Discussion Paper No.328

Controlling Non-point Source Pollution in
Cournot Oligopolies with Hyperbolic Demand

Akio Matsumoto
Chuo University

Ferenc Szidarovszky
Corvinus UniversityDecember 2019

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Controlling Non-point Source Pollution in
Cournot Oligopolies with Hyperbolic Demand
Akio Matsumotoy Ferenc Szidarovszkyz

Abstract
In a case of non-point source pollution, the regulator is unable to monitor the individual emission levels of the …rms, having access only to thetotal industry emission level. An emission standard is de…ned by the regulator and if the industry emission level exceeds this standard, then the…rms are uniformly penalized, otherwise uniformly awarded. This environmental regulation policy is added to a hyperbolic n-…rm oligopoly. The
equilibrium output and emission levels for each …rm and for the industry
are determined and conditions are given under which the regulation policy
can control emission levels.

Keywords: Nonpoint source pollution, Ambient charge, Hyperbolic
price function, Duopoly and triopoly, Complex dynamics

The …rst author highly acknowledges the …nancial supports from the Japan Society for
the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 18K0163) and from Chuo
University (Grant for special research)y Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan. akiom@tamacc.chuo-u.ac.jp
z Depaartment of Mathematics, Corvinus University, Budapest, Fövám tér 8, 1093, Hungary. szidarka@gmail.com

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Introduction

Non point source (NPS) pollution is produced by multiple di¤use sources andin sharp contrast to point source pollution generated from a single source. Today’s main pollutions such as water pollution contaminating water bodies that
include river, lake, ocean and groundwater, air pollution that may cause harm
to humans as well as other living organisms and ocean plastic pollution thatrecently comes to focus are NPS. Its key feature is, due to multiple sources,
non-observability of individual contributions to the total concentration. Consequently, the traditional environmental policy implementation such as emission
charge in form of Piguvian tax, marketable emission permit, direct control onemission level are not applicable. Segerson (1988) suggests ambient based policies involving the use of rewards and penalties to control NPS pollution. Moreprecisely, the regulator selects an emission standard and if the industry emission
level is higher than this standard, then the …rms are uniformly penalized, otherwise, uniformly awarded. This type of ambient charges are included in oligopoly
models in which it strictly depends on the actions of other …rms whether a …rm
receives penalty or award.1
Much of the literature is studying how to address NPS pollution is gametheoretic. This is because it strictly depends on the actions of other agents
whether an agent receives penalty or award. Ganguli and Raju (2012) construct
a Bertrand duopoly and numerically demonstrate an increase of the ambient
charge could increase the total concentration, which they call a "perverse" e¤ect.
Ishikawa et al. (2019) extend their duopoly model to an n-…rm model and showthat the ambient charge is e¤ective in controlling NPS pollution in duopoly andtriopoly models whereas the sign of the e¤ect depends on the number of the…rms and the degree of substitutability in the model with n4. Raju andGanguli (2013) examine the ambient charge e¤ect in a Cournot duopoly withproduct di¤erentiation and numerically show its e¤ectiveness under a two-stagegame. Sato (2017) analytically exhibits that a higher ambient charge reduces thetotal emission in a Cournot market without product di¤erentiation. Matsumoto
et al. (2017) construct an n-…rm Cournot model and reexamine the result ina dynamic framework when the products are homogenous. Those results areobtained in a special circumstance in which the demand functions are linear.
In this study, a linear demand is replaced with a hyperbolic demand and thespecial results derived in hyperbolic duopolies and triopolies of Matsumoto etal. (2019) is generalized to n-…rm hyperbolic models. Conditions are derived
under which the ambient charges can control the emission levels of the …rms aswell as the industry emission.
This paper is organized as follows. In Section 2 the basic model is outlined,
and in Section 3 conditions are derived for the possibility of controlling emission
1 Large number of works have been devoted to the di¤erent variants of the classical Cournot(1838) model including oligopolies with and without product di¤erentiation, multi-productmodels, labor managed oligopolies, market share and contest games among others. Okuguchi(1976) o¤ers a comprehensive summary of the earlier results up to the mid 1970’s, theirmultiproduct generalizations are introduced in Okuguchi and Szidarovszky (1999).

