Extended Liability under double Moral Hazard
J. O. Anyangah
February, 2004
Abstract
This paper considers the impact of extending a lender’s liability for environmental damage caused by its borrower on (a) the design of …nancial contracts, (b) the
precuationary e¤ort decision, and (c) the …rm’s prospects for insolvency. In this
model, both the lender’s action along with the borrower’s investment in precaution
in‡uence the probability of an environmental damage. Adopting a bilateral moral
hazard framework, we …nd that extending the lender’s liability induces a shift toward more debt and more precautionary care in the case where the highest possible
realizations of environmental damage is su¢ ciently large compared to the value of
the …rm. The model predicts that extending the lender’s liability may diminish the
fragility of a …rm that is otherwise weak …nancially.
1. Introduction
The doctrine of extending liability to third parties with whom an injurer transacted
business has long fascinated many researchers, especially following the enactment of the
Comprehensive Environmental Response, Compensation and Liability Act (CERCLA).1
The possibility that a …rm protected by limited liability can cause an environmental
damage and then declare bankruptcy (Shavell, 1986 calls such …rms judgement-proof)
appears to many to be ine¢ cient in that it dilutes the …rm’s incentive to be cautious
ex-ante. Extending liability to third parties with deep pockets (they are not liable to
bankruptcy) such as the …rm’s lenders, is viewed as an e¢ cient regulatory response
to the problem of judgement-proofness. Because they are held liable in the event of
injury, lenders subjected to extended liability will not only have a strong incentive to
monitor more closely how the loaned funds are being utilized, but will also invest in
Department of Economics, University of Alberta, and St. Thomas University. I am grateful to
Henry van Egteren and Yingfeng Xu for their extremely helpful comments.
1
This act empowers courts to surbodinate a lender’s entire claim against the borrower if the lender
exercised control over the borrower …rm. CERCLA appears to have inspired similar legilations in
Canada, the United Kingdom and the broader EU (See Boyer and La¤ont, 1996).
activities designed to ensure that the …rm makes adequate investment in environmental
risk reduction. More still, extending liability in this manner may relax the …rm’s limited
liability constraint, thereby allowing the lenders to internalize the damages (Shavell,
1986; Boyd and Ingberman, 1997; Heyes, 1996; Innes, 1991).
This study examines the impact of extending the lender’s liability for environmental
harm caused by its borrower on the nature of …nancial contracts, the optimal investment in precaution, and the …rm’s prospects for insolvency. In the model, a wealthconstrained owner-manager of a …rm desires to undertake a project that carries the risk
of an environmental accident. To execute the project, he or she must raise …nancing
for the project from an external party. In return for …nancing the project, the lender
receives rights to the project’s returns in a sharing rule that combines a …xed payment
(debt) with a proportional share of the remainder (equity). Additionally, the lender
acquires rights to directly intervene in the operation of the project on an ongoing basis. Such intervention, which can be thought of as productive e¤ort, diminishes the
likelihood that the project will turn out to be damaging. Together, the manager’s unobservable level of care and the lender’s e¤ort in‡uence the probability that a damaging
accident occurs, which in turn determines the …rm’s likelihood of insolvency.
Depending upon the size of the …rm’s residual assets, we show that extending the
lender’s liability may or may not a¤ect the likelihood of the …rm’s insolvency. When
the …rm’s asset value is su¢ ciently pronounced relative to the highest possible damage
realizations, extending liability neither improves nor worsens the induced probability of
insolvency. In this case, the …rm’s unlimited assets relieve the lender of any obligations
toward the victim and make his deep pockets irrelevant. On the other hand, when
the …rm’s assets are more limited, extended liability outperforms the no lender liability
regime in terms of expected damage and the induced probability of solvency. The
reason for this is as follows: In a setting where the lender must not only subordinate his
claim against the …rm, but also indemnify environmental liability, any optimal …nancial
contract must incorporate more debt and less equity for the lender in order to o¤set
the downside threat that the lender faces in the states of the world in which the …rm
is insolvent. In our framework, debt increases the return to the lender’s e¤ort in the
state of the world in which the …rm is solvent, thereby increasing his incentive to reduce
the …rm’s exposure to environmental risk. Furthermore, since more debt is necessarily
accompanied by greater equity (ownership) for the manager, her incentive to work more
diligently is also increased. In short, debt increases the fruits of the lender’s intervention
and also intensi…es the ownership incentive for the manager and therefore her incentive
to invest in environmental risk reduction.
Of course, this is not the …rst attempt to model this relationship. Most prominently,
Pitchford (1995) has shown that when lenders are held liable for environmental damages
that arise from activities of judgement-proof …rms, social damages may actually increase
not increase. This result is fundamentally at odds with the spirit of CERCLA, and
2
also contrasts sharply with Boyer and La¤ont (1997) and Heyes (1996), who …nd that
lender liability may increase the incentive for accident prevention. Pitchford (1995) also
contradicts similar studies by Polinsky (1993), Privileggi et al., (2001), Segerson and
Tietenberg (1992) and Shavell (1997), who analyze the problem of extended liability in
the context of the relationship between a …rm and its manager/employee.
More recently, Balkenborg (2001) has focused on the impact of bargaining power
at the contracting stage in an imperfectly competitive world. He shows that there
is a cuto¤ level of creditor bargaining power, below which increased lender liability
increases accidents, and above which it does not. Lewis and Sappington (2001) considers
a setting with many di¤erent levels of damages rather than a binary-damage technology
assumed by Pitchford (1995), and …nds that the lender’s deep pockets can be valuable in
mitigating the judgement-proof problem. Dionne and Spaeter (2003) use a framework
in which investment in precaution a¤ects the distribution of environmental losses and
operating revenue to show that extending liability may increase the level of precaution.
In all these studies, the lender is portrayed as a "sleeping" investor who interacts
with the …rm at arms length. In other words, the lender is presumed to lack the ability
to perform extensive due diligence in an ongoing basis beyond committing his sunk
investment. Instead, all the e¤ort necessary to minimize the risk of environmental
damages is supplied by the borrower. This depiction of the lender ignores the fact that
the lender may intervene in the project and undertake actions which may in‡uence
the project’s outcome in general and the likelihood of an environmental damage, in
particular. It neglects to consider that the optimal contract may a¤ect the lender’s
precautionary incentive, which may in turn enhance the total expected surplus the two
parties have to share. The present study attempts to …ll this gap by introducing the
lender as an active participant in project activity. More precisely, it is assumed that in
addition to providing his sunk investment, the lender obtains inside management rights
and expends personally costly e¤ort in order to minimize environmental risk and the
…rm’s prospects for solvency. This modi…cation enables us to accord a richer role for
debt and equity in the …nancial contracts.
Clearly, the characterization of lender’s involvement in this manner presupposes that
the lender is not inhibited by any regulatory constraints in its relationship with the client
…rm. This assumption not only …ts in perfectly well with the type of relationship that
typically prevails between a venture capitalist and a …rm, but is also consistent with
the existing and emerging banking practices in a number of countries.2 In the US
where bank-…rm relationships have historically tended to be more limited, banks now
have the freedom to hold equity in client …rms through merchant banking subsidiaries.3
2
Venture …nancing is typically a relationship …nancing. For example, in addition to receiving equity,
a venture capitalist may obtain the right to sit on the …rm’s board of directors or act as an o¢ cier of
the company (Pozdena, 1990).
3
The Gramm-Leach-Bliley Act of 1999, which repealed sections of the Glass Steagall Act allows
3
The Japanese and German …nancial systems are well known for their close bank-…rm
relationship and concentrated ownership. In much of western Europe, banks are allowed
to engage in "universal banking", a practice that essentially gives them the freedom to
own and be owned by non…nancial …rms (Agarwal and Elston, 2001; Cybo-Otton and
Murgia, 2000). Despite the expanded range of activities that a lender can perform, there
is no study, to the best of my knowledge, that has hitherto examined the impact of a
lender’s intervention on the nature of the …nancial contracts, precautionary incentives
and the likelihood of insolvency.
This study is also closely related to the literature that has examined the economic
application of double moral hazard. Demski and Sappington (1991) examine the problem of double-sided moral hazard in the context of a …rm with a risk neutral owner
and a risk averse worker, and show that the problem can be completely and costlessly
resolved if the principal has the option of requiring the agent to purchase the enterprise
at a pre-negotiated price. Cooper and Thomas (1985, 1988), Dybvig and Lutz (1993),
Emons (1988) and Manns and Wissink (1988) examine the nature of the optimal warranty when both the producer and the consumer of a product take privately observed
actions that a¤ect the quality of the product. Agrawal (1999), and Eswaran and Kotwal (1985) examine double-sided moral hazard in the context of agricultural contract.
Lafontaine and Shaw (1996) examine royalty contracts in franchising. Romano (1994),
and Bhattacharyya and Lafontaine (1995) examine models of double-sided moral hazard to explain the prevalence of linear contracts. Kim and Wang (1998) extend the
same theme by examining the robustness of linear contracts and show that the optimal
contract is generally not linear if the agent is risk averse.
The rest of this paper is organized as follows. Section 2 develops the central elements
of the model. In section 3, we characterize the optimal contract under full information.
Section 4 introduces double moral hazard and analyzes the impact of extended liability
on the …rm’s insolvency. In section 5, we provide some implications and concluding
remarks.
2. Basic model
In the spirit of Pitchford (1995), we consider the relationship between three risk-neutral
actors: an owner-manager, an outside …nancier, and a victim.4 The manager would like
to undertake a project that is socially bene…cial, but carries the risk of an environmental
accident.5 The project is known to require a …xed amount I in order to be undertaken,
but the available initial wealth that the manager can invest is less than the required
…nancial holding companies to provide equity …nancing to non…nancial …rms for a limited time period
(Barth et al., 2000; Furlong, 2000; Schmid, 2001).
4
The assumption of risk-neutrality is necessary to eliminate any risk-sharing concerns.
5
Throughout, we will use the term ‘manager’, ‘entreprenuer’and ‘…rm’interchangeably.
4
investment outlay. Thus, for the project to be executed, further funding must be secured
from the …nancier.6 Assume that the owner cannot divert any borrowed funds to
…nance perquisite consumption. Thus, with outside funds K provided by the …nancier,
the manager’s own contribution to the new investment is w = I K, which we assume
is costlessly veri…able by the …nancier.7 Throughout, we take K (and therefore w) as
given, not to be determined in the model.
The nature of the project is as follows. There is a single period, which is divided
into two points of time: t=0 (beginning) and 1 (end). No one discounts between the
beginning and the end of the period. At time t =0, the manager makes a decision on the
level of investment in care to undertake in order to minimize the risk of environmental
accident. The monetary equivalent disutility of this investment is represented by e
where e 2 <+ .
A special focus of this paper is on the …nancier’s ongoing intervention in the project,
especially his role in reducing the project’s exposure to environmental risk. Poor environmental performance by the …rm may increase the likelihood of environmental legal
liability and impede the …nanciers’s ability to recoup the loaned funds.8 Thus, in addition to funding the project, the …nancier commits resources at time t = 0 to ensure that
the project is operated in an environmentally sound manner.9 The monetary cost of
these resources is represented by a 2 <+ . Under a will be subsumed factors that can be
instrumental in limiting the project’s exposure to environmental risk such as consulting,
advising, monitoring, oversight and due diligence, performed by the …nancier beyond
committing the sunk investment. In the following, we simply refer to a as the lender’s
‘e¤ort’.
