; D

In choosing the joint distribution, the regulator is implicitly choosing the
probability of assignments, (30), and the …ne schedule, (31). Still, there is a
technological relationship between the return and the investment strategy that
must not be violated. In particular, the identity
¼ i(g; r; s; ¾; D) = ¼ i (gjr; s; ¾; D)fi(rjs; ¾)! i(s; ¾ ; D )
must hold. This identity can be guaranteed to hold by the system of linear
equations
X
X
_
_
_
8i; D; r ; s; ¾;
¼ i (g; r ; s; ¾; D) = fi ( r js; ¾)
¼ i(g; r; s; ¾; D):
(33)
g

g ;r

The next set of constraints are the moral-hazard constraints. These con­
straints guarantee that the bank takes the recommended investment strategy

25

conditional on truthfully reporting its type, as follows:20 8i; D; s; ¾; s;
b ¾;
b
X
X
¼ i (g; r; s; ¾; D)(r ¡ D ¡ g) +
¼ i(g; r; s; ¾; D)(¡°s)
¸

X

(34)

g;r

g;r ¸D

¼ i (g; r; s; ¾; D)

g;r ¸D

X
b ¾)
b
f i(r js;
(r ¡ D ¡ g) +
¼ i (g; r; s; ¾; D )(¡°s)
b
f i(r js; ¾)
g ;r

The left-hand side of (34) is the utility from taking the recommended action
bb
while the right-hand side is the utility from taking deviating strategy (s;
¾).
Both sides are weighed by the marginal distribution ! i(s; ¾; D ). Notice that
the ½D and T terms have been dropped as they cancel out on both sides of the
constraint.
The last set of constraints are the truth-telling constraints. These constraints
are the most problematic ones for computational purposes. First, we write these
constraints in the same form as used in Myerson (1982). Next, we write them
in a form that is more useful for computational purposes.
Let ± denote a function mapping the set of possible action pairs (s; ¾) into
itself. Intuitively, if the regulator recommends action pair (s; ¾), a possible
deviating action pair would be ± (s; ¾) (so the deviating screening level would
be ±1 (s; ¾) ; and the deviating risk level would be ± 2 (s; ¾)). Using this notation,
one can write the truth-telling constraints in the following way, as in Myerson
(1982):
X X
X
¼ i (g; r; s; ¾ ; D )(r ¡ D ¡ g) +
¼ i(g; r; s; ¾; D)((1 + ½)D ¡ °s)
g ;s;¾; D r¸D

g;s;¾ ;D;r

(35)

¸
+

X

X X

¼ j (g; r; s; ¾; D)

g ;s;¾ ;D r¸ D

g;s;¾ ;D;r

¼ j (g; r; s; ¾; D)

f i(r j± 1 (s; ¾) ; ± 2 (s; ¾))
(r ¡ D ¡ g)
f j (rjs; ¾)

fi (rj±1 (s; ¾) ; ± 2 (s; ¾))
((1 + ½)D ¡ °s);
b 8±; j 6= i
f j (rjs; ¾)

In words, the left-hand side of (35) gives the expected value to a type-i bank of
truthfully reporting its type and selecting the recommended action pair (s; ¾).
The right hand side gives the expected value if that bank misrepresents itself
as a type-j bank, and then, if it receives a recommended action (s; ¾), the bank
actually chooses the deviating actions ± (s; ¾). The key point to note about
this constraint is that it must hold for all possible functions ± . These func­
tions must specify all of the possible o¤-equilibrium strategies that a bank can
take in response to a recommended (s; ¾) pair. There are a huge number of
these functions. Since these recommendations may be random, each possible
o¤-equilibrium strategy needs to include a response to each possible recommen­
dation. In particular, if there are ns possible screening levels and n¾ possible
2 0 While equations (34) - (37) are written di¤erently than the corresponding constraints in
the Regulator’s problem, they are equivalent. They just have not been algebraically manipu­
lated to break out the deposit insurance put option as a separate term.