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levels of the …rms and the industry total emission. Concluding remarks andfurther research directions are presented in Section 4.

2

Basic Model

Consider an n-…rm oligopoly without product di¤erentiation. Output of …rm iis xi and let Q be the total demand. The price function is hyperbolic,
1Q

p=(1)

Pn
where the total demand is supposed to be equal to the industry output, i=1 xi .
Let ci be marginal cost and ei the emission technology coe¢ cient of …rm i;
implying that ei xi is the emission level. If E is the emission standard and thepenalty or reward fraction speci…ed by the regulator,2 then the payo¤ of …rm iis01nX
xi@ci xiej xj E A .
(2)
i = Pnj=1 xjj=1

Firm i maximizes its payo¤ with given value of the outputs of the other …rms.
Solving the …rst-order condition
Pnxi
j=1 xjciei = 0(3)
2Pnxj
j=1
yields the best reply of …rm i as

xi =ryiciyi(4)where yi and ci are the output of the rest of the industry and the extended
marginal cost, which are de…ned asyi =nX

xj and ci = ci + ei :j6=i

If Q =Pn

j=1 xj , then from equation (4)Q2 =yiQ xi=:cici(5)Adding equation
ci Q2 = yi2 E and m are the strategic variables of the regulator and should be determined so as tomaximize some form of a welfare function. In the present paper these are assumed to be given
exogeneously, however their optimal determination will be considered in our future studies.

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for all i with C =Pn

j=1 cj yields

CQ2 = (nthenQ=1) Qn1C:(6)

From equation (5)xi = Q

ci Q2 :

Substituting equation (6) presents the Cournot level of …rm i’s production
xci =(n1) C (nC21)ci:(7)

The second factor of the numerator should be positive to have positive production at the Nash equilibrium. It can be written as
PP(n 1)ci +(n 1)ei
C (n 1)ci =j=1 cj
j=1 ej
=nc

ci +

1
ci
n+e

ei +

1ei
n

where c and e are the average production cost and the average abatement technology de…ned asPP
j=1 ej
j=1 cjand e =:c=nn
If …rm i has more e¢ cient production cost and abatement technology thanthe industrial averages, then its optimal production is positive. Further, itcan be said that positive production is possible in case of (c > ci ; e < ei ) or(c < ci ; e > ei ) or (c < ci ; e < ei ) if the di¤erences are small enough. We makethe two assumptions.
Pn
Assumption 1. j=1 cj (n 1)ci 0 for i = 1; 2; :::; n:PnAssumption 2. j=1 hj (n 1)hi 0 for i = 1; 2; :::; n; where
ei = hi e1 with hi

0 for i = 2; 3; :::n; e1 > 0 and h1 = 1:

Pn
Notice that at least one of the values j=1 cj (n 1)ci is positive which
canPn be seen by adding these expressions for all i. Same holds for expressions(n 1)hi . Assumption 1 concerns the marginal production costs andj=1 hj
makes the marginal production costs not be much di¤erent from each other.
Needless to say, it is satis…ed if ci = c 0 for all i. Assumption ensures thatthe abatement technologies of the …rms are close to each other. The Cournot
outputs can be rewritten as01
nX
(n1)
e1 @xci =ai +hj (n 1)hi A(8)
C2
j=1
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withai =The following is clear:Pn

j=1 cj(n1)cie1:

Theorem 1 Given Assumptions 1 and 2, the optimal production level for each…rm is positive.

3Ambient Charge E¤ect

We now consider the e¤ect caused by changing the value of m on the total levelof emission. The total level of emission at Cournot point is de…ned as
EC =nXei xci

i=1Using equation (8), the right hand side is rewritten asihPPn
Pn
n 1 nhPn
nec(n1)c+(nijii=1j=1i=1 ei
j=1 ejC2The …rst square bracketed term of the last line can be summarized as
hPiPnnec(n1)c= e1 S1iji
i=1
j=1
where

S1 =nX
i=12nX4hicj32)ci 5 > 0:

(nj6=i1)Pn2
i=1 ei(9)

