selective penalization of polluters: an inf-convolution approach

DOI: 10.1007/s00199-003-0433-9Economic Theory 25, 421–454 (2005)

Selective penalization of polluters:an inf-convolution approach

Ngo Van Long1 and Antoine Soubeyran2

1 Department of Economics, McGill University, 855 Sherbrooke St W., Montreal,Qc, H3A 2T7, CANADA (e-mail: [email protected])

2 GREQAM, Universite de la Mediterrane, Chateau La Farge, Route des Milles, Les Milles, FRANCE

Received: June 12, 2002; revised version: September 4, 2003

Summary. We consider an asymmetric polluting oligopoly. We demonstrate thatoptimal tax rates per unit of emission are not the same for all firms. We call thisproperty selective penalization. Our Optimal Distortion Theorem states that theefficient tax structure requires that high cost firms pay a higher tax rate. Our Pro-concentration Motive Theorem states that optimal taxes increase the concentrationof the industry, as measured by the Herfindahl index. Our Magnification Effectindicates that the variance of marginal costs is magnified by a factor which dependson the marginal cost of public funds.

Keywords and Phrases: Pollution, Environmental regulation, Oligopoly.

JEL Classification Numbers: Q20, D60, D63.

1 Introduction

When production generates pollution as a by-product, competitive firms over-produce in the sense that marginal social cost exceeds price. Under perfect compe-tition, a Pigouvian tax equal to marginal damage cost is called for. This rule applieswhether firms are identical or not. Thus, under perfect competition, all pollutingfirms are treated fairly by a regulator who seeks to achieve efficiency: the tax rateper unit of emission is the same for all firms, even under heterogeneity of produc-

We wish to thank Peter Neary, Kim Long, Raymond Riezman and a referee for very helpfulcomments. Financial support from SSHRC and FCAR are gratefully acknowledged. We would like tothank Hassan Benchekroun, Kim Long, and Koji Shimomura for discussions and comments.Correspondence to: N. Van Long

422 N. Van Long and A. Soubeyran

tion costs. When the market is not competitive,1 however, as we will show below,it is no longer true that efficiency can be achieved by a uniform tax rule.

In this paper, we consider an asymmetric polluting oligopoly: firms have differ-ent production costs, and their pollution characteristics may also be different. Wewill demonstrate that, in this case, optimal tax rates per unit of emission are not thesame for all firms. We call this property “selective penalization”, or “favoritism inpenalties.” Thus, the “efficiency” objective may be served only at the expense of“fairness.” We do not propose to resolve the issue of fairness, or equity, here. Ourobjective is to analyze the direction of “favoritism in penalties” and to establishprecisely how the efficiency-inducing firm-specific tax structure depends on thestructure of (heterogenous) production costs, and heterogenous pollution-outputratios. We determine which types of firms (low cost, or high cost firms) should bepenalized more.

Asymmetry is important, because it is a prevalent real world feature, and be-cause it introduces another source of distortion: in a Cournot equilibrium, marginalproduction costs are not equalized across firms, resulting in production inefficiencyat any given total output. In this context, pollution taxes or pollution standards mustseek to remedy both the environmental problem and the intra-industry productioninefficiency problem. With asymmetric oligopoly, the regulator would want to beable to correct distortion on a firm-specific basis. While de jure differential treat-ments of firms in the same industry (in the sense that different standards applyto different firms), may be politically unacceptable in most economies, de factodifferential treatments (e.g., different degrees of enforcement and verification) maybe feasible. In what follows, whenever the terms “firm-specific tax rates”, or “firm-specific pollution standards” are used, they should be interpreted in the de factosense.

The objective of this paper two-fold. Our first aim is to characterize the structureof optimal firm-specific emission tax rates, and to provide an intuitive explanation ofour results on differential treatments. Here, we go a step further than just establishingthe conditions for unequal treatment of equals, which have been provided elsewhere(Salant and Schaffer, 1996, 1999; Long and Soubeyran, 1997a,b, 2001a,b). Wefully characterize the direction of bias. Our second aim is to highlight the conflictbetween efficiency and equity. We show that, in the case of taxation, firms that areequally polluting (i.e., their emissions per unit of output are identical) may be taxeddifferently, when their production costs are different. In a sense, this is inequitable.In the case where regulation is by means of pollution standards, we again show thatefficiency may require that different standards be imposed on heterogenous firms.

One of our main results is the Optimal Distortion Theorem. We show thateven in the case where the rates of emission per unit of output are identical forall firms, the efficient tax structure requires that high cost firms pay a higher tax

1 Buchanan (1969) and Barnett (1980) have shown that the optimal tax per unit of emission undermonopoly is less than the marginal damage cost (and it can be negative). Katsoulacos and Xepapadeas(1995), consider the case of a symmetric polluting oligopoly (i.e., they assume that firms are identical)and show that if the number of firms is endogenous and if there are fixed costs, the optimal Pigouviantax could exceed the marginal damage cost, because free entry may result in an excessive number offirms.