26

(n n )

risk levels, then there are (ns n¾ ) s ¾ constraints of the form (35) per (i; j)
pair, deposit combination.2 1
Fortunately, this serious curse of dimensionality can be dealt with by reformulating the truth-telling constraint. In this reformulation, an additional
choice variable w(s; ¾; D; i; j) is introduced that keeps track of the maximum
o¤-equilibrium utility a type-i agent can receive if he reports his type as j and
is recommended (D; s; ¾). 22 This solution strategy was anticipated in the way
we wrote equation (13) in the Regulator’s Problem, where the utility from o¤­
equilibrium strategies was dealt with by using the max operator rather than
enumerating all the possible o¤-equilibrium strategies. The o¤-equilibrium util­
b ¾;
b D; i; j 6= i :
ity constraints are 8s; ¾; s;
w(s; ¾; D; i; j)

¸

XX
g

X

¼ j (g; r; s; ¾; D)

r¸D

¼ j (g; r; s; ¾; D)

g ;r

f i( rjsb; ¾
b)
(r ¡ D ¡ g) + (36)
fj (r js ; ¾ )

fi (rjsb; ¾
b)
((1 + ½)D ¡ °s):
b
f j ( rjs; ¾ )

These constraints give the most utility a type-i bank can receive if it reports
that it is a type-j bank and is assigned (D; s; ¾ ). This utility is weighed by
! j (s; ¾; D): The o¤-equilibrium utility can now be used to guarantee truthtelling. The truth-telling constraints are
X X
8i; j 6= i;
¼ i(g; r; s; ¾; D)(r ¡ D ¡ g)
(37)
+

X

g ;s;¾ ;D;r

g ;s;¾ ;D r¸ D

¼ i (g; r; s; ¾; D )((1 + ½)D ¡ °s) ¸

X

w(s; ¾; D; i; j):

s; ¾;D

The left-hand side is the utility from telling the truth and taking the recom­
mended action while the right-hand side is the utility the agent would receive
from lying and then taking the best o¤-equilibrium strategy possible.
The result of this reformulation is that for the example in the paper with
2
eight di¤erent investment strategies we only need (ns n¾ ) + 1 constraints per
(i; j) pair, deposit combination to satisfy the above truth-telling condition. This
substantial reduction in the size of the linear program made it feasible for us to
study the problem in this paper.
The program is
2
3
X
X
max
hi 4
¼ i (g; r; s; ¾; D)(r ¡ ¿ g + ½D ¡ s) 5
¼i(¢ )¸0; wi( ¢)

i

g;r;s;¾ ;D

2 1 In

the examples of section 4 there are four possible sc reening levels and two possible risk
le vels, so the total number of constraints (35) pe r (i; j) pair, deposit combination would equal
16,777,216.
2 2 This strategy is based on the one used by Prescott (2003) to deal with a similar mode l
where the shock was to an agent’s marginal disutility of e¤ort.

27

subject to probability measure constraints (32), technology constraints (33),
moral-hazard constraints (34), o¤-equilibrium incentive constraints (36), and
truth-telling constraints (37).
The program is a linear program. There is a …nite number of constraints
and a …nite number of choice variables: ¼ i(g; r; s; ¾; D) and w(s; ¾; D; i; j) for
each type i, j 6= i and each point in the (g; r; s; ¾; D ) grid. We wrote our
code for creating the linear programming coe¢cients in Matlab. The linear
program was then solved by calling Minos, a Fortran program solver developed
at the Stanford Systems Optimization Laboratory. Minos was called using the
TOMLAB optimization library. To check the accuracy of the code we also
independently programmed the problem in the GAMS programming language,
and then called Minos from GAMS.

28

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30

Table 1: Private and Social Value of Unregulated Banks
Private Value

Social Value

Type = low
(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (
----------(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

0.20 ,
0.30 ,
0.40 ,
0.50 ,

0.60 )
0.60 )
0.60 )
0.60 )

1.1219
1.1725
1.1853
1.1600

0.9246
1.0707
1.1199
1.1105

0.20 ,
0.30 ,
0.40 ,
0.50 ,

1.00 )
1.00 )
1.00 )
1.00 )

1.3220
1.3520
1.3441
1.3050

0.9246
1.0707
1.1199
1.1105

0.20 ,
0.30 ,
0.40 ,
0.50 ,

0.60 )
0.60 )
0.60 )
0.60 )

1.5596
1.6864
1.7254
1.7276

1.5246
1.6707
1.7199
1.7105

0.20 ,
0.30 ,
0.40 ,
0.50 ,

1.00 )
1.00 )
1.00 )
1.00 )

1.6873
1.7865
1.8231
1.8126

1.5246
1.6707
1.7199
1.7105

Type = high
(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (
----------(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

Notes: Table 1 displays the value of the bank’s objective (“Private Value”) and the regulator’s objective (“Social Value ”) as a function of type
(low or high), risk level s and screening level s, when capital and fines are both set to zero. The private value is exclusive of the lump sum tax
T. The parameters correspond to the baseline parameterization in Section 4.2: mmin = {0, 0.6}, mmax = {1.7, 2.3}, a = {5, 5}, r = 0.05, t = 0.01,
g=1.0.

