Tohoku University Policy Design Lab Discussion Paper

TUPD-2022-005

Non-point source pollution control policy for stochastic but
constant environmental damage

Eiji Sawada
Faculty of Economics, Kyusyu Sangyo University

Graduate School of Economics and Management, Tohoku University

Takayoshi Shinkuma
Faculty of Economics, Kansai University

Graduate School of Economics and Management, Tohoku University

Akira Hibiki
Graduate School of Economics and Management, Tohoku University

March 2022

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Non-point source pollution control policy for stochastic but
constant environmental damage
Eiji SAWADA1,4,∗, Takayoshi SHINKUMA2,4 and Akira HIBIKI3
1

Department of Economics, Kyushu Sangyo Univ, 2-3-1, Matsukadai Higashi-ku, Fukuoka, 813-8503,
Japan.

2

Department of Economics, Kansai Univ.

3

Graudate School of Economics and Management, Tohoku Univ.

4

Policy Design Lab (Graudate School of Economics and Management), Tohoku Univ.

Abstract The interest of this paper is in the area of non-point source pollution, especially that
produces a certain large environmental damage in a stochastic manner. Most previous studies on
non-point source pollution control policies have implicitly assumed that the abatement eﬀorts
of economic agents as a whole can be estimated by assuming environmental damages whose
magnitude depends on the abatement eﬀorts of economic agents. When assuming environmental
damage that occurs at a certain magnitude in a stochastic manner, the policies proposed byprevious studies may achieve eﬃciency but do not prevent collusion. Therefore, this paper
designs new eﬃciency and collusion-proof policies that work even when not only individualabatement eﬀorts cannot be observed, but furthermore, when the total abatement eﬀort cannot
be estimated. Our policy only requires an honest reportig as a whole. By remaining room foradjustment in the reported amount of individual economic agents, our policy is also shown toachieve an equal burden among eocnomic agents.
Keywords: Non-point source pollution control policy; stochastic constant environmental damage; self-reporting; relative evaluation; collusion-proof; equal burden.

∗

Corresponding author. Email address: e.sawada@ip.kyusan-u.ac.jp1

1

Introduction

In environmental economic research, non-point source pollution is characterized as pollutionin which the behavior of individual economic agents is unobservable and is highly aﬀected by
the uncertainty of natural factors. Among them, it is particularly diﬃcult to apply traditional
environmental economic policies to non-point source pollution due to the moral hazard problem
caused by the unobservable behavior of individual economic agents (Holmstöm,1982; Segerson;1988). Policies to control non-point source pollution require a fundamental rethinking of thedesign of environmental policies, and various approaches have proposed policies that can achieveeﬃciency without requiring information on the behavior of individual economic agents (Xepapadeas, 1992; Hansen, 1998; Horan, Shortle and Abler, 1998; Shortle, Horan and Abler, 1998;
Hansen and Romstad, 2007).
The interest of this paper is in the area of non-point source pollution, especially that produces
a certain large environmental damage in a stochastic manner. Prior studies have deﬁned themagnitude of environmental damage as dependent on the behavior of economic agents. Thisis a very natural deﬁnition, and similar deﬁnitions are widely used in environmental economic
research, not only in the area of non-point source pollution research. However, when we look at
actual environmental problems, there are many environmental problems, especially those cause
extensive damage, that once they occur, the magnitude of the damage cannot be controlled.
In recent years, such environmental problems have been on the increase due to the eﬀects ofclimate change. The only thing we can do for such environmental problems is to make eﬀorts
to prevent the occurrence of environmental damage.
Can we apply the non-point source control policies, which have been developed by previous
studies, to environmental damage of a certain magnitude? If not, how can the policies proposedby previous studies be modiﬁed? This is the research question of this paper.
The structure of this paper is as follows. First, in Section 2, as a reference to clarify the
problem of this paper, we review the non-point source pollution control policies that assume
environmental damage dependent on abatement eﬀorts. In Section 3, we conﬁrm that the policies
proposed by previous studies can achieve eﬃciency but cannot prevent collusion when assuming
environmental damage that occurs at a certain magnitude in a stochastic manner. Section 42and Section 5 propose a new non-point source pollution control policy that improves on thepolicies proposed in the previous studies: the eﬃciency policy in Section 4 and the collusionproof policy in Section 5, respectively. As an eﬃciency policy, this paper proposes a policy that
introduces self-reporting. Within Sections 4 and 5, it is shown that the proposed policies workeﬀectively without requiring the honest declaration of individual economic agents for both theeﬃciency policy and the collusion-busting policy. Finally, in Section 6, we summarize the paperand discuss the limitations and remaining issues of our analysis.