Selective penalization of polluters: an inf-convolution approach 423

rate on emissions. Our result implies that the efficient tax structure favors theefficient firms, but the magnitude of the favors is a decreasing function of themarginal cost of public funds.2 More generally, we define the Pigouvian distortionfor firm i as the difference between the tax rate on its output and the adjusted3

marginal pollution damage caused by an extra unit of its output, and show that theoptimal Pigouvian distortion for firm i is greater than that for firm j, if and onlyif the marginal social cost of firm i is greater than that of firm j. (Marginal socialcost is the sum of marginal production cost and adjusted marginal damage cost.)Another characterization of optimal tax structure is our Pro-concentration MotiveTheorem. Optimal taxes penalize the inefficient firms more, and thus increases theconcentration of the industry, as measured by the Herfindahl index. In fact, we showthat the variance of the distribution of the firms’ tax-inclusive marginal costs afterthe imposition of efficient taxes exceeds the variance that would be obtained if therewere no taxes. We call this the Magnification Effect: the variance of marginal costsis magnified by a factor which depends on the marginal cost of public funds. Froma mathematical point of view, our result has a nice geometric interpretation. Weshow that optimal taxes can be obtained as a solution of minimizing the distancebetween a hyperplane and a reference point. Our approach is an application of theduality theory using conjugate function and inf-convolution.

Our main focus in on firm-specific taxation on emissions. A brief section onfirm-specific standards is included to show the general applicability of our method.4

Our derivation of optimal taxes and optimal standards (in slightly different models)shows that there is a unified framework for analyzing firm-specific penalties.

In the models we present below, we use a two-stage game framework. In thefirst stage, the regulator sets firm-specific emission taxes or standards. In the secondstage, firms compete in the final good market. To fix ideas, we focus on the casewhere firms produce a homogenous good, and compete a la Cournot. However, ouranalysis can easily be adapted to deal with other cases, such as Bertrand competitionwith differentiated products, spatial competition (as in the Hotelling model), andeven markets in which some firms are Stackelberg leaders.

The games considered in this paper belong to the class of games called “costmanipulation games with costs of manipulating”(see Long and Soubeyran, 2001a).The regulator uses the chosen policy instrument to affect, on a discriminatory basis,the marginal costs of individual firms. This in turn affects their equilibrium outputsand market shares. The costs of manipulating can take different forms. In the caseof taxation, when the marginal cost of public funds exceeds unity, these costsinclude the loss of tax revenue when the regulator changes the tax structure. Inthe case of standards, firms are induced to acquire costly equipment to reduce the

2 See Browning (1976) and Ballard et al. (1985) for exposition of this concept, and for empiricalestimates of the marginal cost of public funds for the US economy. See also Section 2.3 below.

3 An adjustment factor is applied to account for the marginal cost of public funds.4 We do not seek to compare taxes to standards, because there exists already a large literature on that

subject. In fact, the model we develop to analyze optimal standards is quite different from the modelwe use to analyze optimal taxes, and therefore it would not make sense to discuss, using our results, therelative attractiveness of these two policy measures.


pollution generated by their production process. Such equipment alters the marginalproduction costs.

We are able to provide a unified treatment of firm-specific pollution policiesbecause we transform variables in such a way that different modes of intervention(in distinct models of emission generation) can be seen to have the same basicstructure. We show that maximizing the first stage objective with respect to oneof the environmental instruments (such as Pigouvian taxes, specific pollution stan-dards) is equivalent to choosing Cournot equilibrium quantities. This is because thediscriminatory use of policy instruments in the first stage amounts to the same thingas manipulating the marginal costs of production which in turn affect second-stageequilibrium outputs.

2 Selective penalization by Pigouvian taxes

In this section present our basic model of an asymmetric polluting oligopoly, andderive the optimal firm-specific Pigouvian taxes. Here, our major task is to showhow the optimal firm-specfic taxes are related to the structure of heterogenouscosts and heterogenous emission-output ratios. Our main results are summarizedin Proposition 3 (Optimal Distortion Theorem). According to this theorem, in aoligopoly consisting of firms with non-identical costs, the optimal Pigouvian taxfor each firm must deviate from its adjusted marginal damage per unit of output,and such deviations vary among firms: the deviation should be the greatest for themost inefficient firm, and lowest for the most efficient firms. In general, inefficientfirms are penalized more, relative to efficient firms.

2.1 The basic model

We consider a polluting oligopoly consisting of n non-identical firms producing ahomogenous final good. Let I = 1, 2, ..., n. The total output of the final good isQ =

∑i∈I qi. The inverse demand function for the final good is P = P (Q) where

P ′(Q) < 0. In this model, emission Ei is proportional to output: Ei(qi) = eiqi.In general, ei = ej . This is the first source of heterogeneity. Firm i has productioncost ci(qi). The subscript i in ci(.) indicates that in general firms are heterogenousalso in production cost. Firms sell their good in the same market place, but they arelocated at different points. We represent this third source of heterogeneity (distancefrom the central market place) by assuming that firm i must incur a transport costdi per unit of output (in general di = dj). We assume that firm i must pay a tax tiper unit of its emission. (The regulator must set the tax rates optimally.) The profitfunction is

πi = P (Q)qi − ci (qi) − tieiqi − diqi

We define

τi ≡ tiei (1)


so that τi is the tax per unit of output of firm i. It is convenient to define ωi ≡ di+τi.The profit function becomes

πi = P (Q)qi − Ci(qi, ωi)

where Ci(qi, ωi) is the (tax-inclusive) total cost function:

Ci(qi, ωi) ≡ ωiqi + ci (qi)

The (tax-inclusive) average variable cost function is:

ρi(qi, ωi) = ωi +ci (qi)

qi(2)

and the (tax-inclusive) marginal cost function is

θi(qi, ωi) = ωi + c′i(qi) (3)

The difference between θi and ρi , defined as µi, measures the degree of convexityof the cost function. We have

µi = θi − ρi = c′i(qi) − ci(qi)

qi

If ci(.) is linear, then µi equals zero identically. If ci(.) is strictly convex, then µi

is positive for all positive output levels.

