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Limited liability and incentives when firms can inflict damages greater than net worth
Limited Liability and Incentives when Firms Can Inflict Damages Greater than Net Worth
Ir is well known that, under certain assumptions, strict liability causes a firm to internalize the social cost of accidents, so it chooses the socially optimal activity and preventive cat-e levels.’ A familiar modification of’this result is made in models that alloy bankruptcy. When potential damages to third pat-ties exceed the fit-m’s net worth, part of‘ the risk is externalized. Consequently, the firm externalizes part of‘thc benefit of- restraining its activity Icvel and increasing the level of‘ preventive care. This at‘implies that limited liability gument, which has been emphasized in the literature. causes the firm’s activity level to be too high and the level of‘ preventive care to he too low.’ This note emphasizes another consideration that points in the opposite direction. E:xpcttditures on preventive care reduce the firm’s wealth, thus reducing the size of a court judgement that it can pay. The reduction in the size of the jutlgement that it cm pay increases the cost of bankruptcy borne by potential creditors, including tort victims. ‘Thus limited liability shif’ts part of’ the expected cost of‘ preventive care to the victims of’ tort. A similar argument applies to the firm’s activity level. A lower activity level often itnplics :I reduction in the firm’s stock of capital. A reduction in the fit-m’s stock of‘ capital reduces the firm’s wealth, which increases the cost of’ bankruptcy borne by potential creditors. Thus limited liability shifts part 01‘ the expected cost of’ towering activity levels to the victitns of tort. The general conclusion cart be summarized as fdlows. Limited liability has two effects. First, it externalizes the firm’s expected cost of accidents, which causes a higher activity level and a lower level of’ preventive cat-e. Second, it externalizes pat-t of the firm’s expected cost of cat-e and undercapitalization, which causes a higher level of’ care and a lower level of activity. In general, one effect is not necessarily larger than the other. ‘This note highlights this argument with an exatnple and then proves it mathematically. An Example: Generators and Disposers of Toxic Waste Firtns that generate ages that are greater
or dispose of’ toxic waste often than their ability to compensate
have the potential to inflict datnvictims. Environmental law has
326
Limited liability for damap
evolved to a standard of strict liability as a means of internalizing social costs. But limited liability prevents firms from bearing the full consequences of their actions. Although this reduces the incentive to keep the probability of an accident down, the above discussion indicates that such a firm may actually overinvest in preventive care, or operate on a smaller than optimal scale and distribute profits that might be used more efficiently, from society’s perspective, to increase the productive capacity of the firm. In the hazardous waste area, legislation aimed at mitigating the distortions caused by limited liability includes the Kesource Conservation and Recovery Act (RCKA) of 1976 and the Comprehensive Environmental Response Act (Superfund) of 1980. Together, these acts require firms that treat, store, dispose, or transport hazardous waste to show financial responsibility for third party damages. If insurance is not purchased, a firm must show a net worth above a specified lower limit as a means of providing proof of financial responsibility. The difficulty of obtaining pollution liability insurance following the KCKA and Superfund legislation has made this net worth requirement an important option for meeting the financial responsibility requirements of these acts:’ By placi?! a floor on initial net worth, this requirement focuses on the fact that limited liablhty externalizes part of the expected costs of an accident, thereby reducing the incentive to keep the probability of an accident down. In general, as I will show, raising the level of initial net worth simply provides a greater incentive to reduce the probability of an accident. By encouraging a Firm to increase its initial net worth, the requirement encourages it to increase its investment in preventive care and/or to reduce its activity level. The fact that limited liability reduces the expected cost of preventive care and undercapitalization is not considered. If the level of preventive care is above, or the activity level is below, the socially optimal level as a result of this second effect, then the net worth requirement will exacerbate the problem. Therefore, without knowing which of these two effects of limited liability is dominant, it is impossible to determine whether firm decisions will become more or less socially efficient as a result of the net worth requirements set forth in the RCRA and Superfund legislation.
