Can Allowing to Trade Permits Enhance Welfarein Mixed Oligopoly?

Kazuhiko Kato

Received August 3, 2005; revised version received April 4, 2006Published online: July 31, 2006

� Springer-Verlag 2006

We compare the effects of tradable emission permits (TEP) and non-tradableemission permits (NTEP) in a mixed oligopoly, where public firms and privatefirms compete in a product market. If all technologies and initial endowmentsof emission permits are symmetric among public and private firms and if theemission constraint is exogenous and binding, social welfare is greater (resp.smaller) under TEP than under NTEP when the weight of social welfare ineach public firm’s objective function and the degree of convexity of theproduction cost function and that of the abatement cost function are small(resp. large).

Keywords: oligopoly, mixed market, tradable emission permits.

JEL Classification: L33, Q58.

1 Introduction

In 1997, COP3 adopted the Kyoto Protocol which contained tradableemission permits (TEP) as a method of controlling global warming andwent into force on the 16th of February in 2005.1 Prior to this, the United

1 The 3rd Session of the Conference of the Parties to the United NationsFramework Convention on Climate Change.

Vol. 88 (2006), No. 3, pp. 263–283DOI 10.1007/s00712-006-0206-6 Journal of Economics

Printed in Austria

States brought TEP into operation (Hahn, 1989).2 TEP has been imple-mented experimentally in Britain since 2002 and the European Unionintroduced it on January 1, 2005. Many countries may follow suit toadopt TEP as one of the main environmental regulations in the future.This paper examines the effects of TEP in a mixed market, where publicfirms and private firms compete.3

Examples of mixed markets are numerous in those countries that haveratified the Kyoto Protocol. Norway’s Statoil run by a public owner, andFrance’s Renault jointly run by private owners and a public owner,compete with private firms in the same market. Furthermore, althoughsome countries have not participated in the Kyoto Protocol yet, there is apossibility that they may use TEP for the self-restriction on their emis-sions or participate in the Kyoto Protocol. In China, for instance, where alarge amount of greenhouse gases is discharged, mixed markets are fairlycommon, such as in energy industries and energy-intensive industries. IfChina takes its environmental problems in the future more seriously andviews TEP as a successful international experiment, it may adopt TEP orparticipate in the Kyoto Protocol. These cases illustrate why it isworthwhile to examine the effects of TEP in a mixed market.

There are two representative cases showing how economists model theobjectives of public firms. The first case is that a public firm’s objective ispure social welfare maximization (De Fraja and Delbono, 1989; in recentyears, White, 1996, Fjell and Pal, 1996, Mujumdar and Pal, 1998, Pal,1998). The second case is an extension of the first: the public firm is onlya partial social welfare-maximizer (Bos, 1991; in recent years, Matsum-ura, 1998; Barcena-Ruiz and Garzon, 2003; Matsumura and Kanda,2005). In this paper, we use the framework of the second. This frameworkis more general than the first since the second includes the first as aspecial case.

This paper employs a model composed of �N firms: n0 public firms andn1 private firms. Each public firm maximizes the weighted sum of socialwelfare and its own profit and each private firm maximizes its ownprofit. They produce output and discharge emissions. The government hasa responsibility to regulate emissions, whose level is binding and

2 This case dealt not with the global warming problem but with other detri-mental externality problems, such as water pollution and air pollution.3 A public firm is a firm that is either wholly run by the public owner, or jointly

owned by private and public owners.

264 K. Kato

exogenously given. In order to control emissions, the government canonly choose whether to allow the firms to trade emission permits or not.

We suppose that all technologies and initial endowments of emissionpermits are symmetric among public firms and private firms. If theemission constraint is binding, (i) under TEP, public (resp. private) firmsare buyers (resp. sellers) of emission permits, (ii) when the degree ofconvexity of each firm’s production cost function and that of each firm’sabatement cost function are small and the weight of social welfare in eachpublic firm’s objective function is small (resp. large), social welfare isgreater (resp. smaller) under TEP than under non-tradable emission per-mits (NTEP), which does not allow firms to trade emission permits, (iii)the larger the degree of convexity of each firm’s production cost functionor that of each firm’s abatement cost function is, the narrower the range ofparameters in which TEP is superior to NTEP in terms of social welfare.If the degree of convexity of either the production cost or abatement costfunctions is sufficiently large, TEP is inferior to NTEP in terms of socialwelfare regardless of how social welfare is weighted in public firms’objective functions.

The intuition behind result (i) is as follows. When each public firm is apure profit-maximizer, the social welfare level under TEP is the same asthat under NTEP. When each public firm takes social welfare into accountin addition to its own profit, TEP no longer entails the same welfare levelas NTEP. Each public firm produces more output than each private firmunder both regulations, because the output is underproduced in the oli-gopoly market. However, the more output each public firm produces, themore emissions it discharges. Each public firm has the responsibility ofmaking more abatement effort under TEP than under NTEP. Both themarginal abatement cost and the shadow price of the emission constraintof each public firm are larger than those of each private firm under NTEP.Therefore, under TEP, public firms buy the emission permits from privatefirms.