Remark. Our total tax-inclusive cost function Ci(qi, ωi) has the property that∂Ci(qi, ωi)/∂ωi = qi. This property is crucial for the analysis that follows. Therecan be situations in which the transport cost or taxes are not proportional to output.In such cases, our results would have to be modified.5

We will show how the government can optimally manipulate the tax-inclusivecosts of the firms so as to maximize social welfare. To do this, we set up the problemas a two-stage game. In the first stage, the government sets firm-specific taxes, andin the second stage, firms compete as Cournot rivals, taking tax rates as given. Asusual, to solve for the optimal taxes, we must first analyse the equilibrium of thegame in stage two.

2.2 Stage two: Cournot equilibrium given tax rates

To ensure the existence and uniqueness of Cournot equilibrium, we make the fol-lowing assumptions:6

5 We thank a referee for pointing this out.6 These assumptions are identical to those made by Gaudet and Salant (1991). Assumption 4 states

that if the marginal cost function has a negative slope, that slope is less negative than the slope of thedemand curve. Asumptions 4 and 5, taken together, ensure that each firm’s objective function is strictlyconcave in its output. As they pointed out, Assumption 5 is overly strong, in that condition (3) of theirtheorem only requires that the sum of responses of firm outputs to industry output (i.e., the sum ofthe right-hand sides of our Equation (6) over all active firms) is less than unity, while Assumption 5ensures that each term in the sum is non-positive (the denominator of each term being strictly positive byAssumption 4). The theorem of Gaudet and Salant applies to the linear cost case, as well as non-linearones. For the case of strictly convex cost, there is an alternative proof using contraction mapping, seeLong and Soubeyran (2000). For a theorem that ensures that all firms produce strictly positive outputsunder non-identical constant marginal costs, please see Appendix A4.


Assumption 1. There exists a Q ∈ (0,∞) such that P (Q) > 0 for Q ∈ [0, Q),

and P (Q) = 0 for Q ∈ [Q,∞).Assumption 2. P (Q) is twice-continuously differentiable and P ′(Q) < 0 forQ ∈ [0, Q

).

Assumption 3. ci(qi) is twice-continuously differentiable and c′i(qi) > 0 for all

qi > 0, i ∈ I .

Assumption 4. For all Q ∈ [0, Q)

and i ∈ I , there exists some η > 0 (possiblydependent on Q and i) such that c′′

i (qi) − P ′(Q) ≥ η > 0 for all qi ≥ 0.

Assumption 5. P ′(Q) + qiP′′(Q) ≤ 0 for all Q ∈ [0, Q

)and qi ∈ [0, Q].

The first order condition for an interior equilibrium for firm i is

∂πi

∂qi= P ′(Q)qi + P (Q) − ωi − c′

i(qi) = 0, i ∈ I (4)

It is convenient to express the equilibrium output of firm i, denoted by qi,i ∈ I ,as a function of the equilibrium output of the industry, Q, and of the parameters offirm i’s (tax-inclusive) cost function:

qi = qi

(Q, ωi

)(5)

Assumption 4 ensures that qi

(Q, ωi

)has continuous derivative with respect to Q

for Q ∈ [0, Q)

and it is given by

∂qi

∂Q=

P ′(Q)

+ qiP′′(Q)

c′′i (qi) − P ′

(Q) , i ∈ I (6)

Inserting (3) and (5) into (4), we obtain

P ′(Q)

qi

(Q, ωi

)+ P

(Q)

= ωi + c′i

[qi

(Q, ωi

)](7)

Summing (7) over all i, we obtain the identity

P ′(Q)

Q + nP(Q)

= nωI +∑i∈I

c′i

[qi

(Q, ωi

)](8)

where ωI ≡ (1/n)∑

i∈I ωi. Equation (8) indicates that the equilibrium output canbe determined from the knowledge of the ωi’s. Thus we write

Q = Q(ω) (9)

where ω ≡ (ω1, ω2, ..., ωn).We now express the equilibrium profit of firm i as follows

πi =(P − ρi

)qi =

[(P − θi

)+(θi − ρi

)]qi


=[−P ′

]q2i + µi (qi) qi

=[−P ′

]q2i + qic

′i(qi) − ci(qi) (10)

where we have made use of the Cournot equilibrium condition

P − θi =[−P ′

]qi (11)

Expression (10) deserves some comments. Since the profit expression in (10) incor-porates the Cournot equilibrium condition (11), it indicates that, in the first stage ofthe game, while the government can manipulate the qi and Q via the choice of thepolicy parameters τi, it cannot violate the Cournot equilibrium condition. (Techni-cally, this is very much like the incentive compatibility constraint in principal-agentproblems: the principal cannot ignore economic agents’ equilibrium conditions.)We now turn to a complete analysis of the first stage of the game.

2.3 The first stage: optimization by the government

The objective of the government is to maximize a weighted sum of profits, con-sumers’surplus, and tax revenue, minus the damage cost caused by pollution. Sinceonly relative weights matter, we normalize by setting to unity the weight assignedto profits.The weight given to consumers’ surplus is β where β ≥ 0. For example,if the output of the industry is exported, then a fraction of the consumers’ surplusaccrues to foreigners, and β would be smaller, the greater is the fraction exported.The weight γ ≥ 1 is a measure of the marginal cost of public funds.

