Discussion Paper No.330

The effectiveness of Ambient Charges in ann-firm Cournot Oligopoly
Yuichi Sato
Chuo University
Economic Reserch Institute,Associate ResearcherJune 2020

INSTITUTE OF ECONOMIC RESEARCH
Chuo UniversityTokyo, Japan

The Effectiveness of Ambient Charges in ann-firm Cournot Oligopoly
Yuichi Sato∗

Abstract
This study investigates the effectiveness of ambient charges for Nonpoint Source (NPS) pollution in an n-firm Cournot oligopoly framework.
Previous research showed that ambient charges can decrease NPS pollution in a two-firm Cournot duopoly model. We extend this analysis todemonstrate that ambient charges can abate NPS pollution in an n-firm
Cournot oligopoly case. We use standard market demand functions forn-firm Cournot cases in the same industry. The results show that a reduction in NPS pollution occurs not only in a Cournot duopoly case, butalso in general Cournot oligopoly cases.
Keywords: Nonpoint Source pollution; Ambient charges; Cournot
oligopoly;
JEL classification: L13∗ Chuo University Economic Reserch Institute, Associate Researcher1

1Introduction

This paper examines whether ambient charges can reduce Nonpoint Source(NPS) pollution in an n-firm Cournot oligopoly framework. NPS pollution— that is, pollution from many diffuse sources as opposed to one single source— accounts for the majority of the pollution levels today1 . This pollution
cannot precisely be observed at the level of abatement or from the dischargeof any individual suspected polluter. These unobservable features prevent us
from reducing NPS pollution. In order to abate this pollution, Segerson (1988)
describes ambient charges — charges based on the total amount of pollution
irrespective of specific origin —- as a possible instrument of NPS pollution control. In her scheme, the regulatory authority considers emission cut-off levels,
not for the individual firms in an industry, but for all the firms belonging to arelevant industry. If the emission standards set by government for NPS pollution of all firms in an industry are compared based on actual levels of emissions,those firms exceeding the cut-off levels can be identified and forced to pay thesame fines as each other. By imposing cut-off levels on all the firms in an industry, we can control the total amount of NPS pollution. This payment scheme
can be described as ambient charges. According to Segerson (1988), there aremany studies on ambient charges. Poe et al. (2004) and Suter et al. (2008)
discuss ambient charges in terms of experimental research. Ganguli and Raju
(2013) analyze the outcome of ambient charges in a Cournot duopoly market
under constant and decreasing returns to scale. Guerrini et al. (2018) examines
ambient charges in a delay Cournot duopoly model. Sato (2017) uses ambientcharges to reduce NPS pollution in a static Cournot duopoly case.
1 Ganguli and Raju(2013)2

This paper extends the research conducted by Sato (2017) to an n-firm
Cournot oligopoly framework, and suggests that ambient charges can decrease
NPS pollution in the n-firm Cournot oligopoly case in the same way as in a
Cournot duopoly case.
This paper is organized as follows. The methodology is described in Section
2, where we demonstrate that ambient charges can reduce NPS pollution in ann-firm Cournot oligopoly case. In the final section, we present the concluding
remarks.

2An n-firm Cournot oligopoly modelThis section demonstrates that ambient charges can reduce NPS pollution in
an n-firm Cournot oligopoly market. It shows that this reduction occurs not
only in a Cournot duopoly case, but also in general Cournot oligopoly cases.
These cases have not previously been explored in this context, so we believe
this research provides further insight on the effectiveness of ambient charges for
reducing NPS pollution. First, we set the market demand function for firm i (i= 1,2 . . . , n):p=a−bnXqi ,f or

i = 1, 2, . . . , n.

(1)

i=1

Assume that there are n firms in the same industry that produce a homogeneousproduct. In an n-firm Cournot oligopoly market, p stands for a product price,which is assumed to be the same in all firms in the same industry. a stands
for the choke-off price, which indicates that the market price is non-zero and apositive constant. b is the slope of the inverse demand curve of Equation (1)3and a positive constant. qi is the quantity of firm i ’s production.

Pni=1 qi shows

the total quantity of n-firm production. It is also assumed that n firms havethe same production technology. Next, we set the profit function to obtain theoptimum quantity of each firm’s production:nX
πi = pqi − cqi − m(ei qi − Ē)
i=1

=a−bn
X!
qi − c − mei qi − m

i=1nX!
ej qj − Ē .

j=1,j6=i(2)
The marginal cost of a firm i is a positive constant c. ei represents the ratioof each firm’s emissions. Firm i emits pollutants ei qi in connection with theseproductions. The government can measure an industry’s total emission quantity
asPn

i=1 ei qi . The environmental standard is Ē, and it is provided exogenously.

