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Green Consumerism and Pollution Control

P. Chander1 and S. Muthukrishnan

2

Abstract

This paper shows that, like taxes and subsidies, collective action by environmentally

aware/green consumers can reduce pollution and improve welfare. In a model with two

competing firms each producing a good of different environmental quality and two types

of consumers with different preferences for quality, we show that collective action by the

high-type consumers results in an equilibrium with not only lower pollution and higher

social welfare, but also higher prices and profits for the firms.

JEL Classification Numbers: H41, L13, H23, Q50.

Keywords: collective action, public good, vertically differentiated duopoly,

environmentally aware consumers.

1National University of Singapore. Email: ecsparka@nus.edu.sg. 2 National University of Singapore. Email: subha.m6@gmail.com.

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Green Consumerism and Pollution Control

1. Introduction

Consumption of certain goods generates both private and public benefits. For instance, a

consumer benefits directly from consuming organic food because it is more nutritious and

healthier with fewer risks to personal health from pesticides and herbicide residues.

However, organic farms are also more sustainable and environmentally better than

conventional farms because they do not release synthetic pesticides or herbicides into the

environment. Thus, consumption of organic food not only directly benefits a consumer,

but also helps indirectly in preserving and sustaining the ecosystem which benefits all

consumers.3 Similar positive relationship between private and public benefits also holds

if the consumers perceive the quality of goods produced with the use of less polluting

inputs or cleaner technologies to be higher, though there may be no real difference. For

instance, electricity produced from renewable energy may be perceived as better than that

produced from coal, though there is no real difference in its quality when consumed.

More generally, a consumer may drive additional utility (i.e. private benefits) from the

consumption of a good simply from knowing that it will contribute less to pollution. This

is known as the “warm glow” effect (see Andreoni (1989) and Ribar and Wilhelm

(2002)).

It is well-known that higher private benefits from the consumption of a good, whether

real as in the case of organic food or altruistic as in the case of green electricity, can

induce consumers to pay more for it and firms to invest in cleaner technologies, see e.g.

Arora and Gangopadhyay (1995).

3 Other interesting examples include coffee grown under the canopy of tropical forests, rather than in open

deforested fields, which tastes better as well as helps preserve forests. Some additional examples are

discussed in the concluding section.

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Cremer and Thisse (1999), Bansal and Gangopadhyay (2003), and Erikkson (2004)

among others consider models of price competition and product differentiation when

consumers are environmentally aware. These models address many important questions

concerning the impact of green consumerism on market equilibrium and the role of

various economic instruments such as taxes and subsidies for bringing reduction in

pollution. But they all consider individual action by consumers and accordingly assume

that each consumer takes the level of pollution as exogenously given. However, if some

consumers come together and decide collectively which good to buy, then they can

influence the level of pollution. For instance, if all consumers sharing a common

economic-ecological system decide collectively to buy only organic food, then, besides

the private benefits from the consumption of organic food, each consumer will also

benefit from a better preserved and more sustainable ecosystem which is free from

pesticides and herbicides.4 The Organic Consumers Association (OCA) in the US is one

such example of mobilization of hundreds of thousands of consumers who buy only

organic food.5

Economic implications of collective action by green consumers have not been studied

previously. This paper begins analyzing the impact of collective action by green

consumers on the prices of goods, level of pollution, and social welfare. We consider a

model consisting of two competing firms each producing a good of different

environmental quality and two types of consumers with different preferences for quality.

Since the consumers who form a coalition and decide collectively which good to buy can

influence the level of pollution, we treat the level of pollution as a choice variable in the

utility maximization problems of the consumers in the coalition. We show that the

coalition, once formed, is self-sustaining and does not collapse, since the resulting

equilibrium prices are such that no consumer individually has incentive to deviate and

“free-ride”.

4 The underlying assumption here is that the consumers share a common environment in which both

consumption and production take place. 5 It is an association of consumers to promote a more responsible and sustainable approach to food

production.