2Constant environmental damage whose probability of occurence
depends on abatement eﬀorts

Consider polluting economic activities conducted by N risk-neutral economic agents i = 1 . . . N .
PLet xi be the abatement eﬀort of economic agent i and X = N
j=1 xj be the total abatement
eﬀort of economic agents. The pollution assumed in this paper is non-point source pollution.
Therefore, throughout this paper, policy makers cannot observe abatement eﬀorts of individual
economic agents {xj }N
j=1 .

In this section, we consider again the same problem as in the previous section, assuming
constant environmental damage whose probability of occurence depends on abatement eﬀorts.
Let D > 0 be the environmental damage of constant magnitude and P (X) ∈ [0, 1] be theprobability of occurence of environmental damage, and assume P ′ (·) < 0. The only diﬀerence
from the previous section is that the expected environmental damage is speciﬁed and expressed
as P (X)D.
Let Ci (xi ) be the abatement cost function of economic agent i and assume C ′ (xi ) > 0 and
C ′′ (xi ) > 0.
From the above, we can formulate the expected social cost minimization problem as following:


N
X
min 
Cj (xj ) + P (X)D  .
(1){xj }N
j=1

j=1

3

The ﬁrst-order conditions for this problem are
Ci′ + P ′ (X)D = 0 for i = 1 . . . N.

(2)

The solution to this problem is denoted by superscript ∗. In this case, the expected value of thetotal burden on economic agents isNX

Cj (x∗j ) + P (X ∗ )D.

(3)

j=1

In designing environmental policies in the subsequent paper, (6) is the criterion for evaluating
the total burden on economic agents. That is, if the total burden on economic agents exceeds(6) when an environmental policy is introduced, such a policy is undesirable because it would
be an excessive burden on economic agents.
Without any environmental policy in place, private economic agents will not willingly pay theabatement costs and will choose zero as their abatement eﬀort. The simplest non-point pollution
control policy that can provide appropriate pollution abatement incentives to economic agentsis to make each economic agent bear the full environmental damage. That is, each economic
agent is required to pay the realized value of D(X, ε) when environmental damage occurs.
The problem faced by economic agent i after such policy is introduced ismin ( Ci (xi ) + P (X)D ) .
xi(4)

The ﬁrst-order condition for this problem isCi′ + P ′ (X)D = 0.

(5)

From (2) and (5), we see that the ﬁrst-order conditions are equal. While this approach canachieve eﬃciency through a very simple policy, it is known to have two major problems. The
ﬁrst problem is the excessive burden of economic agents. The expected value of the total burden
on economic agents is expressed as
NX

Cj (x∗j ) + N P (X ∗ )D.

(6)

j=1

Since everyone bears the full environmental damage, the expected value of the tax should beequal to the actual environmental damages times the number of economic agents. Needless4

to say, (6) exceeds (3). As the number of economic agents increases, the deviation from (3)increases, and the policy become less realistic.
The second problem, often pointed out, is that economic agents have an incenteive to collude
when introducing the full burden of environmental damage. When the magnitude of environmental damage depends on the abatement eﬀorts of economic agents, their abatement eﬀorts
reduce not only their own burden but also that of all other economic agents. By colluding
to minimize the total cost of several economic agents, the economic agents can then reduce
the overall cost of the economic agents participating in the collution compared to before thecollusion.
In this paper, to understand the workings of the collusion-proof policy, we consider only thecollusion of N eocnomic agents as a collusion. The problem of collusion when the full burden ofenvironmental damage is introduced ismin{xj }N
j=1

N
X( Cj (xj ) + P (X)D ) .

(7)

j=1

The ﬁrst-order conditions for this problem isCi′ + P ′ (X)D + (N − 1)P ′ (X)D = 0 for i = 1 . . . N.