At time t = 1, the project realizes an exogenously given net return v, from which all
payments are drawn. We assume that v is costlessly veri…able. In addition, the project
generates a stochastic environmental damage ~l, which has support in the interval [0; L].
Denote by F (l=e; a) the cumulative distribution function of l given e¤ort levels a and e
by the lender and the manager, respectively. f (l=e; a) > 0 is the corresponding density
6
Throughout we use the term ‘…nancier’and ‘lender’interchangeably. The lender could be thought
of as a bank, a venture capital …rm or any other creditor. As we show below, the distinguishing feature
of our model is that the lender is portrayed as an active participant rather than a sleeping investor.
7
We make this assumption for expositional ease only. It is well known that individuals more often
than not possess private information about their limited wealth; they may engage is a variety of strategies
aimed at concealing the true value of their assets (See, for example, Lewis and Sappington, 2000). Thus,
an expanded model could allow the owner-manager to have private knowledge about her limited wealth,
and assign all the bargaining power to the …nancier.
8
Evidence of a positive correlation between environmental performance and …nancial performance
has been provided by a number of studies, including Hamilton (1995), Hart (1995), and Blacconierre
and Patten (1993).
9
For instance, the …nancier, perhaps a venture capitalist, a bank or his representative may sit on
the board of directors of the …rm or otherwise explicitly exercise control over the …rm and improve the
…rm’s performance (See, for example, Sahlman, 1990; and Kroszner and Strahan, 2000).
5
function.10 Thus, unlike other studies that have examined the role of the …nancier
in similar settings, herein we assume that the …nancier’s intervention may actually
increase the …rm’s end-of-period cash ‡ow by reducing the likelihood of an environmental
accident.11 We make the following assumptions with respect to the distribution of l.
Assumption 1. For any l 2 [0; L], f (l=e; a) is twice di¤erentiable in a and
e.
1
Assumption 2. F (l=e ; a)
1
2
F (l=e; a2 ) 8 l; e; a
a .
Assumption 3. Fee (l=e; a)
1
2
F (l=e2 ; a) 8 l; a, e
0 and Faa (l=e; a)
0.
e and F (l=e; a1 )
12
Assumption 4. lime#0 Fe (l=e; a) = 1 and lima#0 Fe (l=e; a) = 1 .
Assumption 5. lime#1 Fe (l=e; a) = lima#1 Fe (l=e; a) = 0 .
Assumption 1 is necessary for the existence of an optimal solution. Assumption 2
indicates that higher e¤ort levels renders lower environmental damage l more likely in
the sense of stochastic dominance. Assumption 3 indicates stochastically diminishing
marginal productivity of e¤ort levels supplied by the two parties. Assumptions 3, 4 and 5
are necessary to guarantee the uniqueness of the lender and the manager’s e¤ort choices.
We assume further that Fe (0=e; a) = 0, Fe (L=e; a) = 0, Fa (0=e; a) =0, Fa (L=e; a) = 0
and .
A contract requires the …nancier to provide the sum of K dollars as a loan, and it
speci…es a sharing rule for the …nal project return. We restrict ourselves to a class of
sharing rules that have both debt and equity components. Thus, the manager makes
two kinds of payments to the lender: will denote the debt payment that the manager
makes to the lender; this payment is made regardless of the project outcome. (1
)
will denote a contingent dividend or the lender’s equity stake in the project. This is
the fraction of the residual cash left in the …rm after the …rm’s obligations, including
debt and environmental legal liability, have been paid o¤ that accrues to the lender (
2 [0; 1].)13 Let r denote the return per dollar invested elsewhere. Thus, rK is the
opportunity cost of the lender’s invested capital K. Following standard practice, we
assume that the manager faces limited liability. This implies that the manager cannot
lose more than her equity in the event of insolvency.
10
This formulation follows that of Demski and Sappington (1991).
Besanko and Kanata (1993), and Diamond (1984, 1993) have examined the role of banks as delegated
monitors. Besanko and Kanata (1993) is closer to our study since it assumes that monitoring increases
the entrepreneur’s e¤ort, which in turn improves the likelihood of the …rm’s success.
12
Throughout subcripts will denote partial derivatives.
13
Dionne and Spaeter (2003) focus on a standard debt contract, thereby precluding any role for equity
participation and the lender’s intervention.
11
6
At the end of the period, the owner-manager receives a dividend of
times the
residual cash left in the …rm after the environmental legal liability, l and the lender’s
debt, k have been paid o¤. If the …rm is bankrupt at the end of the period, then the
manager receives nothing and the lender becomes the residual claimant of the …rm’s
assets after the victim has been compensated. For a given net return v and debt
obligation , there is a critical amount of environmental liability l at which the manager
is just able to meet her legal obligations; That is, for positive levels of debt and net …rm
value
v
l
0.
(2.1)
Equation (2.1) de…nes the solvency condition, which occurs when l = l . Clearly, this
equation de…nes l as a function of :
l = l ( ).
(2.2)
Thus, l is the threshold such that if l
l , the manager can meet all her payment
obligations. On the other hand, if l > l , the manager cannot meet her payment
obligations and the lender and the victim become the residual claimants. It follows that
Zl
the probability that the …rm is solvent is Pr ob(l l ) = f (l=e; a)dl.
0
The manager decides the amount to invest in care, e. The lender chooses the amount
to invest in e¤ort a. Thus the optimal levels of a and e are endogenously derived in
this framework. The owner’s objective is to maximize her end-of-period cash ‡ow, U o ,
which can be written as follows:
Uo =
l Z( )
[v
l] f (l=e; a)dl
e.
(2.3)
0
The …rst term on the right hand side of equation (2.3) represents the payo¤ to the
manager if the project remains solvent at the end of the period. Recall that this will
occur when l l . Given limited liability, only the equity of the manager is exposed to
tort risk. The last term on the RHS (2.3) is the cost of undertaking e¤ort on the part
of the manager.
The lender’s expected payo¤, U B in the absence of extended liability can be written
as
7
U
B
= (1
)
l Z( )
[v
[v
l] f (l=e; a)dl
l] f (l=e; a)dl + F (l =e; a)
(2.4)
0
+
minfv;Lg
Z
a.
l ( )
The …rst integral on the RHS of equation (2.4) denotes the expected value of the lender’s
payo¤ if the project is solvent. The second term represents the expected value of the
promised principal that the lender would receive as a lender to the project (F (l =e; a)] =
l Z( )
f (l=e; a)dl is the probability of solvency.) The second integral term on the RHS
0
denotes the expected value of the lender’s cash ‡ow if the project is insolvent. Note
that this value is evaluated over the interval [l ( ); minfv; Lg] implying that in the
absence of extended liability, the victim cannot receive more than the value of …rm’s
assets. The last term is the monetary cost of e¤ort expended by the lender.
The framework outlined above emphasizes an important dimension to the relationship between the lender and the owner-manager. On the one hand, the manager’s
expected payo¤ crucially depends on the level of e¤ort undertaken by the lender, a. On
the other hand, the ability of the lender to recoup the loaned funds depends on the level
of investment in precaution undertaken by the manager. The following analysis will attempt to examine how such interdependence a¤ects the nature of the optimal …nancial
contracts between the lender and the …rm and their incentive to exercise precaution.
The sequence of events is as follows: In stage 1, the regulator publicly announces
the liability rule. In the second stage, the owner o¤ers the …nancier a contract. This
contract stipulates how she will compensate the lender for the K dollars he supplies to
the …rm. In the third stage the lender accepts or reject the contract proposed by the
owner.14 If the contract is accepted, the lender is allocated control rights over the …rm’s
operations. In stage 4, the lender and the manager simultaneously choose their e¤ort in
order to maximize their returns given { ; g and the liability rule. Final project output
is observed and compensation made according to the contract selected by the manager
in the second stage.
Following Pitchford (1995), we examine three regimes of liability. Strict liability
for the manager refers to the case where the lender contributes nothing towards the
compensation of the victim in the event of insolvency. In such a situation, insolvency
14
In the following, we assume that the project is always undertaken. Thus, the owner o¤ers a contract
that guarantees the lender at least his outside opportunity pro…t.
8
results in uncompensated liability since courts cannot take away the …rm’s assets. Full
lender liability refers to the case where the lender is required to compensate any residual
liability arising from …rm’s limited assets. Partial lender liability refers to the case in
which the lender compensates only a fraction of the residual liability.
3. Full information solutions
Before proceeding to characterize the …rm’s private optimum under bilateral information
asymmetry, we consider two benchmark cases. The …rst case assumes a setting where
no moral hazard problem arises because both the lender’s and the manager’s e¤orts are
observable, and the optimal level of precautionary e¤ort is prescribed by a regulator
maximizing social welfare. In this case, the net value of the project is v, the expected
ZL
damage from the project is lf (l=e; a)dl and the costs of undertaking e¤ort on the part
0
of the lender and the …rm are, respectively, a and e. Thus, the expected social value
ZL
of the project is S(e; a) = v
lf (l=e; a)dl a e. The social problem is therefore
0
maxe;a S(e; a). It is straightforward to see that the concavity of f (l=e; a) in a and
e guarantees the existence and uniqueness of the socially e¢ cient e¤ort combination
(e; a) such that
ZL
0
s
s
Fe (l=e ; a )dl =
ZL
Fa (l=es ; as )dl = 1
(3.1)
0
where the …rst two terms represent the social marginal return from the e¤ort.
3.1. No lender liability
In the second benchmark case, there is still full information, but the optimal contract is
determined through private interactions between the owner and the lender. We assume
that the manager holds all the bargaining power. This may be a reasonable assumption
where there is no competition among borrowers, for example, if there is only one suitable
project that the lender can …nance or where there is intense ex-ante competition among
potential lenders. In this case, the manager will extracts all the surplus, leaving the
lender’s pro…t identically equal to zero. Under conditions of no lender liability, the
9
optimal …nancial contract is determined by the solution to the following problem [NF]
l Z( )
max
f ; ;a;eg
[v
l] f (l=e; a)dl
e
(3.2)
0
subject to
U
B
= (1
)
l Z( )
[v
[v
l] f (le; adl
l] f (l=e; a)dl + F (l =e; a)
(3.3)
0
+
minfv;Lg
Z
a
rK.
l ( )
Equation (3.3) is the lenders participation constraint, which ensures that the lender
receives his outside opportunity payo¤, rK. Replacing as a function of U B in the
owner’s objective function (3.2), the owner’s problem can be rewritten as:
max U o =
a;e
l Z( )
[v
l] f (l=e; a)dl +
0
minfv;Lg
Z
[v
l] f (le; a)dl
a
UB
e
(3.4)
l ( )
subject to the lenders participation constraint U B
rK. Since U o is decreasing in
U B and the owner has all the bargaining power, the participation constraint U B rK
can be replaced by equality without loss of generality. Hence, the optimal contract is
characterized by the following problem:
o
max U =
a;e
l Z( )
[v
l] f (l=e; a)dl +
0
minfv;Lg
Z
[v
l] f (le; a)dl
rK
a
e.
