Green consumers and the choice between

emission and output taxes

Christos Constantatos

Department of Economics, University of Macedonia

Christos Pargianas

Department of Economics, Brown University

Eftichios S. Sartzetakis∗

Department of Economics, University of Macedonia

Abstract

TThe present paper examines the choice of environmental policy in

the presence of green consumers within a framework of imperfect compe-

tition and endogenous choice of abatement level. We first confirm that in

the absence of policy intervention, there exists a level of consumers’ en-

vironmental responsiveness beyond which welfare is decreasing as market

imperfections become more prominent relative to environmental concerns.

We also find that both output and emission taxes are welfare superior to

the free-market case. What is surprising however, is that the welfare per-

formance of an optimally chosen emissions tax is monotonically decreasing

in consumers’ environmental sensitivity. while the opposite is true for an

output tax. At low levels of consumers’ environmental consciousness an

emissions tax is welfare superior, but eventually, there is a level of envi-

ronmental consciousness beyond which an output tax welfare dominates

an emissions tax. Therefore, an emissions tax is better suited to societies

that have not yet developed high levels of environmental awareness, while

societies characterized by high levels of environmental awareness should

prefer an output tax.

JEL classifications: D62, H21, L13

∗Corresponding author: Eftichios Sartzetakis, Department of Economics, University ofMacedonia, 156 Egnatia Str., Thessaloniki 54006, Greece. E-mail address: esartz@uom.gr.

1

1 Introduction

Over the past few decades there has been clear evidence of a continuous increase

in consumers’ environmental awareness. Based on the emergence of "green con-

sumerism", a substantial literature has been developed examining its effect on

private sector’s voluntary environmental actions and on the design of environ-

mental policy. The majority of the literature is based on the premise that "green

consumers" differentiate between products on the basis of their environmental

attribute. However, many markets do not offer "green" alternatives and when

they do, it is not clear that consumers differentiate mainly on the basis of the

product’s "greenness". Furthermore, there is evidence that consumers are wor-

ried about the level of air and water pollution, they feel well-informed and they

believe they can play a role in protecting the environment (see for example Eu-

robarometer (2014)). Thus, even in markets that "green" alternatives are not

present, consumer’s choice is influenced by the level of pollution.

The present paper develops a framework in which consumer’s choices depend

on each product’s level of pollution and examines the effect of the level of con-

sumer’s awareness in designing optimal environmental tax policy. We examine

the case where firms can reduce their pollution through fixed investments, such

as pollution filters, adoption of a new production technology, etc. Since fixed in-

vestments go in pair with imperfect competition, we focus on cases where firms

have market power. We first examine the case without any government inter-

vention and we find that increasing consumers sensitivity and reaction is not

monotonically related to social welfare: beyond a certain point, further increases

in environmental sensitivity may be harmful. This apparently strange result is

due to the fact that consumers "punish" polluting firms by reducing their con-

sumption in a market where consumption is already too low, due to imperfect

competition. Thus, any benefits from further internalizing the externality must

be weighted against the reduction of consumption benefits. This suggests that

there is an optimal amount of environmental consciousness-building that stops

before equating the marginal reduction of environmental damage to the mar-

ginal cost of education or advertising.

We then examine the optimal choice of emissions and output tax. For a

given level of environmental investment, whether the tax is per unit of output

or pollution is of little importance, since the relation between the two is fixed.

When, however, the choice of such investment is endogenous the two types of

taxes create different incentives for environmental investment and may affect

consumption, pollution, and finally total welfare, in very different ways.1 When

consumers’ behavior is not affected by environmental considerations, simple

intuition suggests that an optimally chosen emissions tax cannot be welfare-

inferior to an optimally chosen tax on output, since it aims directly at the de-

sired target, and therefore produces greater investment incentives; as a result,

the emissions tax provides greater output and cleaner environment. However,

in the presence of environmentally aware consumers the relative performance of

1See Crémer and Thisse (1999) and Constantatos and Sartzetakis (1999) on how the tax-

ation per unit of output affects product specification, which is based on fixed costs.

2

the two taxes becomes more complex. We show that increases in consumers’ en-

vironmental awareness affect negatively the welfare performance of an emissions

tax, while on the contrary, they increase welfare when an output tax/subsidy is

used. As a result, while for low levels of environmental sensitivity an emissions

tax produces higher welfare, at high such levels the output tax must be pre-

ferred. The intuition of these results is simple: as consumers take increasingly

care of the environment by changing their consumption pattern, reductions in

the output tax rate are more effective in increasing total quantity, thus leading

to a) greater consumption value, and b) greater incentives for fixed investments

in pollution reduction.

The discussion of negative externalities in most textbooks leads to the Pigou-

vian tax (one that equates the marginal private benefit of emissions in produc-

tion to the marginal social damage of emissions) levied either on emission or

output, assuming, explicitly or implicitly, that output and emission taxes are

equivalent. This equivalence is based on the assumption that the amount of

emission produced per unit of output is immutable, ignoring the realistic pos-

sibilities of engaging in abatement, employing cleaner production technology or

substituting dirty inputs. Recognizing that a given level of output may yield

different levels of emissions, breaks up the equivalence of output and emission

taxes. A significant literature has been developed examining the optimal choice

of environmental tax instrument in different settings. Schmutzler and Goulder

(1997) compare emissions and output taxes in a partial equilibrium framework

and in the presence of imperfect monitoring of emissions, while Fullerton et

al. (2001) and Cremer and Gahvari (2002) in a general equilibrium framework.

Goulder et al. (1997) examine the interactions with pre-existing distortionary

taxes.