In the public finance literature, the concept of marginal cost of funds plays animportant role in the evaluation of public projects. For example, if γ = 1.17, thena public project must produce marginal benefit of more than $1.17 per dollar ofcost if it is to be welfare improving. This is because each dollar of revenue raisedthrough distortive taxation (such as income tax) creates a deadweight loss of $0.17.Similarly, since a dollar of revenue raised by tax on pollution can be used to reducedistortive income tax by a dollar, and thus reduce the associated deadweight lossby $0.17, a dollar of revenue raised by pollution tax is worth $1.17.

The estimates of the marginal cost of public funds vary. Ballard et al. (1985),using a dynamic sequenced numerical general equilibrium model of the US econ-omy, empirically estimated γ for the US; they found that γ lies between 1.17 and1.56; earlier estimates by Browning (1976) were the range 1.09 to 1.16 since heused a more conservative estimate of labour supply elasticity; for Sweden, wherethe income tax rates were very high, Hansson and Stuarts (1983) found that γexceeded 1.69.

We will restrict attention to the empirically relevant range of γ : 1 ≤ γ ≤ 2.Thus, welfare is

W =∑i∈I

πi + βS + γ∑i∈I

tiEi − D(E), where 1 ≤ γ ≤ 2, (12)


where E =∑

i∈I Ei , D(E) is the damage cost, and S is the consumers’ surplus

S =∫ Q

0P(Q)

dQ − P (Q)Q

In what follows, we assume that the damage cost function is linear, D(E) = σE >0.

Then, using (12) and (14), the social welfare at a Cournot equilibrium may bewritten as

W =∑i∈I

πi + βS(Q)

+ γ∑i∈I

(τi − δi)qi

(Q, ωi

)(13)

where Q = Q(ω), πi is given by (10), and δi is defined as

δi ≡ σei/γ (14)

Here, δi may be called the adjusted marginal damage per unit of output of firm i,where the adjustment factor is 1/γ ≤ 1. The difference between τi, the optimaltax per unit of firm i’s output, and δi, its adjusted marginal damage per unit ofoutput, will be called the optimal distortion for firm i. We will show that (i) if firmsare identical, the optimal distortions are equal for all firms, and (ii) if firms areheterogenous, the optimal distortions are no longer equal: higher cost firms will bepenalized more, relative to lower cost firms.

Note that the first term on the right-hand side of (13) contains tax payments byfirm (in the expression πi) and the third term contains social value of tax revenueγτiqi. These two tax terms do not cancel each other out when γ = 1.

From expression (13), we see that welfare can be maximized by an appropriatechoice of the firm-specic tax rates τi. However, as we demonstrate below, it isanalytically much more convenient to solve the welfare maximization problem byusing the equilibrium outputs qi as choice variables, and afterward infer the optimaltaxes. The two methods yield the same solution. We now transform variables sothat the τi’s are no longer explicitly present in the objective function. We willshow below how to replace the τi’s in (13) by equilibrium quantities. From theequilibrium condition (4),

P ′qi + P = di + τi + c′(qi)

we get

τi − δi = qiP ′ +(P − di − δi

)− c′(qi) (15)

Substituting (15) into (13), we can decompose welfare into two terms, as shown

below. The first term,F(Q)

, is a performance measure at the industry level: it does

not contain firm-specific informations. The second term is a sum of firm-specificterms.

W = F(Q)

−∑i∈I

fi

(qi, Q

)(16)


where

F(Q)

≡ βS(Q)

+ γP Q (17)

fi

(qi, Q

)≡ (di + δi)γqi + (γ − 1)[−P ′]q2

i + φi (18)

φi ≡ (γ − 1)qic′i(qi) + ci(qi)

Thus, the welfare expression consists of a “benefit” term, at the industry level,minus a sum of “cost” terms, that are firm-specific. Note that for given Q, fi isstrictly convex in qi if γ > 1, c′′

i ≥ 0 and c′′′i ≥ 0. Expression (16) shows that

welfare is directly dependent on the qi’s. The tax rates do not (explicitly) appear inthis expression. Thus W can be maximized by the direct choice of the equilibiumoutputs. Afterwards, the taxes can be inferred from (15).

We have thus obtained a very useful lemma:

Lemma 1. In the welfare maximization problem, there is a one-to-one correspon-dence between determining firm-specific emission tax rates to maximize welfare,expression (13), and determining Cournot equilibrium outputs to maximize welfare,expression (16).

Remark. In deriving Lemma 1, we have relied on the assumption that the total(tax-inclusive) cost function is linear in ωi.

We note that taxing firms is a way of manipulating their marginal costs. Thus, ourPigouvian taxation problem lies within the framework of the “cost manipulationapproach” that we have explained in our previous work (Long and Soubeyran,1997a, 1997b, 2001a).

2.4 A benchmark case: perfect competition

Before solving for optimal emission taxes in an asymmetric oligopoly, it is useful toconsider a benchmark case, with perfect competition. We assume in this subsectionthat the marginal cost of public fund is unity, γ = 1, and the weight given toconsumers’ surplus is also unity, β = 1. In this case, we obtain the well-knownformula for optimal tax τi = δi, i.e., ti = σ for all i ∈ I . (See Appendix A1 fordetails.)