(8)

The diﬀerence between (2) and (8) is the presence of the third term on the left-hand side of (8).
The third term on the left-hand side of (8) represents the magnitude of the reduction in theburden on N − 1 economic agents other than eocnomic agents i due to the additional abatement
eﬀorts of eocnomic agent i. Because the marginal beneﬁt of the abatement eﬀort of economic
agents i increase by this term, eocnomic agent i choose an excess abatement eﬀort over x∗i . The
collusion of economic agents will result in a loss of the eﬃciency of the policy.
To prevent collusion, the policy should be modiﬁed so that the burden does not decrease
even if excessive abatement eﬀorts are chosen. For example, the following procedure can be used
to decouple excessive abatement eﬀorts from reduced burdens.
At ﬁrst glance, it appers that collusion prevention is possible in a simple way, but thisapproach assumes that the total abatement eﬀorts of economic agents can be estimated. And
this is only possible when the magnitude of environmental damage depends on the abatement
eﬀorts of economic agents. Only when the environmental damage and the abatement eﬀorts
5

of economic agents is one to one relationship can X be estimated backward from the observed
ε and D(X, ε) realizations. By assuming a abatement eﬀort-depndent environmental damage
function, it can be said that the previous studies assumed that policy makers can estimate thetotal abatement eﬀorts of economic agents.
The ﬁrst-order conditions for expected social cost minimization problem are thenCi′ + P ′ (X)D = 0 for i = 1 . . . N.

(9)The expected value of the total burden on economic agents isNX

Cj (x∗j ) + P (X ∗ )D.

(10)

j=1

Since the environmental damage is only speciﬁed, the results shown in the previous section
regarding (i) that the full burden of environmental damage can achieve eﬃciency, (ii) that thefull burden of environmental damage is an excessive burden on economic agents, and (iii) thateﬃciency is undermined by the collusion of economic agents are similarly established. The
qeustion is whether collusion can be prevented in the same way as in the previous section. If theabatement eﬀorts of economic agents can be estimated, the same approach as in the previous
section can be applied, but if they cannot be estimated, another approach must be required.
Here, the realized value of environmental damage that the policy maker can observe isalways D. Therefore, even if the environmental damage can be observed, the total abatement
eﬀorts of economic agents cannot be estimated from it. Since the probability of occurrenceof environmental damage depended on the abatement eﬀorts of economic agents, if the trueprobability of occurrence could be known, the total abatement eﬀorts of economic agents could
be estimated from it. However, once an occurrence is observed, it is not possible to know towhat extent the actual probability of occurrence has descreased due to abatement eﬀorts. If thefrequency of occurrence is very high, it may be possible to obtain an approximation of the trueprobability of occurrence within a short period of time, but at least as far as the environmental
issues on which this paper focuses, a long period of time is required.

6

3

Non-point source pollution control policy based on self-reported
abatement eﬀorts

In this section, we propose a new non-point source pollution control policy based on self-reported
abatement eﬀorts. The policy maker requires economic agents to self-report their abatementPN
eﬀorts. Let ri be the reported amount of economic agents i and R =j=1 rj be the totalreported amount of economic agents. This reported amount does not necessarily coincide withthe true abatement eﬀorts of economic agents. The policy maker can design a policy using the
newly available reported amount of eonomic agents {rj }N
j=1 and the relative reported amountr{ Rj }N
j=1 ， as follows.
Eﬃciency policy (P1):
1. Pay an incentive of ti to all eoconomic agent i = 1, . . . , N for every one unit of reported
amount.
2. In case of environmental damage, collect a lump-sum tax {T̄j }N
j=1 from only one economic
agent.
3. The economic agent that bears the lump-sum tax is detemined stochastically according tor

the relative reported amount { Rj }N
j=1 .
The structure of the eﬃciency policy (P 1), in which only one economic agent bears a large
burden, may be similar to the all-or-nothing policy proposed by Xepapadeas (1992). In thepolicy of Xepapadeas (1992), only one eocnomic agents determied stochastically bore the fullburden of environmental damage in order to achieve both eﬃciency and an appropriate burden
on economic agents at the same time. However, the policy proposed by Xepapadeas (1992)
diﬀered in that the economic agents bearing the burden were chosed randomly, whereas in ourpolicy, the economic agents with large relative reported amount have a higher probability of
being chosen. Moreover, as will be shown in the proposition that follown, our policy can adjust
not only the total burden but also the individual burdens.
The expected value of the burden on economic agent i due to introducing the eﬃciency policy7

