There have been opposing conclusions in the literature as to whether lender liability increases the probability of environmentally harmful accidents. This paper shows that s multiplicity of equilibria is the key to solve such contradictory conclusions.Environmental Risk, Lender Liability

Maeda, Yasuo

Ono, Tetsuo

English

Research Papers in Economics

Hitotsubashi University Repository
TitleLender Liability for Environmental Risk Revisited
Author(s) Ono, Tetsuo; Maeda, Yasuo
Citation
Issue Date 2003-05
Type Technical Report
Text Version publisher
URL http://hdl.handle.net/10086/14469
RightMay 29, 2003
Lender Liability for Environmental Risk Revisited∗
Tetsuo Ono
Insitute of Policy and Planning Sciences, University of Tsukuba
Yasuo Maeda
Department of Economics, Keio University
Abstract
There have been opposing conclusions in the literature as to whether lender liability increases the
probability of environmentally harmful accidents. This paper shows that s multiplicity of equilibria is
the key to solve such contradictory conclusions.
Keywords: Environmental Risk, Lender Liability.
JEL Classiﬁcation: K32.
∗This paper was presented at APFA Finance Conference in 2002 held in Tokyo. For helpful comments on a
previous version, authors wish to thank Gordon Roberts.1 Introduction
There have been opposing conclusions as to whether lender liability increases the probability
of environmentally harmful accidents. Pitchford (1995) has shown that an increase in lender
liability for environmental risk would lead to an increase in the probability of environmental
accidents. On the other hand, Heyes (1996) and Boyer and Laﬀont (1997) have shown the
contradictory result. These two opposing results raise the following question: how do such
seemingly contradictory conclusions emerge from otherwise very similar formal models?
Balkenborg (2001) has recently answered the above question by showing that the distribution
of bargaining power between the lender and the owner is the key to the puzzle. Pitchford
conclusion holds when the credit market is competitive (i.e., the bargaining power of the lender
is not too high), while it does not when the market is imperfectly competitive (i.e., the power of
the lender is high).
This paper provides a diﬀerent framework from Balkenborg (2001), and then show two op-
posing consequences of lender liability under a perfectly competitive credit market. The key to
the puzzle is a multiplicity of quilibria. The eﬀect of lender liability on the accident probability
is entirely diﬀerent between the two equilibria.
2 The Model
A wealth-constrained owner with no initial wealth conducts a project yielding a gross return V
after an initial investment K where V> K> 0. To ﬁnance the project the owner needs a loan
from a lender in a competitive credit market, whom we assume to have deep pockets, i.e., not to
be wealth-constrained. Both agents are risk neutral.
The project may cause an accident that generates total damage cost h to anonymous victims.
1The probability of an accident depends on the owner’s unobservable eﬀort choice to prevent it.
The owner can reduce the accident probability by expending nonpecuniary (or eﬀort) costs
according to a cost function φ(p). The cost function φ :[ 0 ,1] →  + is twice continuously
diﬀerentiable on (0,1), and satisﬁes the following properties.
Assumption 1: φ  < 0,φ    > 0,limp→1 φ =0 ,limp→0 φ = ∞,limp→1 φ  =0 , and limp→0 φ =
−∞.
Figure 1 depicts the graph of the function φ. The meaning of Assumption 1 is that, if a
reduction in p is desired, cost rises. As p gets close to zero, cost becomes prohibitively high.
Assumption 1 leads to the following result which will be used for showing the non-negative proﬁt
of the owner.
Lemma 1: −(1 − p)φ (p) − φ(p) > 0∀p ∈ (0,1).