Mathrmatical
Proof
The above arguments will now be proven mathematically. Assume that the firm is an expected value maximizer and has an initial net worth of W. The probability that an accident occurs and third parties suffer injuries of a fixed amount, L, is p(x,y), where x is the level of preventive care and y is the level of risky activity. The probability of an accident is decreasing in x and increasing in y, implying that pX < 0 and p, > 0. It is assumed that pX, > 0, py, s 0, and p(x,O) = 0. Activity of level y results in a net benefit to the firm of B(y) before the effect on expected losses is considered. Here, B, > 0, B,, < 0 and B(0) = 0. Preventive care is assumed to cost $1 per unit. A rule of strict liability is employed, so the firm is responsible for all damages up to the net worth at the end of the period, W + B(y) - x; liability insurance 1s not available. It is assumed that, for all values of y, B(y) is below I. - W for any feasible value of W. This implies that L is greater than any potential value of net worth at the end of the period.
L.L.
327
POSEY
The socially optimal levels of preventive care and activity imize total social benefits minus total social costs:
Max,u, The
first order
conditions
E =
are those
w +
R(J) - x - P(x,y)L.
-P,L
=
levels that max-
(1)
are
1
(14
and R, = p,,L. In the case of limited
liability,
the firm solves
M~%X.,, E = W + B(y) The
first order
conditions
equate
-jD,[W
marginal
+ B(y) and B,[l
First,
consider
(111)
x - f(x,y)[W benefits
xl = [l
-
+ H(y) -
with marginal
(2)
cost:
(2a)
PI
- p] = p,[W + H(y) -
x].
x].
(2b)
the case where the activity level is fixed at y,,. In choosing the level of expenditure on preventive care, the firm wilt equate the expected marginal benefit of such expenditure with its expected marginal cost in Equation (2a). Comparing the left-hand sides of Equations (la) and (2a) shows that, for each level of x, the marginal benefit of investing in preventive care to reduce the probability of an accident is tower under limited liability, since the firm’s post-accident liability is reduced from total damages, L, to net worth, [W + B(y,,) -xl. But, spending more on preventive care further reduces the firm’s post-accident liability. The expected cost of an additional unit of preventive care is thus reduced to the unit cost weighted by the probability that an accident does not occur [compare the right-hand sides of (la) and (‘La)]. This is because, in the event an accident does occur, the cost of the additional unit is offset by an equivalent reduction in the amount of’ damages that must be paid out to victims, resulting in a net cost of zero. I‘his reduction in the expected cost of preventive car-c provides an incentive to increase the amount invested in suct~ care. If this tatter effect dominates, then the level of preventive care chosen by the firm wilt be greatel than the socially efficient level. ‘rhis result can be illustrated graphically. The first order conditions for both the socially optima1 preventive care level and the preventive car-c level chosen by a firm under limited liability are depicted in Figure 1.’ As the figure shows, the level chosen under limited liability, x’, may be either above or below the socially optimal level, x*. Next, assume that the level of preventive care is fixed at x,, = 0 and the activity tevct is chosen. As before, limited liability reduces the incentive to keep the probability of an accident down. ‘l-he marginal cost of’ the risky activity is reduced f’rom p,L in Equation (lb) to p,[W + H(y,,)] in Equation (2b), providing an incentive to increase the activity level. But, increasing the activity level wilt increase the amount of funds available to pay victims if’ an accident occurs. ‘l‘his results in a decrease in the expected marginal benefit of’ the risky activity. The expected marginal benefit becomes the marginal benefit, B,. weighted by the probability that an accident does not occur, since if an accident does occur there wilt be an offsetting increase in the amount of the court judgement the firm must pay. ‘l‘his provides an incentive to reduce the level of aciivity. As a result, the firm may actually underinvest in capital and operate on a smaller scale than would be socially efficient.