The increasing importance of green consumerism has raised the question of

the appropriate adjustments to the traditional environmental tax and subsidy

policies and also introduced information campaigns and advertising as an addi-

tional policy instrument (see for example Petrakis (2005), Nyborg (2006) and

Sartzetakis (2012)). The literature has approached these issues using different

frameworks. Most of these models assume that green consumers differentiate

products based on their environmental attributes inducing some firms to pro-

duce a greener variety of the product. This differentiation has been examined

mainly within a framework of vertical differentiation (Bansal and Gangopad-

hyay (2003), Garcia-Gallego and Georgantzis (2009), Bansal (2008) and Doni

and Ricchiuti (2013)), and less within a framework of horizontal differentiation

(Conrad (2005)). Alternatively, Gil-Molto and Varvarigos (2013) examine the

case in which environmental consciousness leads consumers to devote resources

to reduce pollution (participation in carbon offsetting schemes, donations to

NGOs, etc). Some of these papers show that a increasing consumers’ responsi-

bility is not always welfare improving and does not even improve environmental

quality.

This paper contributes to the above two parts of the literature by exam-

ining how environmental tax policies perform when consumers are sensitive to

environmental issues. The main contribution of this paper is that it examines

3

the effect of the degree of consumers’ environmental awareness on the choice

of environmental policy. We find that for societies with low levels of environ-

mental sensitivity an emissions tax produces higher welfare, while societies that

have developed high levels of environmental awareness, should prefer an output

tax. Although some other papers also examine the welfare effects of govern-

ment policies in the presence of environmentally conscious consumers,2 they

do comparative analysis with respect to the damage parameter associated with

environmental externality and not the level of environmental awareness.

The rest of the paper is structured as follows. Section 2 presents the model

with an extensive discussion on the construction of social welfare and also lays

the structure of the game. Section 3 presents the firms’ choice of output and

abatement, while Section 4 presents the benchmark free market equilibrium.

Section 5 derives the optimal output and emissions taxes. Section 6 compares

the optimal values of output, abatement and welfare under the two tax regimes.

Section 7 concludes the paper.

2 The model

2.1 Production and pollution

Consider a differentiated product available in two types, each one produced

by a different firm; let = 1 2, denote the output of firm 1 and 2, respec-

tively. Marginal production costs of either type are constant, and for simplicity

normalized to zero. The production of each unit of generates units of some

harmful pollutant. Each firm can abate part (or the entirety) of its pollutant’s

emissions. Let indicate each firm’s total abatement, then firm’s net emissions

are = − ≥ 0. Without loss of generality we set = 1, therefore

∈ [0 ], = 1 2. The cost of abatement is quadratic in the amount of abatedunits of pollutant, = (2) 2 , ≥ 1. The total environmental damage

generated form pollution is,

=

2

⎛⎝X=12

⎞⎠2

(1)

where is a parameter transforming units of emission into environmental dam-

age, that is reductions in social welfare.3

2See for example Bansal and Gangopadhyay (2003), Moraga-Gonzalez and Padron-Fumero

(2002) and Bansal (2008).3The parameter can be seen as either a "social preference" parameter determining how

much society dislikes emissions, a "technical" parameter determining the "objective" damage

caused by emissions, or a combination of both.

4

2.2 Preferences

The pure consumption-utility of the representative consumer from good is

=X=12

− 12

⎛⎝X=12

2 + 212

⎞⎠+ , (2)

where 0, ∈ [0 1) is the degree of substitutability between the two producttypes, and represents the amount of the numeraire-good consumed.4 Con-

sumer income is assumed to be large enough as to rule out corner solutions at

the choice of each type of good , therefore, the presence of will be ignored

hereafter.

We assume environmentally aware consumers taking into account environ-

mental damage. The representative consumer’s willingness-to-pay for polluting

firm’s product is a decreasing function of that firm’s total net emissions and her

utility function is,

e =X=12

(− ) − 12

⎛⎝X=12

2 + 12

⎞⎠=

X=12

[+ ] − 12

⎛⎝X=12

(1 + 2) 2 + 212

⎞⎠ (3)

where ≥ 0 is a taste parameter expressing consumer’s aversion for pollution.5When = 0, consumers do not react to pollution and thus, e = . When

0, e allows for partial internalization of the environmental externality

by adding a vertical differentiation element to consumers’ behavior.6 Given

that is determined taking into account consumer’s preferences, that is, , we

should specify their relationship. It is clear that we cannot allow consumers to

internalize more than 100% of the environmental damage, otherwise it is as if

4When = 0, the two products are completely independent and each firm is a monopolist

in its own market. As → 1, the Lerner index of each firm approaches zero, as in perfect

competition.5The parameter is measured in monetary units per squared units of output (e.g., $2).

Technically, the presence of abatement possibilities corresponds to the introduction of an

element of "product quality": at the expense of some investment (fixed in terms of output)

a firm can increase (strictly speaking: "decrease the reduction of") the consumers’ per-unit

appreciation of its product.6The coomon way to introduce a vertical-differentiation dimension in this type of utility

function is to write it as =

=12

(+ ) − 12

=12

2 − 212where represents

product ’s quality. In a previous version of this work we have used this perceived utility func-

tion with = , i.e., in the eyes of the consumer, total abatement corresponds to an increase

of the quality of a firm’s product. The problem with this formulation is that abatement is

valued for its own sake. For instance, an equal increase in production and abatement provides

greater consumer satisfaction even if the additional quantity thus produced is not consumed

(wasted).

5

they hate emissions for their own sake.7 This means that the private utility

from any quantity consumed may not be inferior to the gross social value of the

same quantity (consumption value minus the value of environmental damage it

creates), i.e., e ≥ −(2), which, in the symmetric case where 1 = 2 = ,

and 1 = 2 = , simplifies to,8

≤ −

≡ e 1 (4)

In the sense that the above condition restricts consumer’s disutility from emis-

sions to the highest damage they can inflict, hereafter we will term it as the

no-prozac condition. We consider e as an upper bound to the admissible

values of . Besides , this upper bound depends upon the equilibrium values

of consumption and abatement; obviously, a necessary condition for e is .