(3.5)
l ( )
Clearly, in this …rst-best world, the optimal contract is one that maximizes the expected
total (private) surplus. The …rst-order conditions associated with the optimal level of
precautionary e¤ort undertaken by the manager and the level of expenditure on e¤ort
by the lender reduce to
minfv;Lg
Z
Fe (l=e ; a )dl
0
10
1=0
(3.6)
and
minfv;Lg
Z
Fa (l=e ; a )dl
1 = 0.
(3.7)
0
where (e ; a ) denotes the solution to problem [NF]. The next lemma describes the e¤ort
combination under the optimal contract.
Lemma 1. The optimal contract under full information and no lender liability
speci…es e¤ ort combination (e ; a ).
Proof : See the appendix
The idea behind lemma 1 is straight forward. Since e¤orts are observable, the
manager simply usurps all the surplus from the lender while ensuring that they both
undertake e¤orts that maximize the expected total gain from the project. This is
achieved by setting (e; a) = (e ; a ). Thus, the marginal return from e¤orts, which
occurs in terms of the marginal improvement in distribution of damages must be just
equal to the marginal disutility of e¤ort. Observe that in the optimal contract, the
marginal bene…t from both e¤orts are evaluated, not surprisingly, over all possible states
of nature; that is, both in the states of the world in which the …rm is solvent, l 2 [0; l ]
and in the insolvency states, l 2 [l ; minfv; Lg]. In short, e¤orts are rewarded according
to the total (private) gains that accrue to both parties.
Due to limited liability on the part of the owner, the level of the accident cost
internalized by the manager depends on the magnitude of v, the …rm’s asset value
relative to the harm. More precisely, if v is su¢ ciently large; that is, if v > L such
that minfv; Lg = L , e¤ort will be rewarded according to the total social gains (the
accident cost will be fully internalized). In this case, the optimal e¤orts undertaken will
be socially optimal. If v < L, however, the level of expenditures on e and a will fall short
of the socially optimal levels. Intuitively, the protection a¤orded the manager by limited
liability prevents her from fully internalizing the environmental damages generated by
the project, thereby resulting in too little e¤ort levels being exerted relative to the social
optima.
3.2. Optimal contracting under extended liability
The preceding section highlighted the fact that a combination of potentially huge environmental damage and exceedingly tight asset bounds on the part of the …rm can
result in uncompensated liability. This suggests that …rms which can cause accidents
and become insolvent will have diluted incentives to be cautious. Extended liability is
therefore a natural policy response to “judgement-proofness" of …rms without su¢ cient
11
assets to pay for the cost of indemnity. Because they are held liable in the event of injury, third parties subjected to extended liability will have stronger incentives to invest
in activities designed to ensure that the …rm exercises due care. This section examines
the impact of extending the liability to the lender on the nature of …nancial contract
that the manager will adopt under full information.
3.2.1. Full lender liability
When the responsibility for the victim’s compensation is extended to the …nancier in
full, the lender must take into account the severe downside risk he faces in the event of
insolvency in deciding whether or not to …nance the project. Assuming, as before, that
the manager holds the balance of the bargaining power and can therefore appropriate
all the surplus, the manager’s problem [FF] will read:
max
f ; ;a;eg
l Z( )
[v
l] f (l=e; a)dl
e
(3.8)
0
subject to
~ (e; a; ; ) = (1
U
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(3.9)
0
+
ZL
[v
l]f (l=e; a)dl
a
rK:
l ( )
Equation (3.9) is the lender’s participation constraint under full lender liability. It
ensures that the lender gets at least as much as his outside opportunity pro…t. The …rst
integral on the left-hand side of this equation denotes the expected value of the promised
dividend that the lender would receive as a shareholder if the project is solvent. The
second term represents the expected value of the promised principal that the lender
would receive as a lender to the project. The second integral term on the left-hand side
of (3.9) denotes the expected value of the lender’s cash ‡ow if the project is insolvent.
Note that this value is evaluated over the interval [l ( ); L]. Thus, should the …rm’s
operating pro…t v be insu¢ cient to compensate the victim, the lender’s unbounded
wealth will still ensure that the residual liability is fully indemni…ed.15 The last term
on the LHS is the monetary cost of e¤ort expended by the lender. The only term on
the RHS is the opportunity cost of the lender’s invested capital rK.
15
This can be construed to mean that the lender has unlimited sizable assets that can be attached by
the court.
12
Making use of the binding participation constraint in the objective function, we can
recast the lenders problem into an unconstrained maximization problem of the form
max
fa;eg
l Z( )
[v
l] f (l=e; a)dl +
0
ZL
[v
l]f (l=e; a)dl
rK
a
e.
(3.10)
l ( )
Let (~
e ;a
~ ) denote the unique solution to the manager’s problem under full lender
liability derived from the …rst-order conditions for problem [FF]. Then we can now
state the following lemma (The proof is similar to that of lemma 1 and we therefore
omit it).
Lemma 2. The optimal e¤ ort combination under full information and full lender
liability (~
e ;a
~ ) is characterized by:
l Z( )
Fe (l=~
e ;a
~ )dl
1=0
(3.11)
0
ZL
l ( )
l Z( )
ZL
Fa (l=~
e ;a
~ )dl
1=0
(3.12)
Fe (l=~
e ;a
~ )dl +
and
0
Fa (l=~
e ;a
~ )dl +
l ( )
Equations (3.11) and (3.12) give the e¢ ciency conditions for e¤ort supply when the
lender is fully accountable for any residual liability that is uncompensated due to the
…rm’s limited assets. The …rst term in (3.11) gives the marginal bene…t of e¤ort in
terms of the marginal improvement in the distribution of l in the state of the world
in which the project is solvent. The second term displays the marginal bene…t from
investing in care in terms of the marginal improvement in the distribution of l when the
…rm’s operating pro…t is insu¢ cient to compensate the harm. Note that this last term is
calculated over the interval [l ( ); L], thereby ensuring that there is no uncompensated
liability. Thus, in this framework, the lender’s deep pockets prove valuable in ensuring
that entire environmental damage is fully internalized. Expression (3.12) has a similar
interpretation.
13
3.2.2. Partial lender liability
Suppose now that the legal liability is modi…ed in such a manner as to provide the victim
or courts with a recourse to the lender’s wealth, albeit partially, in the event that the
…rm’s assets are inadequate to compensate the victim.16 Formally, partial lender liability
requires the lender to pay a fraction 2 (0; 1) of the damages whenever l exceeds v.
According to this rule, the maximum loss that the lender can su¤er in insolvency, for a
given net value of the …rm v, is given by minfL; v + (L v)g. This limit collapses to
L when = 1 (full lender liability) and to minfL; vg (no lender liability) when = 0.
When L > v, we have L > v + (L v) > v and minfL; v + (L v)g = v + (L v). On
the other hand, when L < v, we have L < v + (L v) < v and minfL; v + (L v)g = L.
For ease of exposition, we assume throughout that v + (L v) is su¢ ciently large such
that Fe (v + (L v)=e; a)=0 and Fa (v + (L v)=e; a) = 0. The lender’s payo¤ can
be written as
U (e; a; ; ) =(1
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(3.13)
0
+
minfL;v+
Z (L v)g
[v
l]f (l=e; a)dl
a.
l ( )
The only di¤erence between (3.13) and (2.4) is the second integral term on the RHS of
(3.13), and in particular, the upper limit of integration for this term min fL; v+ (L v)g.
It is evident that the maximum loss that the lender can su¤er in the insolvency states
crucially depends on the di¤erence L v, which is the magnitude of the environmental
damage relative to the net value of the …rm. As we show below, this di¤erence ultimately
determines the extent to which the environmental damage is internalized.
The optimal levels of e¤ort supplied by the lender and the owner are determined by
the solution to the following owner’s problem [PF]: Maximize (2.3) subject to U (e; a; ; )
rK. Assuming an interior equilibrium, the …rst-order conditions are
l Z( )
0
Fe (l=e ; a )dl +
minfL;v+
Z (L v)g
Fe (l=e ; a )dl
1=0
(3.14)
l ( )
and
16
This may also be interpreted as a situation in which the lender is held strictly liable, but exogenous
imperfections in the liability system lead the lender to compensate only a fraction of the harm done by
the …rm’s activity.
14
l Z( )
Fa (l=e ; a )dl +
0
minfL;v+
Z (L v)g
Fa (l=e ; a )dl
1=0
(3.15)
l ( )
where (e ; a ) denote the unique e¤ort combination yielded by the solution to problem
[PF].
From equation (3.14), we see that if L v; that is, if the …rm’s residual assets are
adequate to compensate the victim for all possible levels of environmental damage, then
the e¤ort level e will equal the e¤ort level es ; that is, the e¤orts supplied under private
contracts will be socially optimal. If, on the other hand, L > v so L > v + (L v) > v
and minfL; v + (L v)g = v + (L v), then the environmental harm is clearly not
fully internalized even though the lender is subjected to extended liability. Intuitively,
holding the lender only partially liable for the damage e¤ectively puts an upper bound
on the loss that he can compensate ex-post, thereby reducing the value of the lender’s
deep pockets. The interpretation of (3.15) is analogous. Hence, we can state:
Corollary 1. Suppose that assumptions 1-6 hold. Then full lender liability has
e~ = es and a
~ = as . Partial lender liability has e < es and a < as if L > v; it has
e = es and a = as if L v.
Proof : See the appendix
4. Double moral hazard
Next we discuss the optimal …nancial contract under bilateral informational asymmetry:
We assume that the manager’s e¤ort e is neither observable nor veri…able by the lender
(and therefore not contractible). We also introduce a moral hazard incentive on the
part of the lender. The assumption of nonobservability of the lender’s e¤ort can be
construed as a situation in which the manager cannot assess the value of the lender’s
input ex-ante. In short, there exists a "double-sided" moral hazard problem and the
optimal contract has to take into account both the lender’s own incentive provision as
well as the manager’s incentive provision.17 The assumption that the lender needs to
be given adequate incentives to expend appropriate level of e¤ort may on the surface
look uncommon, but it is nevertheless a reasonable one. Apart from the fact that e¤ort
is personally costly for the lender, this point can also be understood by realizing that
a lender who has invested in a portfolio of projects may have an incentive to spend too
little time on a particular project if his expected return from that project is low.
17
The framework we follow below is analogous to that employed in the literature on team production.
In this setting one player designs the compensation scheme as a Stackelberg leader, and thereafter
participates in the production as a team member (McAfee and McMillan, 1991).
15
The following discussion is organized in two parts. In the …rst part, we discuss the
solution that occurs when there is no lender liability. The optimal outcome in this
setting is then compared, in the subsequent part, to the situations that develops when
the liability can be extended to the lender.