The intuition behind the results (ii) and (iii) is as follows. Comparedwith NTEP, TEP has two positive effects and one negative effect on socialwelfare. One positive effect is that the total abatement cost is sociallyminimized at given emission levels under TEP. The other positive effect isthat the total output is larger under TEP than under NTEP. Under NTEPthe abatement cost of each public firm is larger than that of each privatefirm. Under TEP, however, they are equalized. Thereby each public firmhas an incentive to produce more under TEP than under NTEP. The

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 265

negative effect is that TEP causes the inefficient reallocation of produc-tion among public firms and private firms; under NTEP each public firmproduces more than each private firm, and under TEP the difference in theoutput among them is enlarged. Thus, whether TEP is superior to NTEPor not in terms of social welfare depends on the relative size of thepositive and negative effects. To examine the superiority of TEP or NTEP,we investigate the relationship among the demand and cost structures.

Our investigation is based on some earlier works on TEP and mixedmarkets. Studies on the effects of TEP in markets that consist only ofprivate firms include Malueg (1990) and Sartzetakis (1997; 2004).4 Theycompare TEP with NTEP and show that the superiority of TEP or NTEPin terms of social welfare is determined by the difference in the tech-nologies or the initial endowments of emission permits between regulatedfirms.5 Without these differences, TEP is equivalent to NTEP in terms ofsocial welfare. In a mixed market, however, even without such differ-ences, the social welfare levels between these two regulations differ. Thisis caused by the difference in objectives between the public firm and theprivate firm. Usually, TEP is considered to work when there are differ-ences with respect to the abatement technologies among firms. Even ifthose differences do not exist, however, firms have an incentive to tradeemission permits and such trading brings benefits to this industry whenthe cost function is nonlinear. When the output levels of public andprivate firms are different, their emissions are also different. Therefore,their marginal abatement costs are different. This asymmetry is caused bythe difference in their objectives even if the technologies are the same.Focusing on the efficiency of the abatement cost at given emission level,we find that TEP creates benefits because it causes the equalization of themarginal abatement costs among firms.

This paper is organized as follows. The next section describes our model.Section 3 derives the characteristics of the equilibrium under NTEP andSect. 4 derives those under TEP. Section 5 compares NTEP and TEP andhighlights the main results of this paper. Section 6 concludes the main text.

4 With respect to the case where firms have a market power in the emissionpermits market, see Hahn (1984). As for the case where firms have a power inboth the product market and emission permits market, see von der Fehr (1993).5 Hung and Sartzetakis (1998) show that TEP is inferior to NTEP in terms of

social welfare if the government can learn the technologies of firms and decidethe initial endowments of emission permits.

266 K. Kato

2 The Model

Consider a representative industry consisting of �N firms, n0 public firmsand n1 private firms, all of which have the same technologies and thesame initial endowments of emission permits, but different objectivefunctions. We assume that �N ¼ n0 þ n1, n0; n1 � 1. They produce ahomogeneous good and simultaneously compete with quantity in theproduct market. The inverse demand function is given by p ¼ A� Q,where p is the price of the good, Q is the total output and

Q ¼Pn0

i¼1 q0i þPn1

j¼1 q1j, where q0i denotes the output of public firm 0i(i ¼ 1; . . . ; n0) and q1j denotes that of private firm 1j (j ¼ 1; . . . ; n1). Weassume that A is positive and sufficiently large. The production costfunction is given by cq2

l=2 l ¼ 01; . . . ; 0n0; 11; . . . ; 1n1ð Þ, where c > 0.Emissions are discharged through production. Each firm can reduceemissions either by increasing abatement efforts or by reducing output.An emission function of firm l is Eðql; alÞ ¼ qql � al, where q > 0 andal is the abatement effort of firm l. The abatement cost function is givenby ka2l=2, where k > 0.6 We assume that this industry is given the totalamount of emission permits �E and each firm is equally endowed with�E= �N exogenously, and this is strictly smaller than the unregulatedlevel.7;8 The emission constraints are always binding for them, and thus,they have a responsibility to abate emissions. The profit of firm l is givenby9

pl ¼ A� Qð Þql �c2

q2l �k2

a2l : ð1Þ

Social welfare is the sum of the consumer’s surplus, producer’s surplus,and environmental damage. It is given by

6 We assume that the emission function and the cost function are separablewith respect to the output and the abatement effort. As the example of CO2

emissions shows, a firm can reduce emissions, even more than it discharges by itsown production, by acquiring emission permits by planting trees.7 A possible reason for dividing �E equally for each firm is due to the same

technologies of the emission and abatement among firms.8 In the equilibrium under no regulation, the abatement effort of each firm

whose objective is to maximize its own profit is zero.9 Under TEP, the profit of firm l is pl plus the revenue or expenditure of trading

emission permits. To simplify the explanation of each firm’s objective function,we call pl the profit of firm l.

