Asian Journal of Law and Economics
Volume 1, Issue 1 2010 Article 2
Environmental Citizen Suits with Pigovian
Punitive Damages
Sung-Hoon Park, Gyeonggi Research Institute
Jason F. Shogren, University of Wyoming
Recommended Citation:
Park, Sung-Hoon and Shogren, Jason F. (2010) "Environmental Citizen Suits with Pigovian
Punitive Damages," Asian Journal of Law and Economics: Vol. 1: Iss. 1, Article 2.
DOI: 10.2202/2154-4611.1001
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Environmental Citizen Suits with Pigovian
Punitive Damages
Sung-Hoon Park and Jason F. Shogren
Abstract
Federal environmental laws encourage private citizens to act like "private attorney generals"
and to sue a firm. This citizen group competes over the rewards of levels of regulation and
enforcement. The firm can reduce its output to curtail the likelihood of losing the contest. This
paper explores whether one can combine citizen suits with Pigovian punitive damages to equate
private and social incentives. We show: (i) without punitive damages, the level of output of the
firm is only optimal in a special case; (ii) with punitive damages given to the citizen group,
Pigovian punitive damages can be found, but it could be negative—the citizen group might have
to compensate the firm; (iii) the ideal level of punitive damages can be achieved if a government
takes it; and (iv) punitive damages which are taken by the citizen group induces more effort
expended in the conflict relative to when the government receives the punitive damages.
KEYWORDS: citizen suits, environmental damage, marginal environmental damage, marginal
net benefit, Pigovian punitive damages, total legal expenditures
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
1. Introduction
In the United States, regulators promote the use of citizen suits to help enforce
federal and state environmental laws (see Naysnerski and Teitenberg, 1992; Settle
et al. 2003). Regular citizens are encouraged to act like “private attorney
generals,” seeking out polluters and then suing them for damages in an
environmental conflict. Citizen suits are said to be a deterrent for potential
polluters. The threat of a suit could cause the polluter to account for the damages
before the suit.
The existing literature exploring the nature of such conflicts has focused
on the social costs of expending scarce resources fighting over a prize (e.g., Baik
and Shogren, 1994; Liston-Heys, 2001). The open questions are whether the
existence of the citizen suit provision alone is enough to induce the firm to select
the socially efficient level of output, or whether the regulator should introduce an
additional penalty provision, namely punitive damages. Punitive damages are
frequently used to balance concerns for economic efficiency, retribution, and
other rationales (see Luban and Eisenberg, 1998; Diamond, 2002). Now a
victorious citizen group would receive payments in terms of damages suffered
plus any punitive damages deemed warranted by the court.
Herein we develop an environmental conflict model to consider whether a
regulator can use citizen suits with punitive damages to create a Pigovian
instrument that could equate private and social incentives. We consider two-
player (a firm and a citizen group) contests with citizen suits as punitive damages.
The players are risk-neutral, and have equal legal ability. Events occur as follows.
In the first stage, the firm maximizes its payoff by choosing what output to
produce. The firm then announces its production plan to the public. In the
second stage, after both parties know the projected output, the citizen group and
the firm compete simultaneously and independently by irreversible effort level to
win their prizes in the second stage. Using subgame-perfection as the solution
concept, our results suggest that (i) without punitive damages, the level of output
of the firm is only optimal in a special case; (ii) with punitive damages given to
the citizen group, Pigovian punitive damages can be found, but it could be
negative—the citizen group might have to compensate the firm; (iii) the ideal
level of punitive damages can be achieved if a government takes it; and (iv)
punitive damages which is taken by the citizen group induces more effort
expended in the conflict relative to when the government receives the punitive
damages.
In next section, we set up the two-stage game without punitive damages.
In Section 3, we consider punitive damages for the citizen group. Section 4
presents the actual situation that exists in federal U.S. environmental law—the
1
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
government takes punitive damages if the citizen group wins the suit. Finally, we
offer our concluding remarks in Section 5.