(P 1) isri
E[Ti ] = ti ri + P (X) T̄i .
R(11)The problem faced by the economic agent i ismin ( Ci (xi ) + E[Ti ] ) .
xi ,ri(12)First let us see that the eﬃciency policy (P 1) can achieve eﬃciency for the case when economic
agents do not consider collusion. In the proposition, it is shown that the achievement of eﬃciency
does not require an honest report. Consequently, a new policy parameter will be added: whatlevel should be targeted as the amount of self-report. Here, denoting {gj }N
j=1 be the targets ofPN
reported amount from economic agents, G = j=1 gj be the sum of the individual target andP
G−i = Nj̸=i gj be the sum of the individual target other than economic agent i.
Proposition 1. The eﬃciency policy (P 1) choosing the policy parameters as∗ G−i
ti {gj }N
D for i = 1, . . . , N,
j=1 = −P (X )Ggi GT̄i {gj }N
D for i = 1, . . . , N
j=1 =gi
can achieve the eﬃciency under the given targets of reported amount {gj }N
j=1 .
Proof. The ﬁrst-order conditions for the problem (12) areCi′ +∂E[Ti ]= 0,∂xi
∂E[Ti ]
= 0,
∂ri

(13)
(14)

where

and R−i =Pri∂E[Ti ]
= P ′ (X) T̄i = 0,∂xi
R∂E[Ti ]
R−i
= ti + P (X) 2 T̄i ,
∂ri
R

(15)
(16)

j̸=i rj is the sum of the reported amount other than economic agent i. For (9) and(13) to coincide, we must have
T̄i =

R
D.
ri

8

(17)

Substituting (17) into (14) and rearranging for ti , we obtainti = −P (X)

R−i
D.
Rri

(18)

When choosing {T̄i }N
i=1 according to (17), eﬃciency can be achieved independent of the reported
amount. Therefore, the policy parameters of the eﬃciency policy (P 1) to achieve the eﬃciency
can be expressed as∗ti ({gj }Nj=1 ) = −P (X )T̄i ({gj }Nj=1 ) =

G−i
D,
Ggi

(19)

G
D.
gi

(20)

Next, let us see that the eﬃciency policy (P 1) does not excessively burden economic agents.
Let E[Ti ({gj }Nj=1 )] be the expected value of the burden on economic agent i when the eﬃciency
policy (P 1) is introduced and the policy parameters are chosen as (19) and (20).
Proposition 2. The eﬃcient policy (P 1) choosing policy parameters as (19) and (20) canachieve appropriate burden of economic agents. In other words, it satisﬁes thatNX∗
E[Tj ({gj }Nj=1 )] = P (X )D,

j=1

under the given targets of reported amount.
Proof. Substituting xi = x∗i , ri = gi , (19) and (20) into E[Ti ({gj }Nj=1 )], we obtain∗
E[Ti ({gj }Nj=1 )] = −P (X )

gi G
gi
G−i
Dgi + P (X ∗ )D = P (X ∗ )D .
Ggi
G gi
G

(21)

Summing this over the entire eocnomic agents, we obtain
N
X

PN
∗
E[Tj ({gj }Nj=1 )] = P (X )D

j=1 gjG

j=1= P (X ∗ )DG
= P (X ∗ )D.
G(22)

By proposition 1 and proposition 2, both eﬃciency and an appropriate burden on economic
agents can be achieved under any given reported amount. It is not the case, then, that thetargets of reported amount {gj }N
j=1 has no eﬀect anywhere. From (21), we know that the targets
9

of reported amount {gj }N
j=1 changes the expected value of the burden on individual economic
agents. Therefore, the target of reported amount {gj }N
j=1 allows the policy maker to adjust the
expected value of the burden on individual economic agents while keeping the expected value ofthe burden on economic agents as a whole at an appropriate level.
One special allocation is the equal burden among economic agents achieved by determing{gj }N
j=1 to satisfy
PN
E[Ti ({gj }Nj=1 )] =j=1 E[Tj ]Nfor i = 1, . . . , N.