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 267

SW ¼ZQ

0

ðA� sÞds�X c

2q2l �

X k2

a2l � D �Eð Þ: ð2Þ

The damage function is DðEÞ with D0ðEÞ > 0. E represents theaggregate emission level of this economy. As mentioned before, theemission constraint �E is binding in this industry. The environmentaldamage is a fixed value, Dð �EÞ.

Next we define the objective function of each firm. The objectivefunction of public firm U0i and that of private firm U1j are given by

U0i ¼ hSW þ ð1� hÞp0i; h 2 ½0; 1�; ð3ÞU1j ¼ p1j: ð4Þ

When h ¼ 0, public firm 0i is a pure profit-maximizer, and it is a puresocial welfare-maximizer when h ¼ 1. h is understood as the shareholding of the public sector and 1� h is that of the private sector.10

The government must regulate emissions to protect the environmentbecause of international agreements or the results of consultation withvarious interest groups. There are two regulations that the government canchoose. One is non-tradable emission permits (NTEP), under which firmscannot trade emission permits. The other is tradable emission permits(TEP), under which firms can trade emission permits. The governmentimposes one of these regulations on firms at the outset. �E is the sameunder two regulations.

Under TEP we assume that all firms are price takers in the emissionpermits market. Because of this setting, some readers may think that it isnatural for the price of an emission permit, pe, to be exogenous since allplayers are price takers in the emission permits market. In this paper,however, we consider pe to be endogenous in order to clarify the effectsof the public firm’s objective on pe.11 Suppose that there are infinitely

10 For a rationalization of this objective function, see Bos (1991). h can beconsidered to be dependent on the share holdings of the government and thenumber of the executives who come from government agencies because theirmagnitude could influence the public firm’s objective.11 Suppose pe is exogenous. Then, the welfare comparison of TEP and NTEP

is complex because the total emission of the industry becomes endogenous underTEP. Therefore, we have to classify the cases by the magnitude of pe and theeffect of the environmental damage. We leave this case to future research.

268 K. Kato

many markets which are identical to this industry in an economy. In thiscase, each product market is a mixed oligopoly. On the other hand, in theemission permits market, there are many groupings which consist of n0public firms and n1 private firms from all industries. In other words, weconsider a representative market with n0 representative public firms andn1 representative private firms. In addition, we consider the followingsituation; the government decides the degree of privatization or nation-alization of public firms across nation by, for example, a 5-year plan. Ifwe regard five years as one period, our model is consistent. Thus, weassume that the shift of the proportion of public firms or private firmsoccurs not only in a single market but also in the rest of all industries.

In our model there is no market power in the emission permits market.If there were market power in an emission permits market in addition tomarket power in the product market, it would be difficult to identifywhich effects cause the difference in social welfare between TEP andNTEP and to what extent each effect causes on the difference.

The timing of decision makings of the government and firms is asfollows. First, the government chooses one of the environmental regu-lations, NTEP or TEP. Then, all firms simultaneously compete withquantity, deciding ql and al. In the following, we confine our analyses tosymmetric equilibria where all public firms choose the same output andabatement effort level, and all private firms do the same.

3 Non-tradable Emission Permits

In this section, we derive the equilibrium of NTEP. NTEP prohibits eachfirm from trading emission permits. Each firm can discharge emissions aslong as it obeys its own emission constraint. Maximization problems ofpublic firm 0i and private firm 1j are given by

maxq0i;a0i

Unt0i ¼max

q0i;a0ihSW þ ð1� hÞp0if g s.t.

�E�N¼ qq0i � a0i; ð5Þ

maxq1j;a1j

Unt1j ¼ max

q1j;a1jp1j� �

s.t.�E�N¼ qq1j � a1j: ð6Þ

Untl represents the objective function of firm l under NTEP. Let kl be the

shadow price of the emission constraint of firm l. Solving the maximi-zation problem of each firm, we derive the following equilibrium out-comes under NTEP.

