Discussion Paper No.275
Extended Oligopolies with Pollution Penalties
and Rewards
Akio Matsumoto
Chuo University

Ferenc Szidarovszky
University of Pécs

Hirokazu Takizawa
Chuo University
March 2017

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Extended Oligopolies with Pollution Penalties
and Rewards
Akio Matsumotoy, Ferenc Szidarovszkyzand Hirokazu Takizawax

Abstract
An extended n-…rm oligopoly with product di¤erentiation is considered. It is assumed that the government selects an emission standard forthe industry and based on the output and technology of each …rm it selects a maximum allowed amount of emission for each …rm. If the actual
amount is higher than the allowed maximum, then the …rm has to pay a
constant multiple of the excess to the government, otherwise it is rewarded
similarly based on the saved emission amount. The existence of the uniqueinterior equilibrium is …rst proved, and then time delay is introduced into
the penalties the …rms have to pay and into the rewards the …rms receive.
In analyzing the stability of the equilibrium both continuous and discrete
time scales are considered. For mathematical simplicity the case of symmetric …rms is analyzed. In the continuous case the equilibrium is either
always stable or stable if the delay is su¢ ciently small and at the criticalvalue Hopf bifurcation occurs. In the discrete case the delay is assumed
to be unity. The equilibrium is stable if either the total industry output
is su¢ ciently large or the common speed of adjustment of the …rms issu¢ ciently small. The e¤ect of the level of penalty or reward and that of
the emission standard on the industry output and therefore on the total
emission level is also examined.
Keywords. Pollution penalties, Nonpoint source pollution, Policy
lag, Cournot competition

The authors highly appreciate the …nancial supports from the MEXT-Supported Program
for the Strategic Research Foundation at Private Universities 2013-2017, the Japan Society
for the Promotion of Science (Grant-in-Aid for Scienti…c Research (C), 25380238, 26380316,
16K03556) and Chuo University (Joint Research Grant). The usual disclaimer applies.
y Professor, Department of Economics, Senior Researcher, International Center for further
Development of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, akiom@tamacc.chuo-u.ac.jp
z Professor, Department of Applied Mathematics, University of Pécs, Ifjúság u. 6., H-7624,
Pécs, Hungary, szidarka@gmail.com
x Professor, Department of Economics, Senior Researcher, International Center for further
Development of Dynamic Economic Research, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, taki@tamacc.chuo-u.ac.jp1

1

Introduction

The e¤ect of di¤erent environmental regulation policies has been investigated bymany researchers (Downing and White, 1986; Jung et al., 1996; Montero, 2002
among others). Mostly single …rms were considered in relation to environmental
R&D, and very few works were devoted to the extension of oligopoly models inthis direction. Montero (2002) examined the e¤ect of R&D investment for pollution abatement technology with di¤erent environmental policies in duopolistic
product markets. Static models were investigated in earlier stages, the existenceof equilibrium in Cournot oligopoly with pollution treatment cost sharing wasproved by Okuguchi and Szidarovszky (2002), which work was further extended
by including emission standard and R&D in the oligopoly model (Okuguchi
and Szidarovszky, 2007). If the government is unable to assess the individual
emission levels of the di¤erent …rms, then it can measure only the total level ofpollution, and ambient charges are introduces. In this policy (Segerson, 1988)
the government de…nes a cut-o¤ for the total pollution level and regardless ofthe speci…c emission level of each …rm, all are equally punished or rewarded.
Ganguli and Raju (2012) demonstrated that in a Bertrand duopoly increasing ambient charges could lead to greater pollution, however Raju and Ganguli(2013) showed the opposite e¤ect in a Cournot duopoly framework when theincrease of ambient charges reduce pollution. This result was further generalized by Matsumoto et al. (2017) for n-…rm oligopolies where the stability of the
dynamic model with naive expectation was also examined. If the governmentis familiar with the used technology and production output of each …rm, thenit is able to assess the proportion of each …rm from the total pollution level.
Therefore each …rm can be punished or rewarded according to its assessed individual emission level compared to its allowed proportional maximum from thegovernment de…ned cut-o¤ threshold. In this paper this idea will be elaborated.
After the formulation of the mathematical model the existence of the static
equilibrium will be proved. Assuming gradient adjustment of the …rms, dynamic models will be developed with both continuous and discrete time scales,
and conditions will be derived for the stability of the equilibrium. The stability
conditions will be then analyzed and compared. And …nally the e¤ect of penalty
and reward parameters on the industry output and the total pollution level will
be investigated.