Segerson (1988) shows that ifPn

i=1 ei qi ≥Ē, then the government will levy all

firms the same penalties, amounting to m times the difference between thetotal emission quantity and the environmental standard. This difference allows
us to provide the form of automatic, required payments in an n-firm Cournot
oligopoly model. Differentiating Equation (2) for qi , we obtain the best response
function for an n-firm Cournot oligopoly case:nX∂πi= a − 2bqi − bqk − c − mei = 0.
∂qi(3)

k=1,k6=i

In Equation (3), n firms maximize their profits. The second-order condition ofEquation (2) for qi is:∂ 2 πi= −2b < 0.
∂qi24(4)Equation (4) shows the value at which profits are at their maximum. Next,
we show the existence of n-firm Cournot equilibrium for Equation (3), which
consists of simultaneous equations for n firms. A series of these simultaneousequations is summarized in vector form A, q and B, which are:

Aq = B.

(5)

whereA = (Aij )(n,n) ,q = (qi )(n,1) ,with

Aii = 2band

Aij = bf ori 6= j,

B = (a − c − mei )(n,1) .

Now, we calculate the existence of n-firm Cournot equilibrium for Equation (5).
Here, we can consider |A| 6= 0 2 :

A−1 = (xij )(n,n) .

(6)

Considering the identity matrix I for A−1 A, we obtain the following vectorform:

q = A−1 B.

Calculating Equation (7) for the existence of A−1 3 , we can write as:
2 See Appendix A.1
3 See Appendix A.2

5(7)





 q1


 q
 2

 .
 ..


 q
 i

 .
 ..


qn
 n




 −1




 .

 ..

1
=

 (n + 1)b  .

 ..




 .

 ..





−1





. . . −1   a − c − me1. . . . . . −1   a − c − me2


..
..
..
.. .
.
.
. ..
..  n.
.   a − c − mei..
.. . .
.. .
.
.
. 
a − c − men... ... n.








−1

−1

n−1

..
.

..

..
.

..
.

..
.

..
.

−1

−1...

.

(8)
Arranging Equation (8), we obtain the optimum quantity of production for nfirms:qi∗ =a − c − nmei +Pn

j=1,j6=i mej(n + 1)b.

(9)

Moreover, we define the industrial emission function E (m) in an n-firm market,
which is:E(m) =nXei qi∗ .

(10)

i=1

Substituting Equation (9) into Equation (10), we obtain4 5"n−1
#nnn
XXXX
122E(m) =(a − c)ei − m(ei − ej ) +ei(n + 1)bi=1
i=1 j=2
i=1

f ori, j ∈ N,(11)Differentiating Equation (11) for E (m), we obtain:Pm
4 We define
i=m ai ≡ am .
5 See Appendix A.3 for the calculations for Equation (9) and Equation (10).

6i 6= j.

E 0 (m) = −n−1nnXXX
1(ei − ej )2 +e2i
f or(n + 1)b i=1 j=2i=1i, j ∈ N,i 6= j.
(12)Equation (12) indicates the derivative of NPS pollution in an n-firm Cournot
oligopoly case. n is a positive constant, so E 0 (m) is negative. Hence, we obtain:
Theorem 1.
E 0 (m) < 0.

This theorem shows that when we impose ambient charges on an n-firm Cournot
oligopoly market, the total amount of NPS pollution is abated. This explanationsuggests that ambient charges are effective in reducing the quantity of NPS
pollution in an n-firm Cournot oligopoly case in the same way as in a two-firm
Cournot duopoly case.

3Conclusion

This paper examines the effectiveness of ambient charges decreasing NPS pollution in a Cournot oligopoly market. We also show the abatement of ambient
charges in a two-firm Cournot duopoly case. In this case, the total emissionfunction is:E(m) =2X
i=1ei qi∗ =(a − c)(e1 + e2 ) − m((e1 − e2 )2 + (e1 )2 + (e2 )2 ).
3b7(13)

P2∗∗∗i=1 ei qi (=e1 q1 + e2 q2 ) can be denoted as a function E(m), which indicates

the total amount of NPS pollution in a two-firm Cournot duopoly case. By
differentiating it, the decrease in pollution is expressed as E ’(m), which is:

E 0 (m) = −(e1 − e2 )2 + (e1 )2 + (e2 )2< 0.
3b(14)

Equation (14) is the same as Sato(2017)’s formula of two-firm Cournot duopoly
model. This conclusion also identifies the outcome of an two-firm Cournot
duopoly case at which ambient charges can abate NPS pollution. Hence, weshow that this paper generalizes Sato (2017).
This paper is only a static model, and does not consider dynamic or incomplete competitive models. In order to extend this research, we must examine
dynamic models to more precisely outline the point at which the NPS pollution
can be reduced.