2. Citizen Suit without Punitive Damages
We first establish a benchmark model—a two-stage citizen suit game without
punitive damages. Consider an environmental contest in which a firm and a
citizen group expend effort competing with another to win their prize or rent.
The firm expends observable and irreversible effort x to win profit G(Q), which is
a function of its output Q. Output, however, causes external environmental
damage, D(Q). The citizen group expends effort y to avoid the damages
associated with output. Following contest theory, the probability the firm wins
is px(x, y), for x + y > 0, and px = ½, for x + y = 0; where ∂px/∂x > 0 and ∂px/∂y < 0,
and ∂2px/∂x2 < 0 and ∂2px/∂y2 > 0. Assume the players have the same legal ability
in the contest, i.e., px(x, y) = 1 – py(x, y). Assume all information is common
knowledge.
We now solve for the subgame perfect equilibrium of the benchmark
contest. Consider first the second stage. Let L represent the expected loss for the
citizen group. The citizens choose their effort y to minimize expected
environmental damage D plus the legal expenditures:
min (y) L = pxD(Q) + y
s.t. y ≥ 0,
(1)
and the firm selects effort x to maximize its expected payoff independently:
max (x) π = pxG(Q) – x
s.t. x ≥ 0,
(2)
yielding the first-order conditions:1
(∂px/∂y)D(Q) + 1 = 0,
(3)
(∂px/∂x)G(Q) – 1 = 0.
(4)
1 The second-order conditions are satisfied in both cases with ∂2px/∂x2 < 0 and ∂2px/∂y2 > 0.
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
We obtain a unique Nash equilibrium of this second-stage subgame, x =
x(G(Q), D(Q)) and y = y(G(Q), D(Q)), the probability-of-winning for the firm, px
= px(G(Q), D(Q)), and the firm’s indirect expected payoff, π = π(G(Q), D(Q)).2
In the equilibrium of this second-stage subgame, we obtain a result which says
the effect on the firm does not consider environmental damage.
Proposition 1. Suppose the firm does not consider the environmental damage.
(a) The firm increases its legal expenditures if and only if it is the contest favorite,
i.e., has a greater than 50 percent chance of winning at the Nash equilibrium. (b)
The citizen group increases its legal expenditures regardless of whether the
group is the favorite or not.
Proof. See Appendix A.
Now because the citizen suit exists, the firm accounts for the external
damages in its indirect payoff function. The question is whether the existence of
a citizen suit triggers partial or full internalization of the external damages by the
firm.
We now work backwards and consider the first stage—the firm
maximizes its expected payoff π(G(Q), D(Q)) with respect to the output Q by
solving:
argmax (Q) π(G(Q), D(Q))
s.t. Q ≥ 0.
(5)
The first- and second-order conditions are:
∂π ∂G ∂π ∂D
+ = 0 ,
∂G ∂Q ∂D ∂Q
(6)
and
2 2 2∂ π ⎛⎜ ∂G ⎞⎟ ∂π ∂
2G ∂ 2π ⎛ ∂D ⎞ ∂π ∂ 2 D
2 ⎜ ⎟ + 2 + 2 ⎜⎜ ⎟⎟ + 2 < 0 . ∂G ⎝ ∂Q ⎠ ∂G ∂Q ∂D ⎝ ∂Q ⎠ ∂D ∂Q
(7)
2 Assume ∂x/∂G > 0, ∂x/∂D (≥ or ≤) 0, ∂y/∂G (≥ or ≤) 0, ∂y/∂D > 0, ∂px/∂G > 0, ∂px/∂D < 0, and
∂π/∂G > 0, and ∂π/∂D < 0.
3
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
The term ∂G/∂Q in expression (6) is marginal net benefit, MB (= MR – MC);
∂D/∂Q is marginal environmental damage, MD. Expression (7) says that the
sufficient condition to be satisfied with the second-order condition is ∂2π/∂D2 < 0.
Expression (6) can be rewritten as:
MB ∂π ∂π= −{( ) /( )}MD.
∂D ∂G
(8)
Expression (8) shows the relationship between marginal net benefit and marginal
environmental damage. Without the citizen suit in the second stage, the firm
would select its output such that marginal net benefit is zero, MB = MR – MC = 0.