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Our analysis shows that collective action by consumers with higher willingness-to-pay

for environmental quality reduces competition and leads to higher prices for goods of

both qualities. That is because the consumers who form a coalition and decide

collectively which good to buy take into account not only their private benefits from

consuming the good but also the favorable impact of their decision on pollution which

enables the firm producing the cleaner good to extract a higher price. As a result the other

firm is also able to charge a higher price. Furthermore, our analysis shows that, like taxes

and subsidies, collective action by consumers with higher willingness-to-pay reduces

pollution and improves welfare. In fact, it can even lead to optimal control of pollution if

the difference in the willingness-to-pay of the two types of consumers is sufficiently

large.

The paper is organized as follows. Section 2 describes the model. Section 3 characterizes

the market equilibrium under the assumption that the consumers act individually and each

firm maximizes its profit, taking the price of the other firm as given. We use that as the

benchmark equilibrium. Section 4 characterizes the equilibrium when the high-type

consumers form a coalition and decide collectively which good to buy taking into

account the impact of their decision on pollution. Section 5 compares the two equilibria

and studies how collective action by consumers impacts the market equilibrium,

pollution, and social welfare. Section 6 draws the conclusion. All the proofs are gathered

in the Appendix to the paper.

2. The model

A simple model of preferences for a vertically differentiated product was developed by

Mussa and Rosen (1978) and applied to the analysis of vertically differentiated product

markets by Gabszewicz and Thisse (1989) among others. An abundant literature

developed from these applications. The present paper is an extension of the Mussa-Rosen

model in that it introduces the idea that some agents may form a coalition and decide

collectively which product to buy.

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Our model consists of two firms each producing a good of different environmental

quality. The environmental quality of a good is positively related to the cleanliness of the

technology used to produce it - the cleaner the technology, the higher the quality of the

good.

We assume that the cost of producing one unit of a good of environmental quality s is

).(sc A higher means a less polluting good. To keep matters simple, we assume that

goods can be produced in only two environmental qualities: high, to be denoted by ,Hs

and low, to be denoted by .Ls Let ),( HH scc ),( LL scc and LH cc . We assume

that firm 1 produces the good of low quality and firm 2 of high quality.

We consider a population of consumers (who share the same economic-ecological

system) with different preferences/willingness-to-pay for goods available in different

environmental qualities. We assume that each consumer buys either none or one unit of a

good. The consumers are environmentally aware and willing to pay a higher price for a

good if it is of higher environmental quality.

The utility of a consumer who buys one unit of the good of quality s is SpsU

where p is the price of one unit of the good of quality ,s is consumer’s preference or

willingness-to-pay for quality, and S is the amount of pollution abated. To keep matters

simple, we assume that there are only two types of consumers with preferences H and

L for quality where H > L . We shall refer to them as the high- and low- type

consumers.

Let denote the proportion of low-type consumers and thus 1 of high-type

consumers. If r and t are the qualities of the goods purchased by the low- and high- type

consumers, respectively, then the amount of pollution abated is .)1( trS The

amount of pollution abated is thus determined by the aggregate consumption of all

consumers, but the impact of an individual consumer’s consumption on it is negligible.

By assumption, }.,{, HL sstr Let Sptu HH and Spru LL denote the

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utilities of the high- and low type consumers, respectively. The variable S appears in the

utility of each consumer as it represents a public good. To highlight the generally low

concern for the environment of a low-type consumer, we assume that

3. Individual action and equilibrium prices

In this section, we adopt the standard assumption in this literature that each consumer

acting individually takes the amount of pollution abatement as given and thus ignores it

from his utility maximization exercise, even though his utility depends on it. As noted in

the Introduction, consumption of a good by a consumer generates both private and public

benefits as it reduces pollution. However, when consumers act individually, each one

thinks that the impact of his consumption on pollution is negligible - it is the aggregate,

rather than the individual consumption, that affects the pollution level. A consumer’s

decision to buy a good is then motivated entirely by his own private benefit from

consumption of the good. Accordingly, in this section, we ignore the amount of pollution

abatement S from the utility maximizing exercise of both types of consumers.

Let Lp and Hp denote prices of the goods of low and high environmental qualities,

respectively. We assume that each firm sets the price of the good produced by it so as to

maximize its profit, given the price of the other firm and preferences of the consumers.