Table 2: Optimal Capital Regulation in the Baseline Case
Bank Type: Private or
Public Information?

Public

High type %:

NA

Probability of
Contract Assignment:
Assignment Screening Level (s):
Risk Level (s ):
1
Capital Level (1-D):

Private

1% - 85%
1
0.4
1.0
14%

1
0.4
1.0
14%

86%

87% - 99.9%
0.07
0.4
1.0
14%

1
0.3
1.0
0%

Low Type
Contract
Assignment
2
(if
applicable)

Probability of
Assignment:
Screening Level (s):
Risk Level (s ):
Capital Level (1-D):
Probability of
Assignment:
Contract Screening Level (s):
High Type
Assignment Risk Level (s ):
Capital Level (1-D):

0.93
0.3
1.0
0%
1
0.4
1
0%

1
0.4
1.0
14%

1
0.4
1.0
1%

1
0.4
1.0
0%

Notes for Table 2: This table gives the details of the optimal contracts for the low type bank and the high type bank in the baseline case when
fines are prohibited, so the regulator only uses capital requirements. The parameterization used is: r = 0.05, g = 1.0. The first two panels,
labeled “Low Type,” give the details of the contract for the low type bank. (The second panel is only used if the optimal contract for the low
type randomizes between two contract assignments.) The third panel gives the contract for the high type bank. Each panel gives the
probability of the assignment (1.0 unless a random contract is used), and the assigned screening level, risk level, and capital level. The first
column, labeled “Public”, gives the optimal contract when type is public information. The remaining three columns give the optimal contract
when type is private information and the percentage of high type banks takes three different ranges. The parameters correspond to the baseline
parameterization in Section 4.2: mmin = {0, 0.6}, mmax = {1.7, 2.3}, a = {5, 5}, r = 0.05, t = 0.01, g=1.0.

Table 3: Optimal Contracts in the Baseline Case with Capital and Fines
Bank Type: Private or Public
Information?
High type %:
Probability of Assignment:
Screening Level (s):
Risk Level (s ):
Capital Level (1 -D):
Contract Fines
Return = 0.04:
Assignment
Return = 0.5:
1
Return = 1:
Return = 1.5:
Return = 2:
Return = 2.5:
Return = 2.96:

Public

Private

NA

1% - 88%

89%

99.5%

1
0.4
0.6
0%

1
0.4
0.6
0%

.62
0.4
0.6
0%

1
0.3
1
0%

0
0
0
0
0
0
1.8910

0
0
0
0
0
0
1..8910

0
0
0
0
0
0
1.8910

0
0
0
0
0
0
0

Low Type

Contract
Assignment
2
(if
applicable)

High Type

Probability of Assignment:
Screening Level (s):
Risk Level (s ):
Capital Level (1 -D):

.38
0.4
1.0
14%

Fines
Return = 0.04:
Return = 0.5:
Return = 1:
Return = 1.5:
Return = 2:
Return = 2.5:
Return = 2.96:
Probability of Assignment:
Screening Level (s):
Risk Level (s ):
Capital Level (1 -D):

1
0.4
1
0%
0
0
0
0
0
0
0

Fines
Contract
Assignment Return = 0.04:
Return = 0.5:
Return = 1:
Return = 1.5:
Return = 2:
Return = 2.5:
Return = 2.96:

1
0.4
1
0%

1
0.4
1
0%

0
0
0
0
0
0
0
1
0.4
1
0%

0
0
0
0
0
0
0

0
0
0
0
0.8917
0.4321
0

0
0
0
0
0.8812
0
0

Notes for Table 3: This table gives the details of the optimal contracts for the low type bank and the high type bank under the baseline parameterization: t = 0.01, r = 0.05, g = 1.0. The first two panels, labeled “Low
Type,” give the details of the contract for the low type bank. The second panel is only used if the optimal contract randomizes between two contract assignments. The third panel gives the contract for the high type
bank. (In the baseline parameterization, the high type bank is never assigned a randomized contract.) Each panel gives the probability of the assignment (1.0 unless a randomized contract is used), the assigned
screening level, risk level, and capital level, and the amount of fines on each of the seven return levels. The first column gives the optimal contract when type is public information. The remaining three columns give
the optimal contract when type is unobservable and the percentage of high type banks takes three different values or ranges. The parameters correspond to the baseline parameterization in Section 4.2: mmin = {0, 0.6},
mmax = {1.7, 2.3}, a = {5, 5}, r = 0.05, t = 0.01, g=1.0.