Limited liability for damages
328
PX L
- pX[W+B(yd-xl
X
x’ x1 x’ FIG. I. First order
conditions
for preventive
care choices
The first order conditions for both the socially efficient level of- activity and the level of activity chosen under limited liability are depicted graphically in Figure 2.’ It is clear from the graph that the level of activity chosen under limited liability, y’, may be either above or below the socially efficient level, y*. These various incentive effects exist when preventive care and activity levels are chosen simultaneously or when loss severity is stochastic.” Under any of these assumptions, it is possible to find firms spending more on preventive care, or spending less on capital, than would be socially efficient. Finally, the claim that increasing initial net worth provides an incentive to increase care and to decrease the level of activity is proven mathematically. Assume that y is fixed at y(). The first order condition for x, Equation (2a), can be rewritten as E, = 0. The implicit function theorem implies that
dx’ _=_. dW
-ET,,, E,,
The sign of E,, is negative by the second order condition for the choice of x, so the sign of this derivative is equal to the sign of E,,\. But E,,, = - pX > 0, so the level of preventive care chosen increases with the initial net worth of the firm. A graphical analysis can lend some insight. Looking at Figure 1, it is obvious that an increase in W shifts - p,[W + B(y,J - x] upward and has no effect on [ 1 - p]. Thus, raising W further internalizes the expected cost of an accident, but has no impact on the effect that limited liability has of externalizing some of the expected
329
L.L. POSEY
$
PYL
PYW PYW
’
’
,
I
:
:
’
’
I
:
!
;Y*Y'
Y
FIG. 2. First order conditions
cost tive A that
for activity
level chokes
of preventive care. The resulting increase in care is beneficial only if the prevencare level was below the socially efficient level. similar analysis can be used for the case where activity levels are chosen. Assume x is fixed at x,, = 0. Using the implicit function theorem, we can show that d?r’
dW
=-. -E,u. E,y
Since E,, is negative by the second order condition for the choice of y, the sign of this derivative is equal to the sign of E,,. Since E,, = - p, < 0, increasing initial net worth reduces the firm’s activity level. As Figure 2 shows, an increase in W shifts the curve p,[W - B(y)] upward and has no ef‘fect on B,[ 1 - p]. The cost of an accident is further internalized by bringing net worth closer to the potential loss, L, thus providing an incentive to reduce the probability of an accident. If the activity level was already below the socially optimal level, then the outcome is more inefficient once initial net worth is increased. References and Notes (1980) 9 Journal of Lqpl Sturlia l1. See S. Shavell, “Strict Liability Versus Negligence,” 25. “‘The Case of- the Disappearing Defendant: An Eco2. See, for example, J.S. Summers, nomic Analysis,” (1983) 132 iJrLiurr,Gty of Permsy1uarG.u Laur Reuirw 145-185; S. Shavell, “The Judgment Proof Problem,” (1986) 6 Intrmational Rrzww ofLaw and I5conomic.t 4%
Limited
330
3.
58; W.M.
Landes
Personal
Injuries,”
The
of Riyk
right-hand
I. [ 1 -
5.
( 1984) und
I~.wran~~
side (rhs)
p] has a positive
closer
to
below
the Ihs of (la),
a constant
for
function.
6.
becomes
A mathematical quest.
closer
‘fhe
simplicity,
75-l
When
Studir.s
Regime
for Catastrophic
4 17-440.
and <:atastrophic
Environmental
Risk,”
(1988)
00.
(Ya),
derivative, left-hand that
They
rhs of’ (2b) has a positive fore,
first
Law as a Regulatory
oj Legal
I.iability
- pYL. Both
first derivatives.
“Tort
of Equation
1 as x increases.
It is assumed, negative
Posner, 13 ,J ownal
“Pollution
See M.‘1‘. Katzman, 55 Journal
4.
and R.A.
for damap
liability
[ 1 - p], is below the rhs of Equation
a negative side
111s expressions p\, =
are equal
0. The
second
(Ihs) of (Za), have
-p,[W
negative p(x.0)
and a negative
+
B(y,,)
-
x], is
(lb) and (2b) have
= 0. The
to p,W since second
(la),
and becomes
first derivatives.