To ensure greater competition among the firms, we rule out cases in which a firm may

never find it profitable to sell its product to a consumer. Accordingly, we make the

following assumption:

A1: 0 LLL cs and .0 HHL cs

Since ,LH assumption A1 also implies 0 LLH cs and 0 HHH cs . It is

convenient to note some additional implications of assumption A1. If ,0 psL

0 psH , which means that if a low-type consumer is willing to buy a good of quality

,s then so is a high-type consumer. Thus, it is never the case that a low-type consumer

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buys the good of some quality, but a high-type consumer does not buy the good of any

quality. Since LH and LH ss , the two inequalities LLLHHL psps and

HHHLLH psps can never hold at the same time for any prices Lp and .Hp

Thus, it is never the case that a low-type consumer buys the good of high quality,

but a high-type consumer buys the good of low quality.

Several types of equilibria are possible. There are nine types in total. E.g., one is that no

one buys the good of either quality. Another is that the low-type consumers buy the good

of high quality and the high-type consumers buy the good of low quality or the low-type

consumers buy the good of high quality, but the high-type consumers do not buy the good

of any quality and so on. Assumption A1 rules out all but three types of equilibria, which

are the following: (1) both types of consumers buy the good of low quality, (2) the low-

type consumers buy the good of low quality and the high-type consumers of high quality,

and (3) both types of consumers buy the good of high quality. In order to ensure the

existence of the equilibrium we will focus on, we make a further assumption:

A2.1: .)1(LH

LHLH

ss

cc

Since ,LH assumption A2.1 is satisfied only if LHLHH ccss )( and thus

,)( LHLHL ccss i.e., if the cost of production increases by more than the

willingness-to-pay of either type of consumer. As argued at the end of this section,

assumption A2.1 ensures that implications of collective action can be studied in a model

with only two goods. The assumption is sufficient for the existence of the type 1

equilibrium, but rules out the existence of type 2 and 3 equilibria in which the high-type

consumers buy the good of high quality. That is seen as follows.

Since each firm maximizes its profit given the price of the other firm, existence of a type

3 equilibrium requires LLLLLLHHL cspsps or )( LHL ss LH cp

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,LH cc which is clearly ruled out by assumption A2.1. We show that assumption A2.1

also rules out the existence of type 2 equilibria and then prove the existence of a unique

type 1 equilibrium.

A type 2 equilibrium, if one exists, is a pair )~,~( HL pp such that

0~ LLL ps , ,0~ HHH ps ,0~ LL cp ,0~ HH cp (1)

,~~LLHHHH psps (2)

.~~HHLLLL psps (3)

Inequalities (1) denote the participation constraints of the consumers and the firms.

Inequalities (2) and (3) represent the self-selection constraints of the high- and low-type

consumers, respectively.

Since )~,~( HL pp is an equilibrium, it should not be possible for firm 1 to lower its price

such that, besides the low-type consumers, the high-type consumers also prefer to buy the

good of low quality and the profit of firm 1 is higher. Thus, if LL pp ~ and

,~HHHLLH psps then ).~( LLLL cpcp This means that )~,~( HL pp must be

such that

.)1(~~LLLHHHH cpsps (4)

Similarly, firm 2 should not be able to lower its price such that the low-type consumers

will also buy the good of high quality and its profit is higher. That is, if HH pp ~ and

,~LLLHHL psps then ).~)(1( HHHH cpcp This means that )~,~( HL pp

must be such that

.~)1(~HHHLLLL cpsps (5)

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Inequality (2) is weaker than inequality (4), since .~LL cp Similarly, (3) is weaker than

(5), since .~HH cp Since each firm maximizes its profit, taking the price of the other

firm as given, (4) and (5) must hold with equality in equilibrium. However, equalities (4)

and (5) have a unique solution which is

2

2

1

)1())1)(((~

HLLHLH

L

ccssp , (6)

.1

)1())((~2

2

HLLHLH

H

ccssp (7)

Therefore, if )~,~( HL pp is indeed a type 2 equilibrium, then it must satisfy inequalities

(1), and equalities (6) and (7). However, substituting from (7), )~( HH cp

,0]1/[)])(1())([( 2 LHLHLH ccss by assumption A2.1. Thus,

)~,~( LH pp cannot be an equilibrium as it does not satisfy the inequality .~HH cp This

rules out existence of type 2 equilibria.