Solving the consumer’s maximization problem, yields the following system

of inverse demand functions,

= (− )− − = (+ )− (1 + ) − = 1 2 6= (5)

With 0 it is only when firm abates all the pollution it creates, = , that

consumers are willing to pay for the good according to the pure-consumption

benefits they receive. Solving the above system for , gives a system of

linear demand functions for = 1 2, 6= ,

=(1 + 2) (+ − )− (+ − )

(1 + 2− )(1 + 2+ ) (6)

2.3 Social welfare

In constructing the social welfare function we assume that environmentally con-

scious consumers do no derive private benefits from acting responsibly. They

sacrifice some of their private benefits in order to reduce environmental degrada-

tion.9 Thus, in formulating social welfare we use (2) instead of (3), subtracting

environmental damage from net emissions and abatement cost,

= − −

= X=12

−⎛⎝12

X=12

2 + 212

⎞⎠−

2

⎛⎝X=12

⎞⎠2

−

2

X2 (7)

7 If this constraint is not satisfied, even if we rule out negative emissions, superfluous (in

terms of cost) abatement may take place, purely for consumer satisfaction.8We do not impose symmetry in the solution, but as it turns out, the solution is symmetric.

In the case of asymmetric solution, (4) would read as ≥ 2[1(1−1)+2(2−2)](1+2−1−2)2 .

9 Independently of its motivation, such behavior may be necessary (and to some extent

"rational", despite the free-riding problem) when government intervention is very limited or

completely absent (the free market case).

6

where =P

=12. A value of 2 implies that, compared to consumption,

environmental damage is of lower significance. As it turns out, parameter

enters multiplicatively in all expressions and plays no role in any of the results,

it is therefore normalized hereafter to = 1, without loss of generality.

Alternatively, in constructing the social welfare function (7), could be

replaced by e . In supporting such formulation one could argue that consumers’"green" behavior is related to private damage from consuming the polluting

product (for instance, personal health damages related to the use of pesticides

used in non-organic produce).10 In such case, the regulator must take into

account the private disutility from contributing to the environmental damage,

separately from the external damage captured by . Compared to that in (7),

this formulation assumes reduced consumption value, it therefore places more

importance on the environmental condition, and as a result, it adds an a priori

bias towards the use of an emissions tax. Despite that bias, all the results in this

paper hold under this alternative specification as well.11 Since the formulation

in (7) clearly avoids any "double accounting" of environmental damages (once as

a reduction of private consumer utility, and once as reduction of social welfare),

in what follows we adopt (7).

The solution of the above model yields very involved expressions and beyond

a certain point, we can only solve numerically. For this reason, hereafter we

assume = 1 and = 0. The first specification constitutes an arbitrary choice

of a parameter’s value with almost no consequences. Setting = 0, implies

that the two goods are in the eyes of the consumer neither substitutes nor

complements. This corresponds to having two unrelated monopolies, thus ruling

out strategic effects between the two firms. Under this assumption, there is no

reason having two firms and thus, hereafter we assume one single monopolist.

Although we are able to check the robustness of our main results with respect

to changes of each one of these parameters, it is impossible to obtain analytical

proofs.

2.4 Game Structure and Government Intervention

We represent the market outcome as a perfect information game between three

strategic players, the regulator and the two monopolists. Each firm is making

two decisions, one about its output and the other about its level of abatement.

We assume that abatement decisions are of long-run nature, and therefore always

precede pricing or output decisions.

10The case where consumers’ conscious behavior is motivated by "warm glow" feelings,

constitutes a special case of private benefits.11The case where consumers’ conscious behavior is motivated by "warm glow" feelings,

constitutes a special case of private benefits. However, the use of (3) instead of (2) in the

social welfare function becomes in this case problematic. When social welfare is computed

using (2), the first best values of and are independent of When using (3) instead, all first

best values depend on The latter implies that the regulator does not decide the optimal

amount of damage strictly based on the objective cost-benefit analysis, but tries also to satisfy

consumers’ desire to "punish" polluting products.

7

The regulator aims at maximizing social welfare by imposing a tax, either

directly on net emissions (emissions tax), or on the total output produced (out-

put tax); we do not allow for a combination of both. Under the assumption

that the production of one unit of output creates one unit of emissions, for

any given level of abatement (exogenous qualities) the two types of taxation are

equivalent, since the effect of one can be replicated by the other through an

appropriate adjustment of the tax-rate. This equivalence may no longer hold

when the abatement decision becomes endogenous.12

Obviously, the government’s commitment power to its policy is of paramount

importance for the result of the market game. We assume that the regulator

can commit to a tax rate before firms make their abatement decisions.13 Hence,

at the first stage the government imposes an emissions or output tax at a given

rate, at the second stage firms decide their abatement level, and at the third

their output or price.

2.4.1 First best

Before turning our attention to the market game, we present the benchmark

case of the first best, in which a benevolent regulator determines both quantity

and abatement level. The socially optimal quantity and abatement (denoted by

a superscript "∗") are obtained through direct maximization of (7),14

∗ =1 + 2

1 + 4 ∗ =

2

1 + 4 (8)

Since for all finite values of , ∗ ∗, environmental damage at the socialoptimum is positive,

∗ =4

(1 + 4)2 (9)

Finally, social welfare at the first best is,

∗ =1 + 2

1 + 4= ∗ (10)

3 Firms’ choice of output and abatement

Assuming a tax rate on output, each firm’s objective function is,

= (− ) − 122 (11)

12See Cremer and Thisse and Constantatos and Sartzetakis for the case of output taxation

in markets described by pure vertical-differentiation models.13The case where firms commit to abatement before the regulator announce the tax rate is

examined in a companion paper.14 It can be easily verified that

22

= −2(1 + 2) 0 and

22

=

−2 (1 + 2) 0

8

while in case of an emissions tax, the corresponding function is,

= − ( − )− 122 (12)

In the "free-market" case where taxes are zero, the two functions above coincide.