With the citizen suit, however, the firm now considers both marginal net benefit
and a weighted fraction of the marginal damage, –{(∂π/∂D)/(∂π/∂G)}MD > 0.
Now increasing output has two effects—a direct profit effect that increases the
firm’s payoff, and a weighted indirect citizen suit effect that decrease its payoff
since greater damages trigger more effort by the citizen group. The weighting
term captures at the margin the expected odds of victory in the contest given
relative damages and profits, i.e., the marginal loss in payoffs due to litigated
damages relative to the marginal gains in payoffs due to direct profits.
In general, the weighted indirect citizen-suit effect causes the firm to
reduce output (i.e., Q < Q**, MB > 0). But we achieve the socially optimal level
of output (Q = Q*) if and only if the weighting term equals unity, –
{(∂π/∂D)/(∂π/∂G)} ≡ θ = 1. If the weighting term is less than unity (θ < 1), the
firm reduces output but it still overproduces relative to the social optimum, Q* <
Q < Q**. If the opposite holds (θ > 1), the firm actually underproduces relative to
the social optimum, Q < Q*. The firm reduces its output as long as its expected
payoff is positive. Proposition 2 summarizes the choice of output in the first
stage.
Proposition 2. Assuming the firm’s payoff is concave in its output, the existence
of a citizen-suit mechanism (a) causes the firm to reduce its output, because now
it internalizes how its output affects the citizen group, and therefore reduces its
likelihood of losing the suit. (b) The magnitude of the reduction in output,
however, depends on the relative magnitude of the marginal indirect damage
effects –(∂π/∂D) and marginal direct profit effect, (∂π/∂G). (c) A citizen suit
induces a socially efficient level of output only in a special case of equal
marginal direct-profit effects and marginal indirect-damage effects. (d)
Otherwise, a firm overproduces (under-produces) output relative to the social
optimum if the indirect-damage effect is small (large) relative to the direct-profit
effect.
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
We gain additional insight into our result by reframing the general model
using the classic Logit contest success function used throughout the contest
literature.3 Now the probability of the firm winning is px = x/(x + y), and the
citizen group winning is py = (1 – x/(x + y)). Let the firm’s profit function be G =
P(Q)Q – C(Q). In the second stage, the citizen group selects a level of contest
effort y to minimize expected environmental damage D plus legal expenditures,
and the firm maximizes its expected payoff simultaneously and independently
(see expressions (1) and (2)). By doing so, we obtain reaction function for the
firm and the citizen group. Solving the two reaction functions jointly, we obtain a
unique Nash equilibrium in the second stage of this subgame.
Lemma 1. At the Nash equilibrium in the second-stage of the subgame, the
effort levels of the players are xN = G2D/(G + D)2 and yN = GD2/(G + D)2, the
firm’s expected payoff is πN = G3/(G + D)2, and the citizen group’s expected loss
is LN = GD(G + 2D)/(G + D)2.
Lemma 1 says that environmental damage is (partially) internalized when the
firm maximizes its payoff in the first stage. In the first stage, the firm chooses its
output to maximize the expected payoff:
arg max(Q) π N {P(Q)Q −C(Q)}
3
=
{P(Q)Q −C(Q) + D(Q)}2
s.t. Q ≥ 0.
(9)
The first-order condition for maximizing the firm’s expected payoff πN gives
MB ={2G /(G +3D)}MD.
(10)
Examining expression (10), we see the Logit equivalent of expression (8), in
which the weighting term takes the form –(∂π/∂D) = 2G and (∂π/∂G) = (G + 3D),
which we summarize in Corollary 1.
Corollary 1. (a) If the firm’s profit is three times environmental damage G = 3D,
the citizen suit induces the firm to select the socially optimal level of output; (b) if
G < 3D, which implies a relatively steep marginal damage curve, the citizen suit
3 The Logit function is used extensively in the contest literature, see, for example Tullock (1980),
Dixit (1987), Baik and Shogren (1994), Heyes (1997), Hurley and Shogren (1997), and Liston-
Heyes (2001).