Table 4: Private and Social Value of Regulated Banks
Optimal Contract of Low-type Banks
Private Value
Social Value

Optimal Contract of High-type Banks
Private Value
Social Value

Type = low
(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

0.20 ,
0.30 ,
0.40 ,
0.50 ,

0.60 )
0.60 )
0.60 )
0.60 )

1.11999
1.16939
1.17924
1.14933

0.92458
1.07064
1.11986
1.11034

1.00512
0.98048
0.93554
0.87430

0.92343
1.06875
1.11743
1.10759

(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

0.20 ,
0.30 ,
0.40 ,
0.50 ,

1.00 )
1.00 )
1.00 )
1.00 )

1.17924
1.13388
1.04524
0.94440

0.92317
1.06850
1.11694
1.10684

1.17204
1.17924
1.16832
1.13100

0.92310
1.06895
1.11817
1.10871

Type = high
(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

0.20 ,
0.30 ,
0.40 ,
0.50 ,

0.60 )
0.60 )
0.60 )
0.60 )

1.53659
1.57752
1.47614
1.31935

1.52437
1.66958
1.71743
1.70637

1.23150
1.30321
1.35410
1.47319

1.52132
1.66684
1.71621
1.70791

(s, s) = (
(s, s) = (
(s, s) = (
(s, s) = (

0.20 ,
0.30 ,
0.40 ,
0.50 ,

1.00 )
1.00 )
1.00 )
1.00 )

1.23535
1.09230
0.96035
0.84346

1.52008
1.66373
1.71130
1.70076

1.51925
1.64260
1.69761
1.69761

1.52292
1.66924
1.71867
1.70931

Notes: Table 4 displays the value of the bank’s objective (“Private Value”) and the regulator’s objective (“Social Value ”) for the optimal lowtype contract and the optimal high-type contract as a function of type (low or high), risk level s and screening level s. The optimal contract for
a low-type bank has zero capital and a fine of 1.8910 on the highest return. The optimal contract for a high-type bank has zero capital and a
fine of 0.8917 on the return equal to 2.0 and a fine of 0.4321 on the return equal to 2.5. The parameters correspond to the baseline
parameterization in Section 4.2: mmin = {0, 0.6}, mmax = {1.7, 2.3}, a = {5, 5}, r = 0.05, t = 0.01, g=1.0.

Table 5: Comparison of Social and Private Values of Contracts for the Baseline Case

Bank Type: Private or Public
Information?

High type %:

Public

NA

Private

1%

88%

89%

99.5%

Social value of contract:

1.11987

1.11987

1.11987

1.11720

1.07068

Private value of contract before taxes:

1.17924

1.17924

1.17924

1.22470

1.35196

Private value of contract after taxes:

1.11987

1.12067

1.19065

1.20066

1.24785

Social value of contract:

1.71993

1.71867

1.71867

1.71903

1.71993

Private value of contract before taxes:

1.82314

1.69761

1.69761

1.73275

1.82315

Private value of contract after taxes:

1.71993

1.63904

1.70902

1.70971

1.71904

Low Type

High Type

Notes to Table 5: This table gives the social value (the value in the regulator’s objective function) and the private value (the value in the
bank’s objective function) of optimal contracts both before and after taxes. The parameterization for the baseline case is: t= 0.01, r=0.05, g =
1.0. The first column, labeled “Public” gives these valuatio ns for the optimal contract when bank type is observable. The remaining four
columns give the values for the optimal contracts when type is unobservable, and the percentage of high types takes on four values. The
parameters correspond to the baseline parameterization in Section 4.2: mmin = {0, 0.6}, mmax = {1.7, 2.3}, a = {5, 5}, r = 0.05, t = 0.01, g=1.0.