(23)

Proposition 3. The eﬃcient policy (P 1) choosing policy parameters as (19) and (20), andchoosing the targets of reported amount to satisfy (23) can simultaneously achieve eﬃciecny andequal burden.

4

Policy amendments to prevent collusion

The propositions about the properties of the eﬃciency policy (P 1), derived in the previous section, did not consider collusion-proofness. In fact, if economic agents collude, all the propositionsof the previous section would not hold. Therefore, in this section, we attempt to modify thepolicy to prevent collusion. Speciﬁcally, we add the collusion-proof policy (P 2) to the eﬃciency
policy (P 1), proceeding as follows. The information available to the policymaker for designing
the policy is the individual abatement eﬀorts and their sum, Xi∗ and X ∗ ; the individual reported
amount and their sum, ri and R; and the observations about the occurrence of environmental
damages. Policy makers can know whether environmental damage has occurred or not. The
occurrence or non-occurrence is denoted by a random variable n that takes on a value of either
0 or 1. Using all of this available information, we can design a collusion-proof policy as follows.

10Collusion-proof policy (P2):
1. The policy maker continues to observe whether environmental damage occurs for a certain
period of tim (e.g., one year).
2. Calculate the square of the diviation between the eﬃcient probability of occurrence forsociety as a whole, and the realized value of n.
3. Calculate the square of the deviation between the self-report based probability of occurrence and the realized value of n.
4. Collect he diﬀerence between 3. and 4. multiplied by the policy parameter αi from
economic agent i.
5. Furthermore, collect the diﬀerence between the total eﬃcient abatement eﬀort and thetotal reported amount multiplied by another policy parameter βi from economic agent i.

From the above procedure, the additional burden Vi of economic agent i due to the collusionproof policy (P 2) can be expressed asnoVi = αi (P (X ∗ ) − n)2 − (P (R) − n)2 + βi (X ∗ − R)2 .

(24)

Since, n is a random variable, Vi is also a random variable. The expected value of Vi can becalculated as followsnoE[Vi ] = αi (P (X ∗ ) − P (X))2 − (P (R) − P (X))2 + βi (X ∗ − R)2 .

(25)

When the collusion-proof policy (P 2) is added, the problem faced by economic agent i ismin (Ci (xi ) + E[Ti ] + E[Vi ]) .
xi ,ri(26)

The problem of the collusion of all economic agents isminN{xj }N
j=1 ,{rj }j=1

N
X

(Cj (xj ) + E[Tj ] + E[Vj ]) .

j=1

11

(27)

The addition of the collusion-proof policy (P 2) would be useless if it undermines the properties of the eﬃciency policy (P 1) shown in the previous section when economic agents do notcollude. Therefore, let us ﬁrst focus on the problem (26) and see whether the propositions in
the previous section can be replicated when the eﬃciency policy P (1) and the collusion-proof
policy (P 2) are introduced simultaneously.
Proposition 4. The eﬃcient policy (P 1) choosing the policy parameters as
G−i{D + 2αi (P (G) − P (X ∗ ))}Ggi− 2αi P ′ (G) (P (G) − P (X ∗ )) + 2βi (X ∗ − G) for i = 1, . . . , N,G
{D + 2αi (P (G) − P (X ∗ ))} for i = 1, . . . , N
T̄i ({gj }N
j=1 , αi ) =giNti ({gj }j=1
, αi , βi ) = − P (X)and the collusion policy (P 2) can achieve eﬃciency under the given targets of reported amountNN{gj }Nj=1 and the given policy parameters of collusion-proof policy (P 2) {αj }j=1 and {βj }j=1 .

Proof. The ﬁrst-order conditions for the problem (26) areCi′ +∂E[Ti ] ∂E[Vi ]
+
= 0,
∂xi
∂xi
∂E[Ti ] ∂E[Vi ]
+
= 0,
∂ri
∂ri

(28)
(29)

where
∂E[Vi ]
= −2αi P ′ (X) (P (R) − P (X ∗ )) ,∂xi
∂E[Vi ]
= 2αi P ′ (R) (P (R) − P (X)) − 2βi (X ∗ − R) .
∂ri

(30)
(31)

For (9) and (28) to coincide, we must have
T̄i =

R
D + 2P ′ (X)αi (P (R) − P (X ∗ )) .
ri

(32)