Appendix
A.1

Calculation for the nonsingularity of |A|

The determinant |A| of A in an n-firm Cournot oligopoly model can be defined
as:

|A|=

2bb...

bb2b...

b..
.

..
.

..

.

bbb...

2b8(A.1)

We consider two cases for (A.1). When n=2, (A.1) can transform

2bbb2b|A|== 2b × 2b − b × b = 2 × 2b2 − (2 − 1)b2 = 3b2 > 0.

(A.2)

When n ≥ 3, we consider a cofactor expansion for a determinant of vector A.
Considering this expansion for a row i in the determinant of vector A, we obtain:|A|= ai1 |Ci1 |+nX

aij |Cij |= 2 × nbn − (n − 1)bn = (n + 1)bn > 0.

(A.3)

j=2

Hence, (A.2) and (A.3) have the nonsingularity of |A|.
Q.E.D.

A.2

Calculation for the inverse matrix of A

Here, we calculate A−1 , which is the inverse matrix of A. In order to show A−1 ,we use Equations (5) and (6). We pick up an i ’s row and all of the columns inEquation (5), and calculate simultaneous equations:92bxi1 + bxi2 + . . . + bxin = 0








bxi1 + 2bxi2 + . . . + bxin = 0

.

 ..

 bxi1 + bxi2 + 2bxii−1 + bxii + . . . + bxin = 0(B.1 a)(B.1 b)(B.1 c)bx + bxi2 + bxii−1 + 2bxii + . . . + bxin = 1

 i1







 bxi1 + bxi2 + bxii−1 + bxii + 2bxii+1 + . . . + bxin = 0



.



 ..







bxi1 + bxi2 + . . . + 2bxin = 0.

(B.1 d)(B.1 e)(B.1 f)

Subtracting Equation (B.1 a) from Equation (B.1 b),

bxi1 − bxi2 = 0.

wherexi1 = xi2 .

(B.2)

Likewise, subtracting another upper equation from another lower one in theprevious simultaneous equations, we obtain:

xia = x,f ori, a ∈ N,a = 1, 2, . . . , n,i 6= a.

(B.3)

Substituting Equation (B.3) into Equations (B.1 a), (B.1 b), (B.1 c), (B.1 d),(B.1 e) and (B.1 f), we obtain a pair of simultaneous equations:10 nbx + bxii = 0(n − 1)bx + 2bxii = 1.

(B.4 a)(B.4 b)from which,

1x = − + xii .
b(B.5)

Substituting Equation (B.5) into Equation (B.4 a), we obtain:

xii =n.
(n + 1)b(B.6)

x=−1.
(n + 1)b(B.7)

Hence, we obtain:

Q.E.D.

A.3

Arrangement for Equation (11)Substituting Equation (9) into Equation (10), we obtain:"
1
E(m) =(a − c)(e1 + e2 + · · · + en ) − m e1 (ne1 − e2 − · · · − en )(n + 1)b
#+ e2 (−e1 + ne2 − · · · − en ) + · · · + en (−e1 − e2 − · · · + nen ) .

Arranging this equation, we obtain:

11"# Xnnn−1nnXX
XX122E(m) =(a − c)ei − m (i − 1)ei − 2ei ej +ei .
(n + 1)bi=1
i=1
i=1 j=2i=1(C.1)

Transforming (C.1), we obtain Equation (11).
Q.E.D.

References[1] Guerrini, L., Matsumoto, A. and Szidarovszky, F., Delay Cournot Duopoly
Models Revisted, Institute of Economic Research Chuo university, 298, 2018.
[2] Poe, G.L., Schulze, W.D., Segerson, K., Suter, J.F., and Vossler, C.A., Exploring the Performance of Ambient-Based Policy Instruments When Nonpoint Source Polluters Can Cooperate, American Journal of Agricultural
Economics, 86(5), 1203-1210, 2004.
[3] Raju, S. and S. Ganguli, Strategic firm interaction, returns to scale, environmental regulation and ambient charges in a Cournot duopoly, Technologyand Investment, 4, 113-122, 2013.
[4] Sato, H., Pollution from Cournot duopoly industry and the effect of ambientcharges, Journal of Environmental Economics and Policy, 6(3), 305-308,
2017.
[5] Segerson, K., Uncertainty and incentives for nonpoint pollution control,
Journal of Environmental Economics and Management, 15, 87-98, 1988.

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[6] Suter, J.F., Vossler, C.A., Poe, G.L., and Segerson, K., Experiments on
Damage-Based Ambient Taxes for Nonpoint Source Polluters, American
Journal of Agricultural Economics, 90(1), 86-102, 2008.

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