Ihs of Equations
at y = 0 since
y = 0, the rhs of (2b) is equal first derivative
derivative,
rhs of B(0)
derivative
(I II) is
= 0. ‘l‘he mcl,
there-
to p,L as y increases.
proof
for
each of these
cases
is available
fr-om the author
upon
I-C-
Ir is well known that, under certain assumptions, strict liability causes a firm to internalize the social cost of accidents, so it chooses the socially optimal activity and preventive cat-e levels.’ A familiar modification of’this result is made in models that alloy bankruptcy. When potential damages to third pat-ties exceed the fit-m’s net worth, part of‘ the risk is externalized. Consequently, the firm externalizes part of‘thc benefit of- restraining its activity Icvel and increasing the level of‘ preventive care. This at‘implies that limited liability gument, which has been emphasized in the literature. causes the firm’s activity level to be too high and the level of‘ preventive care to he too low.’ This note emphasizes another consideration that points in the opposite direction. E:xpcttditures on preventive care reduce the firm’s wealth, thus reducing the size of a court judgement that it can pay. The reduction in the size of the jutlgement that it cm pay increases the cost of bankruptcy borne by potential creditors, including tort victims. ‘Thus limited liability shif’ts part of’ the expected cost of‘ preventive care to the victims of’ tort. A similar argument applies to the firm’s activity level. A lower activity level often itnplics :I reduction in the firm’s stock of capital. A reduction in the fit-m’s stock of‘ capital reduces the firm’s wealth, which increases the cost of’ bankruptcy borne by potential creditors. Thus limited liability shifts part 01‘ the expected cost of’ towering activity levels to the victitns of tort. The general conclusion cart be summarized as fdlows. Limited liability has two effects. First, it externalizes the firm’s expected cost of accidents, which causes a higher activity level and a lower level of’ preventive cat-e. Second, it externalizes pat-t of the firm’s expected cost of cat-e and undercapitalization, which causes a higher level of’ care and a lower level of activity. In general, one effect is not necessarily larger than the other. ‘This note highlights this argument with an exatnple and then proves it mathematically. An Example: Generators and Disposers of Toxic Waste Firtns that generate ages that are greater
or dispose of’ toxic waste often than their ability to compensate
have the potential to inflict datnvictims. Environmental law has
326
Limited liability for damap
evolved to a standard of strict liability as a means of internalizing social costs. But limited liability prevents firms from bearing the full consequences of their actions. Although this reduces the incentive to keep the probability of an accident down, the above discussion indicates that such a firm may actually overinvest in preventive care, or operate on a smaller than optimal scale and distribute profits that might be used more efficiently, from society’s perspective, to increase the productive capacity of the firm. In the hazardous waste area, legislation aimed at mitigating the distortions caused by limited liability includes the Kesource Conservation and Recovery Act (RCKA) of 1976 and the Comprehensive Environmental Response Act (Superfund) of 1980. Together, these acts require firms that treat, store, dispose, or transport hazardous waste to show financial responsibility for third party damages. If insurance is not purchased, a firm must show a net worth above a specified lower limit as a means of providing proof of financial responsibility. The difficulty of obtaining pollution liability insurance following the KCKA and Superfund legislation has made this net worth requirement an important option for meeting the financial responsibility requirements of these acts:’ By placi?! a floor on initial net worth, this requirement focuses on the fact that limited liablhty externalizes part of the expected costs of an accident, thereby reducing the incentive to keep the probability of an accident down. In general, as I will show, raising the level of initial net worth simply provides a greater incentive to reduce the probability of an accident. By encouraging a Firm to increase its initial net worth, the requirement encourages it to increase its investment in preventive care and/or to reduce its activity level. The fact that limited liability reduces the expected cost of preventive care and undercapitalization is not considered. If the level of preventive care is above, or the activity level is below, the socially optimal level as a result of this second effect, then the net worth requirement will exacerbate the problem. Therefore, without knowing which of these two effects of limited liability is dominant, it is impossible to determine whether firm decisions will become more or less socially efficient as a result of the net worth requirements set forth in the RCRA and Superfund legislation.