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 269

qnt0i ¼

cþ k þ 1

XM ; qnt

1j ¼cþ k þ 1� h

XM ;

ant0i ¼

cþ k þ 1

XqM �

�E�N; ant

1j ¼cþ k þ 1� h

XqM �

�E�N;

knt0i ¼

cþ k þ 1

XkqM � k �E

�N; knt

1j ¼cþ k þ 1� h

XkqM � k �E

�N;

where k ¼ kq2, M ¼ Aþ kq �E= �N , X ¼ ðcþ k þ 1þ n1Þ ðcþ k þ 1þ�N � n1 � hÞ � ð �N � n1Þn1.Then, social welfare under NTEP is

SW nt ¼n1ðcþ kþ 2þ n1Þh2� 2ðcþ kþ 1Þf �Nðcþ kþ 1þ n1Þþ n1gh2X 2

M2

þ�Nðcþ kþ 1Þ2ðcþ kþ 2þ �NÞ

2X 2M2� k �E2

2 �N�Dð �EÞ: ð7Þ

From the results, we can easily check qnt0i > qnt

1j and ant0i > ant

1j and

qq0i � ant0i ¼ qq1j � ant

1j ¼ �E= �N . The equilibrium output of public firm 0iis larger than that of private firm 1j because public firm 0i’s objectiveincludes consumer’s surplus partially. To satisfy the emission constraint foreach firm, the equilibrium abatement effort of public firm 0i is larger thanthat of private firm 1j. From the above, we find that the shadow price of theemission constraint of public firm 0i is larger than that of private firm 1j.

4 Tradable Emission Permits

Public firm 0i’s and private firm 1j’s maximization problems are given by

maxq0i;a0i

U t0i ¼ max

q0i;a0ihSW þ ð1� hÞp0i þ pe

�E�N� qq0i þ a0i

� �� �

; ð8Þ

maxq1j;a1j

U t1j ¼ max

q1j;a1jp1j þ pe

�E�N� qq1j þ a1j

� �� �

: ð9Þ

Utl represents the objective function of firm l under TEP. As they are price

takers in the emission permit market, they trade emission permits, takingthe permit price pe as given. The market clearing condition is

270 K. Kato

�E ¼X

qql � alð Þ: ð10Þ

We find that the equilibrium outcomes under TEP are

qt0i¼

�Nðcþ1ÞY

M ; qt1j¼

�Nðcþ1�hÞY

M ;

at0i¼at

1j¼�Nðcþ1Þ�n1h

YqM�

�E�N; pe¼

�Nðcþ1Þ�n1hY

kqM�k �E�N;

where

Y ¼ k �Nðcþ 1Þ � n1hf g þ �N cþ 1þ �Nð Þðcþ 1Þ � ðcþ 1þ n1Þhf g.Then, social welfare under TEP is

SW t¼fn1kþ�Nðcþ2þn1Þgn1h2�2ðcþ1Þf �Nðcþ1þn1Þþn1ðkþ1Þg �Nh

2Y 2

�NM2þðcþ1Þ2ðcþ kþ2þ �NÞ2Y 2

�N3M2�k �E2

2 �N�Dð�EÞ: ð11Þ

The equilibrium output of public firm 0i is larger than that of private firm 1jfor the same reasons as in Sect. 3.With respect to the equilibrium abatementeffort, we can find the difference between NTEP and TEP. By tradingemission permits, firms can abate emissions among them. Trading emissionpermits continues until each firm’s marginal abatement effort level isequalized. In this paper, as the technologies of all firms are the same, theabatement effort level is the same among them. Note that the price of theemission permit is equal to the marginal abatement effort of each firm and itis not influenced by the initial allocation of each firm’s emission permits.12

5 Comparison of NTEP and TEP

In this section we examine how the differences between the two envi-ronmental regulations influence the equilibrium outcomes and the socialwelfare levels. First, we find the following proposition.

Proposition 1: (i) When h ¼ 0, the NTEP equilibrium outcomes are thesame level as the TEP ones. (ii) When 1 � h > 0, public firms are buyersand private firms sellers of emission permits. (iii) The following

12 See Appendix A.

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 271

relationships summarize the comparison of NTEP to TEP equilibriumoutcomes:

ð1Þ qt0i � qnt

0i; qt1j � qnt

1j;

ð2Þ at0i � ant

0i; at1j � ant

1j;

ð3Þ Qt � Qnt;P

atl �

Pant

l ;

where strict inequalities hold for all h 2 ð0; 1�.

Proof: (i) Setting h ¼ 0 in equations in Sects. 3 and 4 yields qtl ¼ qnt

l ,at

l ¼ antl . (ii) From their respective equilibrium values we find

knt0i � pe � knt

1j and thus, each private firm sells permits to public firms sincethe permit price exceeds the marginal abatement cost of each private firm.(iii) A simple comparison of the equilibrium values of the choice variablesin Sect. 3 to those in Sect. 4 yields the results in Proposition 1. h

When h ¼ 0, firms do not have incentives to trade emission permitsunder TEP since knt

0i ¼ pe ¼ knt1j. Therefore, the equilibrium outcomes

under NTEP are the same level as those under TEP. When h 2 ð0; 1�,public firms (resp. private firms) are buyers (resp. sellers) of emissionpermits since knt

0i > pe > knt1j.