4.1. No lender liability
Suppose the regime of liability holds the manager strictly liable in the event of an
accident. Assigning the manager the responsibility for full compensation of the victim
may not necessarily be feasible in this situation. Given the protection a¤orded the …rm
by limited liability, the manager cannot have any negative cash ‡ows regardless of the
level of environmental damage. Thus, if the project is solvent, the manager pays o¤
all her obligations and shares the residual dividends with the lender according to the
terms of the …nancial contract. In the event of insolvency (Recall that [1-F (l =e; a)] =
l Z( )
1
f (l=e; a)dl is the probability of insolvency), the manager loses only her equity
0
stake in the project. On the other hand, the lender receives all of the …rm’s residual
assets after the victims have been compensated18 . If the …rm’s assets are less than the
level of damages; that is, if v < l, the lender gets nothing and the damages are not fully
indemni…ed. Thus, if the lender accepts contract f ; g such that v > l, then even when
the …rm is insolvent the lender need not, and the expected payo¤ to the lender is
U B (e; ; ; a) = (1
)
l Z( )
[v
[v
l] f (l=e; a)dl
l] f (l=e; a)dl + F (l =e; a)
(4.1)
0
+
minfv;Lg
Z
rK
a.
l ( )
The second integral term on the right-hand-side of equation (4.1) captures the well
known fact that the victim’s compensation is generally a senior claim in bankruptcy.
Thus, in the event that the owner is unable to pay o¤ all her obligations, the lender
can receive the control rights over the …rm’s assets only after the victim has been fully
indemni…ed. Thus, if v < l, the …rm has no residual assets and the lender pays nothing
to the victims. In the case of a successful project, the entrepreneur receives the bene…ts
minus the costs of the loan.
18
This assumption implies that the victim has an inviolable senior claim on the …rm’s asset in the
event of insolvency.
16
An important consideration is the lender’s preferences over ( , ) pairs.19 This can
be illustrated by characterizing the lender’s indi¤erence curves. Fixing the lender’s
expected payo¤ at U B (e; ; ; a) = rK, the indi¤erence curves give the locus of the
combinations of and for which the lender will be indi¤erent. Implicit di¤erentiation
of U B (e; ; ; a) = rK yields upward sloping indi¤erence curves,
@
@
=
@U B (e;
; ; a)=@
@U B (e; ; ; a)=@
=
l Z( )
[v
l] f (l=e; a)dl
0
l Z( )
> 0.
f (l=e; a)dl + [v
(4.2)
l ]f (l =e; a)
0
Interpreted, condition (4.2) implies that an increase in , by increasing the manager’s
stake in the project, e¤ectively reduces the lenders expected return in the state of the
world in which the …rm is solvent, and therefore requires an increase in to maintain
the the lender at his reservation payo¤.
Knowing this fact, we now characterize the lender’s and the owner’s e¤ort choices.
At the e¤ort selection stage of the game, the manager takes as given the lender’s e¤ort
a, and the …nancial contract ( ; ) from stage 2 of the game. Thus, her incentive
compatibility condition is given as follows:
o
max U (e; ; ; a) =
e
l Z( )
[v
l] f (l=e; a)dl
e
(4.3)
0
Di¤erentiating (4.3) with respect to e yields the …rst-order condition that determines
the manager’s optimal level of precautionary e¤ort:
Ueo (e;
; ; a) =
l Z( )
Fe (l=e; a)dl
1 = 0.
(4.4)
0
This says that at the optimum, the manager’s return from investment in precaution
over the states of nature for which the …rm is solvent must be just equal to the marginal
disutility of undertaking precaution. Note that the concavity of Fe (l=e; a) ensures that
19
Dionne and Spaeter (2003) restrict themselves to a debt contract as the only optimal mechanism to
compensate the lender for the funds that he supplies.
17
the second-order condition associated with the choice of precaution
SOCe =
o
Uee
(e;
; ; a) =
l Z( )
Fee (l=e; a)dl < 0
(4.5)
0
holds globally.
Of particular interest is the question of how the manager’s choice of care e will
change in response to changes in her equity stake and the level of debt used to
…nance the project; that is, what are the signs of dde and dde ? The …rst-order condition
(4.4) implicitly de…nes e = ( ; )
e( ; ). Thus, the answer requires the total
di¤erentiation of the …rst order condition (4.4). From the total di¤erentiation of the
…rst-order condition (4.4), we obtain,
l Z( )
de/ d =
Fe (l=e; a)dl
0
,
SOC e > 0,
(4.6)
and
de/ d =
Fe (l =e; a)/ SOC e < 0.
(4.7)
Equation (4.6) says that holding the level of debt constant, an increase the manager’s
stake in the project increases the manager’s level of investment in precaution. This
is because an increase in increases the fruits of the manager’s e¤ort in the states of
the world in which the …rm is solvent. On the other hand, the sign of de/ d is clearly
negative. The reason is that a higher level of debt, by reducing the …rm’s residual assets,
increases the likelihood of insolvency, thereby reducing the range of states of the world
in which the manager receives a return from her e¤ort. This causes a decrease in the
marginal return from precaution, which in turn reduces the level of precaution on the
part of the manager.
The lender’s problem at the e¤ort selection stage is analogous. He chooses a to
maximize his expected payo¤, taking e, and as given; that is, the lender’s incentive
compatibility condition is
B
max U (a; ; ; e) = (1
a
)
l Z( )
[v
[v
l] f (l=e; a)dl
l] f (l=e; a)dl + F (l =e; a)
0
+
minfv;Lg
Z
l ( )
18
a.
(4.8)
The …rst-order condition with respect to a yields
2
6
UaB (a; ; ; e) = 4(1
)
l Z( )
Fa (l=e; a)dl +
0
minfv;Lg
Z
l ( )
3
7
Fa (l=e; a)dl5
1=0
(4.9)
The …rst-order condition (4.9) implicitly de…nes a = ( ; )
a( ; ). Compared
to the manager’s e¤ort choice decision given by (4.4), the lender evaluates the marginal
bene…t of undertaking e¤ort over all states of the world, not only for states of the
world in which the project is insolvent. Thus, the marginal bene…t from e¤ort can
be decomposed into two: First, a higher level of e¤ort reduces the probability of an
environmental accident, thereby increasing the states of the world in which the …rm is
solvent. This is captured by the …rst term in parenthesis. In addition, a higher level
of e¤ort, by reducing the probability of loss, increases the lender’s ability to recoup the
loaned funds in the states of the world in which the …rm is insolvent. This e¤ect is
represented by the second term in the parenthesis. Again the concavity of F (l=e; a) in
a ensures that the second-order condition holds globally.
Before proceeding to characterize the manager’s optimum under no liability setting,
it is useful to derive the e¤ect of debt and equity on the level of e¤ort undertaken by
the lender. From the total di¤erentiation of the …rst order condition given by (4.9), and
upon rearrangement of terms, we obtain,
da/ d =
l Z( )
Fa (l=e; a)dl
0
,
SOC a < 0
(4.10)
and
da/ d =
Fa (l =e; a)/ SOC a > 0.
(4.11)
Equation (4.10) records the e¤ect of equity on the lender’s incentive to exert e¤ort.
It shows that an increase in the manager’s equity stake results in a diminution in the
lender’s incentive to work on the project. The reason for this is that an increase in
decreases the return to the lender’s e¤ort in the states of the world in which the …rm
is solvent, thereby dulling his incentive to work diligently for the project. The e¤ect of
debt on the lender’s incentive is unambiguously positive. This is because a higher level
of debt increases the average return to the lender in states of the world in which the
…rm is solvent and thus it increases the lender’s incentive to undertake e¤ort.
We now move on to the penultimate stage of the game in which the manager determines the level of and . Unlike under full information, the optimal …nancial contract
under double-sided moral hazard will maximize the manager’s expected pay-o¤ subject
19
to the lender’s participation constraint and two incentive constraints: The lender’s own
incentive compatibility constraint and the manager’s incentive constraint. Thus, the
manager solves the following program [NL]:
max
f ; g
l Z( )
[v
l] f (l=e; a)dl
e
(4.12)
0
subject to
e
e( ; ) as de…ned by 4.4
(4.13)
a
a( ; ) as de…ned by 4.9
(4.14)
and
U
B
= (1
)
l Z( )
[v
[v
l] f (l=e; a)dl
l] f (l=e; a)dl + F (l =e; a)
(4.15)
0
+
minfv;Lg
Z
a
rK.
l ( )
Constraints (4.13) and (4.14) are the owner’s and the lender’s incentive compatibility
constraints, respectively. They ensure that the two parties choose their e¤ort optimally
given any debt-equity combination ( ; ). Notice that since U o and U B decrease and
increase, respectively with , the participation constraint (4.15) is always binding.20
Making use of this condition, the manager’s problem can be simpli…ed further as:
max
f ; g
l Z( )
0
[v
l] f (l=e; a)dl +
minfv;Lg
Z
[v
l] (l=e; a)]dl
rK
a
e
(4.16)
l ( )
subject to (a; e) satis…es (4.13) and (4.14) for ( ; ). Proposition 1 below gives the key
characteristic of the optimal contract under no lender liability.
20
This is essentially a zero-pro…t constraint on the part of the lender. It underpins our assumption of
perfect competion in the loanable funds market.
20
Proposition 1. Under no lender liability, the optimal contract, denoted by { ; g, is
characterized as:
1
=
1+
a + rK
(1
)
[v
l] f (l=e; a)dl
0
=
where r0 = 1
Zl
Fe (l =e;a)
Fa (l =e;a) r0
minfv;Lg
Z
l
F (l =e; a)
minfv;Lg
Z
and
(4.17)
[v
l] f (l=e; a)dl
(4.18)
Fa (l=e; a)dl.
l ( )
Proof : See the appendix
Equation (4.17) gives the optimal equity stake while (4.18) is the …xed (debt) payment in a world where the lender’s assets cannot be used to compensate the victim. In
the following, we brie‡y explain the characteristics of and then dedicate the remainder
of this subsection to the implication of equation (4.18). The (expected) …xed payment
contains four terms: The …rst two guarantee the lender his outside opportunity payo¤
and a compensation for his e¤ort. The last two components, which enter the expression
for in a negative fashion, reduce the lenders return by his expected dividend (which
accrues in the solvency states) and the expected value of residual assets left after the
victim has been compensated (in the insolvency states). Thus, if the lender is willing
to invest in the project as a shareholder, he must expect a lower …xed (debt) payment
is return.
Let us now move on to the equity component of the lender’s compensation. Comparing (4.17) with the analogous condition obtained for the problem of single moral
hazard (for example, see Pitchford 1995, equation 8) we see that the characterization
of the double moral hazard contract must be di¤erent from the characterization of the
optimal contract under double moral hazard. In the case of one sided moral hazard
where the lender does not make any direct contribution toward reducing the environmental damage, Fe (l =e; a)/ Fa (l =e; a) = 0 since lime#0 Fa (l =e; 0) = 1 (assumption
4) and it is straightforward that zero equity for the lender and 100 percent debt would
be the optimal solution. Since Fa (l =e; a) > 0 when a > 0, however, equation (4.17)
suggests that 100 debt is not generally optimal in context of double-sided moral hazard.
The key di¤erence is that when the lender can in‡uence damage realizations and the
…rm’s prospects for insolvency, the optimal contract must simultaneously take account
21
of incentives for both parties. In this case, making the borrower the residual claimant
strengthens his incentives but results in incentive loss on the part of the lender.