Mathrmatical
Proof
The above arguments will now be proven mathematically. Assume that the firm is an expected value maximizer and has an initial net worth of W. The probability that an accident occurs and third parties suffer injuries of a fixed amount, L, is p(x,y), where x is the level of preventive care and y is the level of risky activity. The probability of an accident is decreasing in x and increasing in y, implying that pX < 0 and p, > 0. It is assumed that pX, > 0, py, s 0, and p(x,O) = 0. Activity of level y results in a net benefit to the firm of B(y) before the effect on expected losses is considered. Here, B, > 0, B,, < 0 and B(0) = 0. Preventive care is assumed to cost $1 per unit. A rule of strict liability is employed, so the firm is responsible for all damages up to the net worth at the end of the period, W + B(y) - x; liability insurance 1s not available. It is assumed that, for all values of y, B(y) is below I. - W for any feasible value of W. This implies that L is greater than any potential value of net worth at the end of the period.
L.L.
327
POSEY
The socially optimal levels of preventive care and activity imize total social benefits minus total social costs:
Max,u, The
first order
conditions
E =
are those
w +
R(J) - x - P(x,y)L.
-P,L
=
levels that max-
(1)
are
1
(14
and R, = p,,L. In the case of limited
liability,
the firm solves
M~%X.,, E = W + B(y) The
first order
conditions
equate
-jD,[W
marginal
+ B(y) and B,[l
First,
consider
(111)
x - f(x,y)[W benefits
xl = [l
-
+ H(y) -
with marginal
(2)
cost:
(2a)
PI
- p] = p,[W + H(y) -
x].
x].
(2b)
the case where the activity level is fixed at y,,. In choosing the level of expenditure on preventive care, the firm wilt equate the expected marginal benefit of such expenditure with its expected marginal cost in Equation (2a). Comparing the left-hand sides of Equations (la) and (2a) shows that, for each level of x, the marginal benefit of investing in preventive care to reduce the probability of an accident is tower under limited liability, since the firm’s post-accident liability is reduced from total damages, L, to net worth, [W + B(y,,) -xl. But, spending more on preventive care further reduces the firm’s post-accident liability. The expected cost of an additional unit of preventive care is thus reduced to the unit cost weighted by the probability that an accident does not occur [compare the right-hand sides of (la) and (‘La)]. This is because, in the event an accident does occur, the cost of the additional unit is offset by an equivalent reduction in the amount of’ damages that must be paid out to victims, resulting in a net cost of zero. I‘his reduction in the expected cost of preventive car-c provides an incentive to increase the amount invested in suct~ care. If this tatter effect dominates, then the level of preventive care chosen by the firm wilt be greatel than the socially efficient level. ‘rhis result can be illustrated graphically. The first order conditions for both the socially optima1 preventive care level and the preventive car-c level chosen by a firm under limited liability are depicted in Figure 1.’ As the figure shows, the level chosen under limited liability, x’, may be either above or below the socially optimal level, x*. Next, assume that the level of preventive care is fixed at x,, = 0 and the activity tevct is chosen. As before, limited liability reduces the incentive to keep the probability of an accident down. ‘l-he marginal cost of’ the risky activity is reduced f’rom p,L in Equation (lb) to p,[W + H(y,,)] in Equation (2b), providing an incentive to increase the activity level. But, increasing the activity level wilt increase the amount of funds available to pay victims if’ an accident occurs. ‘l‘his results in a decrease in the expected marginal benefit of’ the risky activity. The expected marginal benefit becomes the marginal benefit, B,. weighted by the probability that an accident does not occur, since if an accident does occur there wilt be an offsetting increase in the amount of the court judgement the firm must pay. ‘l‘his provides an incentive to reduce the level of aciivity. As a result, the firm may actually underinvest in capital and operate on a smaller scale than would be socially efficient.