An intuition behind Proposition 1 (iii) is as follows. The abatementeffort of public firm 0i (resp. private firm 1j) is smaller (resp. larger)under TEP than under NTEP because of trading emission permits. Sincethe emission constraint under TEP is more relaxed than under NTEP,public firm 0i can increase its own output more under TEP than underNTEP. On the contrary, the output of private firm 1j decreases more underTEP than under NTEP. The increase in public firms’ output is more thanthe decrease in private firms’ output. As the total output is larger underTEP than NTEP, more emissions are discharged under TEP than NTEP.To satisfy the total emission constraint, therefore, the total abatementeffort is greater under TEP than under NTEP.

Second, we examine how the shift in the proportion among privatefirms and public firms affects the equilibrium outcomes under NTEP andunder TEP.

Proposition 2: If n1 increases with �N fixed,

NTEP: qnt0i;q

nt1j;a

nt0i;a

nt1j;k

nt0i; and knt

1j increase; Qnt andX

antl decrease;

TEP: qt0i; and qt

1j increase; Qt;X

atl;a

t0i;a

t1j; and pe decrease:

272 K. Kato

Proof: Simple differentiation of the equilibrium values of the choicevariables in Sects. 3 and 4 yields the results in Proposition 2. h

Under both regulations, when the number of private firms increaseswith the total number of firms fixed, each public firm’s output increasesand each private firm’s output increases. This is because each private firmhas an incentive to produce less output than each public firm and thestrategic substitution effect works. By changing a public firm into a pri-vate firm, its firm’s output decreases. A decrease in its output is largerthan an increase in other firms’ output, and then the total outputdecreases. Under NTEP, an increase in the output of each firm leads to anincrease of the emission. Therefore, the abatement effort of each firmincreases to satisfy the emission constraint and its shadow price ofemission constraint increases. Under TEP, however, the abatement effortand the price of the emission permit decrease because the total output, andtherefore total emission decreases more than before.

Third, we examine how an increase in the weight of social welfare inpublic firms’ objectives affects the equilibrium outcomes under NTEPand under TEP.

Proposition 3: If h increases,

NTEP: qnt0i;a

nt0i;Q

nt;X

antl ; and knt

0i increase; qnt1j;a

nt1j; and knt

1j decrease;

TEP: qt0i;a

t0i;a

t1j;Q

t;X

atl; and pe increase; qt

1j decreases:

Proof: Simple differentiation of the equilibrium values of the choicevariables in Sects. 3 and 4 yields the results in Proposition 3. h

As an increase in h means that each public firm gives more weight tosocial welfare, it increases its output more than before. This makes eachprivate firm’s output decrease because of a strategic substitution effect.This is the same under NTEP and TEP. With respect to the abatementefforts, however, a difference between NTEP and TEP arises. UnderNTEP, as each firm must obey each emission constraint, public firm 0imakes more abatement effort and private firm 1j makes less abatementeffort than before. And then, shadow price of emission constraint forpublic firm 0i increases and that for private firm 1j decreases. On thecontrary, under TEP, marginal abatement costs are equal among firmsbecause they are price takers in the emission permit market. The larger the

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 273

total output is, the larger the total emission is. The price of the emissionpermit increases to satisfy the total emission constraint �E. Thus an in-crease in h causes an increase in their abatement efforts and the price ofan emission permit.

Fourth, we compare social welfare under NTEP with that under TEP. Itis difficult to calculate the condition analytically. Therefore, we confinethe analysis in the case c � 1 and we compare the two social welfarelevels.13 We find the following lemma:

Lemma 1: Define �h ¼ b�ffiffiffiffiffiffiffiffiffiffib2þacp

a . This is the only value of h 2 ð0; 1� atwhich the social welfare under TEP equals that under NTEP.

Proof: See Appendices B and C where the values of a; b and c aredefined. h

From Lemma 1, we can establish the following proposition.

Proposition 4:

ð1ÞWhen 0 � k � / c; �Nð Þ; SW t � SW nt if 0 � h � �h;SW t < SW nt if �h < h � 1:

ð2ÞWhen k > / c; �Nð Þ; SW t � SW nt for h 2 ½0; 1�;

where/ c; �Nð Þ ¼ � 2c2 þ �N � 1ð Þc� 3

� ��þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9c2 þ �N �Nc� 2c� 2ð Þ þ 18f g

p

cþ 9i.

2c and @/ c; �Nð Þ=@c < 0.

An intuition behind Proposition 4 is as follows. TEP has two positiveeffects and one negative effect in comparison to NTEP. One of thepositive effects is to increase total output, which correspondinglyincreases consumer’s surplus. The other positive effect is to minimize thesocial abatement cost, given the emission level is fixed. The negativeeffect is the inefficient reallocation of production. Thus, the superiority ofTEP or NTEP in terms of social welfare depends on the relative size ofthe positive effects and the negative effect.