Proposition 1: Suppose assumptions A1 and A2.1 hold. If the consumers act

individually, then there exists a unique equilibrium which is of type 1, i.e., both

types of consumers buy the good of low quality.

Notice the role played by assumption A2.1, which implies ),()( LHHLH sscc in

the proof of Proposition 1. The assumption implies a relatively higher equilibrium price

for the good of high quality ( HH cp at the minimum) such that the price-quality

tradeoff offered by the good of high quality is not good enough even for the consumers

with higher willingness-to-pay. This intuitively explains why, under assumptions A1 and

A2.1, no type 2 equilibrium exists.

We are interested in the type 1 equilibrium because if the initial equilibrium is of type 2

(the usual equilibrium considered in the literature), then it is not possible to study the

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impact of collective action on pollution in a two-good model. In type 2 equilibrium, the

high-type consumers buy the good of highest quality and, therefore, collective action by

them cannot lead to their buying a cleaner good which results in lower pollution. To keep

the model simple, we start with the type 1 equilibrium. If we start instead with a type 2

equilibrium (i.e. high-type consumers buy the good of high quality and the low-type of

low quality), then to study the impact of collective action we would need to modify our

model by including an additional good of higher quality than those demanded in type 2

equilibria. Such an extension of the model does not lead to additional insights in

understanding how collective action impacts pollution and welfare, but it would certainly

complicate the analysis. Similarly, there is really no difference between type 1 and type 3

equilibria except that then also the model will have to consist of an additional good of

higher quality. Starting with the type 1 equilibrium as the benchmark provides the

simplest way to analyze the impact of collective action on pollution and welfare.

4. Collective action and equilibrium prices

Having characterized the equilibrium for the case when the consumers decide

individually, we consider next the case in which some consumers form a coalition and

decide collectively which good to buy. As noted in the Introduction, each consumer in

the coalition in that case realizes that the level of pollution depends on which good he and

other members of the coalition would buy. In particular, if the high-type consumers form

a coalition and decide collectively which good to buy, then each of them realizes that

their buying the good of higher environmental quality will lower pollution. If each of

them takes that into account, then each of them would be willing to pay a higher price for

the good of higher environmental quality.

To keep matters simple, we assume that while the high-type consumers form a coalition

and engage in collective action, but the low-type consumers, in view of their low concern

for the environment, continue to act individually. We verify below that the resulting

equilibrium prices are such that the high-type consumers indeed have no incentive to

deviate. In other words, the coalition, once formed, is self-sustaining.

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Since the high-type consumers form a coalition and decide collectively which good to

buy, they no longer take the amount of pollution abated S as exogenously given, and it

enters the utility maximization exercise of each high-type consumer as a choice variable.

By definition, the amount of pollution abated is ,)1( trS if r and t are the

qualities of the goods purchased by the low- and high- type consumers, respectively. Let

p and q be the prices of the goods of qualities r and ,t respectively. Then, the utility of

a low-type consumer who buys one unit of the good of quality r is .SprU LL

Since each low-type consumer is assumed to act individually, he must take the amount of

pollution abated S as exogenously given. We can thus take the utility of a low-type

consumer, who buys one unit of the good of quality ,r simply as .~

prU LL The

utility of a high-type consumer who decides collectively to buy one unit of the good of

quality t is SqtU HH rtqtH )1( where r is the quality of the good

bought by the low-type consumers. Since the high-type consumers have no control over

the decisions of the low-type consumers, each high-type consumer must take r as

exogenously given. We can thus take the utility of a high-type consumer who decides

collectively to buy one unit of the good of quality t simply as .)1(~

tqtU HH

We do not drop the term t)1( from the utility of a high-type consumer, since the high-

type consumers decide collectively which good to buy and thus can influence the level of

environmental pollution, taking as given the environmental quality of the good bought by

the low-type consumers.

To sum up, we can take, for the purpose of characterizing the equilibrium, the utility of a

low-type consumer who decides individually to buy one unit of the good of quality r

simply as prU LL ~

and that of a high-type consumer who decides collectively to buy

one unit of the good of quality t as ,))1((~

qtU HH where p and q are the

prices of goods of qualities r and ,t respectively.6 Comparing the utilities of the two

6 For welfare comparison, of course, we take into account the impact of both r and ,t i.e., of

.)1( trS

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types of consumers, we can interpret collective action by high-type consumers as

equivalent to their having a higher willingness-to-pay, i.e., )1( H instead of .H

With this interpretation of collective action in mind, we make the following assumption.