Proposition 1 reveals that the manager’s equity stake crucially depends on Fe (l =e; a)/
Fa (l =e; a), which is the ratio of the marginal (probabilistic) productivity of e to the
marginal productivity of a in the states of the world in which the …rm is solvent. The
greater the value of Fe (l =e; a) relative to Fa (l =e; a), the lower the level of and the
higher will be lender’s equity participation. The reason for this observation is the following. Equity serves to induce the owner to undertake appropriate precaution. Thus,
the more productive is the manager’s e¤ort in terms of reducing the probability of insolvency, the lower the level of attention that therefore needs to be paid to the incentives
created for the manager under the contract. Given the positive relationship between
and , proposition 1 leads to the intuitively appealing prediction that the lender will
be o¤ered a lower …xed (debt) payment, and will secure a relatively large stake in the
venture when the manager has a higher marginal productivity of e¤ort. This result
parallels that found in Lewis and Sappington (2000). In their model, however, agents
with higher marginal productivity end up with lower stakes in a rational attempt to
increase their chance of operating a project.
Conversely, the larger the relative value of Fa (l =e; a), the stronger will be the incentives awarded to the manager and the more diluted will be the incentives awarded to
the lender. In the limit as Fa (l =e; a) ! 1, we get the corner solution = 1; that is,
if the lender’s marginal productivity is su¢ ciently high, the owner may o¤er the lender
zero equity and the project would be a 100 percent debt-…nanced.
A point of interest is how varies with l; that is, how do changes in the distribution
of environmental damages a¤ect the optimal equity stake? In general, we can examine
two types of changes in the distribution of environmental damages. First, we can analyze
how changes in the interval [0; L] in‡uence the optimal equity stake. Second, we can
examine shifts in the distribution of F while the interval [0; L] remains unchanged.
In this sense, an increase in l can be interpreted as a shift (probabilistic) in F that
puts more weight on major environmental damage in the sense of …rst-order stochastic
dominance.
Corollary 2. Changes in the distribution of damage realizations imply
d
lmax
= 0 if lmax = L ,
d
lmax
> 0 if lmax = v and
d
>0
l
where lmax =min fv; Lg.
Proof : See the appendix.
A change in the distribution of environmental risk, as represented by an increase in
implies that a major accident is more likely. If the …rm’s asset bounds
lmax =min fv; Lg
22
are exceedingly tight; that is, if lmax =min fv; Lg = v, then the borrower’s optimal equity
stake will increase. On the other hand, if the …rm’s assets are su¢ ciently large such that
lmax =min fv; Lg = L, then the borrower’s optimal equity stake will not be a¤ected. On
the contrary, if the the distribution of F worsens in the sense of stochastic dominance,
the borrower unambiguously substitutes debt for equity. In other words, the lender is
less inclined to invest in the project as an insider if the environmental (project) risk is
great.
4.2. Full lender liability
In this section, we examine the properties of the corresponding solution when the regime
of liability holds the lender liable for the full amount of harm done by the project in the
event that the …rm’s residual assets are insu¢ cient to compensate the victim. As stated
before, we assume that the manager is protected by limited liability. On the other, the
lender exercises managerial control over the …rm and is therefore subjected to extended
liability. The key point to note here is that the lender’s unbounded assets and active
involvement in the project ensures that no liability is uncompensated, ex-post. With
the manager holding all the bargaining power, her problem [FL] reads:
max
f ; g
l Z( )
[v
l] f (l=e; a)dl
e
(4.19)
0
subject to
e 2 arg max
l Z( )
[v
l] f (l=e; a)dl
e
(4.20)
0
a 2 arg max(1
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
0
+
ZL
[v
l]f (l=e; a)dl
a
l ( )
23
(4.21)
and
~ (e; a; ; ) = (1
U
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(4.22)
0
+
ZL
[v
l]f (l=e; a)dl
a
rK.
l ( )
Equations (4.20) and (4.21) are the incentive compatibility constraints, ensuring that
each party selects the level of care optimally given the …nancial arrangement. Equation
(4.22) is the participation constraint, which ensures that the lender’s expected payo¤ is
at least as high as his outside opportunity payo¤. The following proposition summarizes
the solution to the manager’s problem [FL]. The proof is analogous to that of proposition
1 and is therefore omitted.
Proposition 2. Under full lender liability and limited information, the optimal contract, denoted by { ~ ; ~ g is characterized as:
1
~=
1+
a + rK
(1
[v
l] f (l=e; a)dl
0
~=
where r1 = 1
~)
l Z( )
Fe (l =e;a)
Fa (l =e;a) r1
(4.23)
ZL
0
F (l =e; a)
ZL
.
[v
l] f (l=e; a)dl
(4.24)
Fa (l=e; a)dl.
l (
Proposition 2 suggests a relationship between the owner’s equity stake and the ratio
that is generally not analogous to that identi…ed under no lender liability.
Comparing expression (4.17) and (4.23), one can see that in moving from strict liability
=e;a)
to full lender liability, the term with which FFae (l
(l =e;a) is multiplied has been modi…ed.
The critical question therefore is the direction of this change. Under full lender liability,
the upper limit of integration for the integral term in r1 is always L. Under no lender
liability, however, the victim has no recourse to the lender’s wealth in the event that
the …rm’s residual assets are inadequate. Consequently, the upper end point for the
Fe (l =e;a)
Fa (l =e;a)
24
integral term in r0 is constrained by the …rm’s residual assets. Thus, when L < v such
that minfv; L) = L, equations (4.17) and (4.23) coincide. That is to say that when
the scale of the …rm’s asset value is su¢ ciently large relative to the highest possible
damage realization, then extending liability to the lender does not alter the character
of the …nancial arrangement; the optimal equity stake under full lender liability will be
as great as when the lender is not liable. On the other hand, when L > v such that
minfv; L) = v, then equations (4.17) and (4.23) are no longer identical. In this case
r1 < r0 and equations (4.17) and (4.23) suggest that ~ > .
To see the intuition for this observation, recall that when v < L, then no lender
liability results in uncompensated liability in the event that the worst possible damage
is realized since the lender is under no obligation to compensate the victim. However,
under full lender liability, the …nancier’s unlimited wealth ensures that there is no uncompensated liability such that v < L implies an increase in the range of states of the
world in which the lender incurs a negative cash ‡ow. Hence, the optimal contract under
no lender liability cannot be identical to the optimal contract under full lender liability
when v < L. More precisely, the optimal contract under full lender liability requires
that the lender be promised a higher expected return in the states of world in which the
…rm is solvent in order to compensate him for the downside risk he faces in the adverse
states. This is done by raising the debt component of the compensation relative to its
value under no lender liability. In other words, the owner commits to a relatively large
debt repayment schedule in order to o¤set the severe downside threat that the lender
faces in the event of insolvency. Only by structuring the contract in this manner can
the lender feel appropriately compensated to accept any …nancial arrangement under
extended liability.21 But if is now higher relative to its level under no lender liability,
then must also be higher. The latter follows immediately from the positive relationship between and , and the fact that the optimal contract always holds the lender
to his outside opportunity pro…t.
As a result of these considerations, when v < L, the manager’s optimal equity stake
must be higher when the lender bears the full burden for victim compensation than
when he does not bear any responsibility. The manager substitutes debt for equity,
while holding the lender to his outside opportunity payo¤.
4.3. Partial lender liability
We now look at the case of partial lender liability under limited information in order
to analyze any additional considerations that may bear on the nature of the optimal
21
Note that the only other apparatus that the manager can manipulate to make the contract attractive
to the lender is to increase latter’s equity stake or dividends (1- ). However, as we now know, raising
(1- ) has a double-edged feature here in that it increases the lender’s incentive to work but at the same
time dulls the manager’s incentive for e¤ort choice.
25
contract. Our primary objective here is document the impact of partially extending
liability on equity participation and the debt component of the compensation contract.
To examine how holding the lender only partially liable for the victim’s compensation
a¤ects the character of …nancial contracts, we simply add to the owner’s unconstrained
maximization problem de…ned by equations (4.19) and (4.20), the following constraints:
a 2 arg max(1
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(4.25)
0
+
minfL;v+
Z (L v)g
[v
l]f (l=e; a)dl
a
l ( )
and
U (e; a; ; ) =(1
)
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(4.26)
0
+
minfL;v+
Z (L v)g
[v
l]f (l=e; a)dl
a
rK.
l ( )
Call this modi…ed maximization program, problem [PL]. Constraints (4.25) and (4.26)
are the lender’s incentive and the participation constraints, respectively. Note that the
incentive condition on the part of the manager is unchanged from the full lender liability
setting since limited liability implies that the owner can lose no more than her equity
no matter what the regime of liability is in place. The solution to problem [PL] is
summarized in the following proposition:
Proposition 3. Under partial lender liability, the optimal contract, denoted by f ; g,
is characterized as
1
=
1+
a + rK
=
(1
)
l Z( )
[v
(4.27)
Fe (l =e;a)
Fa (l =e;a) r2
l] f (l=e; a)dl
0
minfL;v+
Z (L v)g
l ( )
F (l =e; a)
26
[v
l] f (l=e; a)dl
(4.28)
where r2 = 1
minfL;v+
Z (L v)g
Fa (l=e; a)dl.
l (
Proof : See the appendix
Expressions (4.27) and (4.28) can be interpreted in the same manner as (4.23) and
(4.24).
4.4. Extended liability and …nancial structure
So far we have derived the optimal …nancial arrangements under alternative schemes of
liability allocation. In this section, we attempt to compare the three liability schemes in
terms of their impact on the optimal equity stake. Obviously, the basis of comparison
is a world in which the double moral hazard problem created by unobservable e¤ort is
present, but in which extended liability is absent. One can then ask what e¤ect the
introduction or extending the lender’s liability makes to the character of the optimal
…nancial structure.
It turns out that the di¤erence L v, the size of the highest possible damage realization relative to the …rm’s asset value, is at the core of the relative ranking of the
equity stakes induced by the alternative liability regimes. To see how, …rst, consider
a situation where v=L. In this case, a direct comparison of (4.17), (4.23) and (4.27)
indicates that extending the lender’s liability does not alter the level of equity stake
chosen by the manager. In all the three cases, the top tail/ceiling of the integral terms
in r0 ,r1 and r2 are identical. Intuitively, when v=L, limited liability implies that the
owner can never lose more than her equity. In addition, the …rm’s unbounded assets
insulate the lender’s exogenous wealth from being used for the victim’s compensation.
Consequently, both parties will be indi¤erent between no lender liability and extended
liability (both full and partial) and the nature of the …nancial contract adopted by the
owner will be identical across the three regimes; that is, = = ~ and = = ~ .
Now consider a situation where the value of the …rm’s assets v is su¢ ciently large
such that minfv; Lg = L. In this case, (4.17) and (4.23) indicate that
= ~ . In
addition, since v > v + (L v) = v(1
) + L > L, it is evident that minfL; v +
(L v)g = L. This implies that r0 = r1 = r2 and so we again have = ~ = and
= = ~ precisely for the reasons cited above.
Finally, suppose that the value of the …rm’s residual assets v is su¢ ciently small
such that minfv; Lg = v. This can be interpreted as a situation in which the …rm either
faces a potentially huge environmental damage or its asset bounds are exceedingly tight.
