overpenalizes these firms when damages are stochastic. It is precisely when this occurs— when the damage threshold rule tends to overpenalize low-damage firms and under-
penalize high-damage firms—that this rule will be optimal. In general terms, a DTL rule is distinguished from other liability functions by the
property that it maximizes liability when damages are high and minimizes liability when damages are low. Among liability rules that deliver a common level of precautionary
incentive to some low-damage firm, DTL penalizes higher damage firms—those firms that have higher probabilities of creating high damages—as much as possible. More-
over, limited liability often precludes the assessment of sufficient accident penalties on the high-damage injurers—those that have higher probabilities of creating accident
damages above the asset bound. In such cases, care-exertion incentives can be enhanced by the use of a DTL rule that, although providing a requisite precaution incentive to
low-damage firms, gets the high-damage injurers expected liability as high as possible and, hence, as close as possible to their expected damages.
In practice, there are many examples of liability rules akin to that characterized in this paper. For instance, in the United States, both federal and state environmental laws
require firms to report their own violations, providing liability protections when they do and imposing nonreporting penalties when they do not. In general, however, the
liability protections are only available when the violations do not involve serious harm to public health or to the environment; in essence, the self-reporting laws exempt “small
damage” violations from liability and do not exempt “large damage” violations.
7
Simi- larly, CERCLA distinguishes between “major” and “minor” accidents when it assesses
liability for hazardous substance releases. Major accidents are subject to more onerous damage assessments that include lost passive use values; minor accidents are exempted
from both exhaustive damage assessments and from liability for passive-use damages that often represent the bulk of environmental harm.
8
In regulating agricultural pol- lution, Arizona laws also distinguish between “egregious violations” that are subject to
asset-liquidating penalties and nonegregious violations that are exempted from the penalties, with an “egregious violation” judged by the extent of harm that it causes
[Cory 1997]. In all of these cases, liability assessments exhibit the central qualitative attribute of this paper’s damage threshold rule, subjecting violators with damages above
a threshold to large liability, while treating violators with damages below the threshold to much lower, and often negligible, penalties.
The balance of the paper is organized as follows. Section II develops the model framework and is followed by derivations of the paper’s main results in Section III.
Section IV discusses a variety of extensions and implications of the analysis, including the scope for ex ante safety standards to improve on liability regulation alone.
II. The Model
Consider a set of risk-neutral firmsagents that engage in activities that can lead to damaging accidents. Firms are distinguished by their type, t, with higher t levels
associated with higher accident damages in a sense made precise below. In the popu-
7
For example, see von Oppenfeld 1996 for discussion of Federal law, and Anderson 1996 and ADEQ 1996 for relevant discussions of State law.
8
In the Exxon-Valdez case, for example, economists estimated lost active use values to be approximately 4 million and lost passive use values to be in the range of 3 to 5 billion [Randall 1993]. See Grigalunas and Opaluch 1988
for a detailed discussion of damage assessment procedures under CERCLA.
184 Optimal liability
lation of firms, t has the relative frequency density qt, which has positive support on [t, t]. Firms have private information about their type.
A type t firm either engages in the accident-related activity or does not operate. If a firm operates, it can exercise care that reduces the probability that an accident will
occur. The level of care will be denoted by x, which has the nonempty and compact feasible set, X 5 [0, x]. The probability that a type t firm has an accident is px, t, where
p
x
, 0 and p
xx
. 0 care reduces accident risk at a decreasing rate. Excluding accident-related costs, a type t firm obtains a net expected benefit of px, t, where
p
x
, 0 and p
xx
¶ 0 for all x . 0 care is increasingly costly.
9
p and p are both twice continuously differentiable, with an arbitrary dependence on the firm type
parameter t.
10
When an accident occurs, damages to other parties are d, where d is the realization of a random variable that has the positive and twice continuously differentiable density
function hd; t on the support [d, d]. Higher t types have “higher” damage distribu- tions in the sense of the monotone likelihood ratio property MLRP [see Milgrom
1981]
11
: ]
]d
S
h
t
~d; t h~d; t
D
. 0 ; ~d, t. 1
Condition 1 implies that expected damages, Ed; t, are increasing in t. When an accident occurs, the courts will assign liability to the injurer that is assumed
to depend only upon ex-post damages, ld. A type t firm’s expected accident liability thus equals px, t Eld; t, where E is the expectation operator over damages.
Given the liability rule ld, each firm chooses its care level to maximize its net benefits:
max x [ X
p~ x, t 2 p~ x, t E~l~d; t 2
The solution to 2 will be denoted by xl, t. For simplicity, it will be assumed that xl, t is in the interior of X, the feasible care-choice set, for all firm types and all feasible
nonzero liability functions ld.
12
A firm will operate if and only if its net expected benefit from operation—the maximal net profit in equation 2—is non-negative.
Under “pure” strict liability, ld equals d and, hence, each firm will pay the true ex post damages; optimal first-best care choices and operation decisions will thus ensue.
The main purpose of this analysis is to study implications of limited firm liability for the
9
Any victim precaution measures are assumed to be fixed at their optimal levels to focus attention on injurer incentives for accident prevention. For the same reason, I abstract from multimarket effects and potential efficiency
costs of imperfect competition [e.g., see Sunding and Zilberman 1996].
10
When turning to entry and entry-deterrence effects of optimal liability in the third part of Section IV below, I will consider possible restrictions on the nature of the dependence of p and p on t.
11
The MLRP condition 1 holds for a wide class of distributional specifications and is a somewhat stronger condition than first-order stochastic dominance FOSD of higher t versus lower t damage distributions [Milgrom
1981]. The MLRP is closely related to FOSD in the following sense: The MLRP holds if and only if FOSD holds for any possible conditional distribution for d [see Whitt 1980]. The MLRP also implies that the outcome variable d can
serve as a signal of firm type in that higher realizations of d imply a higher damage type [Milgrom 1981].
12
A nonzero liability function assesses strictly positive liability on a nondegenerate interval of the damage support, [d, d]. With such functions, sufficient conditions for interior care choices, xl, t [ 0, x, are: p
x
0, t 5 0 infinitesimal care is approximately costless and lim
x 3 x
p
x
x, t 5 2` the marginal cost of maximal care is arbitrarily large.
185 I
NNES
structure and effects of an optimal legal rule in the presence of asymmetric information. Therefore, it is assumed that strict liability cannot always be assessed because ex post
damages sometimes exceed the firm assets that are available for damage payment:
y 5 level of firm assets , d.
3 Although 3 defines a nonstochastic asset level, y can be interpreted as the mean
level of ex post assets. With this interpretation, the following analysis constrains assets to be stochastically independent of damages. In a related vein, it is assumed that y is
invariant to the firm’s type t. This assumption is made to focus the analysis on a legal rule that depends only on the damage assessment d and not on firm assets, y, except to
the extent that y limits the firm’s liability.
13,14
If the firm’s monetary profits are included in its assets, this specification requires that net firm benefits px, t equal the difference
between an exogenous profitasset measure and a nonmonetary effort cost cx, t.
15
Given the asset level y, limited liability constrains the legal rule to satisfy: ld ¶ y d. Hence, by the assumed inequality in 3, pure strict liability is infeasible. It is also
assumed that ld must be everywhere non-negative so that injurers cannot be re- warded for accidents and piecewise continuous to ensure existence of the expecta-
tion, Eld; t.
16
The feasible set for ld is thus defined as L 5 {1:0 ¶ 1d ¶ y and l is piecewise continuous over the domain, [d, d]}.
III. The Optimal Legal Rule