5
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
induces the firm to reduce its output but it still overproduces relative to the social
optimum, Q* < Q < Q**; and (c) if G > 3D—which implies a relatively flat
marginal damage curve, the firm underproduces output relative to the social
optimum, Q < Q*.
3. Pigovian Punitive Damages for the Citizen Group
With citizen suits but without punitive damages, the level of output of the firm is
optimal only under specific conditions. We now explore whether adding punitive
damage could be used to equate private and social incentives (see for example
Diamond, 2002). Let PN denote the level of punitive damages paid to the citizen
group selected by a regulator.
In the second stage, the citizen group chooses a level of contest effort y to
minimize the expected loss:
min (y) L = px(D + PN) – PN + y
s.t. y ≥ 0,
(11)
and the firm’s problem is to maximize its expected payoff by
max (x) π = px(G + PN) – PN – x
s.t. x ≥ 0.
(12)
The first-order conditions for minimizing and maximizing each player’s expected
loss L and payoff give:
(∂px/∂y)(D + PN) + 1 = 0,
(13)
(∂px/∂x)(G + PN) – 1 = 0.
(14)
This subgame has a unique Nash equilibrium in the second stage. We
denote it by {x(G(Q) + PN, D(Q) + PN), y(G(Q) + PN, D(Q) + PN)}.
At the second stage, having perfect sight about expected payoff, π(G(Q) +
PN, D(Q) + PN; PN), the firm seeks to maximize its expected payoff choosing its
output:
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
argmax (Q) π(G(Q) + PN, D(Q) + PN; PN)
s.t. Q ≥ 0.
(15)
The first- and second-order conditions for maximizing the firm’s expected payoff
π give:
∂π ∂G ∂π ∂D
+ = 0 ,
∂(G + PN ) ∂Q ∂(D + PN ) ∂Q
(16)
and
∂2
2 2
π ⎛⎜∂G⎞⎟ ∂π ∂
2G ∂2π ⎛∂D⎞ ∂π ∂2D
⎜ ⎟ + + ⎜ ⎟ + <0.∂(G+PN)2 ⎝∂Q⎠ ∂(G+PN) ∂Q2 ∂(D+PN)2 ⎜ ⎟ 2⎝∂Q⎠ ∂(D+PN) ∂Q
4
(17)
Expression (16) can be rewritten as:
MB = −{( ∂π ) /( ∂π )}MD.
∂(D + PN) ∂(G + PN)
(18)
Expression (18) shows the increase in output is affected by both ∂π/∂(G + PN)
and ∂π / ∂(D + PN ) . The reason is that punitive damages have two properties:
an opportunity cost in rents for the firm and an extra prize for the citizen group.
Whether the regulator can use punitive damages to induce a firm to reduce its
output to optimal levels depends on both the numerator and denominator in (18):
even if the numerator is reduced, the denominator is also reduced.
Proposition 3. The optimal punitive damages would set PN so ∂π/∂(G + PN)
equals |∂π/∂(D + PN)|, such that MB = MD in expression (18).
4 The sufficient condition to satisfy the second-order condition is ∂2π/∂(D + PN)2 < 0.
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
Punitive damages as a Pigovian tool can be further understood using the
Logit contest success function. Lemma 2 summarizes the unique Nash
equilibrium in the second stage of this subgame.
Lemma 2. At the Nash equilibrium in the second stage of the subgame, the
effort levels of the players are x = (G + PN)2(D + PN)/(G + D + 2PN)2 and y = (G
+ PN)(D + PN)2/(G + D + 2PN)2, the firm’s expected payoff is π = (G + PN)3/(G
+ D + 2PN)2 – PN, and the citizen group’s expected loss is L = {(G + PN)(D +
PN)(G + 2D + 3PN)}/(G + D + 2PN)2} – PN.