Proof: −(1 − p)φ (p) − φ(p) is strictly decreasing in p ∈ (0,1) since d[−(1 − p)φ (p) −
φ(p)]/dp = −(1 − p)φ  (p) < 0∀p ∈ (0,1). In addition, limp→1[−(1 − p)φ (p) − φ(p)] = 0. Thus,
−(1 − p)φ (p) − φ(p) > 0∀p ∈ (0,1). 
No eﬀort will be exerted unless liability is imposed. In this paper, we study the consequences
of a joint and strict liability rule which requires the owner and the lender jointly to pay liability
c ∈ [0,h]. We assume that the owner is made liable with all his wealth ﬁrstly and that the lender
has to add the remainder. Because the lender has deep pockets, he cannot evade the liability
payment.
We assume that it is not possible for the owner and the lender to write a contract specifying
p, because the lender lacks the appropriate monitoring technology, or more generally because it
is too costly to describe in a contract the factors that determine p. We consider (pure) debt
2contracts, where the owner borrows an amount K and is required to make a repayment RK
where R is a gross rate of interest on loan.
The time structure of the model is as follows. (1) A social planner sets the joint and strict
liability c ∈ [0,h]. (2) The lender and the owner make a contract to ﬁnance the investment in
a competitive credit market. (3) The owner decides on the level of probability of accident, p.
(4) The net returns from the project are realized. (5) An accident may or may not occur. If an
accident occurs, the owner and the lender have to pay liability c.
In closing this section, we characterize a socially optimal level of the accident probability p∗
such that social surplus is maximized. The expected social surplus SW(p)i s
SW(p)=V − K − ph − φ(p),
where V −K is the net social value of the project, ph is the expected cost of accident, and φ(p)i s
the cost of eﬀort. The socially optimal level of the accident probability maximizing social surplus
is p∗(h) which is the solution for φ (p)=−h. We assume that SW(p∗) > 0, which implies that
the project should be ﬁnanced from the social point of view.
3 Contract and Equilibrium
3.1 Contract
This section considers the contract under joint and strict liability, and shows its consequences.
The owner achieves the value of the project V and discharges RK to the lender. The owner
is made liable with his wealth V − RK if the accident occurs. He can owe a full liability c if
V − RK − c ≥ 0. On the other hand, if he cannot pay a full amount of liability, a lender has to
3add the remainder c − (V − RK). Thus, the expected proﬁt of the owner EΠi s
EΠ=V − RK − pmin{c,V − RK}−φ(p). (1)
The ﬁrst term is the net value of the project for the owner,1 the second term is the expected
amount of liability, and the third term is the cost of eﬀort. Given R, the owner chooses the
probability p to maximize the expected proﬁt.
The lender conjectures that the (unobservable) eﬀort exerted by the owner will be pE (where
the superscript E means the lender’s expectation). He lends to the owner provided that he ﬁnds
it individually rational.
(R − 1)K − p
E max{o,c − (V − RK)}≥0. (2)
This is the participation constraint of the lender, which says that an expected proﬁt is non-
negative. The ﬁrst term, (R − 1)K, is the net return of the loan, and the second term is the
expected payment of liability. If the amount of liability c is greater than the net value of the
project for the owner, V − RK, then the lender must pay the remainder c − (V − RK). In a
competitive credit market, the participation constraint (2) should be met with equality.
3.2 Equilibrium
Based on the behavior of the owner and the lender explained above, we introduce the concept of
equilibrium as follows.
Deﬁnition 1: An equilibrium in this model is a pair (¯ p, ¯ R) such that
(i) the owner’s expected proﬁt is maximized at p =¯ p given R = ¯ R and is non-negative;
(ii) the lender’s participation constraint holds with equality when pE =¯ p and R = ¯ R.