In terms of equation (4.27), it implies that minfL; v + (L v)g = v + (L v) since
L > v + (L v) = v(1
) + L > v. Comparing r0 , r1 and r2 , one can see that
r0 > r2 > r1 from which it follows that ~ > > . Furthermore, since and must
27
move in the same direction in order to hold the lender to his outside opportunity payo¤,
we can infer that ~ > > .
This result is revealing: It shows that even though extending liability induces a
substitution of debt for equity when v < L, this shift is stronger under full lender
liability than it is under partial lender liability. Intuitively, partial lender liability causes
an increase in the range of the states of the world in which the lender face negative cash
‡ows. However, this increase is not as pronounced as it is under full lender liability.
Because the lender is not held liable for the full amount of the harm, partial lender
liability e¤ectively imposes an exogenous limit on the downside risk that the lender
faces in the states of the world in which the …rm is insolvent. Indeed, so long as
L v > 0, the victim is never fully compensated under partial liability in the event that
the worst possible disaster manifests itself. Hence, the lender’s ability to get the owner
to commit to a larger debt repayment is diminished under partial liability than under
full lender liability. Thus, from propositions 1, 2 and 3 we have:
Corollary 2: The relationship between the regime of liability allocation and the
lender’s equity participation is ambiguous. More speci…cally:
(i) If v L, then extending liability has no e¤ ect on the lender’s equity;
(ii) If v < L, extending liability induces a substitution of debt for equity.
4.5. Extended liability and the …rm’s insolvency
In our model the …rm is insolvent only if the environmental damage is very high, i.e. if
and only if l > l ( ). Therefore, the probability that the …rm fails depends exclusively
on its exposure to the environmental risk, which in turn depends on the level of care
l Z( )
exercised by the owner and lender’s e¤ort. Recall that 1-F (l =e; a) = 1
f (l=e; a)dl
0
is the probability of insolvency. Hence, higher e¤orts imply a lower likelihood of insolvency. In this section, we examine the relationship between the scheme of liability
allocation, e¤ort levels and the probability of the …rm’s insolvency.
The proposition below, summarizes the impact of extended liability on the project’s
prospects for solvency.
Proposition 4. The relationship between the regime of liability and the probability of
insolvency is ambiguous. More speci…cally:
(i) If v L, then extending liability has no impact the …rm’s probability of failure;
(ii) If v < L, then extended liability induces a higher solvency rate.
28
Proof : See the appendix
Proposition 4 suggests a relationship between the regime of liability allocation and
project failure analogous to that between the format of liability allocation and the
optimal equity participation. The key point to note here is that when v
L, that
is, when the …rm’s asset value is su¢ ciently large, the lender’s wealth is e¤ectively
insulated from the victim no matter what regime of liability allocation is in place. This
implies that the lender’s need for incentives to expend e¤ort will be identical across the
three systems and so will the levels of , a and e. In short, we have [1 F (l =~
e; a
~)]
= [1 F (l =e; a)] = [1 F (l =e; a)].
However, when v falls below L; that is, when the …rm’s assets are su¢ ciently low
or the worst possible environmental accident is major in magnitude, our model predicts
that the character of the …nancial arrangement will be sensitive to the regime of liability allocation. In particular, while no lender liability continues to ensure that the
lender’s assets are immunized from the courts, full lender and partial lender liabilities
imply that the lender faces the prospect of su¤ering a wealth loss beyond his equity
in the insolvency state. To induce the lender to …nance the project requires that any
optimal contract guarantee him a higher return in the good state; that is, in the good
state, the lender must be promised a higher …xed (debt) payment. Since the outcome
under extended liability promises the lender a higher expected return than under no
lender liability, increasing the likelihood of solvency, therefore, becomes attractive for
the lender. Consequently, the level of care is higher under extended liability than under
no lender liability.
But how does the higher level of …xed (debt) payment necessitated by extended
liability a¤ects the manager’s incentive to exercise care? It turns out that the higher
debt level actually induces higher precaution incentives on the part of the manager as
well. One might wonder how to reconcile the apparent increase in the owner’s incentive
to exert care with the debt overhang: For if there is a shift toward more …xed payment,
does it not follow that the owner will become a residual claimant over fewer states of
nature? What then induces the owner to undertake more precaution? The key point
here is that and necessarily move in the same direction as the owner adjusts the
…nancing arrangement in order to hold the lender to his outside opportunity payo¤ level.
Hence, when a …rm raises its leverage in order to compensate the lender for the higher
downside risk he faces in the severe state, this must be o¤set by a lower equity stake for
the lender just enough to keep his expected payo¤ unchanged. Hence, is necessarily
higher under extended liability than it is under no lender liability and so is the owner’s
incentive to apply care. To sum up, the induced levels of e¤ort, a and e, are higher
under extended liability than under no lender liability and so we have [1 F (l =~
e; a
~)]
>[1 F (l =e; a)] and [1 F (l =e; a)] >[1 F (l =e; a)].
The results derived here contrasts sharply with those of Pitchford (1995) and Dionne
29
and Spaeter (2003). In Pitchford (1995), the larger the …xed payment the owner is
required to make, the less incentive the owner has to reduce an accident. Consequently,
lender liability does not reduce the likelihood of damages. In the same vein, Dionne and
Spaeter (2003) argue that extending liability to the lender, by increasing the face value
of debt, unambiguously increases the …rm’s probability of failure. In all these models,
however, the lender plays a purely passive role in the a¤airs of the client …rm. Hence,
it is not feasible in these settings to consider a richer role for debt, in particular, the
e¤ect of higher debt on the lender’s incentive to engage in more productive intervention
and the impact this might have on …rm’s probability of failure. More precisely, their
frameworks neglect to consider that a shift toward more leverage, by increasing the fruits
of the lender’s e¤ort, may well lead to an increase in the lender’s e¤ort, and therefore
the project’s prospects for solvency.
5. Implications and conclusion
We have explored the nature of …nancial contracts between a risk-neutral owner-manager
of an environmentally risky project and a risk-neutral lender when both parties may
contribute toward reducing the risk of an environmental accident. Our objective was
to examine how extending liability to the lender can a¤ect the nature of the optimal
contract, and how this in turn can a¤ect the induced levels of e¤ort and the …nancial
fragility of the …rm.
The main conclusion is that the relationship between the regime of liability and the
probability of insolvency is ambiguous. Depending upon the size of the …rm’s residual
assets, extending liability may or may not a¤ect the likelihood of the …rms insolvency. If
the …rm’s residual assets are su¢ cient to pay o¤ the damage; that is, if the environmental
harm is minor compared to the …rm’s asset, then the induced probability of insolvency
is insensitive to the liability regimes. On the other hand, if the lender must pay for
the victim’s compensation, then extended liability outperforms the no lender liability
regime in terms of the induced probability of success. Thus, the model predicts that
extending liability may diminish the fragility of …rms that are otherwise weak …nancially.
Our model also predicts a relationship between the marginal productivity of e¤ort
and equity participation consistent with that implied in the signalling literature. In
that literature, high productivity is associated with less equity because equity …nancing
serves as a signal that low return is likely. In our model, however, a more productive
principal o¤ers a lower equity stake for the agent in a rational attempt to induce the
agent to undertake the appropriate level of e¤ort; the higher the productivity of the
manager’s e¤ort relative to the lender, the lower the level of the incentives that need
to be created for the manager under the contract and the higher the level of incentives
that should be created for lender.
The model also yields a number empirically testable hypotheses. First, there should
30
be a relationship between the type of …nancial claim issued by the …rm and its exposure
to environmental risk. More debt should be associated with …rms with large environmental incidents, while more equity should be issued by …rms with minor environmental
infractions. This implies that a variable indicating the type of …nancing arrangement
adopted by the …rm should be included in the regression explaining the probability that
a …rm is cited for an environmental violation.
Second, the model predicts that if a …rm operates an environmentally risky line of
business and the lender has inside management rights, then such a …rm should be more
pro…table than that in which the lender does not intervene. To test this hypothesis,
one could gather …rm-speci…c data on …nancing patterns of …rms engaged in operations
with environmental risks and add environmental screening or lender intervention as
an explanatory variable of the …rm’s pro…tability. For example, a dummy variable
indicating the presence of a lender or his representative on the …rm’s board of directors
could be included in explaining the pro…tability of the …rm.
Thirdly, there should be a relationship between the probability of a …rm’s insolvency
and the type of …nancial claims issued by the …rm. To test this hypothesis, one could
gather data on …rms that actually …led for bankruptcy or bankruptcy protection due to
environmental incidents and add capital structure choice as an explanatory variable of
the …rm’s bankruptcy risk. One would then expect …rms that were highly leveraged to
have declared bankruptcy less often.
There remains some interesting extensions. Our model considered the case where the
owner’s contribution towards the new investment is observable or costlessly veri…able.
A natural extension is to consider an expanded model in which the owner is endowed
with privileged information about her contribution. One could also admit possibility
that the owner may reduce the size of the investment and divert some of the loaned
funds to …nance her perquisite consumption. The lender’s contractual challenge would
then involve inducing the owner to exercise adequate precaution as well as undertaking
the required investment. Another extension might admit alternative speci…cation of
risk preferences. A typical small business may not hold a large portfolio of projects and
may therefore be unable to diversify away all risks associated with a project. It may
therefore be appropriate to model the manager as a risk-averse party.
31
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35
6. Appendix
Proof of lemma 1
Under conditions of no lender liability, the manager’s and the lender’s payo¤ are given,
respectively, by
l Z( )
o
U =
[v
l] f (l=e; a)dl
e
(6.1)
0
and
U
A
= (1
)
l Z( )
[v
[v
l] f (le; adl
l] f (l=e; a)dl + F (l =e; a)
(6.2)
0
+
minfv;Lg
Z
a.
l ( )
From equation (2.4), we have
l Z( )
[v
l] f (l=e; a)dl =
0
l Z( )
[v
l] f (l=e; a)dl + F (l =e; a)
(6.3)
0
+
minfv;Lg
Z
[v
l] f (le; adl
U B.
a
l ( )
Substituting for
Uo =
l Z( )
in equation (6.1) using condition (6.3), we obtain
[v
l] f (l=e; a)dl +
0
minfv;Lg
Z
[v
l] f (l=e; a)dl
a
U B.
e
(6.4)
l ( )
The owner’s problem [NF] can be rewritten as follows:
o
max U =
a;e
l Z( )
0
[v
l] f (l=e; a)dl +
minfv;Lg
Z
l ( )
36
[v
l] f (le; a)dl
a
e
U B.