A2.2: .1)1(

H

LH

LHL

ss

cc

Note that assumption A2.2 is not inconsistent with assumption A2.1, which implies

,)( LHLHH ccss but not .))(1( LHLHH ccss Thus, unlike assumption

A2.1, assumption A2.2 does not rule out existence of type 2 equilibria if the high-type

consumers form a coalition and decide collectively which good to buy. To put it another

way, type 2 equilibria may now exist because it is harder for firm 1 to induce the high-

type consumers to buy the good of low quality if they take into account the adverse

impact of their consumption of the low-quality good on pollution. Therefore, rather than

lowering its price to induce the high-type consumers to buy the good of low quality, firm

1 may now find it more profitable to raise its price and extract more surplus from the

low-type consumers.

As earlier, several types of equilibria are possible under collective action. However,

similar considerations as in Section 3 rule out all but type 2 equilibria. In fact, there

exists a unique type 2 equilibrium.

Proposition 2: Suppose assumptions A1, A2.1, and A2.2 hold. If the high-type

consumers form a coalition and decide collectively which good to buy, then there exists a

unique equilibrium

which is of type 2, i.e., the low-type consumers buy the good

of low quality and the high- type of high quality.

Proposition 1 shows that in equilibrium the price-quality tradeoff offered by the good of

high quality is not good enough for the high-type consumers if they do not take into

account the impact of their individual consumption decisions on the level of pollution.

Proposition 2 implies that this is not so if the high-type consumers form a coalition and

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take into account the favorable impact of their collective decision to buy the good of high

quality on the level of pollution.

Proposition 3: Suppose assumptions A1, A2.1, and A2.2 hold. Then collective action by

high-type consumers leads to higher prices of goods of both qualities, i.e.,

When high-type consumers decide collectively which good to buy, they internalize the

externality associated with their consumption of the good. That impacts the market

equilibrium in two different ways.

First, it allows the firm producing the good of high environmental quality to charge a

higher price. Accordingly, firm 2 earns a positive (i.e. higher) profit, which is equal to

,0))(1( * HH cp as shown in the proof of Proposition 2. Second, it reduces

competition and as a result the firm producing the good of low environmental quality is

also able to raise its price. However, the impact of this on the profit of firm 1 is

ambiguous. On the one hand, the profit of firm 1 should be higher because the price of

the good produced by it is now higher, but on the other hand, the profit should be lower

because fewer consumers now buy it. The profits of firm 1 in the two cases are

))(()(1 LHHLHLL sscccp and )( **1 LL cp which is not less than

)),(( LHLLH sscc since )(*LHLHL sscp as shown in the proof of

Proposition 3. Since ,, LHHL ss and ,1 the profit of firm 1 is higher or lower

depending on the distribution parameter . If the proportion of the high-type consumers

is low (i.e. is large), the profit of firm 1 is likely to be higher. It was shown that

collective action by the high-type consumers leads to a positive (i.e. higher) profit for

firm 2. This means that both firms stand to gain if the high-type consumers decide

collectively which good to buy, especially if the proportion of the low-type consumers is

sufficiently high.

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We assumed so far that none of the high-type consumers who form a coalition and decide

collectively which good to buy considers deviating and to “free ride”. We now check

whether in the resulting equilibrium under collective action a high-type consumer will

really have an incentive to “free ride” and buy instead the good of low quality thinking

that no one will notice and it will have no or little impact on the level of environmental

pollution. Since a coalition must take into account the adverse impact of its deviation on

environmental pollution, the incentive to deviate for an individual high-type consumer is

stronger than for a coalition of high-type consumers.7 It is therefore sufficient to rule out

deviations by only individual high-type consumers.