The inequality is due to Assumption 1. The second square bracketed term canbe rewritten as
PnPnPn(n 1) i=1 e2i = e21 S2
i=1 eij=1 ejwhereS2 =nX
i=10nXhi @hjj=1(n11)hi A(10)which is positive due to Assumption 2. With the new parameters, S1 and S2 ;the total emission level is written as(n 1) e1
(S1 + e1 S2 ) > 0:
EC =C2Di¤erentiating the last equation of E c with respect to gives"#nX(n 1) e1
@E C=
e1 S2 C 2 (S1 + e1 S2 )
ei :
@
C3
i=1

Since the second term in the square brackets is always positive, we have thefollowing result,5io:Lemma 1 If e1 S2 C2(S1 + e1 S2 )Pn

i=1 ei0; then

@E C
< 0:
@

From Lemma 1, the following two results can be derived.
Corollary 1 IfPn

Pn

i=1 cii=1 ei ; then

@E C
< 0:@Proof. Notice that simple calculation shows that"nX@E C(n 1) e1eS=(ciei )
1 2
@
C3i=1

2S1nX

ei

i=1

which implies the assertion.
We now summarize the results obtained.
PnTheorem 2 Under Assumptions 1,2 andi=1 cicharge is e¤ ective in controlling the NPS pollution,#Pn

i=1 ei ;the ambient

@E C
< 0:@
We now turn attention to the individual level of emission value of …rm i,
although the regulator is unable to observe this e¤ect. Di¤erentiating equation(7) with respect to gives
3120nnX@xcin 1 4@Xej 5 :(11)ej (n 1)ei A C 2 C (n 1)ci=@
C3
j=1
j=1
Lemma 2 If
@xci
< 0:@Pnj=1 ej(n1)ei C

2 C(n1)ciPn

j=1 ej< 0; thenThe bracketed term in (11) can be rewritten asnXej

C + (n1) ci + (nj=11)cinXej(n1)ei C:

j=1

The …rst term is nonpositive, since
C + (n 1) cinX=(cj + ej )j=10= @

nX
j=1

cj

(n(n1) (ci + ei )10nX1)ci A + ej @hj
j=1

6(n11)hi A0

by Assumptions 1 and 2. The second term hs the form
01nnn
X
X
X
(n 1) (ci + ei )ej (n 1)ei @cj +ej A=(n2j=11) 4cinX

ejei

j=1

nX
j=1

So we have the following result.

j=1

3

j=1

cj 5 :

Theorem 3 Under Assumptions 1 and 2, the ambient charge can control theindividual level of ouput (or emition) of …rm i if
i = cin
Xejei

j=1

Notice thatn
X
j=1

j =n
X
j=1

cjn
Xn
X

cj < 0nX

ci(12)

j=1

ej

j=1

i=1

n
X

ei = 0i=1

showing that (12) cannot hold for all …rms. Relation (12) means that the rationof ci and ei is below the ratio of their average.

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Concluding Remarks

This paper examined the e¤ect of ambient charges in n-…rm hyperbolic oligopolies
assuming non-point source (NPS) pollution. Conditions were …rst derived whichguaranteed non-negative equilibrium output levels for the …rms. Then the e¤ectof ambient charges on the emission levels of the individual …rms as well as on
the total emission level of the industry was examined giving conditions underwhich regulation is able to control emission. In this model linear production
cost functions and simple hyperbolic price function were assumed. In our future
research projects more general function types and di¤erent kinds of oligopoly
models will be investigated.

7References
Cournot, A. Researches on the mathematical principles of the theory of wealth,1838.
Ganguli, S., and Raju, S., Perverse environmental e¤ects of ambient charges
in a Bertrand duopoly, Journal of Environmental Economics and Policy,
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Ishikawa, T., Matsumoto, A., and Szidarovszky, F., Regulation of non-point
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Okuguchi, K., and Szidarovszky, F. The theory of oligopoly with multi-product…rms (2nd ed.), Berlin, Springer, 1999.
Raju, S., and S. Ganguli, Strategic …rm interaction, returns to scale, environmental regulation and ambient charges in a Cournot duopoly, Technology
and Investment, 4, 113-122, 2013.
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