At the first stage, the firm chooses its output to maximize expected payoff
argmax (Q) π {P(Q)Q −C(Q)+ PN}
3
= 2 − PN , {P(Q)Q −C(Q)+ D(Q)+ 2PN}
(19)
s.t. Q ≥ 0,
yielding the expression
MB ={2(G + PN) /(G +3D+ 4PN)}MD .
(20)
We solve expression (20) for PN to get Corollary 2.
Corollary 2. Suppose any punitive damages are distributed to the citizen group
if they are victorious in the environmental conflict. The level of Pigovian punitive
damages (when positive) is PNCG = (G* – 3D*)/2. (a) If G* > 3D*, Pigovian
punitive damages are positive; (b) if G* < 3D*, Pigovian punitive damages are
negative.
Corollary 2 illustrates the optimal level of punitive damages that would induce a
firm to set its output level to socially efficient levels. Part (a) says that the
punitive damages would be positive; part (b) of Corollary 2 suggests conditions
exist when punitive damages could be negative (also see Diamond, 2002). With
fewer environmental damages or greater profit, more punitive damages can be
tolerated. Even if punitive damages PNCG induce the firm to produce the socially
optimal level of output, a negative PNCG is unlikely to be applied in practice.
Combining Corollaries 1 and 2, we obtain that exogenous Pigovian punitive
damages taken by the citizen group may be a mis-measurement to set the firm’s
output level to socially efficient levels. The reason is as follows. If the firm
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
underproduces output relative to the social optimum (i.e., Q < Q*), punitive
damages taken by the citizen group could be negative. If the firm overproduces
relative to the social optimum (i.e., Q > Q*), punitive damages taken by the
citizen group would be positive.
Given Corollary 2 and Lemma 2, Lemma 3 summarizes the patterns of
Nash equilibrium effort levels, expected payoff, and expected loss.
Lemma 3. Given punitive damages taken by the citizen group, the effort level of
the players are xCG = 9(G* – D*)/32 and yCG = 3(G* – D*)/32, the total legal
expenditures are xCG + yCG = 3(G* – D*)/8, the firm’s expected payoff is πCG =
(69D* – 5G*)/32, and the citizen group’s expected loss is LCG = (D* – G*)/32.5
Lemma 3 says the effort levels are positive since G* > D*, even though the firm’s
expected payoff is likely to be negative.
Next consider the case in which punitive damages are endogenous in the
level of environmental damages; PN(D). Given expression (19), we obtain (21):
MB =θ cg (G, D, PN;∂PN cg / ∂D)MD.
(21)
Given MB = MD (i.e., θcg = 1) and evaluated at PN = 0, expression (21)
yields
(∂PN cg /∂D) = (G* )2 (3D* −G*) /{2(G* )3 + (D* )2 (3G* + D*)}.
PN=0
(22)
The limitation of applying the endogenous variables as punitive damages is that
the firm’s expected payoff may not be concave in its output in the first stage—the
sign of the second-order condition depends on ∂PN/∂D and ∂2PN/∂D2. This
implies that we may face multiple Nash Equilibria or we may not find any Nash
equilibria in the subgame. Another limitation is conducted from expression (22).
The expression shows that, with no punitive damages, the level of output of the
firm is only optimal in a special case, i.e., 3D* – G* = 0.
5 Throughout, we use an * to emphasize that the Nash equilibrium is from the optimal level of
output of the firm.
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
4. Pigovian Punitive Damages to the Government
Now consider the actual situation that exists in federal U.S. environmental law—
the government takes punitive damages if the citizen group wins the suit.
Consider the second stage of the game. The citizen group’s problem is to
minimize the expected loss in expression (1), and the firm maximizes its expected
payoff independently by
max (x) π = px(G + PN) – PN – x
s.t. x ≥ 0.
(23)
The first-order conditions are
(∂px/∂y)D + 1 = 0,
(24)
(∂px/∂x)(G + PN) – 1 = 0.
(25)
This subgame has a unique Nash equilibrium in the second stage, denoted
by {x(G(Q) + PN, D(Q)), y(G(Q) + PN, D(Q))}.