13 See more detail for the case of c < 1 in Appendix B. c � 1 does not implyany special meaning. The reason for using the case of c � 1 is to enable us tocompare social welfare under NTEP and that under TEP.

274 K. Kato

First, we consider the case where k is small and c is nearly equal to 1.By trading emission permits, for a small h, the increase in consumer’ssurplus is larger than the decrease in the producer’s surplus because theoutput each public firm produces is slightly greater than the output eachprivate firm produces. Therefore, the inefficient reallocation of productionis small. In this case, the positive effects overcome the negative effect.However, for a large h, the negative effect exceeds the positive effects.The output each public firm produces is much greater than the output eachprivate firm produces and the inefficient reallocation of production is highenough to outweigh the positive effects.

Second, we consider the case where either the degree of convexity ofthe production cost function or that of the abatement cost function islarge. In this case, the inefficient reallocation of production is high, andthen the range of the weight that TEP is superior to NTEP in terms ofsocial welfare is narrow or even vanishes.

Note that from Proposition 4, when h ¼ 1, that is, public firms are puresocial welfare maximizers, social welfare is always smaller under TEPthan under NTEP in any value of c ð� 1Þ and k. In this case, we find thatpublic firm 0i’s incentive to produce more output than private firm 1jcreates the enormous loss to social welfare by trading emission permits.

In fact, c � 1 is a sufficient condition for the social welfare functionunder each regulation to be strictly concave in h. When c � 1, if thegovernment can choose the value of h which maximizes social welfarebefore the quantity competition under each regulation, we can calculatethe optimal h for each regulation, which we call h?nt for NTEP and h?t forTEP respectively and find that h?nt > h?t, where h?nt ¼ ðcþ k þ 1Þ2=fðcþ k þ 1Þ2 þ ðcþ kÞð �N � n1Þg and h?t ¼ �Nðcþ 1Þ2=f �Nðcþ 1Þ2

þcðk þ �NÞð �N � n1Þg. From the results, we can see that the enlargementof the inefficient reallocation of production by trading emission permitssignificantly does harm on social welfare.

Figures 1 and 2, and Table 1 illustrate the result of the simulation of �hin c and k 2 ½0; 2� and n0 ¼ n1 ¼ 1.14 Figure 1 illustrates the relationshipsamong c; k; and �h from overall and Fig. 2 represents them in contour. InFig. 2, in the range that c and k 2 ½0; 2�, the number on each contour lineshows the value of �h. The clear space to the left-hand side of the contourline of ‘‘1’’ represents the pairs where social welfare is always greater

14 In Table 1, we illustrate �h for all ranges to enhance comprehension of

Proposition 4 well. In Fig. 1 and Fig. 2, we confine the value of �h to be ½0; 1�.

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 275

under TEP than under NTEP for any h. The clear space to the right handside of the contour line of ‘‘0’’ represents the pairs where social welfare isalways smaller under TEP than under NTEP for any h. By looking atthem, particularly from Fig. 2, we can easily find that social welfare underTEP is always greater than that under NTEP when the degree of con-vexity of the production cost function and that of the abatement costfunction is small enough. The above result is reversed, however, whenthey are large.

Finally, we analyze how the shift in the proportion among private firmsand public firms affects �h.

Proposition 5:@�h@n1

< 0 .

Proof: Simple differentiation of the equilibrium values of choice vari-ables yields the results in Proposition 5.15 h

When the number of private firms increases with the total number offirms fixed, an inefficient reallocation of production increases more thanbefore. Therefore, the threshold �h decreases with an increase in thenumber of private firms n1.

10.80.60.40.2

00

0.5

1

1.5

2 0

0.5

1.5

1

2

c

k

−q

Fig. 1. The relationship among c, k, and �h: Overall shot

15 In Proposition 5, we only deal with the case where �h 2 ð0; 1�. In fact, if�h < 0, we find @�h=@n1 > 0.

276 K. Kato

TEP can achieve total cost minimization of the abatement cost at agiven emission level, so it seems to be superior to NTEP in terms of socialwelfare. When there is a distortion in the product market, however, TEPmay cause the social welfare level to be lower than NTEP. This result has