Substituting (32) into (29) and rearranging for ti , we obtainti = −P (X)R−i{D + 2αi (P (R) − P (X ∗ ))} − 2αi P ′ (R) (P (R) − P (X)) + 2βi (X ∗ − R) . (33)
Rri

Again, the achievement of eﬃciency does not depend on the reported amount. In addition,
it does not depend on the policy paramters of the collusion-proof policy (P 2). Therefore, the12policy parameters of the eﬃciency policy (P 1) to achieve the eﬃciency can be expressed as
G−i{D + 2αi (P (G) − P (X ∗ ))}Ggi− 2αi P ′ (G) (P (G) − P (X ∗ )) + 2βi (X ∗ − G)GT̄i ({gj }N{D + 2αi (P (G) − P (X ∗ ))} .
j=1 , αi ) =giti ({gj }N
j=1 , αi , βi ) = − P (X)

(34)
(35)

Proposition 5. When sum of the targets of the reported amount is set to be equal to the sum of
the eﬃcient abatement eﬀorts, the policy parameters of the eﬃciency policy (P 1) that achieveseﬃciency when only the eﬃciecny policy (P 1) is introduced coincide with those when eﬃcient
policy (P 1) and the collusion-proof policy (P 2) are introduced under the given policy parametersof the collusion-proof policy (P 2).
In other words, if G = X ∗ , then it satisﬁesNti ({gj }Nj=1 )|G=X ∗ = ti ({gj }j=1 , αi , βi )|G=X ∗ ,N
T̄i ({gj }Nj=1 )|G=X ∗ = T̄i ({gj }j=1 , αi )|G=X ∗N
under given {αj }N
j=1 and {βj }j=1 .

Proof. When G = X ∗ ，(34) and (35) areti ({gj }N
j=1 , αi , βi )|G=X ∗ = −P (X)T̄i ({gj }N
j=1 , αi )|G=X ∗ =

G−i
D,
Ggi

G
D.
gi

(36)
(37)

(36) and (37) are equal to (19) and (20) respectively.
Proposition 6. Set the sum of the targets of reported amount to the sum of their total eﬃcientabatement eﬀorts. Then, the eﬃcient policy (P 1) choosing policy parameters as (36) and (37),and the pollution-proof policy (P 2) can achieve appropriate burden of economic agents under
given policy parameters of collusion-proof policy (P 2).
In other words, if G = X ∗ , then it satisﬁes thatNX∗
E[Tj ({gj }N
j=1 , αi , βi )|G=X ∗ ] = P (X )D

j=1
N
under given {αj }N
j=1 and {βj }j=1 .

13

Proof. From Proposition 4, X = X ∗ , because eﬃciency could be achieved by adding thecollusion-proof policy (P 2) to the eﬃciency policy (P 1). Now, setting G = X ∗ for the sumof the targets of reported amount, and then we have X = R = X ∗ . Substituting X = R = X ∗into (25), the expected value of the additional burden due the collusion-proof policy (P 2) is
E[Vi ]|X=R=X ∗ = 0 for i = 1, . . . , N.

(38)

What remains is the expected value of the burden of the eﬃciency policy (P 1), E[Ti ], but when
G = X ∗ from proposition (5), the policy parameters of the eﬃciency policy (P 1) that achieveseﬃciency with and without the collusion-proof policy (P 2) did not change. Thus, the magnitudeof E[Ti ] does not change with the addition of the collusion-proof policy (P 2). Naturally, thePsize of the sum, N
j=1 E[Tj ], also remains unchanged. From Proposition 2, since the expectedvalue of the total burden of economic agents when only the eﬃciency policy (P 1) is introduced
coincides with the expected value of environmental damage, the expected value of the total
burden of economic agents after adding the collusion-proof policy (P 2) also coincides with the
expected value of environmental damage.
Comparing propositions 2 and 6 in the previous section, we ﬁnd that the economic agentswere able to achieve an appropriate burden at any target of reported amount in proposition 2,
whereas in proposition 6, which considers collusion, the appropriate burden on economic agents
could not be achieved unless the sum of targets was the sum of eﬃcient abatement eﬀorts.
However, there is still room to adjust the targets of reported amount {gj }N
j=1 to the extent thatthe condition G = X ∗ is satisﬁed. This room for adjustment is suﬃcient to achieve equal burden
among economic agents.
Proposition 7. Set the sum of the targets of reported amount to the sum of their total eﬃcientabatement eﬀorts. Then, the eﬃcient policy (P 1) choosing policy parameters as (19) and (20),and the pollution-proof policy (P 2) can simultaneously achieve eﬃciecny and equal burden under
the given policy parameters of collusion-proof policy (P 2).
Proof. When the eﬃciency policy (P1) and the collusion-proof policy (P 2) are introduced and(34) and (35) are chosen as policy parameters of the eﬃciency policy (P1) to achieve eﬃciency,