2

Model and Cournot-Nash Equilibrium

Consider n …rms in an oligopoly with di¤erentiated products. Let qk be output
of …rm k: The price of the product of …rm k is seen as
pk =

k

qk

k

n
X

qi

(1)

i6=k

with n2 and k > 0 and 0 < k < 1: Firm k emits pollution ek qk in
connection to its production with ek > 0. The government can measure the2total emission quantity and has an exogenously selected environmental standard
E. So the maximum allowed emission of …rm k is clearlye q
Pn k k E:
i=1 ei qi(2)

If a …rm exceeds this amount then it has to pay a penalty of m times theexceeded amount, and if its emission amount is below the maximum allowed
amount, then the …rm is rewarded by m times the saved emission amount. So
the payo¤ of …rm k becomesk = (pk

Kk ) q k

m ek qke q
Pn k k E
i=1 ei qi(3)

where Kk is the marginal cost of …rm k. Using equation (1),
1
0
n
X
mek
@ k qkq i Kk A qk
mek PnE qk :k =ki=1 ei qi

(4)

i6=k

The …rst term corresponds to revenue and production cost and the second term
refers to emission penalty or reward. Assuming interior optimum, the …rst ordercondition implies that
1
0
!
Pn
n
X
mek i6=k ei qi@ kq i Kk A
mek= @ k 2qk
= 0: (5)
Pn
k
2 E
@qk
( i=1 ei qi )i6=kNotice that (5) strictly decreases in qk with …xed values of qi (i 6= k): At qk = 0;
its value is
n
X
mek
E:
qi Kk mek + Pnkk
i6=k ei qi
i6=kIf this value is nonpositive, then qk = 0 is optimum, which is not interior. As
qk ! 1, the value of (5) tends to 1, so there is always a unique best response.
For mathematical simplicity, let us assume symmetric …rms in the sensethat 1 = 2 = ::: = n = and e1 = e2 = ::: = en = e, that is, …rms haveidentical substitutability and technology in emission production. Then equation(5) becomes
k

2qk

n
X

qi + q k

Kk

me +

i=1

and with notation Q =
k

Q

Pn

i=1 qi , we have

Kk

me +

Pn
m ( i=1 qi qk )E=0
Pn2
( i=1 qi )mE
= qk 2
Q

3

+

mE
Q2

:

So
qk =

Kk me) Q2
Q3 + mEQ:
2(2) Q + mE( k

(6)

By adding these equations for k = 1; 2; :::; n and dividing by Q;
!
n
n
X
X
(2
) Q2 + mE =Kk nme Q n Q2 + nmE
k
k=1

k=1

which is a quadratic equation for Q;
[(2

n
X

2

) + n ]Q

n
X

k

k=1

Kk

!

nme Q + (1k=1n)mE = 0:(7)

At Q = 0; the left hand side is negative and as Q ! 1; it converges to +1, so
there is real root. Since the constant term is negative, one root is positive andthe other is negative. So only the positive root has economic meaning. Thenthe corresponding equilibrium levels of the …rms are given by (6).

3

Dynamic Extensions and Stability Analysis

Assume the government has a time delay > 0 in posing penalty or giving
reward to the …rms, and the …rms use gradient adjustments in continuous timescales. Then the resulting dynamic system becomes
qk (t
)2
(8)
It is a nonlinear delay di¤erential equation. Let gk denote the right hand side,then@gk= Sk (2) < 0;@qk (t)q_k (t) = Sk ( k

2qk (t)Q(t) + qk (t)@gk@qk (t

)

@gk
@Q(t

)

=

mE (Q(t
)
Q(t

me

Sk mE
< 0;
Q(t
)2

=

@gk
=
@Q(t)
and

Kk )

Sk < 0Sk mE [2qk (t
Q(t

)

Q(t
)3

)]

< 0:

if there is no dominant term. Let these derivatives be denoted by Ak ; Bk ; Ck
and Dk ; respectively, then the linearized equation has the form, where qk and
Qk are now their distances from equilibrium levels:
q_k (t) = Ak qk (t) + Bk qk (t) + Ck Q(t) + Dk Q(t

4

):

(9)

))

:

By looking for the solution as usual,qk (t) = e t uk ;we have
uk

Ak

n
X

Bk e

ui Ck + Dk e

= 0:

(10)

i=1

For the sake of simplicity, we assume in addition that S1 = S2 = ::: = Sn = S;
the equilibrium is symmetric and the initial output levels are identical. Thenthe coe¢ cients Ak ; Bk ; Ck and Dk are also identical. Then the uk coe¢ cients
must be also the same, therefore we have a delay equation:
A

nC

Be

nDe

=0

(B + nD)e= 0:

or(A + nC)(11)

At the equilibrium
A + nC =

S [(n

and
B + nD =As1) + 2] < 0

SmE(1
Q2

n)

< 0:

= 0; when the system is without delay,= A + nC + B + nD < 0;

implying that the system is asymptotically stable. Stability switch occurs if= i! with some ! > 0. With the notation
U = A + nC < 0 and V = B + nD < 0;
we havei!