Thus, under perfect competition, all firms are treated equally. (We will latershow that this “equal treatment” result does not apply to the oligopoly case.) Eventhough this result is well known, it is useful for future reference to state it as aproposition:

Proposition 1 (Benchmark Pigouvian Tax: fair treatment). Under perfect com-petition, the optimal Pigouvian tax ti (per unit of emission) is the same for all firmsand equal to the marginal damage σ:

ti = σ for all i ∈ I. (19)

Thus all firms are treated fairly.


Recall that by definition, τi = eiti. Thus, Proposition 1 implies that, underperfect competition, the optimal tax per unit of output is τB

i = σei where thesuperscript B indicates the optimal value for the benchmark case.

2.5 An oligopoly with constant marginal costs

Now we turn to the case of an oligopoly with constant marginal costs: ci(qi) = αiqi.(The increasing marginal cost case is treated in Sect. 4.) In this case, (18) becomes

fi

(qi, Q

)≡ (di + δi + αi)γqi + (γ − 1)

[−P ′

]q2i (20)

We define the marginal social cost of firm i’s output as

si ≡ di + δi + αi (21)

Thus, marginal social cost consists of production cost, αi, transport cost, di, andadjusted marginal damage, δi.

We consider two sub-cases: (a) γ = 1 and (b) γ > 1.

2.5.1 Sub-case (a): γ = 1

In this sub-case, the marginal cost of public funds is unity. It is easy to show thatthe optimal policy is to design taxes so that only the firm with the lowest marginalsocial cost will produce (see Appendix A2).

2.5.2 Sub-case (b): γ > 1

In this sub-case, (18) becomes

fi

(qi, Q

)≡ (di + δi + αi)γqi + (γ − 1)

[−P ′

]q2i ≡ γsiqi + b

(Q)

q2i (22)

where

b(Q)

≡ (γ − 1)[−P ′

].

For given Q, the function fi

(qi, Q

)is quadratic and strictly convex in qi. A very

effective way to characterize the optimal outputs is to use a two-step procedure.(See Appendix A2).

Our results can be summarized as follows:(i) The optimal industry output is given by Q which is the solution of the

following first order condition (please see Appendix A2 for details):

WQ = F ′(Q)

− γsI − 1n

b′(Q)

Q2 − 2n

b(Q)

Q +b′(Q)

4b2(Q)∑

i∈I

γ(si − sI)2

= 0


where

F ′(Q)

=[−P ′

](β − γ)Q + γP

and

sI ≡ 1n

∑i

si

Thus both β and γ have impacts on the optimal industry output. In particular, thegreater is the weight attached to consumers’ surplus, the greater is the optimaloutput:

∂Q

∂β= − WQβ

WQQ= −

Q[−P ′

]WQQ

> 0

where WQQ < 0 because of the second order condition. Similarly, if b′ = 0, anincrease in γ will reduce optimal industry output, because the regulator will havea greater incentive to raise revenue by raising emission tax rates.

(ii) The optimal output for firm i is

q∗i =

Q

n+

γ(sI − si)

2b(Q) for all i ∈ I (23)

and this implies that the optimal firm-specific tax rate firm i is given by

τi = q∗i P ′

(Q)

+(P(Q)

− di

)− αi (24)

where q∗i is given by (23).

(iii) Let

∆i ≡ τi − δi (25)

denote the gap between the optimal firm-specific output tax τi under oligopoly andthe adjusted damage cost δi . Then

∆i = q∗i P ′

(Q)

+ P(Q)

− si =(2 − γ)si

2(γ − 1)+ P

(Q)

−λ(Q)

2(γ − 1)(26)

where

λ(Q)

= γsI +2b(Q)

Q

n

We thus can state:

Proposition 2 (Selective penalization). For any pair of firm (i, j), we have

∆i − ∆j = (τi − δi) − (τj − δj) =2 − γ

2(γ − 1)(si − sj) . (27)


Proof. This follows from Equation (26).To understand the formula (27), consider first the special case where ei = ej ,

so that the marginal damage per unit of output is the same for both firms δi = δj .Then, if si < sj , the optimal selective tax rates are such that τi < τj , provided1 < γ < 2. That is, the more efficient firm pays a lower tax rate if 1 < γ < 2.However, note that (2 − γ)/2(γ − 1) is a decreasing function of γ; therefore, ifsi−sj is independent of γ, the gap τi− τj becomes narrower as γ increases (becausewith a greater γ, revenue considerations become more important, and therefore thegovernment increases the tax rate on the low-cost firms). When γ = 2, the tax ratesare equal7.

The intuition behind Proposition 2 is as follows. Recall that in an oligopolywith firms having different production costs, it is in general not optimal to taxfirms equally for their pollution. This is because the Pigouvian taxes now serve twopurposes: correction for pollution externalities, and correction for market powerand for production inefficiency (because oligopolists do not equalize marginal pro-duction costs among themselves) while taking into account the marginal cost ofpublic funds ( γ > 1). We now seek to characterize the optimal departure from thebenchmark Pigouvian taxes obtained in Proposition 1.