14the expected value of the burden on economic agent i is
E[Ti ] = P (X ∗ )D

gi
.
X∗

(39)

The only diﬀerence in the magnitude of the burden among economic agents is gi . Therefore,
equal burden can be achieved by setting the total target of reported amount to X ∗ and dividing∗

it equally among N eocnomic agents to determine gi = XN .
Finally, let us see that the collusion-proof policy (P 2) can infact prevent collusion. The
ﬁrst-order conditions of the problem (27) areCi′ +∂E[Ti ] ∂E[Vi ] X
+
+
∂xi
∂xi
j̸=i

∂E[Ti ] ∂E[Vi ]
+
+
∂ri
∂ri



∂E[Tj ] ∂E[Vj ]
+
∂xi
∂xi

X  ∂E[Tj ]
j̸=i

∂ri

+

∂E[Vj ]
∂ri


= 0,

(40)

= 0,

(41)



where
∂E[Tj ]
rj
= P ′ (X) T̄j ,∂xi
R∂E[Tj ]
rj
= −P (X) 2 T̄j ,∂ri
R∂E[Vj ]
= −2αj P ′ (X) (P (X ∗ ) − P (R)) ,∂xi∂E[Vj ]
= −2αj P ′ (R) (P (R) − P (X)) − 2βj (X ∗ − R) .
∂ri

(42)
(43)
(44)
(45)

# N
N
Denote xi and ri as {x#i }i=1 and {ri }i=1 that satisfy (40) and (41). The fourth term in the(40) left-hand side and the third term in the (41) left-hand side are the new terms added bycollusion, respectively. (42) represents the eﬀect of the additional abatement eﬀort on reducing
the probability of environmental damage, thereby reducing the expected value of the burden
on economic agents other than economic agent i. (43) represents the eﬀect of additional selfreporting reducing the expected value of the burden on all other economic agents by reducing
the probability of economic agents other than economic agent i paying lump-sum taxes. The
addition of these terms distorts the incentives for both abatement eﬀorts and self-reporting.
On the other hand, (44) and (45) represent the eﬀect of additional abatement eﬀorts andself-reporting by economic agent i on the additional burden of the collusion-proof policy (P 2)of economic agents other than economic agent i. (44) and (45) contain the policy parameters αi15and βi of the collusion-proof policy (P 2). The following proposition shows that by choosing αiand βi appropriately, the fourth term in (40) left-hand side and the third term in (41) left-handside, which represent the distortion of incentives due to collusion, can be cancelled out.

16

Proposition 8. The eﬃcient policy (P 1) and the collusion-proof policy (P 2) can achieve eﬃciency with or without collusion of economic agents under the given targets of reported amount{gj }N
j=1 by choosing policy parameters as
G−i{D + 2αi∗ (P (G) − P (X ∗ ))}Ggi− 2αi∗ P ′ (G) (P (G) − P (X ∗ )) + 2βi∗ (X ∗ − G) for i = 1, . . . , N,G∗
T̄i ({gj }N{D + 2αi∗ (P (G) − P (X ∗ ))} for i = 1, . . . , N,
j=1 , αi ) =gi∗∗ti ({gj }N
j=1 , αi , βi ) = − P (X)and {αj∗ , βj∗ }N
j=1 satisfying (46) and (47).
Proof. By Proposition 4, the eﬃcient policy (P1) can achieve eﬃciency when not condieringcollusion under the given policy parameters of the collusion-proof policy (P2), {αj , βj }N
j=1 . Thechange in the ﬁrst-order conditions of the economic agents problem due to collusion is the
addition of the last term in (40) and (41). If these additional terms can be eliminated byproperly choosing {αj , βj }N
j=1 , then eﬃciency can be achieved even if economic agents collude.
We show that one can actually choose such {αj , βj }N
j=1 .
The last term in (40) and (41) include {T̄j , αj , βj }N
j=1 as policy parameters. SubstitutingNN
the eﬃcient policy parmeter {T̄j }N
j=1 when not considering collusion, {T̄j ({gj }j=1 , αj )}j=1 intoN{T̄j }N
j=1 in the last terms in (40) and (41). Then, there we can ﬁnd {αj , βj }j=1 satisfying