Table 1. The relationship among c, k, and �h

kc 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0 1.25 1.24 1.24 1.23 1.23 1.23 1.23 1.23 1.23 1.23 1.230.2 1.18 1.15 1.13 1.11 1.09 1.07 1.05 1.03 1.01 1.0 0.980.4 1.10 1.06 1.02 0.99 0.96 0.92 0.89 0.86 0.83 0.80 0.770.6 1.01 0.96 0.91 0.87 0.83 0.78 0.74 0.70 0.66 0.62 0.580.8 0.91 0.85 0.80 0.74 0.69 0.64 0.59 0.54 0.49 0.44 0.401.0 0.80 0.74 0.68 0.62 0.56 0.50 0.44 0.38 0.33 0.27 0.221.2 0.68 0.62 0.55 0.48 0.42 0.35 0.29 0.23 0.16 0.10 0.041.4 0.56 0.49 0.42 0.34 0.27 0.20 0.13 0.07 0.00 )0.07 )0.131.6 0.44 0.36 0.28 0.20 0.13 0.05 )0.02 )0.09 )0.17 )0.24 )0.301.8 0.30 0.22 0.14 0.06 )0.02 )0.10 )0.18 )0.26 )0.33 )0.41 )0.482.0 0.17 0.08 )0.01 )0.09 )0.17 )0.26 )0.33 )0.42 )0.50 )0.58 )0.66

2

1.5

1

0.5

021.510.50

0

0.25

0.5

0.75

1

c

k

Fig. 2. The relationship among c, k, and �h: Topographical plot

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 277

already been found in earlier works.16 The difference with this paper isthat attention is paid to the degree of difference in the objective functionsamong public firms and private firms. The demand structure and the coststructure determine whether TEP is superior to NTEP in terms of socialwelfare.

6 Concluding Remarks

This paper examines the effects of a government’s choice of environ-mental regulations in a mixed market. So far, either environmental reg-ulation problems without a mixed market or the mixed market withoutenvironmental problems have been analyzed in earlier works. There arefew papers that examine the environmental problem in a mixed market.17

We show under which conditions the government should choose TEP andNTEP when the situation for public and private firms is symmetric, withthe exception of the differences in their objectives. TEP leads to theequalization of the marginal abatement costs between the firms and to anoverall increase in the total output, but it also causes increased ineffi-ciency in the reallocation of production among public and private firms.The magnitude of the weight that public firms put on social welfareand the demand and cost structures determine whether positive effectsoutweigh the negative effects or not.

We discuss the implication of the results in the paper. If mixedmarkets where public firms’ objectives are to maximize social welfareare widely spread, it is desirable that the government does not allowfirms to trade emission permits. For example, China has imposed theenvironmental regulation based on non-tradable emission permits in-stead of adopting tradable emission permits.18 It may be said that Chinahas made correct choices for an environmental regulation. However, ifpublic firms are partially privatized and their objectives are not onlysocial welfare maximization but also profit maximization, then there is apossibility that tradable emission permits is better than non-tradable

16 See Malueg (1990) and Sartzetakis (2004).17 Barcena-Ruiz and Garzon (2001) examine the interaction with environ-

mental tax and privatization in a mixed oligopoly.18 In China there is an emission permits system in which firms have to apply to

the environmental protection administrative organization for the emission permitand follow all rules stated in the emission permit, which include the totalemission restriction. In this system firms cannot trade the permits between firms.

278 K. Kato

emission permits type regulation in terms of social welfare. In this case,if firms consider the environment more than they used to because ofpublic opinion, energy saving, or cost reducing and they invest indecreasing emission per output q and the coefficient of the abatementcost function k, the government may as well examine the introductionof tradable emission permits. The government needs to pay attention tothese factors when it decides whether allowing firms to trade emissionpermits or not.

To simplify the analysis and to see the effect of public firms’ objectiveson the price of the emission permit, we assume that the price of theemission permit is endogenous and the total emission is exogenous. Tofocus on the market power of the product market, we also assume thatunder TEP firms behave as price takers in the emission permits market.We do not consider the cases where firms can buy or sell emission permitsfrom other countries and where firms have market power in the emissionpermits market. In addition, we confine our analysis to quantity compe-tition and do not consider price competition.19 We wish to investigatethese cases in future research.

Appendix A

The Calculation of the Price of the Emission Permit pe

The first-order conditions of maximization of the objectives of each firmare as follows. Note that pe is given in the calculation.

@U0i

@q0i¼ A� 1� hþ cð Þq0i �

Xn0

i¼1q0i

�Xn1

j¼1q1j � qpe ¼ 0; for i ¼ 1; . . . ; n0; ð12Þ

@U0i

@a0i¼ �ka0i þ pe ¼ 0; for i ¼ 1; . . . ; n0; ð13Þ

19 Dastidar (1995; 1997) analyzes the case in which private firms compete inprices in a homogeneous product market with a convex cost function and showsthe existence of multiple equilibria. There are few studies which analyze the samesituation in a mixed oligopoly. In addition, there are also few papers whichanalyze price competition of the differentiated goods in a mixed oligopoly.