1 Note that V −RK is the net value of the project for the owner whereas V −K is the net social value of the
project.
4The form of the contract in this paper is diﬀerent from that in Pitchford (1995) and Balken-
borg (2001). They assume that the lender and the owner choose a pair of the accident (or the
safety) probability and the amount of repayment, which maximizes the joint surplus (Pitchford
(1995)) or the generalized Nash product (Balkenborg (2001)). Contrary to them, we assume that
the amount of repayment (i.e., a gross rate of interest R on loan) is determined in a competitive
credit market (Rajan (1992)). The lender decides whether to hold a contract to the owner, de-
pending on the expectations of the unobservable accident probability chosen by the owner. This
diﬀerence in the form of contracts will lead to a result that are not found in the previous studies.
In what follows, we consider the equilibrium under the following two cases: c ≤ V − K and
c>V− K. In the case of c ≤ (>)V − K, the net social value of the project is greater than or
equal (less than) the joint liability.
Case: c ≤ V − K
We ﬁrst consider the case of c ≤ V − K: the net social value of the project is greater than or
equal to the joint liability.
Proposition 1: Suppose that c ≤ V −K holds. There exists an equilibrium such that (¯ p, ¯ R)
satisﬁes −c − φ (¯ p)=0and ¯ R =1 .
Proof: Set ¯ R =1 . Then the lender’s participation condition (2) is satisﬁed for any value of
p with p ∈ [0,1]. Under ¯ R =1 , the borrower’s expected proﬁt (1) becomes
EΠ=V − K − pc − φ(p).
The ﬁrst-order condition of the borrower’s proﬁt maximization problem is calculated as
−c − φ
 (p)=0 .
Let ¯ p be the solution for the above problem. The ﬁnal task is to show that EΠ(¯ p) ≥ 0 under
5conditions c ≤ V − K, −c − φ (¯ p)=0 , and ¯ R =1 . We have
EΠ=( V − K − ¯ pc) − φ(¯ p)
≥ (1 − ¯ p)c − φ(¯ p); since c ≤ V − K
= −(1 − ¯ p)φ
 (¯ p) − φ(¯ p); since − c − φ
 (¯ p)=0
≥ 0; from Lemma 1.
This proves the proposition. 
When c ≤ V −K holds, the owner can pay the full liability c; the lender holds no liability. It
is immediately shown that ¯ p ≥ p∗ holds and that ¯ p is decreasing in c. Thus, if c ≤ h ≤ V − K,
setting c = h leads to social optimum (see Figure 2).
Case: c>V− K
We next consider the case of c>V−K: the net social value of the project is less than the joint
liability.
Proposition 2: Suppose that c>V− K holds. An equilibrium (¯ p, ¯ R), if it exists, is
characterized by
¯ R =
K +¯ p(c − V )
(1 − ¯ p)K
, (3)
−(V − K)+¯ pc =( 1− ¯ p)φ
 (¯ p). (4)
Moreover, it holds that ¯ p>p ∗.
Proof: Guess ¯ R>1. Then, c>V−K implies that c>V− ¯ RK. The participation constraint
(2) is rewritten as
¯ R =
K + p(c − V )
(1 − p)K
. (5)
6It is clear that the value of the above equation is greater than 1 which is consistent with the
initial guess (see Figure 3).2
The expected proﬁt of the ﬁrm becomes EΠ=( 1− p)(V − ¯ RK) − φ(p). The ﬁrst-order
condition for the maximization problem of the owner is −(V − ¯ RK) − φ (p)=0 . With (5), this
ﬁrst-order condition is reduced to
−(V − K)+pc =( 1− p)φ
 (p), (6)
or
(V − K) − pc
1 − p
= −φ
 (p), (7)
which characterizes the equilibrium level of p.
Next, we will show that EΠ ≥ 0 under (5) and (6).
EΠ=( 1 − ¯ p)(V − ¯ RK) − φ(¯ p)
= V − K − ¯ pc− φ(¯ p); since (5)
= −(1 − ¯ p)φ
 (¯ p) − φ(¯ p); since (6)
≥ 0; from Lemma 1.
Finally, we will show that ¯ p>p ∗. With c>V− K, (4) is (1 − ¯ p)φ (¯ p) > −(1 − ¯ p)c or
φ (¯ p) > −c ≥− h = φ (p∗) which implies ¯ p>p ∗. 
Multiple Equilibria Here, we examine the case of multiple equilibria. Deﬁne
F(p) ≡
(V − K) − pc
1 − p
.
Then, we have
F
 (p)=
1
(1 − p)2[−c(1 − p)+{(V − K) − pc}]=
1
(1 − p)2[(V − K) − c] < 0,
2 Under the assumption c>V−K, we have p(c− V + K) > 0 ⇔ K +p(c−V ) > (1− p)K ⇔
K+p(c−V )
(1−p)K > 1.