Maximizing (11) yields firm’s optimal choice in the output-tax case.

Lemma 1 Under an output tax at a given rate , the monopolist’s optimal

decisions on quantity and abatement are:15

2 =1−

2 + (4− ) (13)

and

2 = (1− )

2 + (4− ) (14)

Proof. Setting = 1 and = 0 into (5) yields,

= (1− )− = (1 + )− (1 + ) (15)

Substituting the value of from above into (11) and performing the maximiza-

tion, we obtain straightforwardly the expressions in the lemma.

Similarly, maximization of (12) yields the monopolist’s optimal choices in

the emissions-tax case.

Lemma 2 Under an emissions tax at a given rate , each firm’s optimal deci-

sions on quantity and abatement are:

2 =(1− ) +

2 + (4− )= 2 +

2 + (4− ) (16)

and

2 = (1− ) + 2(2+ 1)

2 + (4− )= 2 +

2(2+ 1)

2 + (4− ) (17)

Proof. Substituting (15) into (12) yields = ( ) which maximized with

respect to and yields the above expressions.

4 The Free-Market case

In this section we assume no government intervention and we examine the im-

pact of environmental awareness on the free market equilibrium. The literature

has examined the case in which the regulator can manipulate the environmental

awareness parameter by providing information at some cost.

In this paper we are only interested in performing comparative statics analy-

sis with respect to .

15The digit "2" in the superscript indicates second-stage-equilibrium values.

9

Setting = 0, the expressions (17), (16) become respectively:

() =

2 + (4− ) (18)

() =1

2 + (4− ) (19)

Substituting (18) and (19) into (4) we obtain e = 1 − which yields the

highest admissible value of ,

() =

1 + 1 (20)

In what follows, it will be convenient to solve the above in order to obtain for

every , the smallest value of that does not violate the no-prozac condition,

=

1− (21)

The following Lemma establishes the properties of output and abatement with

respect to .

Lemma 3 ∀ ∈ £0 ¤, () is increasing concave, () is decreasing con-vex, and () () ≥ 0.

Proof. The above properties are evident from (18) and (19).

Substituting , from (18) and (19) into the social welfare function (7)

we obtain:

(; ) =(3− 2) + 4(2 + )− (3 + 2)2

(2 + (4− ))2

(22)

Note that for = 0, the sign of depends on the value of : if the soci-

ety places significant importance to emissions ( 32) the market produces

negative net social value. The social planner would like to see this market clos-

ing down, however this does not happen since private surplus and profits are

positive.

The following Proposition summarizes the main results concerning social

welfare and environmental damage in the case of no intervention.

Proposition 1 i) For 13, the social welfare function is concave in , and

∃ () ∈ ¡0 ¢, such that ∗ = ³´represents the maximum of

in the admissible interval of . ii) The function () is increasing concave in

, with lim→∞

= 1. iii) As increases, the admissible region where increases

in have adverse effects on welfare shrinks, but remains positive for any finite

value of . iv) ∀ 0, ∃ () ∈¡0

¢, such that

³

´≡ ∗

³

´, where

∗ is the environmental damage at the first best given in (9).

10

The above Proposition states that increases in consumers’ environmental

awareness, , may not always be welfare enhancing. This is because consumers

face a second best problem where by punishing polluting firms they give them a

two-sided incentive to improve their environmental performance: by increasing

their abatement, but also by reducing their quantity. Our analysis shows that

when consumers have already developed high environmental awareness, further

investing in sharpening their response to polluting products may be detrimental

because of the excessive reduction in quantities. Another conclusion is that

for sufficiently high values of , free-market may lead to cleaner environment

compared to the first-best, even if the externality is only partially internalized.

This is again due to the excessive reduction of quantities. We can show, but

only numerically, that the environmental damage at the optimal value may

be higher or lower than the corresponding first-best damage, depending on the

value of .

5 Optimal taxation

In this section we derive the optimal tax rate levied either on output or emis-

sions. In some cases, the optimal tax rate may be negative, implying a subsidy.

While such a case is not a problem when the rate applies on output, an emissions

subsidy is rather unacceptable and thus, ruled out.

Substituting optimal abatement and quantity from either (14), (13) or (17),

(16) into (7) yields social welfare as a function of , in the case of output or

emissions tax respectively. We can express social welfare in both cases as,

2 =0 + 2

1+ 2

2

[(− 4)− 2]2 = (23)

where all the -coefficients are linear functions of defined as follows: 0 =

3 + (8 − 3) − 2(1 − )2 and =¡0 + 1

¢ = 1 2 = with,

10 = −¡1 + 4− 22¢, 11 = 2(1 − )2, 20 = −

¡1 + 2

¢, 21 = −2 (1− )

2,

10 = −£1 +

¡5− 2+ 2

¢¤, 11 = 2(1−) (3 + 2), 20 = −5

¡1 + 2+ 22

¢,

21 = −2(3 + 2)2.Maximizing the above social welfare function, yields the optimal tax rates in

the case of an output and an emissions tax, which are presented in the following

Proposition.

Proposition 2 The optimal tax rate on output is,

=− (1− 2)− 4 (1 + )+ (1− )2

1 + 2 − 4+ (1 + 2)2 (24)

while the optimal emissions rate is

=−1 + 6 − (5− 2)+ 2 (1− 2)2 − 3

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2 (25)

11

Proof. Maximizing 2 and 2, as expressed in (23) we obtain 2 =

1 + 22 = 0⇔ = − 1

22 = , which yields, almost immediately, the

expressions in (??) and (25). Second order conditions hold whether the tax is

on output or emissions, since 2 22 = 22 0, due to the fact that both

components of 2 are negative whether the tax is on output or emissions.