Limited liability for damages
328
PX L
- pX[W+B(yd-xl
X
x’ x1 x’ FIG. I. First order
conditions
for preventive
care choices
The first order conditions for both the socially efficient level of- activity and the level of activity chosen under limited liability are depicted graphically in Figure 2.’ It is clear from the graph that the level of activity chosen under limited liability, y’, may be either above or below the socially efficient level, y*. These various incentive effects exist when preventive care and activity levels are chosen simultaneously or when loss severity is stochastic.” Under any of these assumptions, it is possible to find firms spending more on preventive care, or spending less on capital, than would be socially efficient. Finally, the claim that increasing initial net worth provides an incentive to increase care and to decrease the level of activity is proven mathematically. Assume that y is fixed at y(). The first order condition for x, Equation (2a), can be rewritten as E, = 0. The implicit function theorem implies that
dx’ _=_. dW
-ET,,, E,,
The sign of E,, is negative by the second order condition for the choice of x, so the sign of this derivative is equal to the sign of E,,\. But E,,, = - pX > 0, so the level of preventive care chosen increases with the initial net worth of the firm. A graphical analysis can lend some insight. Looking at Figure 1, it is obvious that an increase in W shifts - p,[W + B(y,J - x] upward and has no effect on [ 1 - p]. Thus, raising W further internalizes the expected cost of an accident, but has no impact on the effect that limited liability has of externalizing some of the expected
329
L.L. POSEY
$
PYL
PYW PYW
’
’
,
I
:
:
’
’
I
:
!
;Y*Y'
Y
FIG. 2. First order conditions
cost tive A that
for activity
level chokes
of preventive care. The resulting increase in care is beneficial only if the prevencare level was below the socially efficient level. similar analysis can be used for the case where activity levels are chosen. Assume x is fixed at x,, = 0. Using the implicit function theorem, we can show that d?r’
dW
=-. -E,u. E,y
Since E,, is negative by the second order condition for the choice of y, the sign of this derivative is equal to the sign of E,,. Since E,, = - p, < 0, increasing initial net worth reduces the firm’s activity level. As Figure 2 shows, an increase in W shifts the curve p,[W - B(y)] upward and has no ef‘fect on B,[ 1 - p]. The cost of an accident is further internalized by bringing net worth closer to the potential loss, L, thus providing an incentive to reduce the probability of an accident. If the activity level was already below the socially optimal level, then the outcome is more inefficient once initial net worth is increased. References and Notes (1980) 9 Journal of Lqpl Sturlia l1. See S. Shavell, “Strict Liability Versus Negligence,” 25. “‘The Case of- the Disappearing Defendant: An Eco2. See, for example, J.S. Summers, nomic Analysis,” (1983) 132 iJrLiurr,Gty of Permsy1uarG.u Laur Reuirw 145-185; S. Shavell, “The Judgment Proof Problem,” (1986) 6 Intrmational Rrzww ofLaw and I5conomic.t 4%
Limited
330
3.
58; W.M.
Landes
Personal
Injuries,”
The
of Riyk
right-hand
I. [ 1 -
5.
( 1984) und
I~.wran~~
side (rhs)
p] has a positive
closer
to
below
the Ihs of (la),
a constant
for
function.
6.
becomes
A mathematical quest.
closer
‘fhe
simplicity,
75-l
When
Studir.s
Regime
for Catastrophic
4 17-440.
and <:atastrophic
Environmental
Risk,”
(1988)
00.
(Ya),
derivative, left-hand that
They
rhs of’ (2b) has a positive fore,
first
Law as a Regulatory
oj Legal
I.iability
- pYL. Both
first derivatives.
“Tort
of Equation
1 as x increases.
It is assumed, negative
Posner, 13 ,J ownal
“Pollution
See M.‘1‘. Katzman, 55 Journal
4.
and R.A.
for damap
liability
[ 1 - p], is below the rhs of Equation
a negative side
111s expressions p\, =
are equal
0. The
second
(Ihs) of (Za), have
-p,[W
negative p(x.0)
and a negative
+
B(y,,)
-
x], is
(lb) and (2b) have
= 0. The
to p,W since second
(la),
and becomes
first derivatives.
Ihs of Equations
at y = 0 since
y = 0, the rhs of (2b) is equal first derivative
derivative,
rhs of B(0)
derivative
(I II) is
= 0. ‘l‘he mcl,
there-
to p,L as y increases.
proof
for
each of these
cases
is available
fr-om the author
upon
I-C-