We show that if the willingness-to-pay for quality of the high-type consumers is

sufficiently higher, then the relative equilibrium prices of the goods under collective

action are such that each high-type consumer prefers individually (as well as collectively)

to buy the good of high quality even when he can buy instead the good of low quality and

“free ride”. In other words, if the willingness-to-pay of the high-type consumers is

sufficiently higher, then the price-quality trade-off offered by the good of low quality in

the equilibrium under collective action is not good enough for the high-type consumers.

Proposition 4: Suppose assumptions A1, A2.1, and A2.2 hold and . LH Then,

the equilibrium prices *Lp and *

Hp under collective action are such that

,**LLHHHH psps i.e., each high-type consumer individually prefers to buy the

good of high quality.8

Proposition 4 shows that if the two competing firms set their prices assuming that the

coalition of high-type consumers will remain formed then in the resulting equilibrium it

will indeed remain formed as no high-type consumer, individually as well as collectively,

will have incentive to deviate and “free ride”.

7 Moreover, deviations by coalitions, unlike those by individuals, can be observed and taken into account

both by the firms and the other consumers. 8 It is easily verified that the set of parameters that satisfies assumptions A1, A2.1, A2.2, and the condition

is non-empty. E.g., all these assumptions and the condition are satisfied if

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Propositions 4 may appear inconsistent with Proposition 1, but it is not. It is easily

verified that if then

i.e., the price quality tradeoff

offered by the good of low quality in the equilibrium under collective action is not as

good as in the equilibrium in the absence of collective action. In other words, the

equilibrium price of the low quality good goes up relatively more under collective action

if the preferences differ sufficiently.

5. Collective action and social welfare

As shown, if the high-type consumers decide collectively, then the prices as well as the

profits of both firms are higher, especially if the proportion of the low-type consumers is

sufficiently high. However, since the high-type consumers switch to the cleaner good,

pollution is lower. This raises the question whether the resulting fall in pollution is

sufficient to outweigh the increase in prices and improve social welfare.

Since in the absence of collective action by the high-type consumers only the good of low

quality is produced and consumed, the amount of pollution abated .LsS The utility of a

low-type consumer is then ,LLLLL spsU and that of a high-type consumer is

.LLLHH spsU Substituting for ,Lp LLL sU LHLHH scss )( and

HU .)( LHHHLHLHHLH scsscsss

If the high-type consumers form a coalition and decide collectively, then, as shown, the

high-type consumers buy the good of high quality and the low-type of low quality. Thus,

the equilibrium level of environmental pollution in that case is HL ssS )1( and

the utility of a low-type consumer is ),)1((**

HLLLLL sspsU where

,*LL pp as shown. Similarly, the utility of a high-type consumer is

LHHHH spsU ** ,)1( Hs where ),(*HHH cpp as shown.

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Proposition 5: Suppose assumptions A1, A2.1, and A2.2 hold. If the high-type

consumers form a coalition and decide collectively which good to buy, then the sum of

utilities of the consumers and profits of the two firms is higher, i.e.,

..)1()1( 21*2

*1

** HLHL UUUU

What is the maximum welfare gain from collective action? If then the

resulting type 2 equilibrium under collective action is in fact optimal. To prove this, we

only need to show that any outcome in which both type of consumers buy the high

quality good is not optimal, since we already know from Proposition 5 that outcomes in

which both type of consumers buy the low quality good is not optimal. This is indeed so

if the following inequality is true

This inequality is equivalent to

Since

Therefore, by Assumption A2.1, the inequality is indeed true if This shows

that the welfare gains from collective action can be substantial and it can even lead to

optimal control of pollution if the willingness-to-pay for quality of the high-type

consumers is sufficiently higher.9

6. Conclusion

This paper begins analysis of collective action by green consumers. It shows that, like

taxes and subsidies, collective action by green consumers, can reduce pollution.

9 That more divergent preferences can lead to more efficient outcomes has been also noted in other contexts

of voluntary provision of public goods by Bardhan, Ghatak, and Karaivanov (2007).

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Our analysis can be applied to many other instances of vertically differentiated duopoly

involving a good with public good characteristics. Consider for instance an antibiotic that

controls a contagious disease faster. Such an antibiotic not only directly benefits the

individual patients being treated, but also indirectly the entire population as its use by

individual patients reduces others risk of exposure to the infection. However, such an

antibiotic may not be produced and consumed unless some patients collectively decide to

buy it taking into account its public benefits. Other examples include networks that differ

in quality. Agents with higher preference for quality are more likely to join a more costly,

but higher quality network if they decide collectively which network to join and take into

account the positive spillover effect of their decision on the quality of the network .