5.1 Tax or subsidy?

Since the optimal tax rate can be positive (tax) or negative (subsidy), the fol-

lowing lemma determines that it is impossible to have a situation where the

choice is between an output tax and an emissions subsidy:16 either both tax-

rates will be of the same sign, or the optimal output tax rate may be negative

while the optimal emissions tax rate positive

Lemma 4 There exist two values of ,

=(1 + 4) + (1− 2)+ 3

2(1− ) [1 + 2 (1 + )] =

¡1 + 4− 22¢2(1− )2

(26)

such that, ∀ ( ) the socially optimal tax rate, whether on output or

emissions, is positive (negative). When ∈ ( ), the optimal output taxrate is negative while the optimal emissions tax rate is positive.

Proof. Evaluating 2 at = 0 yields 2|=0 = 1 which is positive

(negative) iff ≥ (≤) − 1011. In case of output tax, −10

11=(1+4−22)2(1−)2 =

while in case of emissions tax −1011

= 1+5−22+32(1−)(3+2) = In order to have

we need to show that.

−1011

11−10

=(3 + 2)

¡1 + 4− 22¢

(1− )¡1 + 5− 22 + 3

¢ 1⇔ 2 + 10+ 92 − 73 + 4 0

⇔ ¡1 + 3− 2

¢ ¡2 + 4− 2

¢ 0

which holds ∀ ∈ [0 1] Both and can be shown to be monotonically increasing in , there-

fore ∀ it is possible to define ≡ −1 () and ≡ −1 () Concerning

the former, it is easy to show that = 1 −p3 [2 ( + 1)] but the explicit

determination of is omitted since it involves a third degree equation, and is

therefore quite complex.

16Of course, negative emissions tax rates, i.e., emissions subsidies, make little sense. As

shown below, emissions subsidies are always dominated by output subsidies, and this dom-

inance will hold a fortiori if we replace the negative emissions-tax rate by zero. Hence, we

allow negative emissions-tax rates only for the sake of simplifying the presentation, while in

all policy comparisons we take care that the emissions tax rate be bounded at zero.

12

6 Optimal values under the two tax regimes and

comparison

In this section we derive the optimal values of output or emissions, we compare

social welfare under the two tax regimes and investigate the impact of on the

maximized social welfare. Furthermore, we examine the impact of increases in

on abatement, quantity, environmental damage and total welfare, when no-tax,

a tax on output and a tax on abatement are used and compare the relative

efficiency of each situation. Whether the tax is on output or emissions, in all

cases we consider that it has been adjusted at its optimal level.

6.1 Quantity

By introducing the optimal tax rate from (24) and (25) into (13) and (16),

respectively, and simplifying we obtain optimal quantities under each tax type:

Lemma 5 ∀ ≥ 2, ∈ [0 1), the monopolist’s output under an optimally se-lected output- and emissions-tax rate, is,

( ) =1

1 + 2 − 4+ (1 + 2)2 (27)

and

( ) =6 + 3 + (1 + 4)+ 2

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2 (28)

respectively.

Proposition 3 i) ∀ ∈ [0 1), 0, ii) ∀ ∈ £0 ()¤, 0,

and iii) ∀ ∈ [0 1), 0.

Proof. For part i), we see that the numerator of () in the RHS of (19) is

constant in while the denominator is 2 + (4− ) which is increasing in ,

∀ ∈ [0 1). For part ii), from the RHS of (27) we have,

= − −4 + 2 (1 + 2)¡

1 + 2 − 4+ (1 + 2)2¢2 which has the opposite sign from its numerator. The latter is positive only

when ≥ 2 (1 + 2) (1 + ) = (), and negative otherwise. Finally,

the derivative of the RHS of (28) is,

= −25 + 94 + 72

2 + 2¡25 + 66 + 242

¢+ 8

¡3 + 42

¢2¡

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2¢2 0

Monopolist’s output as function of for both types of taxes and the two

benchmark cases is depicted in the following diagram. Although Figure 1 is

13

Figure 1: Equilibrium quantity as function of The Figure has been traced

for the following parameter values: = 1 = 0 = 1 and = 2. It illustrates

the following cases: first-best (green line), free market (brown line), emissions

tax (cyan line), and ouput tax (blue line). All lines are traced up to = = 23.

plotted for = 2, from our results hitherto, the basic structure of the diagram

is independent of .

It is natural to expect that without tax, increases in social responsibility

lead to reduced output in equilibrium. However, such negative relation between

and quantity would be a priori puzzling, since we expect optimal tax rates

—whether on emissions or output— to decrease with increases in . Yet, in the

case of emissions tax, such negative relation is indeed the outcome. To better

understand why, we decompose the change in quantity following a change in

in two terms, the one presenting a direct effect, and the other an indirect effect

through ,

2( ())

=

2( )

(−)

|=() +2( )

(+−)

()

(−)

= (29)

The first term of the right hand side (RHS) of the above equation shows the

direct effect of an increase in on quantity, when taxes are held constant. This

is the only effect present in the free-market case and is expected to be negative,

because an increase in makes the good less attractive and consumers are willing

to consume less of it. The second term of the RHS shows how environmental

consciousness affects quantity through its impact on the optimal tax rate. This

effect can itself be decomposed in two terms, the first showing the sensitivity of

quantity to changes in the tax-rate, and the second showing the effect of on

the optimal tax-rate. Starting from the latter, it can be shown to be negative

14

regardless of the type of tax imposed, as suggested by intuition: an increase

in implies a more substantial internalization of the environmental externality

and therefore, a reduced need for government intervention.17

It is the term 2( ) that makes the difference between the two tax

types and which corresponds to the tax coefficient in the determination of output

in (13) and (16). From (13) and (16) we have,

2( )

= − 2 + (4− )

[2 + (4− )]2 0

and2( )

=

2 (1 + 2)

[2 + (4− )]2 0

In the case of output tax, both terms to the right of the positive sign in (29)

are negative and their product positive, reinforcing the direct effect of In the

case of emissions tax the indirect effect is negative, working against the direct

effect; part ii) in the above Proposition shows the indirect effect to be stronger.