(6.5)
subject to the lenders participation constraint U B rk. Since U o is decreasing in U B
and the owner has all the bargaining power, the participation constraint can be replaced
by equality without loss of generality. Hence, the maximization problem reduces to
max
fa;eg
l Z( )
[v
l] f (l=e; a)dl +
0
minfv;Lg
Z
[v
l] f (l=e; a)dl
rK
a
e
(6.6)
l ( )
which leads to the …rst-order conditions
l Z( )
[v
l] fe (l=e; a)dl
1=0
(6.7)
0
minfv;Lg
Z
l Z( )
minfv;Lg
Z
[v
l] fe (l=e; a)dl
1 = 0,
(6.8)
[v
l] fe (l=e; a)dl +
l ( )
and
[v
l] fe (l=e; a)dl +
0
l ( )
respectively. Now, integrating the last two equations by parts, we obtain, respectively,
[(v
l ( )
l)Fe (l=e; a)]0
+
l Z( )
Fe (l=e; a)dl
(6.9)
0
+ [(v
minfv;Lg
( )
l)Fe (l=e; a)]l
+
minfv;Lg
Z
Fe (l=e; a)dl
1 = 0.
l ( )
and
[(v
l ( )
l)Fa (l=e; a)]0
+
l Z( )
Fa (l=e; a)dl
(6.10)
0
+ [(v
minfv;Lg
l)Fa (l=e; a)]l ( )
+
minfv;Lg
Z
l ( )
37
Fa (l=e; a)dl
1 = 0.
Making use of Fa (L=e; a) = Fa (0=e; a) = 0, eliminating and rearranging terms, we
minfv;Lg
minfv;Lg
Z
Z
obtain
Fe (l=e ; a )dl 1 = 0 and
Fa (l=e ; a )dl 1 = 0, from which it
0
0
immediately follows that
minfv;Lg
Z
Fe (l=e ; a )dl =
0
minfv;Lg
Z
Fa (l=e ; a )dl = 1.
0
Proof of lemma 2
The principal’s unconstrained maximization is as follows
max
fa;eg
l Z( )
[v
l] f (l=e; a)dl
rK
a
v+ Z
(L v)
e
[v
l] f (l=e; a)dl.
v
0
The …rst-order conditions for an interior solution imply that
l Z( )
[v
l] fe (l=e; a)dl
0
v+ Z
(L v)
[v
l] fe (l=e; a)dl
1=0
l ( )
and
l Z( )
[v
l] fa (l=e; a)dl +
0
v+ Z
(L v)
[v
l]fa (l=e; a)dl
1=0
l ( )
Now, integrating the last two expression by parts, we obtain, respectively,
[(v
l)Fe (l=e; a)]v0
+
l Z( )
Fe (l=e; a)dl + [(v
v+ (L v)
( )
l)Fe (l=e; a)]l
0
+
v+ Z
(L v)
Fe (l=e; a)dl
l ( )
38
1=0
(6.11)
and
[(v
l)Fa (l=e; a)]v0
+
l Z( )
Fa (l=e; a)dl + [(v
v+ (L v)
( )
l)Fa (l=e; a)]l
0
+
v+ Z
(L v)
Fa (l=e; a)dl
1=0
l ( )
Making use of assumption 4 and Fe (L=e; a) = Fe (0=e; a) = 0, the …rst-order conditions associated with the optimal level of e¤orts undertaken by the manager and the
lender reduce to
l Z( )
Fe (l=~
e ;a
~ )dl +
0
v+ Z
(L v)
Fe (l=~
e ;a
~ )dl
1=0
(6.12)
1 = 0.
(6.13)
l ( )
and
l Z( )
Fa (l=~
e ;a
~ )dl +
0
v+ Z
(L v)
Fa (l=~
e ;a
~ )dl
l ( )
Proof of corollary 1
We can rewrite the solution to the social problem as
s
s
Se (e ; a ) =
l Z( )
s
s
Fe (l=e ; a )dl +
0
ZL
Fe (l=es ; as )dl
1=0
(6.14)
ZL
Fa (l=es ; as )dl
1 = 0.
(6.15)
l ( )
and
s
s
Sa (e ; a ) =
l Z( )
s
s
Fa (l=e ; a )dl +
0
l ( )
(es ; as )
The proof that
= ( e~ ; a
~ ) is immediate from (3.11), (3.12), (6.14), and (6.15).
Thus, we need only examine the size of (e ; a ) relative to (es ; as ). Recall that the
optimal e¤ort supply functions under partial lender liability are given by
39
l Z( )
Fe (l=e ; a )dl
1=0
(6.16)
0
minfL;v+
Z (L v)g
l Z( )
minfL;v+
Z (L v)g
Fa (l=e ; a )dl
1 = 0.
(6.17)
Fe (l=e ; a )dl +
l ( )
and
Fa (l=e ; a )dl +
0
l ( )
Evaluating Se (e ; a ) we obtain
Se (e ; a ) =
l Z( )
Fe (l=e ; a )dl +
0
ZL
Fe (l=e ; a )dl
1.
(6.18)
l ( )
There are three possible cases to consider. Case 1 : L = v. In this case minfL; v +
(L v)g = L and Se (e ; a ) = 0. Since S is concave in e and a, it follows that e = es
and a = as . Case 2 : L > v. In this case, v + (L v) = v + L(1
) < L such that
minfL; v + (L v)g = v + L(1
) and Se (e ; a ) > 0. Since S is concave in e, it
follows that that e < es . Case 3 : L < v. In this case v + (L v) = v + L(1
)>L
and so minfL; v + (L v)g = L. It follows that Se (e ; a ) = 0 and e = es . The proof
for a is analogous and we therefore omit it.
Proof of proposition 1
Recall from (4.16) that the manager’s problem is
max
f ; g
l Z( )
[v
l] f (l=e; a)dl +
0
minfv;Lg
Z
0
[v
de
l] fe (l=e; a) dl +
d
l] (l=e; a)]dl
rK
a
e.
(6.19)
l ( )
Partially di¤erentiate (6.19) with respect
minfv;Lg
Z
[v
minfv;Lg
Z
to obtain
[v
l] fa (l=e; a)
da
dl
d
0
Integrating the latter most expression by parts, we obtain
40
de
d
da
= 0.
d
(6.20)
[v
l] Fe (l=e; a)
+
minfv;Lg
Z
de
d
l ( )
+ [v
l] Fe (l=e; a)
0
Fe (l=e; a)
de
dl [v
d
l] Fa (l=e; a)
de
d
minfv;Lg
de
d
l ( )
0
+ [v
+
minfv;Lg
Z
l] Fa (l=e; a)
Fa (l=e; a)
da
dl
d
l ( )
0
minfv;Lg
de
d
l ( )
de
d
da
= 0,
d
0
which can be simpli…ed further to obtain
[v
de
l] Fe (0=e; a)
+ [v
d
de
l] Fe (min fv; Lg =e; a)
+
d
minfv;Lg
Z
Fe (l=e; a)
de
dl
d
0
de
l] Fa (0=e; a)
+ [v
d
[v
+
minfv;Lg
Z
Fa (l=e; a)
l] Fa (min fv; Lg =e; a)
da
dl
d
de
d
de
d
da
= 0.
d
0
By assumption, Fe (0=e; a) =0 and [v l] Fe (min fv; Lg =e; a) = 0. Making use of these
conditions and rearranging, the last expression reduces
2
3
2
3
minfv;Lg
minfv;Lg
Z
Z
6
7 de 6
7 da
Fe (l=e; a)dl 15
+4
Fa (l=e; a)dl 15
= 0.
(6.21)
4
d
d
0
0
Similarly, partially di¤erentiating (6.19) with respect
minfv;Lg
Z
[v
l] fe (l=e; a)
de
dl + [v
d
we obtain
l ] f (l =e; a)
@l
(6.22)
0
+
minfv;Lg
Z
[v
l] fa (l=e; a)
da
dl
d
[v
0
41
l ] f (l =e; a)
@l
de
d
da
= 0.
d
Recall that the …rm’s insolvency condition is given by as
v
l = 0.
(6.23)
It follows, therefore, that
@l
= 1.
(6.24)
@
Integrating equation (6.22) by parts, making use of the fact that Fe (0=e; a) =0 and
[v l] Fe (l=e; a)jl=minfv;Lg =0 and rearranging, equation (6.22) simpli…es to
2
minfv;Lg
Z
6
4
Fe (l=e; a)dl
0
3
2
minfv;Lg
Z
7 de 6
15
+4
d
3
7 da
15
= 0.
d
Fa (l=e; a)dl
0
Dividing equation (6.21) by (6.25) we obtain
(6.25)
de da
da de
=
.
(6.26)
d d
d d
Making use of (4.6), (4.7), (4.10) and (4.11), the last expression can be rewritten as
l Z( )
Fe (l=e; a)dl
Fa (l =e; a)
0
SOC e
SOC a
=
l Z( )
Fa (l=e; a)dl
Fe (l =e; a)
0
SOC a
,
(6.27)
Fa (l=e; a)dl.
(6.28)
SOC e
which simpli…es to
Fe (l =e; a)/ Fa (l =e; a) =
l Z( )
Fe (l=e; a)dl
0
, l Z(
)
0
Now, using the last equation and the …rst order conditions (4.4) and (4.9), we obtain
Fe (l =e; a)
(1
=
Fa (l =e; a)
) 1
r0
(6.29)
R minfv;Lg
Fa (l =e;a)
where r0 = 1 l ( )
Fa (l=e; a)dl. Solving for , we obtain = Fa (l =e;a)+F
.
e (l =e;a)r0
Now dividing the numerator and the denominator of the last equation by Fa (l =e; a)
1
which is equation (4.17)of proposition 1. Equation (4.18)
yields
=
Fe (l =e;a)
1+ F
a (l =e;a)
r0
follows immediately from the binding participation constraint (4.15).
42
Proof of corollary 2
From proposition 1, we have:
[ Fe (l =e; a)/ Fa (l =e; a); ]
d
dl
=
d
lmax
=
minfv;Lg
Z
fa (l=e; a)dl
l ( )
[1 + ( Fe (l =e; a)/ Fa (l =e; a); )r0 ]2
[ Fe (l =e; a)/ Fa (l =e; a); ] Fa (l=e; a)jminfv;Lg
[1 + ( Fe (l =e; a)/ Fa (l =e; a); )r0 ]2
> 0, and
.
d
It follows, therefore, that if min fv; Lg = L, lmax
= 0 since Fa (L=e; a) = 0. On the other
d
hand, if min fv; Lg = v, lmax > 0 assuming that Fa (v=e; a) > 0.
Proof of proposition 3
The …rst-order associated with (4.25) is given by
(1
)
l Z( )
Fa (l=e; a)dl +
0
minfL;v+
Z (L v)g
Fa (l=e; a)dl
1 = 0.