At the second stage, having perfect sight about π(G(Q) + PN, D(Q); PN),
the firm chooses the level of output that maximizes its expected payoff:
argmax (Q) π(G(Q) + PN, D(Q); PN)
s.t. Q ≥ 0.
(26)
The first- and second-order conditions for maximizing the firm’s expected
payoff give:
∂π ⎛ ⎞
⎜⎜
∂G ⎟ ∂π ∂D⎟ + = 0 , ∂(G + PN ) ⎝ ∂Q ⎠ ∂D ∂Q
(27)
and
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
2 2
∂ 2π ⎛⎜ ∂G ⎞⎟ ∂π
⎛ ∂ 2G ⎞ 2 ⎛ ⎞ ⎛ 2 ⎞
⎜ ⎟ + ⎜⎜ ⎟
∂ π ⎜ ∂D ⎟ ∂π ∂ D+ + ⎜ ⎟ < 0 .
∂(G + PN )2 ⎝ ∂Q ⎠ ∂(G + PN ) ⎝ ∂Q
2 ⎟ ∂D 2 ⎜⎠ ⎝ ∂Q
⎟ ⎜ 2 ⎟
⎠ ∂D ⎝ ∂Q ⎠
(28)
Expression (27) can be rewritten as follows:
MB = −{(∂π ) /( ∂π )}MD.
∂D ∂(G + PN )
(29)
Now compare expressions (8) and (29). Assume, without loss of
generality, that marginal payoff meets two requirements: it is positive and
decreasing. The requirements show ∂π/∂G > ∂π/∂(G + PN) for PN > 0, and
∂π/∂G < ∂π/∂(G + PN) for PN < 0. Recalling expression (8), the regulator selects
the size of punitive damages such that the firm equates its private goals with the
broader social incentives. Using expression (29), we state the following
proposition.
Proposition 4. If punitive damages are awarded to the regulator, an optimal
level of Pigovian punitive damages can be found to induce a firm to reduce its
output to socially efficient levels.
Proposition 4 says punitive damages can work as a Pigovian tool—provided that
the rewards accrue to the government, not to the citizen group.
Again considering the case of the Logit success function, in the second
stage, the citizen group chooses a level of contest effort y to minimize the
expected loss, and the firm maximizes its expected payoff independently.6 This
subgame has a unique Nash equilibrium in the second stage.
Lemma 4. At the Nash equilibrium in the second stage of the subgame, the
effort levels of the players are x = (G + PN)2D/(G + D + PN)2 and y = (G +
PN)D2/(G + D + PN)2, the firm’s expected payoff is π = {(G + PN)3/(G + D +
PN)2} – PN, and the citizen group’s expected loss is L = (G + PN)(G + 2D +
PN)D/(G + D + PN)2.
At the first stage, the firm chooses output to maximize expected payoffs:
6 The second-order conditions are satisfied in both cases.
11
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
argmax (Q) π G {P(Q)Q −C(Q) + PN}
3
= − PN
{P(Q)Q −C(Q) + D(Q) + PN}2
s.t. Q ≥ 0,
(30)
which yields the condition:
MB ={2(G + PN) /(G +3D+ PN)}MD .
(31)
From expression (31), we obtain Corollary 4.
Corollary 4. The ideal level of Pigovian punitive damages (when positive) is that
PNG = 3D* – G*. (a) If G* < 3D*, Pigovian punitive damages are positive; (b) if
G* > 3D*, Pigovian punitive damages are negative.
From Corollary 4, if firm profits are less than three times the damages, G* < 3D*,
the regulator can impose a Pigovian punitive damages on the firm. But if the
firm’s profits are sufficiently greater than damage, the firm reduces its output
such that Q < Q*. Now again the regulator can only equate private and social
incentives if it subsidizes the firm.
Given Corollary 4 and Lemma 4, Lemma 5 summarizes the patterns of
Nash equilibrium effort levels, expected payoff, and expected loss.
Lemma 5. At the Nash equilibrium in the second stage of the subgame, the
effort levels of the players are xG = 9D*/16 and yG = 3D*/16, the total legal
expenditures are xG + yG = 3D*/4, the firm’s expected payoff isπG = (16G* –
21D*)/16, and the citizen group’s expected loss is LG = 3(4 + D*)/16.