From (23) the optimal tax structure is such that the more efficient firms (thosefirms with low si) always produce more than the less efficient ones. The quantity∆i measures the deviation of optimal firm-specific tax under oligopoly from thefirm-specific marginal damage cost caused by a unit of output of firm i. We call ∆i

the optimal Pigouvian distortion for firm i.We now characterize the deviation of ∆i from the industry average ∆I . From

(27),

∆i − ∆I =2 − γ

2(γ − 1)(si − sI) (28)

where si is the marginal social cost of firm i’s output. This result shows that theefficient tax structure favors the efficient firms, but this favor falls as γ increases.

From this, we can compute the variance of the statistical distribution of thePigouvian distortions:

V ar∆ =[

2 − γ

2(γ − 1)

]2V ar [s] (29)

Proposition 3 (Optimal distortion theorem). Optimal Pigouvian distortions (thegaps between optimal tax and adjusted marginal damage) are not equalized in aheterogeneous oligopoly. In the empirically relevant range of the marginal cost ofpublic funds (1 < γ < 2), if the marginal social cost si of firm i is greater than theindustry average, the Pigouvian distortion for firm i will be greater than averagePigouvian distortion. The optimal tax structure penalizes inefficient firms.

The variance of the distribution of the Pigouvian distortions is given by (29),and it is a decreasing function of γ.

7 For γ > 2, the more efficient firm must pay a higher tax rate. Recall that in our specification ofthe welfare function, we exclude the case γ > 2.


Remark. In the rather extreme case where γ > 2 (which is unlikely from empiricaldata) if the marginal social cost si of firm i is greater than the industry average, thePigouvian distortion for firm i will be smaller than average Pigouvian distortion.It remains true that the optimal solution implies that the more efficient firms havegreater outputs, see (23).

2.5.3 Related results on selective penalization

The Optimal Distortion Theorem provides a link between the ex-ante heterogene-ity of the oligopoly’s cost structure and the ex-post dispersion of the firm-specificPigouvian tax rates. The optimal tax rate per unit of emission is shown to be notequal across asymmetric firms, in contrast to the standard Pigouvian result. Thereare similar results elsewhere, though not in the same framework. Hoel (1993) con-siders a game8 among n asymmetric countries that generate CO2 emissions fromproduction activities and that suffer asymmetrically from the greenhouse effects.He assumes that emission tax revenue collected by an international agency is redis-tributed to the countries according to some exogenously specified revenue-sharingvector β = (β1, ..., βn) where βi > 0 and

∑βi = 1. He shows that in general a

uniform tax rate will not ensure the socially efficient outcome. In fact, he shows thatthe Pigouvian uniform tax rate (that is equal to the sum of marginal environmentaldamages) would be optimal only if the tax shares βj satisfy the properties that

βj =D′

j(E)∑i D′

i(E)

where E is aggregate emissions and Di(.) is the damage function specific to countryi (Hoel, 1993, p. 54). Hoel’s result, that optimal tax rate is in general not uniform,is similar to the result of our model. Both models share the following features.First, the social planner faces a set of n players that play a non-cooperative game.Second, tax revenue considerations play a major role: in Hoel’s model, budgetbalance requires that tax revenue be redistributed to the n countries, and in ourmodel, tax revenue is given the weight that reflects the marginal cost of publicfunds, which normally exceeds unity.

Other related results on asymmetric firms are reported in Collie (1993) and Longand Soubeyran (1997c, 1999), in the context of strategic trade policies, withoutpollution. Assuming cost asymmetry among domestic oligopolists, these authorspoint out that the optimal export subsidy rates are not identical. The first twopapers (Collie, 1993; Long and Soubeyran, 1997c) focus on the implication ofthe uniform export subsidy when firms are asymmetric, while the third one (Longand Soubeyran, 1999) derive the optimal non-uniform subsidies. These models arerelatively simple because the authors postulated that the marginal cost of publicfunds is unity. It follows that the optimal firm specific subsidy rates should serveto equalize marginal production costs in equilibrium (Long and Soubeyran, 1999,p. 246).

8 We are indebted to a referee for this reference, and for encouraging us to add this discussion.


3 Further interpretations

Our results on firm-specific pollution taxes can be given an interesting geometricinterpretation (see the Projection Theorem below) and an industrial organizationinterpretation (see the Concentration Motive Theorem below).

3.1 A geometric interpretation: the Projection Theorem

We now provide a geometric interpretation of the optimal choice of outputs. Con-sider the first step in the two-step procedure explained in Appendix A2. It is equiv-alent to the program of choosing the qi (i ∈ I) to

min∑i∈I

fi

(qi, Q

)subject to

∑i∈I qi = Q (given) and qi ≥ 0. This step can be described by the

following Projection Theorem.

Proposition 4 (Projection Theorem). The determination of the optimal compo-sition of industry output is equivalent to choosing a vector q ≡ (q1, ..., qn) froman n − 1 dimensional simplex S so as to minimize the distance between the vectorq and a reference vector q ≡ (q

1 , ..., qn ) where

qi ≡ − γsi

2b(Q) (30)

and where

S ≡

q ≥ 0 :∑i∈I

qi = Q

Proof.

fi

(qi, Q

)= γsiqi + b

(Q)

q2i = b

(Q)(qi +

γsi

2b

)2− γ2s2

i

4b(Q)2

= b(Q)[(

qi − qi

)2−(qi

)2]

Thus ∑i∈I

fi

(qi, Q

)= b

(Q)

‖q − q‖2 − b(Q)∑

i∈I

(qi

)2

where the second term on the right-hand side depends only on Q, which is fixed,

and the first term on the right-hand side is b(Q)

times the square of the distance


of the point q in the set S(which is an n−simplex) to the given point q. Given

Q, both b(Q)

and q are fixed. It follows that the first step of the program is

equivalent to finding the minimal distance between q and the given point q.The optimal q which achieves the minimal distance ‖q−q‖ is the projection

of q on the n−simplex S. Its components are given by

qi = qI − γ

2b(Q) (si − sI) (31)

The projection q satisfies

q = q +(qI − q

I

)u

where u =(1, 1, ..., 1) where qI = − γsI

2b(Q) .