(6.30)
l ( )
The manager’s unconstrained problem is
max
f ; g
l Z( )
[v
l] f (l=e; a)dl +
0
minfL;v+
Z (L v)g
[v
l] (l=e; a)]dl
rK
a
e
U B , (6.31)
l ( )
subject to (4.20) and (4.25) Partially di¤erentiate (6.31) with respect
l Z( )
[v
de
l] fe (l=e; a) dl +
d
0
l Z( )
[v
l] fa (l=e; a)
da
dl
d
0
+
minfL;v+
Z (L v)g
[v
l]fe (l=e; a)dl
de
d
l ( )
+
minfL;v+
Z (L v)g
[v
l]fe (l=e; a)dl
l ( )
43
de
d
de
d
da
= 0,
d
to obtain
(6.32)
where
de
=
d
l Z( )
Fe (l=e; a)dl
0
,
da
SOCe ,
=
d
lZ(
Fa (l=e; a)dl
0
,
SOCa2 ,
SOCe =
l Z( )
Fee (l=e; a)dl < 0
0
is the second-order condition associated as de…ned by (4.5) and
SOCa = (1
)
l Z( )
Faa (l=e; a)dl +
0
minfL;v+
Z (L v)g
Faa (l=e; a)dl < 0
l ( )
is the second-order condition associated with equation (4.25). Integrating equation
(6.32) by parts, we obtain
[v
l] Fe (l=e; a)
+
l Z( )
l ( )
de
d
+ [v
0
Fe (l=e; a)
de
dl +
d
0
+ [v
l] Fe (l=e; a)
+
de
d
minfL;v+
Z (L v)g
l] Fe (l=e; a)
l Z( )
Fa (l=e; a)
da
d
l ( )
0
da
dl
d
0
minfL;v+ (L v)g
+ [v
l] Fe (l=e; a)
l ( )
Fe (l=e; a)
de
dl +
d
l ( )
minfL;v+
Z (L v)g
da
d
minfL;v+ (L v)g
l ( )
Fa (l=e; a)
da
dl
d
l ( )
de
d
da
= 0.
d
By assumption, Fe (0=e; a) = Fe (L=e; a) and Fa (0=e; a) = Fa (L=e; a) = 0. Assume
further that if the di¤erence L v is positive, then v + (L v) is su¢ ciently large
such that Fe (v + (L v)=e; a)=0 and Fa (v + (L v)=e; a) = 0. Making use of these
assumptions, the last expression can be simpli…ed further to obtain
44
2
l Z( )
6
4
2
0
l Z( )
6
+4
Fe (l=e; a)dl +
minfL;v+
Z (L v)g
3
7 de
15
d
Fe (l=e; a)dl
l ( )
Fa (l=e; a)dl +
0
minfL;v+
Z (L v)g
(6.33)
3
7 da
= 0.
15
d
Fa (l=e; a)dl
l ( )
Similarly, partially di¤erentiating (6.31) with respect we obtain
2
3
l Z( )
minfL;v+
Z (L v)g
6
7 de
Fe (l=e; a)dl +
Fe (l=e; a)dl 15
4
d
0
2
l Z( )
where
6
+4
l ( )
Fa (l=e; a)dl +
0
minfL;v+
Z (L v)g
3
7 da
15
= 0,
d
Fa (l=e; a)dl
l ( )
de
=
d
Fe (l =e; a)
da
and
=
2
SOCe
d
(6.34)
Fa (l =e; a)
.
SOCa2
Dividing equation (6.33) by (6.34) we obtain
de da
da de
=
,
d d
d d
(6.35)
which can be rewritten as
Fe (l =e; a)/ Fa (l =e; a) =
l Z( )
Fe (l=e; a)dl
0
, l Z(
)
Fa (l=e; a)dl
(6.36)
)/ r2
(6.37)
0
Now, from the …rst-order conditions (4.4) and (6.30), we have
l Z( )
Fe (l=e; a)dl
0
where r2 = 1
R minfL;v+
l ( )
, l Z(
)
Fa (l=e; a)dl = (1
0
(L v)g
Fa (l=e; a)dl. Making use of equation (6.36), the latter
most expression can be simpli…ed further to obtain
45
Fe (l =e;a)
Fa (l =e;a)
=
(1
) 1
r2 .
Now, solving
Fa (l =e;a)
for , we obtain = Fa (l =e;a)+F
. Dividing the numerator and the denominator
e (l =e;a)r2
1
of the last equation by Fa (l =e; a) yields =
, which is equation (4.27) of
Fe (l =e;a)
1+ F
proposition 3.
a (l =e;a)
r2
Proof of proposition 4
We prove this proposition by …rst establishing the following. Under no liability, the
lender’s e¤ort choice decision is given by (4.9). Substituting for in (4.9) using (4.17)
we obtain
UaB (a; e) =
l Z( )
Fa (l=a; e)dl +
0
lmax
Z
Fa (l=a; e)dl
l ( )
Fa (l ( )=a; e)
l Z( )
Fa (l=a; e)dl
0
2
6
Fa (l =a; e) + Fe (l =a; e) 41
lmax
Z
l ( )
3
1=0
7
Fa (l=a; e)dl5
where lmax = minfv; Lg. Now, compute the partial derivative
2
@ UaB (a; e)
6
= Fa (lmax =e; a) 41
@lmax
where
Fa (l =e; a)Fe (l =e; a)
2
0
2
6
= Fa (l =e; a) + Fe (l =e; a) 41
show that
Fa (l =e;a)Fe (l =e;a)
2
l Z( )
lmax
Z
l ( )
3
3
7
Fa (l=e; a)dl5 ,
7
Fa (l=e; a)dl5. It is straightforward to
2 (0; 1). Thus, a necessary and su¢ cient condition for
46
@UaB (a;e)
@lmax
> 0 is that
l Z( )
Fa (l=e; a)dl 2 (0; 1). From (4.9), we have
0
lmax
Z
1
(1
)=
Fa (l=e; a)dl
l ( )
l Z( )
Fa (l=e; a)dl
0
=)
l Z( )
lmax
Z
Fa (l=e; a)dl > 1
0
=)
l Z( )
Fa (l=e; a)dl > 1
=)
Fa (l=e; a)dl
0
l ( )
l Z( )
) 2 (0; 1)
l ( )
lmax
Z
Fa (l=e; a)dl since (1
Fa (l=e; a)dl < 1.
0
Hence
@UaB (a;e)
@lmax
> 0. The proof that
@Ueo (a;e)
@lmax
> 0 is analogous, and we therefore omit it.
We now consider three cases.
Case 1 : L < v such that minfv; Lg = L and minfL; v + (L v)g = L. This means
that the lender’s e¤ort choice under full and partial lender liability regimes are given,
respectively, by
~ B (~
U
~) =
a a; e
l Z( )
Fa (l=~
a; e~)dl +
0
Fa (l =~
a; e~)
ZL
Fa (l=~
a; e~)dl
(6.38)
l ( )
l Z( )
0
Fa (l=~
a; e~)dl
2
6
Fa (l =~
a; e~) + Fe (l =~
a; e~) 41
47
ZL
l ( )
3
7
Fa (l=~
a; e~)dl5
1<0
and
UaB (a; e)
=
l Z( )
Fa (l=a; e)dl +
0
ZL
Fa (l=a; e)dl
(6.39)
l ( )
Fa (l =a; e)
l Z( )
0
Fa (l=a; e)dl
2
6
Fa (l =a; e) + Fe (l =a; e) 41
ZL
l ( )
3
1 = 0.
7
Fa (l=a; e)dl5
The e¤ort selection condition under no lender liability setting is given by
UaB (a; e)
=
l Z( )
Fa (l=e; a)dl +
0
Fa (l =e; a)
ZL
Fa (l=e; a)dl
(6.40)
l ( )
l Z( )
0
Fa (l=e; a)dl
2
6
Fa (l =e; a) + Fe (l =e; a) 41
ZL
l ( )
3
7
Fa (l=e; a)dl5
1 = 0.
A direct comparison of the last three expressions yields a = a = a
~. As for e, it
can be easily checked that r0 = r1 = r2 implying that = ~ = . It follows that
when L < v, e~ = e = e. Hence, we conclude that [1 F (l =~
e; a
~)] = [1 F (l =e; a)]
= [1 F (l =e; a)].
Case 2 : v < L. This means minfv; Lg = v and minfL; v + (L v)g = v + (L v).
Note that v + (L v) = v(1
) + L > v. Making use of these observations, it can
48
be shown that the lender’s e¤ort choice under full lender liability satis…es
~aB (~
U
a; e~) =
l Z( )
Fa (l=~
a; e~)dl +
0
ZL
Fa (l=~
a; e~)dl
(6.41)
l ( )
Fa (l =~
a; e~)
l Z( )
0
Fa (l=~
a; e~)dl
2
6
Fa (l =~
a; e~) + Fe (l =~
a; e~) 41
ZL
l ( )
3
1 = 0.
7
Fa (l=~
a; e~)dl5
The e¤ort selection condition under no lender liability setting is given by
UaB (a; e)
=
l Z( )
Fa (l=e; a)dl +
0
Fa (l =e; a)
Zv
Fa (l=e; a)dl
(6.42)
l ( )
l Z( )
0
Fa (l=e; a)dl
2
6
Fa (l =e; a) + Fe (l =e; a) 41
Zv
l ( )
3
1 = 0.
7
Fa (l=e; a)dl5
The corresponding condition under partial lender liability is
UaB (a; e) =
l Z( )
Fa (l=a; e)dl +
0
minfL;v+
Z (L v)g
Fa (l=a; e)dl
(6.43)
l ( )
Fa (l =a; e)
2
l Z( )
6
Fa (l =a; e) + Fe (l =a; e) 41
Fa (l=a; e)dl
0
minfL;v+
Z (L v)g
l ( )
49
3
7
Fa (l=a; e)dl5
1 < 0.
Evaluating UaB (~
a; e~), we obtain
UaB (~
a; e~)
=
l Z( )
Fa (l=~
a; e~)dl +
0
Zv
Fa (l=~
a; e~)dl
(6.44)
l ( )
Fa (l =~
a; e~)
l Z( )
0
Fa (l=~
a; e~)dl
2
6
Fa (l =~
a; e~) + Fe (l =~
a; e~) 41
Zv
l ( )
B
3
1 < 0.
7
Fa (l=~
a; e~)dl5
(a;e)
since v < L and @U@lamax
> 0 . Since U B is concave in a, it follows immediately that
a
~ > a. Now, evaluating UaB (a; e), we obtain
UaB (a; e)
=
l Z( )
Fa (l=a; e)dl +
0
Zv
Fa (l=a; e)dl
(6.45)
l ( )
Fa (l =a; e)
l Z( )
0
Fa (l=a; e)dl
2
6
Fa (l =a; e) + Fe (l =e; a) 41
Zv
l ( )
B
3
1<0
7
Fa (l=a; e)dl5
(a;e)
since v < v + (L v) and @U@lamax
> 0. Since U B is concave in a, it follows immediately
~aB (a; e) it can be seen that
that a > a. We now compare a with a
~. Evaluating U
~ B (a; e)
U
a
=
l Z( )
Fa (l=a; e)dl +
0
Fa (l =a; e)
ZL
Fa (l=a; e)dl
(6.46)
l ( )
l Z( )
0
Fa (l=a; e)dl
2
6
Fa (l =a; e) + Fe (l =a; e) 41
50
ZL
l ( )
3
7
Fa (l=a; e)dl5
1 > 0.
since L > v + (L v). It follows that a < a
~. In sum, when L > v, a
~ > a > a.
Now making use of corollary 2 and equation (4.4), it can be readily checked that that
e~ > e > e. The foregoing yield’s [1 F (l =~
e; a
~)] > [1 F (l =e; a)] > [1 F (l =e; a)].
Case 3 : v = L. In this case, minfv; Lg = L; minfL; v + (L v)g = L and so
a
~ = a = a. Additionally, since
= ~ = , it follows that that e~ = e = e. Thus, we
conclude that [1 F (l =~
e; a
~)] = [1 F (l =e; a)] = [1 F (l =e; a)].
51