Our results are driven by the difference in the willingness-to-pay for quality. If the

difference is small, collective action may not lead to much reduction in pollution or

improvement in welfare. We assumed that all high type consumers form a coalition. Our

results still hold if not all but a large proportion of them do. That is seen by replacing

for such that in equations (12) – (14) and solving for

and

Propositions 3 and 4 hold for sufficiently close to

We assumed that the consumers take the prices of the two goods as given. It is not

unreasonable to think that the consumers who decide collectively will also have some

buyers’ power in terms of negotiating the price. If so, the effect of higher equilibrium

prices we discovered may be mitigated to some extent. But collective action would still

lead to lower pollution and higher social welfare. The same is true if there are more than

two firms each producing a product of different quality as that too would increase price

competition.

Appendix:

Proof of Proposition 1: Define HH cp and ).( LHHHL sscp We claim that

prices ),( HL pp characterize a type 1 equilibrium in which both types of consumers buy

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the good of low quality at price ,Lp from firm 1. By definition, a pair ),( HL pp is an

equilibrium of type 1 only if

,0 LLL ps ,0 LLH ps ., HHLL cpcp (8)

HHLLLL psps and .HHHLLH psps (9)

We show that ),( HL pp satisfies inequalities (8) and (9). Since LH and ,LH ss it is

easily verified that ),( HL pp satisfies inequalities (9). Replacing Hp by Hc in (9) and

using assumption A1, it follows that 0 LLL ps and .0 LLH ps Furthermore,

assumption A2.1 implies .LL cp Hence, ),( HL pp also satisfies inequalities (8).

To prove that ),( HL pp is indeed an equilibrium, we need to show further that neither firm

1 nor firm 2 can obtain a higher profit by raising its price, given the price of the other

firm. This is clearly true in case of firm 2, since any )( HHH cpp will not induce

either type of consumer to buy the good of high quality, if the price of the good of low

quality is .Lp We show that firm 1 also cannot obtain a higher profit by raising its price.

Since at prices ),( HL pp both types of consumers buy the good of low quality, the profit

of firm 1 is LLHHHLL cssccp )(1 ,0)( LHHLH sscc

by assumption A2.1. From the definition of Lp it is seen that if the price of high quality

good is ,Hp then for any price LL pp the high-type consumers will switch to the good

of high quality and at most the low-type consumers will buy the good of low quality.

Thus, if the price of high quality good is ,Hp then for any price LL pp the profit of

firm 1 is at most ).( LL cp We show that LLLL cpcp )( for any .LL pp

Define Lp such that .HHLLLL csps Since LH ss and ,LH .LL pp By

definition, Lp is the highest price at which the low type consumers will buy the good of

low quality, if the price of high quality good is ).( HH cp It is thus sufficient to show

that .)( LLLL cpcp Substituting for Lp and Lp this inequality is equivalent to

)( LHHLLL ccss .LHHHLH ccss That is, ))(( LHLH ss

18

).)(1( LH cc Assumption A2.1 implies that this inequality is indeed true.

Finally, it is straightforward to see that neither firm 1 nor firm 2 can obtain a higher profit

by lowering its price, given the price of the other firm. This proves that ),( HL pp is an

equilibrium in which both types of consumers buy the good of low quality.

We now prove that ),( HL pp is the unique equilibrium. For this we only need to show

that there exists no other type 1 equilibrium. Suppose contrary to the assertion that

),()ˆ,ˆ( HLHL pppp is also an equilibrium of type 2. Since ,HH cp .ˆ HH pp If

,ˆ HHH cpp then, as just shown above, profit of firm 1 is maximized only if

.ˆ LL pp Thus, if ,ˆ HH pp then ,ˆ LL pp which contradicts our supposition that

).,()ˆ,ˆ( HLHL pppp If ,ˆ HH pp then since firm 1 maximizes profit, Lp̂ must be such

that HHHLLH psps ˆˆ and therefore HHLLLL psps ˆˆ , since LH and

.LH ss Thus, there exists a price Hp such that HHH ppp ˆ and

.ˆ HHHLLH psps This means that given the price of firm 1, firm 2 can obtain a

positive (i.e. higher) profit by lowering its price such that the high-type consumers will

switch to the good of high quality. This contradicts our supposition that )ˆ,ˆ( HL pp is an

equilibrium. This shows that our supposition is wrong and ),( HL pp is the unique type 1

equilibrium.