Economic interpretation of the reference vector. The reference vector has ncomponents, each of which is negative. The ith component of this vector is anindex of the comparative disadvantage of firm i, from the social point of view. Alarger si means that either the firm is located far from the market (a larger di), or ithas a more polluting technology (a higher ei), or its marginal production cost, αi,is higher. For any given industry output level, by minimizing the distance betweenthe output vector and the reference vector, the government is in fact minimizing thesocial cost of production.

3.2 The concentration motive

Our result shows that firm-specific Pigouvian taxes in a polluting oligopoly servetwo functions: the usual function of correcting for externalities, and the functionof correcting for production efficiency, while taking into account the marginal costof public funds. For this second function, the optimal tax vector depends on twoelements (i) the degree of unit-cost asymmetry in the oligopoly, and (ii) the marginalcost of public funds. The first element is measured by the variance of the statisticaldistribution of unit costs before and after taxation (this variance is related to theHerfindahl index.) The second element is measured by γ and reflects the trade-offbetween profits and tax revenue.

Does the optimal Pigouvian tax structure increase or decrease the concentrationof the industry? Before answering this question, it is necessary to examine therelationships among the variance of the distribution of the unit costs, the Herfindahlindex of concentration, industry profit, and welfare. We now state a number oflemmas concerning these relationships. First, recall that the Herfindahl index ofconcentration is

H =∑i∈I

[qi

Q

]2


Given that there are n firms, this index attains its maximum value (H = 1) whenone firm produces the whole industry’s output and the remaning n−1 firms producezero output, and it attains its mimimum value (H = 1/n) when each of the n firmsproduces qi = Q/n. Now all firms will produce the same amount of output if theyhave the same tax-inclusive marginal costs.

Lemma 2. For a given output level Q, the Cournot equilibrium industry profit isan increasing function of the Herfindahl index of concentration.

Proof. Recall that at a Cournot equilibrium, firm i’s profit is πi =[−P ′

]q2i +

qic′i(qi) − ci(qi) With ci(qi) = αiqi, the industry profit is

Π =∑i∈I

πi =[−P ′

]Q2H

where

H =∑i∈I

[qi

Q

]2(32)

Lemma 3. Given the output level Q, the Herfindahl index of concentration is anincreasing function of the variance V ar(θ) of the distribution of the tax-inclusivemarginal costs in a Cournot equilibrium.

H =1n

1 +V ar

(θ)

[(−P ′

)qI

]2 ≥ 1

n(33)

Thus any policy that maximizes the variance of the distribution of tax-inclusivemarginal costs will maximize the concentration of the industry, and, for a given Q,maximize the profit of the industry.

Proof. From (4),

qi =P − θi(−P ′

) (34)

we obtain ∑i∈I

q2i =

∑i∈I

1(−P ′

)2

[(P − θI

)−(θi − θI

)]2

=1(

−P ′)2

[n(P − θI

)2+∑i∈I

(θi − θI

)2]

=1(

−P ′)2

[n((

−P ′)

qI

)2+ nV ar

(θ)]

(35)

The result (33) follows from (35) and (32).


We now can state an important result: with γ > 1, the optimal tax structureincreases the concentration of the industry. (See Proposition 5 below for a precisestatement.) This means inefficient firms are penalized more, relative to efficientfirms. Recall that for the case γ = 1, at the social optimum, only the firm with thelowest social cost (s) will produce, giving rise to the maximum level of industryconcentration (i.e., the Herfindahl index will take on its highest possible value, 1).Here, with 2 > γ > 1, tax revenue is an important consideration, and hence thereis a tradeoff between productive efficiency and tax revenue. It is still the case thatthe optimal tax structure increases the Herfindahl index, by increasing the varianceof the distribution of tax-inclusive marginal costs. We call this the Magnificationeffect.The magnification factor, denoted by Ω(γ), is defined by

Ω(γ) ≡[

γ

2(γ − 1)

]2> 1 for 2 > γ > 1 (36)

This magnification factor is a decreasing function of γ. As γ approaches the value2, the magnification factor falls to 1.

Proposition 5 (A pro-concentration motive theorem). Assume that all firmshave the same emission coefficients: ei = e for all i. Given 2 > γ > 1, theoptimal firm-specific Pigouvian tax structure increases the variance of the statisticaldistribution of tax-inclusive marginal costs within the oligopoly relative to thevariance of the statistical distribution of pre-tax marginal costs9. The relationshipbetween the two variances is given by:

V ar(θ)

= Ω(γ)V ar(θ0) (37)

where θ0i = di + αi is the equilibrium marginal cost of firm i in a Cournot equi-

librium where all the taxes are zero.