6.2 Abatement

By introducing the optimal tax rate from (24) and (25) into (14) and (17),

respectively, and simplifying we obtain optimal abatement under each tax type,

presented in the following Lemma.

Lemma 6 ∀ ≥ 2 ∈ [0 1), the monopolist’s abatement under an optimallyselected output- and emissions-tax rate, is,

( ) =

1 + 2 − 4+ (1 + 2)2 (30)

and

( ) =−1 + 6 + (4 − 2)+ 32

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2 (31)

respectively.

Proposition 4 ∀ i) 0 ii) 0 but iii) 0

17The effect of on the optimal emissions-tax rate is:

()

=

22 +1 +0

[2(2+ 3)2 + 5 [2(+ 1) + 1]]2 0

where 0 = (−10) 4 + 23 − 552 − 4+ 15 1 =

(−2) 44 + 243 − 72 + 36+ 68 0 and 2 = (−80) 2 + 3+ 225 0 and

the denominator is obviously positive. It can be shown that this effect is always negative,

again as expected. A full analytical proof is very tedious and consists of showing that one root

of the trionyme of the numerator above is negative ∀ ∈ [0 1) while the other is increasingin thus taking its maximum value at = 1 The corresponding maximimum value of the

root is 20−1−5 +√65

1 hence, ∀ ≥ 2 the trionyme has the sign of 2 i.e., negative.

15

Proof. From (18) we get () = 2¡1 + 2

¢ [2 + (4− )]

2 0. For

part ii), taking the derivative of the RHS of (30) with respect to yields,

=

(2+1)(1−2)(2(−1)2+2+1)2 0. For part iii) the derivative of the RHS of (31) with

respect to yields,

=−82(2+ 3)2 + 4 ¡122 + − 13¢+ 50(+ 1)

(2(2+ 3)2 + 5(2(+ 1) + 1))2

the denominator of which is positive. The numerator can be written as a tri-

onyme Λ (; ) = 2+¡−962 + 4 + 50¢−722−52, where = ¡−322 + 48 + 50¢,

itself a trionyme in . The discriminant of is ∆ = 8704 =¡16√34¢2 and

its roots are 1 2 =14

¡3∓√34¢ with 1 0 2 0. Hence () 0

iff () 14

¡3 +√34¢; when = 1

4

¡3 +√34¢ Λ (; 2) becomes a linear

function of , i.e., Λ (; 2) = −52£2¡41 + 7

√34¢+ 16

√34 + 93

¤ 0. For

6= 2 we solve Λ (; ) = 0, for ; the discriminant of Λ (; ) is ∆Λ =

100¡643 + 1482 + 108 + 25

¢ 0, therefore we have to consider two roots,

Λ1 Λ2 =

µ−25− 2 + 482 ∓

q(482 − 2 − 25)2 + 4(18 + 13)

¶. We

show first that ∀ 0 Λ1 0: its numerator is positive (negative) iff

() 14

¡3 +√34¢, therefore it always has a sign opposite to that of its de-

nominator. The numerator of the second root, Λ2, is positive ∀ 0, thereforethe root has the sign of its denominator, . When 0, Λ2 0, therefore

∀ 0 we are outside of the roots and Λ (; ) has the sign of , i.e., negative.

When 0 0 Λ2 1, hence ∀ ∈ [0 1) Λ (; ) has the opposite sign of i.e., it is again negative. Thus, ∀ ≥ 1 0 ≤ 1 irrespective of the sign of

Λ (; ) 0 and therefore 0

The monopolist’s abatement as function of for both types of tax and the

two benchmark cases is depicted in the following diagram for = 2.

Again we can expect that in both, the no-tax and output-tax cases when

= 0 firms undertake no abatement, but as social consciousness increases, they

do so. In the case of no intervention, abatement is an increasing concave function

of while in the case of output-tax it is increasing convex, since output also

increases with . What is again surprising is the emissions tax case, in which

increasing leads to less abatement. The answer for this paradox is to be found

once more on the reduction of the optimal tax rate, together with the marginal

impact of a change in tax-rate on a monopolist’s abatement decision. Using

(14) and (17), let us write:

2( ())

=

2( )

(+)

|=() +2( )

(+−)

()

(−)

= (32)

16

Figure 2: Equilibrium abatement as function of The Figure has been

traced for the following parameter values: = 1 = 0 = 1 = 2, and

represents the following cases: first-best (green line), free market (brown line),

emissions tax (cyan line), and ouput tax (blue line). All lines are traced up to

= = 23.

We distinguish again a direct and an indirect effect through . The former is

positive, and the only effect present in the free-market case.18 The latter is again

composed of two terms, the second being the same as in (29) and negative, as

shown earlier. Concerning the first term of indirect effect, note that,

2( )

=

3+ 2

2 + (4− ) 0

while2( )

=

−2 + (4− )

0

Again, the indirect effect reinforces the direct effect in the case of an output

tax, making the abatement function increasing at a higher rate than in the free-

market case, while on the other hand makes the response of firms to the optimal

emissions tax decreasing in .

18 In case of an emission tax we have

2( )

|=() =

2(+ 2)(2+ 3) + 33 + + 6

[(4− )+ 2][2(2+ 3]2 + 5 [2(+ 1) + 1]

0

while in case of output tax

17

6.3 Environmental damage

It has been already shown in the previous section that for the relevant range

of ∈ £0 ¤, is decreasing.19 When a tax on output is imposed, both

quantity (i.e., emissions) and abatement are increasing in , while with a tax

on emissions they are both decreasing. Thus, in either case nothing can be said

a priori about the evolution of their difference and the resulting environmental

damage.