7limp→1 F(p)=−∞, F(0) = (V − K), and F(p)=0a tp =
V −K
c < 1. Hence, Figure 4 depicts
the graph of (7) that characterizes the equilibrium level of the accident probability p. Thus, (7)
could have multiple solutions as shown in Figure 4. We denote eL (eH) as the equilibrium with
low (high) p.3
The eﬀect of liability on the accident probability is entirely diﬀerent between the two equi-
libria. When an amount of liability c increases, the function F(p) turns in a clockwise direction,
while the curve −φ (p) is unchanged. Thus, an increase in liability c leads to an increase (a
decrease) in the accident probability at eL (eH) equilibrium; Ritchford’s result is supported at
eL equilibrium but not at eH equilibrium. Moreover, since p∗ < ¯ p holds, an increase in liability
is harmful (beneﬁcial) when the economy attains the eL (eH) equilibrium. Therefore, liability
leads to opposing eﬀects on social surplus as well as the accident probability between the two
equilibria.
3 For example, let us specify the cost function as φ(p)=1 /p − 1. This function satisﬁes Assumption 1. It
is immediately shown that (1 − p)φ (p) is strictly increasing and strictly concave in p. Thus, there could be a
critical level of c,ˆ c ∈ (0,h), such that (6) has no solution for c>ˆ c and two solutions for c<ˆ c. In case of c>ˆ c,
a large amount of liability leads to the collapse of the credit market, which is consistent with the empirical result
(Schidheiny et al. (1998)). In case of c<ˆ c, multiple equilibria emerge.
8References
Balkenborg, D., 2001, How Liable Should a Lender Be? The Case of Judgement-Proof
Firms and Environmental Risk: Comment, American Economic Review 91, 731-738.
Boyer, M., and Laﬀont, J-J., 1997, Environmental Risks and Bank Liability, European
Economic Review 41, 1427-1459.
Heyes, A.G., 1996, Lender Penalty for Environmental Damage and the Equilibrium Cost
of Capital, Economica 63, 311-323.
Pitchford, R., 1995, How Liable Should a Lender Be? The Case of Judgement-Proof Firms
and Environmental Risk, American Economic Review 85, 1171-1186.
Rajan, R.G., 1992, Insiders and Outsiders: The Choice between Informed and Arms-Length
Debt, Journal of Finance, 1367-1400.
Schmidheiny, S., and Zorraquin, F.J.L. with the World Business Council for Sustainable
Development, 1998, Financing Change: The Financial Community, Eco-eﬃciency, and
Sustainable Development, MIT Press.
9Figure 1. Cost of Effort 
 
 
 
 
 
            ) (p φ  
 
 
 
 
 
 
 
 
                                                           p  
                                                   1       
 Figure 2. The Equilibrium in Case of  K V c − ≤  
 
 
 
 
 
            ) ( ' p φ −  
 
  h  
 
  c  
 
 
 
                                                          p  
          
* p     p                                1 
                                                             Figure 3. The Interest Rate in Case of    K V c − >
 
 
  R  
 
 
            
 
   
 
   
 
  1 
 
                                                          
                                             
                                            1                         
 Figure 4. The Equilibria in Case of    K V c − >
 
 
 
 
 
              ) ( ' p φ −  
               
                 
                 
L e  
K V −         
  
                  
              
 
                                           
H e  
                                                                       
                                                  1       p   
                                                                       
 
 
 
 

Lender Liability for Environmental Risk Revisited

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Lender Liability for Environmental Risk Revisited

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