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 279

@U1j

@q1j¼ A�

Xn0

i¼1q0i �

Xn1

j¼1q1j � ð1þ cÞq1j

� qpe ¼ 0; for j ¼ 1; . . . ; n1; ð14Þ@U1j

@a1j¼ �ka1j þ pe ¼ 0; for j ¼ 1; . . . ; n1: ð15Þ

First, from solving the above 2 n0 þ n1ð Þ equations, we obtainq0i ¼ ðcþ 1Þ A� qpeð Þ=Y 0; q1j ¼ cþ 1� hð Þ A� qpeð Þ=Y 0, and a0i ¼a1j ¼ pe=k where Y 0 ¼ cþ 1þ �Nð Þðcþ 1Þ � cþ 1þ n1ð Þh. Next, wesubstitute them for q0i; q1j; a0i; and a1j in the market clearing condition(10), and solve for pe, then we find the following value of pe:

pe¼q �Nðcþ1Þ�n1hf gA� cþ1þ �Nð Þðcþ1Þ� cþ1þn1ð Þhf g �EY

k: ð16Þ

Note that the initial allocation of the emission permits among firms �E= �Ndoes not appear in (10) and (12)–(15).

Then by adding

�Nðcþ 1Þ � n1hf g � �Nðcþ 1Þ � n1hf gY

k2q2 �E�N

ð¼ 0Þ ð17Þ

to pe and summing up, we can obtain the equilibrium value of pe in Sect. 4.Finally, we substitute it for each variable, and then we obtain the equilib-riumoutcome under TEP. The reason for transforming ð16Þ into pe in Sect. 4

is to compare the value of pe with the values of knt0i and knt

1j more easily.

Appendix B

Welfare Comparison with no Limitations in the Values of c and k

We define the values of a; b, and c as follows: a ¼ 2kn1þ ckn1 þ k2n1 þ kn21 þ 3 �N þ 4c �N þ c2 �N þ 2k �N þ ck �N þ 4n1 �N þ 2cn1

�Nþ kn1 �N þ n21 �N > 0, b ¼ kn1 þ ckn1 þ k2n1 þ 3 �N þ6c �N þ 3c2 �Nþ4k �N þ 4ck �N þ k2 �N þ 2n1 �N þ 2cn1

�N þ 2kn1 �N þ 2 �N2 þ2c �N2 þ k �N2þn1 �N 2 > 0, c ¼ �Nð1þ cþ k þ �NÞðc3 þ 2c2k þ ck2þ �Nc2 � c2 þ �Nck�ck � 5c� 3k � 3� �NÞ, @�h=@c < 0, @�h=@k < 0, b=a > 1 and

b2 þ ac > 0.

280 K. Kato

We conduct our analysis with no limitations on the parameters, c and k.The conditions of a threshold h, called h 2 ½0; 1�, are a� 2b � c � 0

because h � 0 implies c � 0 and h � 1 implies c � a� 2b. With respectto the relationship among the parameters’ values and the magnituderelation of social welfare level under NTEP and TEP, we obtain thefollowing conditions:

ð1Þ SW t � SW nt for h 2 ½0; 1� if and only if c > 0;

ð2Þ SW t � SW nt for h 2 ½0; 1� if and only if c < a� 2b;

ð3Þ There exists h 2 ½0; 1� such that;

SW t � SW nt for h 2 ½0; h�;

SW t<SW nt for h 2 ðh; 1� if and only if a� 2b � c � 0;

(

where a strict inequality holds for all h 2 ð0; 1� except for h. Section 5examines the case c � 1. If c � 1, we find that c > a� 2b. Therefore, ifc � 1, the case where TEP is superior to NTEP in terms of social welfarefor h 2 ð0; 1� does not exist. This appendix considers the case, c < 1. Inparticular, we pick up the case where c is nearly equal to zero. In such acase, we find c < a� 2b. In this case the negative effect vanishes becausec ¼ 0 means both firms have the same constant marginal production cost.Therefore, when both c and k are sufficiently small, TEP is alwayssuperior to NTEP in terms of social welfare regardless of h.

Appendix C

Proof of Lemma 1

The difference in social welfare levels between NTEP and TEP is

SW t � SW nt ¼ k �N � n1ð Þn1h2M2

2X 2Y 2ah2 � 2bh� c

:

The values of h that cause the social welfare under NTEP to be at thesame level as that under TEP are

0;bþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ac

p

a;

b�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2 þ ac

p

a:

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 281

The second solution is larger than 1 because b=a > 1. Thus only the thirdsolution has the possibility to be in ð0; 1�. h

Acknowledgements

The author would like to thank Kazuharu Kiyono, Akihiko Matsui, ShigeruMatsumoto, Toshihiro Matsumura, Dan Sasaki, Daisuke Shimizu, Sjak Smulders,and two anonymous referees of this journal for their thoughtful comments andsuggestions.

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282 K. Kato

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Address of author: – K. Kato, Graduate School of Economics, Universityof Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan (e-mail: fwin9096@mb.infoweb.ne.jp)

Can Allowing to Trade Permits Enhance Welfare in Mixed Oligopoly? 283