U

V (cos !

i sin ! ) = 0:

Separating the real and imaginary parts,
U

V cos ! = 0(12)! + V sin ! = 0:

(13)

and
So
!2 = V 2
In our caseVU=

U 2 = (V

SmE(1
Q2

n)

5

U )(V + U ):+ S [(n1) + 2](14)

The …rst term is negative and the second positive. Since V + U < 0, there is no
solution for ! if (14) is nonnegative, and therefore there is no stability switch.
If (14) is negative, then there is a unique value of !;p! = V 2 U 2:

Hopf bifurcation is used to …nd the direction of the stability switch. Let be thebifurcation parameter and consider as function of : = ( ): By implicitlydi¤erentiating equation (11) with respect to ; we have
d
d

d
d

Ve

=0

showing that
d
d

Ve
1+Ve

=

(

=

d
d

=

=

U)
;
U)

1+ (

where equation (11) is used. When

;

= i!;

! 2 + i!U;1
U + i!
(! 2 + i!U ) (1

U

2

(1

U) + (

i! )!)2with real part having the same sign as! 2 (1

U ) + !U ( !) = ! 2 > 0

showing that the sign of the real part changes from negative to positive, so
stability is lost. That is, at the smallest stability switching point, stability
is lost and it cannot be regained later. If there is positive solution for !, thenjV j > jU j ; and since from (12) and (13) we know that sin ! > 0 and cos ! < 0,
the smallest (critical) value of is the following:
=p

1
V2

U2

cos 1

U
V

:

At the critical value of there is the possibility of the birth of limit cycles. From(14), we know that the system is always asymptotically stable if
mE(11) + 2] Q2n) + [(n0:(15)

From (7), the left hand side equals
n
X

k=1

k

n
X

Kk

k=1

6

!

nme Q

(16)

so the system is stable if multiplier of Q is nonnegative. Notice that the multiplier of Q is the total marginal pro…t of the …rms at zero environmental standard
when all …rms have zero production levels.
If discrete time scales are assumed and the government delay is unity, thenthe continuous time scales di¤erential equation (8) is modi…ed to a second orderdi¤erence equation,qk (t+1) = qk (t)+Sk ( k

2qk (t)Q(t) + qk (t)

Kk )

me

mE [Q(t 1) qk (tQ(t 1)2(17)

and after linearization it becomesq(t + 1) = q(t) + [Aq(t) + Bq(t1) + nCq(t) + nDq(t

1)]

(18)

where we again assume symmetric …rms. By introducing the new variables
a(t) = q(t1) and b(t) = q(t);we havea(t + 1)= b(t)b(t + 1)

= (B + nD)a(t) + (A + nC + 1)b(t)(19)
with coe¢ cient matrix

0

0

@

1

1

V

A

U +1

leading to a quadratic characteristic polynomial(U + 1)2

V =(U + 1)

V:(20)

Asymptotic stability is guaranteed by
U +1
U 1

V + 1 > 0;
V + 1 > 0;
V < 1:The second condition is clearly satis…ed. The …rst inequality can be rewritten
as
!
n
n
X
S X
Kk nme < 2
(21)
V U=
k
Q
k=1

k=1

and the third condition has the form
V =

SmE(n
Q2

1)

< 1:

(22)

In comparing the stability conditions in the continuous and discrete case we
notice the following.
7

1)]

Pn
Pn
(a) If k=1 k
nme < 0, then (21) holds and (22) is satis…ed if
k=1 KkQ is su¢ ciently large or the speed of adjustment S is su¢ ciently small.
The continuous system loses stability if is su¢ ciently large.
PnPn(b) If k=1 k
nme 0, then the continuous system is stable.
k=1 KkThe discrete system is stable if Q is su¢ ciently large or the value of S issu¢ ciently small.
An equivalent condition can be given by rewriting (21) and (22) as
!
n
n
X
S X
Q>
Kk nme
k
2
k=1

and

k=1

q
Q > SmE(nLetting"S
Q = max
2

n
X

k

k=1

n
X

1):
! q

Kk

nme ;

k=1

SmE(n

#

1) ;

(23)

the stability condition becomesP (Q ) < 0(24)

where P (Q) denotes the left hand side of equation (7).