Lemma 5 says that the effort levels are positive, and the firm’s outcome is usually
positive. From Lemma 3 and Lemma 5, Proposition 5 characterizes the difference
between two total legal expenditures.
Proposition 5. If the regulator cannot impose negative punitive damages on the
citizen group, total expenditures for punitive damages taken by the citizen group
are always greater than total expenditures for punitive damages taken by the
government.
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
Proof. The proof of the case of G > 3D is straightforward and therefore omitted.
With the case of G < 3D and PNCG = 0, the total expenditures of the citizen group
are GD/(G + D), while the total expenditures of the firm are 3D*/4. Recalling
Corollary 1, if G < 3D, the firm overproduces relative to the social optimum.
This implies G*D*/(G* + D*) < GD/(G + D) since G* < G and D* < D. Next
comparing G*D*/(G* + D*) with 3D*/4, we find that 3D*/4 < G*D*/(G* + D*).
Herein we obtain that 3D*/4 < GD/(G + D).
Proposition 5 says that when the government cannot impose negative punitive
damages on the citizen group, punitive damages which are taken by the citizen
group incur greater total expenditures than punitive damages taken by the
government.
Next consider punitive damages are endogenous in environmental
damages; PN(D). Given expression (30), we obtain expression (32):
MB =θ g (G, D, PN;∂PN g / ∂D)MD.
(32)
Given MB = MD (i.e., θg = 1) and evaluated at PN = 0, expression (32) yields
(∂PN g /∂D) ={(G* )2 (3D* −G* ) /(D*)2 (3G* + D*)},
PN=0
(33)
which induces the results to be same with expression (22): with no punitive
damages, the level of output of the firm is only optimal in a special case, i.e., 3D*
– G* = 0.
From expressions (22) and (33), we obtain Proposition 6.
Proposition 6. Suppose that the punitive damages are endogenous. The punitive
damages are of no use to equate private and social incentives.
5. Concluding Remarks
Federal environmental laws in the United States encourage private citizens to act
like “private attorney generals” by subsidizing their efforts to sue polluting firms.
This paper develops an environmental-conflict model in which a firm and a
citizen group choose the amounts of competition over the rewards of levels of
regulation and enforcement. We have also explored whether one can use citizen
suits with Pigovian punitive damages to equal private and social incentives.
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
Unlike other analyses that simply take rewards as a given, we have explicitly
modeled the expenditure-reward relationship in which the firm’s output is an
endogenous variable.
Using a general probability-of-winning function and a specific logit
function, we have examined the effect of increasing the firm’s output. We have
characterized the subgame-perfect equilibrium and examined whether one can use
citizen suits with Pigovian punitive damages to equal private and social
incentives. Our results suggest that: (i) without punitive damages, the level of
output of the firm is only optimal in a special case; (ii) with punitive damages
given to the citizen group, Pigovian punitive damages can be found, but this
could be negative—the citizen group might have to compensate the firm; (iii) the
ideal level of punitive damages can be achieved if a government takes the
compensation for damages; (iv) punitive damages taken by the citizen group
induce more effort expended in the conflict relative to when the government
receives the punitive damages; and (v) endogenous punitive damages are of no
use to equate private and social incentives.
APPENDIX A
The effects on expenditure of small changes in the explanatory variables are
found by differentiating the first-order conditions in expressions (3) and (4):
G(Q)(∂2px/∂x2)dx + G(Q)(∂2px/∂x∂y)dy + (∂px/∂x)MBdQ = 0,
(3a)
and
D(Q)(∂2px/∂y∂x)dx + D(Q)(∂2px/∂y2)dy + (∂px/∂y)MDdQ = 0.
(4a)
The contest literature suggests that the signs of the slopes of the reaction
functions are critical determinants of players’ strategic behavior in equilibria.