Proof of Proposition 2: By definition, if ),( HL pp is a type 2 equilibrium then it must

satisfy

,0 LLL ps ,0 LL cp ,0 HH cp (10)

,HHLLLL psps (11)

.)1()1( LLLHHHHH spssps (12)

19

Inequality (12) highlights the fact that the high-type consumers who decide collectively

which good to buy take into account the impact of their decision on the pollution level,

taking as given the environmental quality of the good bought by the low-type consumers.

Since the firms engage in price competition, if ),( HL pp is an equilibrium then firm 2

should not be able to lower its price such that, besides the high-type consumers, the low-

type consumers also prefer to buy the good of high quality and its profit is higher. Thus,

if Hp is such that LLLHHL psps , then ).)(1( HHHH cpcp Thus, the

equilibrium prices ),( HL pp should be such that

.)1( HHHLLLL cpsps (13)

Similarly, it should not be possible for firm 1 to lower its price such that, besides the low-

type consumers, the high-type consumers would also decide collectively to buy the good

of low-quality and its profit would be higher, that is, if Lp is such that

HHHHLLLH spssps )1()1( , then ).( LLLL cpcp Thus, the

equilibrium prices ),( HL pp should be such that

LLLLHHHHH scpssps )1()1()1( . (14)

Clearly, inequalities (11) and (12) are weaker than inequalities (13) and (14),

respectively. Thus, an equilibrium of type 2 must satisfy inequalities (10), (13), and (14).

Since the firms maximize profits, inequalities (13) and (14) must hold with equality in

equilibrium. However, equalities (13) and (14) have a unique solution ),( **HL pp such

that

2

2*

1

)1())1)(1)(((

HLLHLH

L

ccssp (15)

20

.1

)1())1)(((2

2*

HLLHLH

H

ccssp (16)

It is easily verified that *

Lp and *

Hp also satisfy the participation constraints (10).

Assumptions A2.1 and A2.2 imply .0** LH pp Substituting from (15) and (16), the

profit of firm 1 is

,01

)())1)(1)((()(

2

**1

LHLHLH

LL

ccsscp

by assumption A2.1, and that of firm 2 is

,01

))(1())1)((()1())(1(

2

**2

LHLHLH

HH

ccsscp

by assumption A2.2. Hence, ),( **HL pp is the unique equilibrium of type 2.

Proof of Proposition 3: Using assumption A2.1,

)1(1

)1()))1()(1)((( 2

*

HLL

LH

LHLLH

L

ccss

ccss

p

)1(1

)1()())1(1()()1( 22

HLLHLLH ccsscc

)( LHLH ssc .)( LLHHH pssc

Thus, .*LL pp Similarly, using assumption A2.2,

21

.)1(1

)1())(1( 2*

HHLLH

H ccccc

p

Thus, .*HHH cpp This completes the proof of Proposition 3.

Proof of Proposition 4: Substituting for *Lp and *

Hp from (15) and (16), the inequality

**LLHHHH psps is equivalent to

.))1(1)((

)1()1(

)1(1

)1()1()1( 222

LH

HLHLLHLHH

ss

cccc

After some algebra and rearranging the terms, this inequality is equivalent to

.0][

H

LH

LHLH

ss

cc

This inequality is true, since LH and HLHLH sscc )/()( by assumption

A2.1. This completes the proof.

Proof of Proposition 5: Substituting for ,,, **LHL UUU ,HU ,*

1 ,*2 ,1 and ,02 we

need to show that LLLL sps (( * )))1( Hs *)(1( HHH ps

)))1(( HL ss )( *LL cp ))(1( *

HH cp )( LLLL sps

))(1( LHHH scs ).( LL cp This inequality is equivalent to

LLHH ssss )1()1( which is clearly true, since .LH ss This completes

the proof.

22

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