X
j̸=i

X
j̸=i

∂E[Tj ]
∂E[Vj ]+
∂xi T̄j =T̄j ({gj }N ,αj )∂xi
j=1∂E[Tj ]
∂E[Vj ]+
∂ri T̄j =T̄j ({gj }N ,αj )∂ri!
= 0 for i = 1, . . . , N,(46)

= 0 for i = 1, . . . , N,

(47)

!

j=1

N
because there are 2N parameters for 2N eauations. Let {αi∗ , βj∗ }N
j=1 be a paremeter set {αi , βj }i=1

satisfying (46) and (47). Corresponding polciy parmeters of the eﬃciency policy (P1) are
∗
∗
N
∗ N
{tj ({gj }N
j=1 , αj , βj ), T̄j ({gj }j=1 , αj )}j=1 .

17

Proposition 9. Introduce the eﬃcient policy (P1) and the collusion-proof policy (P2) as Proposition 8 and set the sum of the targets of reported amount to the sum of their total eﬃcientabatement eﬀorts (G = X ∗ ). Then, economic agents either choose not to collude or all collude.
Proof. Assume that the eﬃcient policy (P1) and the collusion-proof policy (P2) are both introduced as Proposition 8. First, let us calculate the total burden on economic agents when they
∗ N
all collude. By Proposition 8, {xj }N
j=1 = {xj }j=1 under the policy parameters (34), (35) and∗N{αj∗ , βj∗ }N
j=1 for any {gj }j=1 . By Proposition 5 and Proposition 6, if G = X , then the total

burden on ecnomic agents equals to the optimal expected social cost:
N
X

N
 X
∗ ] + E[Vj ]|X=R=X ∗=Cj (x∗j ) + P (X ∗ )D.
Cj (x∗j ) + E[Tj ({gj }N
,
α
,
β
)|
i
i
G=X
j=1

(48)

j=1

j=1

Thus, the total burden is equal when economic agents do not collude and when they all collude∗
under the policy parameters (34), (35), {αj∗ , βj∗ }N
j=1 and G = X .

Next, let us consider the total burden on economic agents when one collusion except forcollusion among all is chosen. We denote the solution to the problem and the corresponding
burden by superscript #. Then, it must be
N 
X

N
 X
#
#
Cj (x∗j ) + P (X ∗ )D,Cj (x#
)
+
E[T
]
+
E[V
]
≥
j
j
j

(49)

j=1

j=1

because the collusion among all minimizes the total burden on economic agents. Economic
agents can improve Nash by either choose not to collude or by all collude. This inequality holds
for any collusion except for collusion among all.

18

5Concluding remarks

This paper proposes eﬃciency and collusion-proof policies that can work even if the total abatement eﬀorts cannot be estimated as well as individual abatement eﬀorts, assuming stochastic
and constant environmental damage caused by non-point source pollution. The policy proposedin this paper can achieve the three goals of eﬃciency, appropriate burden on economic agents,
and collusion-proof. These properties could be shown if economic agents as a whole made honest
reporting and did not require honest reporting by individual economic agents. By remaining
room for adjustment in the reported amount of individual economic agents, our proposed policy
is also shown to achieve an equal burden.
Finally, we discuss the limitations of our analysis in this paper. The most signiﬁcant limitation of the analysis in this paper is that only the collusion of all economic agents is considered
as collusion. In general, heterogeneous N economic agents can be divided into various collusiongroups. Therefore, collusion-proof policy is required to eliminate incentives for any collusiongroups. Among the policies proposed in this paper, further elaborate policy design remains an
issue, especially for the collusion-proof policy.

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