Proposition 5 i) when the tax is on emissions, ∀, () ∗, and () 0; ii) when the tax is on output, ∈ £0 ¤, () ∗, and

∃ ∈ £0 ¤ = e, such that ∀ ≤ (≥) e, () ≥ (≤) 0, and when

= e, () reaches a maximum.

Proof. Substituting (31) and (28) into the definition of , we get

=

Ã2¡4 + 3− 42¢

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2

!2= 2

with− 2 [2(2+ 3)(18+ 7) + 25(2(+ 2) + 1)]£

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2¤2 0

Evaluating |=0 = 8 (18 + 5) √∗ = 2 (4 + 1), and since ∗ does not

depend on ∀ ∗. Next, we show that the environmental impact ofan output tax is not monotonic in . Using and from (30) and (27),

respectively, we can calculate social damage as,

=

µ2(1− )

1 + 2 − 4+ (1 + 2)2¶2= 2

Deriving the above with respect to yields

= −2 (1− 2) + (1 + 2)¡2− 2

¢£1 + 2 − 4+ (1 + 2)2¤2

which is positive when 0 ≤ e = 1 −p2 (2 + 1) = 1 −

q

(1 + ) = 1, ∀ 0 and negative otherwise. Thus, ∀ ∈h0 ei, en-

vironmental damage under the optimal output tax is monotonically increasing,

reaches a maximum at = e and is monotonically decreasing ∀ ∈ he i.2019 It can also be deduced by the previous analysis which shows that in the no intervention

case, as increases, total output decreases while total abatement increases.20 It can be easily shown that environmental damage reaches its first-best level at ie.,

when social welfare using output tax reaches also its first best level. Since lies outside

the range of admissible values of the environmental damage under an output tax is always

larger compared to the first-best solution.

18

Figure 3: Equilibrium social environmental damage, as function of

The Figure has been traced for the following parameter values: = 1 = 0

= 1 = 2, and represents the following cases: first-best (green line), free

market (brown line), emissions tax (cyan line), and ouput tax (blue line). All

lines are traced up to = = 23.

Evaluating |=0 = 4 (2 + 1)2 4(4 + 1)2 = ∗, and |= = 4( +

1)2 [2( + 2) + 1]2 ∗, ∀ 0

The social damage as function of for both types of tax and the two bench-

mark cases is depicted in the following diagram for = 2.

As expected, the emissions tax yields the lowest social damage, which lies

below the first best. The output tax yields higher damages even relative to the

free market case for high values of .

6.4 Social welfare

In this Section we compare social welfare under different regimes. In the free-

market case, social welfare is expressed in (??). Substituting the optimal tax

rates from (24) into the social welfare function (23) yields the maximized welfare

corresponding to the use of an output, or an emissions tax.

Lemma 7 ∀ ≥ 2, ∈ [0 1) social welfare under an optimally selected output-and emissions-tax rate is,

( ) =1

1 + 2 − 4+ (1 + 2)2 (33)

( ) =4 (1 + 2) (1 + ) + 2

5 + 18 + 2 (5 + 12)+ 2 (5 + 4)2 (34)

19

respectively. As expected, is monotonically decreasing in and, while less

obvious, the same can be shown to hold for .

We now turn our attention to the effect of on social welfare and the com-

parison of social welfare under the two tax regimes. Starting with the case of

output tax, the following Proposition summarizes the effect of on .

Proposition 6 Under an output tax regime: i) ∀, () ≥ (); ii)

∃ = 21+2

∈ (0 1) such that, ∀ ≤ (≥) ≥ (≤) 0 and³´=

∗; iii) .

Proof. In order to prove i) we simply manipulate the difference of the corre-

sponding expressions (33) and (??) to obtain,

− =

£(1− 2)− 22(1 + ) + 4(1 + )

¤2£2(1 + 2)− 2

¤2 £2(1− )2 +

¡1 + 2

¢¤ which is obviously positive, unless its numerator becomes zero. It can be shown

that the latter only happens when = , i.e., () = 0. In order to prove ii)

we manipulate the derivative of (33) to obtain:

=

2 [2 − (1 + 2)]£(2 + 1) + (− 4) + 22¤2 ≥ () 0

when ≤ () 2 (1 + 2) ≡ 1 Substituting into in (33) obtains

straightforwardly that ³´= ∗ Part iii) is straightforward.

No matter the degree of social consciousness, the imposition of an output-

tax at the optimally chosen rate improves welfare, compared to no intervention,

unless the optimal tax rate is zero ( happens to be the implicit solution of

the equation = ). While the () does eventually curve down (exactly

like the () curve), if one accepts the no-prozac condition as an implicit

upper bound on , , and therefore lies outside the admissible interval

of -values.21 Within that interval, the efficiency of the optimal output tax in

terms of improving welfare is always enhanced as increases, contrary to both,

the non-intervention case, and the emissions-tax case (see below).

Let us for the sake of argument, ignore the no-prozac constraint, leaving

only bounded by 1 (second order conditions). In this case, an optimally selected

output tax complemented with the appropriate level of social consciousness can

reach first-best welfare. Note that is itself decreasing in which implies

that as consumers become more conscious, the tax rate must be lowered, and

eventually become negative. Indeed , it can be shown that () 0, i.e.,

the optimal level of social consciousness must be complemented by an output

subsidy in order to reach first best. We face here a typical situation where

21Contrary to the free-market case where

20

two instruments, and deal with two targets,22 optimal quantity and optimal

abatement. While this combination works in the case of an output tax, it cannot

work in the case of an emissions tax.