4E¤ect of Penalty or Reward on Pollution Levels

We can analyze how the total production level (and therefore the total emission
level) depends on the penalty factor m. Considering Q as a function of m, andimplicitly di¤erentiating equation (7) with respect to m; we have
!
n
n
X
X
dQ
dQ
+ neQ
Kk nme+ (1 n)E = 0[(2) + n ] 2Q
k
dm
dm
k=1

k=1

implying thatdQ=dm
2Q [(2

neQ + (n 1)EPn
Pn) + n ] ( k=1 k
k=1 KkBased on equation (7), the denominator can be rewritten as
Q [(2)+n ]+

8

(n

1)mE
Q

nme)

:

(25)

which is positive. So the sign of dQ=dm depends on the sign of the numerator.
The …rst term is negative, the second term is positive. Using again equation(7), we see that the m-multiple of the numerator equals
!
n
n
X
X
Kk Q + [(2) + n ] Q2 :
k
k=1

k=1

It is reasonable to assume that the …rst term is negative. Therefore dQ=dm > 0
if and only ifPnPnk
k=1 Kk:(26)
Q > k=1(2)+n
Let Q denote the right hand side of this inequality and P (Q) again the left handside of equation (7), then this is the case when P (Q) < 0 meaning that
Pn
Pn
[(2
) + n ] Q2 ( k=1 knme) Q
k=1 KkE>:(27)(n 1)m
It is easy to see that the numerator is positive. This inequality means that
increase in the value of m has an increasing e¤ect on the total industry output
as well as in the total emission level if the emission standard E is su¢ ciently
large. Otherwise the opposite e¤ect can be observed.
Next we examine the e¤ect of increasing in the value of E. Considering now
Q as function of E and implicitly di¤erentiating equation (7) we have
!
n
n
X
X
dQ
dQ
Kk nme[(2) + n ] 2Q
+ (1 n)m = 0
k
dE
dE
k=1
k=1
showing thatdQ=
2Q [(2dE)+n ](n 1)mPn( k=1 kPn

k=1 Kk

nme)

:

(28)

We already established that the denominator is positive, so dQ=dE > 0 showingthat the increase in the emission standard always has an increasing e¤ect on theindustry output as well as on the total emission level.

5

Concluding Remarks

This paper examined n-…rm oligopolies with product di¤erentiation when the…rms face penalties or rewards depending on the amounts of their pollution
levels. The government selects an emission standard for the entire industry,and based on the speci…c technology and output of each …rm the government
determines its maximum allowed emission level. The amount of penalty or thatof the reward is determined by the di¤erence of the actual emission level and the
maximum allowed amount. This simple mathematical model can be considered9as the counterpart of models with ambient pollution charges discussed earlier
in the literature. The existence of the unique interior equilibrium was …rst
proved, and then dynamic extensions were introduced and the stability of theequilibrium was examined with both discrete and continuous time scales. In
the continuous time scales a positive delay was introduced in the penalty andreward terms and in the case of discrete time scales unit delay was assumed. In
the case of continuous time scales the equilibrium is always stable if the total
marginal pro…t of the …rms at zero output levels is non-negative, otherwise it isstable if the length of the delay is su¢ ciently small. At the critical value of thedelay Hopf bifurcation occurs. In the case of discrete time scales the equilibrium
is stable if either the industry output is su¢ ciently large or the common speedof adjustment of the …rms is su¢ ciently small. We also established that anincrease in the value of m has an increasing e¤ect on the total pollution level
if the emission standard E is su¢ ciently large, otherwise the opposite e¤ectoccurs. An increase in the value of E always has an increasing e¤ect on the
total pollution level of the industry.
Symmetric …rms were assumed for mathematical simplicity, in which casethe equilibrium could be given analytically, however in the non-symmetric case
it is not possible making the further analysis much more complicated. This issue
will be the subject of our next project.

10References
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Ganguli, S. and S. Raju, Perverse environmental e¤ects of ambient charges ina Bertrand duopoly, Journal of Environmental Economics and Policy, 1,
1-8, 2012.
Jung, C., K. Krutilla and R. Boyd, Incentives for advanced pollution abatement
technology at the industry level: An evaluation of policy alternatives,
Journal of Environmental Economics and Management, 30, 95-111, 1996.
Matsumoto, A., F. Szidarovszky and M. Yabuta, Environmental e¤ects of ambient charge in a Cournot oligopoly, mimeo, 2017.
Montero, J. P., Permits, standards, and technology innovation, Journal ofEnvironmental Economics and Management, 44, 23-44, 2002.
Okuguchi, K. and F. Szidarovszky, Existence of Nash-Cournot equilibrium inoligopoly games with pollution treatment cost-sharing, Pure Mathematics
and Applications, 13, 455-462, 2002.
Okuguchi, K. and F. Szidarovszky, Environmental R&D in Cournot oligopoly
with emission or performance standards, Pure Mathematics and Applications, 18, 111-118, 2007.
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.
Segerson, K., Uncertainty and incentives for non-point pollution control, Journal of Environmental Economics and Management, 15, 87-98, 1988.

11