Rearrangement of expressions (3a) and (4a) shows that the slope of the firm’s
reaction function is equal to dx/dy = –(∂2px/∂x∂y)/(∂2px/∂x2), and that the slope of
the citizen group’s reaction function is equal to dy/dx = –(∂2px/∂y2)/(∂2px/∂y∂x).
Since ∂2px/∂x2 < 0 and ∂2px/∂y2 > 0, the two slopes must be of opposite signs
when they intersect. As Katz (1988) suggests, the sign of ∂2px/∂y∂x (= ∂2px/∂x∂y)
implies that an increase in the effort level of the favorite player leads to a
decrease in his opponent’s effort level, and that an increase in the effort level of
the underdog player leads to an increase in its opponent’s effort level. With our
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
assumption that players have the same legal ability for the contest, the positive
sign of ∂2px/∂y∂x means the firm is the favorite, and the negative sign of that
means the firm is the underdog.
Which player between the firm and the citizen group is favored in
equilibrium depends on the value of the exogenous parameter Q in our model.
We now solve for reduced-form expression for the change in the exogenous
variable x and y. Simultaneous solution of expressions (3a) and (4a) yields:
G{∂2px/∂x2 – (∂2p /∂x∂y)2/(∂2p /∂y2x x )}dx
– {G⋅MD(∂2px/∂x∂y)(∂px/∂y)/D(∂2px/∂y2) – MB(∂px/∂x)}dQ = 0,
(3b)
and
D{–(∂2px/∂y2) + (∂2px/∂x∂y)2/(∂2px/∂x2)}dy
– {D⋅MB(∂2px/∂x∂y)(∂px/∂x)/G(∂2px/∂x2) – MD(∂px/∂y)}dQ = 0.
(4b)
A change in the exogenous parameter Q will affect the players’ legal
expenditures directly by changing the marginal value of expenditures and also
indirectly as each player considers marginal net benefit and marginal damage to
the direct change. Expressions (3c) and (4c) show the effects:
dx/dQ =
{G⋅MD(∂2px/∂x∂y)(∂p /∂y)/D(∂2x px/∂y2) – MB(∂px/∂x)}
÷ G{∂2p /∂x2 – (∂2p /∂x∂y)2x x /(∂2p 2x/∂y )},
(3c)
and
dy/dQ =
{MD(∂px/∂y) – D⋅MB(∂2px/∂x∂y)(∂p 2x/∂x)/G(∂ px/∂x2)}
÷ D{(∂2px/∂x∂y)2/(∂2px/∂x2) – (∂2px/∂y2)},
(4c)
where the denominators of the notations at expressions (3c) and (4c) are negative
while the numerators of these depend upon the firm’s profit and environmental
damage (which are the function of the firm’s output), and probability-of-winning
(which is the function of each player’s legal expenditures).
We consider the signs of dx/dQ and dy/dQ when the firm’s marginal net
benefit is zero at Q** in Figure 1. It follows from expression (3c) and (4c) that:
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Asian Journal of Law and Economics, Vol. 1 [2010], Iss. 1, Art. 2
dx/dQ =
MD (∂2p /∂x∂y)(∂p /∂y) ÷D (∂2p /∂y2){∂2p /∂x2 – (∂2p /∂y∂x)2x x x x x /(∂2px/∂y2)},
(3d)
and
dy/dQ =
MD (∂p 2x/∂y) ÷D {(∂ px/∂x∂y)2/(∂2p /∂x2) – (∂2p /∂y2x x )} > 0,
(4d)
where the values of G , D , and MD at expression (3d) and (4d) imply the firm’s
profit, the environmental damage, and the marginal environmental damage when
the firm seeks to maximize its profit at MB = 0. Some observations follow
immediately. As we mentioned early, the denominators of the notations are
negative at expressions (3d) and (4d). Thus, sgn [dx/dQ] = sgn [∂2px/∂x∂y] at
expression (3d): the firm increases its legal expenditures if and only if the
notation of ∂2px/∂x∂y is positive. The sign of the numerator at expression (4d) is
always negative. □
Figure 1: Equilibrium Output
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Park and Shogren: Environmental Citizen Suits with Pigovian Punitive Damages
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