The following Proposition states that an emissions tax always improves wel-

fare relative to the no-tax outcome, but its efficiency is reduced as social con-

sciousness rises.

Proposition 7 ∀, i) () (); ii) ¡ ()

¢ 0.

Proof. Tedious calculations show that,

− =

©2¡22 + − 3¢+ [(− 2)+ 5) + 1]ª2

[(− 4)− 2]2 {2(2+ 3)2 + 5 [2(+ 1) + 1]} 0

For the second part, we can write (34) as,

( ) =4 (1 + 2) + 4 (1 + 2)+ 2

5 + 18 + 2(5 + 12)+ (8 + 10)2

The above expression is the ratio of two trionymes that are monotonically in-

creasing in , and that of the denominator has all its coefficients greater than

the corresponding ones in the numerator. More conventionally,

∝ − ¡322 + 44 − 15¢2−¡642 + 94 + 35¢−2 ¡122 + 16 + 5¢ 0

The striking part of the above Proposition is that as increases, social wel-

fare is not increasing in some concave way, but is decreasing. However, following

the analysis of equilibrium quantity and abatement this should no longer come

as a surprise. Increases in require lowering of , which in turn implies lower

quantity and abatement, in contrast to lowering . Hence, when the tax-rate

is optimally adjusted to increases in individual environmental consciousness,

increases of the latter monotonically reduce social welfare. If the regulator is

determined to use an emissions tax and can choose both the levels of and ,

the optimal solution is first to choose = 0, and then (0 ). When is given

at the moment of the decision about a tax intervention, an emissions tax seems

suitable in situations where social consciousness is small, or non-existent.

Finally, the following Proposition compares social welfare under the two tax

regimes.

Proposition 8 i) ∀ ≥ 2 ∃ =◦ () ∈ ¡0 ¢ such that ∀ ≤ (≥) ◦ () ≥

(≤) () ; ii) provided that is not too high ( ),◦ ≤ min

©

ª

where satisfies with equality the no-prozac condition (4) in the case of an

emissions tax.

22Although we are not explicitly considering the case that the government influences ,

allowing it to vary could be considered as an additional policy instrument.

21

Figure 4: Equilibrium social welfare, as function of The Figure has been

traced for the following parameter values: = 1 = 0 = 1 = 2, and

represents the following cases: first-best (green line), free market (brown line),

emissions tax (cyan line), and ouput tax (blue line). All lines are traced up to

= = 23.

Proof. From (33) and (34) we obtain,

( ) =

=

2(2+ 3)2 + 5(2(+ 1) + 1)¡2(− 1)2 + 2 + 1

¢(8(+ 1) + (+ 2)2)

We can easily check that ∀ ≥ 2 (0 ) = (18 + 5) [(2 + 1)(8 + 4)] 1,

and (1 ) = (50 + 25) [2(16 + 9)] 1. More tedious but still feasible is to

show that

¡

¢=505 + 1854 + 2583 + 1662 + 48 + 5

325 + 1304 + 1883 + 1212 + 36 + 4 1

since all the coefficients of the numerator are greater than the corresponding ones

of the denominator and 0. Since we know that ∀ () is monotoni-

cally decreasing, while ∀ ∈ £0 ¢ () is monotonically increasing, with

(0) (0), and¡¢

¡¢they must have a unique intersection

at =◦ ().

Part ii) of the proposition requires that◦ () also respects the no-prozac

condition in case of emissions tax as well as the positivity of . The analytical

proof of this part is extremely tedious, so we content ourselves to a numerical

proof. Note that for a given the proof is complete in that it accounts for all

possible values of ( ) Let◦ () =

◦−1indicate for any the value of which

22

combined with that yields () = () ; () is defined in (26),

while () = −1 The following figure shows ( ) combinations. The blue

line corresponds to () the cyan line to −1 the red line to () and the

green line to◦ () The lower yellow line corresponds to = 2 (an exogenously

imposed constraint) and the higher yellow line, to i.e., the value of at which

() =◦ () As we can see, ∀ ≥ 2 the green line lies to the left of the blue

line, leaving the set ∆ of values that are simultaneously to the right of the

green line (the output tax yields higher welfare than the optimal emissions tax)

and to the left of the blue line (the no-prozac in the case of no-, or output-tax)

non-empty. We also see that ∀ ≥ 2 points in that set also satisfy the constraintthat the rate of an emissions tax cannot be negative. If we limit ourselves in

∈ [2 ) there is a subset of ∆ with elements that also satisfy the no-prozac

condition in the emissions-tax case, which is the most stringent condition ≥ 2

The above Proposition shows that there are values in the ( ) space, for

which welfare could be higher or lower under an emissions tax. Therefore,

even when respecting the no-prozac condition and the emissions tax is positive,

welfare superiority of either tax depends of the level of consumers’ environmental

consciousness.

7 Conclusions

The present papers examines the effect that the presence of green consumers

has on firms’ choice of output and abatement and the choice of environmen-

tal policy. We first find that even without policy intervention, reaching high

levels of environmental responsiveness by consumers could reduce welfare when

market imperfections are quite prominent. We also find that while an emis-

sions tax increases welfare relative to the free-market case, the overall welfare

performance of an optimally chosen such tax is monotonically decreasing in con-

sumers’ sensitivity to environmental issues. Furthermore, an output tax is also

welfare improving relative to free market, but when it is imposed in markets

where consumers’ environmental consciousness is not developed, its result is

welfare inferior to that of an optimally chosen emissions tax. However, social

welfare under the imposition of an optimally chosen output tax is monotonically

increasing in the level of environmental consciousness. Eventually, in markets

where consumers’ environmental consciousness has reached high levels, social

welfare is higher under the use of an output subsidy instead of an "emissions"

tax (both selected optimally).

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25