Proof. First, note that if all the taxes are zero, then

θ0i − θ0

I = (di − dI) − (dI + αI) (38)

Recall that qi denote the equilibrium Cournot output of firm i given an arbitraryvector of firm-specific taxes, and qi is the equilibrium Cournot output of firm iwhen the taxes are optimized. From (11), and (23), which is true also when thetilda replaces the hat,

qi − qI = −(θi − θI

)[−P ′

] = − γ(si − sI)

2(γ − 1)[−P ′

]Hence

θi − θI =γ

2(γ − 1)[(di + αi) − (dI + αI) + (δi − δI)] (39)

9 In the empirically unlileky case where γ > 2, replace “increases” by “decreases”.


Therefore

θi − θI =γ

2(γ − 1)[(

θ0i − θ0

I

)+ (δi − δI)

]

V ar(θ)

=[

γ

2(γ − 1)

]2 [V ar

(θ0)+ V arδ + 2cov

(θ0, δ

)](40)

If ei = e for all i, then, in view of (1) and (14), Equation (40) reduces to (37). Notethat Ω(γ) > 1 if 1 < γ < 2.

Remark. The intuition behind the pro-concentration motive theorem is as follows.If δi = δ for all i, the marginal cost of public fund is within the empirically likelyrange (1 < γ < 2), then, for any given industry output level, the optimal firm-specific tax structure increases the variance of marginal costs (from V ar

(θ0)

toΩ(γ)V ar

(θ0)) by taxing more efficient firms at a lower rate, see (27), because this

helps the lower cost firms to expand output relative to the higher cost firms, and asa result improves productive efficiency. However, if γ is great, the tax revenue be-comes a very important consideration, and it becomes optimal to tax more efficientfirms at a higher rate, so as to generate more revenue. Take for example the case ofa duopoly, where firm 2 has higher production cost. For a given level of industryoutput Q, we must maintain t1 + t2 = constant, say 2t. From an initial assign-ment (t1, t2) = (t, t), consider deviation of t2 from t, say t2 = t + κ, and hencet1 = t − κ. An increase in κ yields marginal gain in production efficiency, becausethe same level of industry output Q is produced, but the lower cost firm increasesits output and the higher cost firm reduces its output. However, an increase in κ by∆κ implies reduced tax revenue, by approximately (q1 − q2)∆κ,(plus the effectof induced changes in composition of industry output) and this implies increaseddistortion cost, approximately (γ − 1)∆κ(q1 − q2). For a given γ > 1, the opti-mal deviation κ∗ is at the point where the marginal gain in productive efficiencyis equated to the marginal increase in distortion cost. Clearly, a higher γ shifts themarginal distortion cost upwards, implying a smaller κ∗.

4 Selective penalization under non-linear costs

We now examine the the case where ci(.) is strictly convex. In this case, a firmmay have incentive to have more than one plants, so as to reduce the marginal costof any given output of the (multi-plant) firm. However, more plants may imply agreater fixed cost for the firm. For reason of space, we will not deal with the issue ofoptimal number of plants here. In what follows, we treat the number of plants, ui,owned by firm i, as exogenous. The fixed cost is denoted by H(ui). Since all theplants owned by firm i are identical by assumption, if qi denote firm i’s output,the output at each of its plant is simply zi = qi /ui. In this case, the functions

fi

(qi, Q

)become

fi

(qi, Q

)= (di + δi) qi + (γ − 1)

[−P ′

]q2i + φi


where

φi = (γ − 1)uizic′i(zi) + uici(zi) + H(ui)

The first stage of the game can be solved in two steps: In step (i), we solve

maxqi

W = F(Q)

−∑i∈I

fi

(qi, Q

)where

F(Q)

≡ βS(Q)

+ γQP(Q)

(41)

subject to ∑i∈I

qi = Q

where Q is given, and qi ≥ 0. In step (ii), we determine the optimal Q.To solve step (i), we form the Lagrangian

L = F(Q)

−∑i∈I

fi

(qi, Q

)+ λ

[∑i∈I

qi − Q

]or

L = F(Q)

− λQ +∑i∈I

[λqi − fi

(qi, Q

)]We obtain the first order conditions

λ =∂fi

∂qi= γ(di + δi + c′

i(qi/ui)) + (γ − 1)qi

[1ui

c′′i (qi/ui) − 2P ′

](42)

Then

∂2fi

∂q2i

dqi +∂2fi

∂qi∂QdQ = dλ

Let qi denote the optimal solution. Then, assuming that c′′′i ≥ 0, so that ∂2fi/∂q2

i >0, we have

qi = qi

(λ, Q

)with ∂qi/∂λ > 0. The equation

∑i∈I qi

(λ, Q

)= Q determines λ

(Q)

.

The second step:We now determine the optimal Q.We follow the duality methodused in Rockafellar (1970). Following Rockafellar, we define the conjugate function

f∗i

(λ, Q

)= sup

qi

[λqi − fi

(qi, Q

)]


then

f∗i

(λ, Q

)= λqi

(λ, Q

)− fi

(qi

(λ, Q

), Q)

It follows that the optimal value of the Lagrangian (optimized with respect to theqi) is

L(Q)

=[F(Q)

− λ(Q)

Q]

+∑i∈I

f∗i

(λ(Q)

, Q)

Differentiating L(Q)

with respect to Q and equating it to zero yields

F ′(Q)

− Qdλ

dQ− λ

(Q)

+∑ ∂f∗

i

∂λ

dλ

dQ+∑ ∂f∗

i

∂Q= 0 (43)

Since

∂f∗i

∂λ= qi

(λ, Q

)+[λ − ∂fi

∂qi

]∂qi

∂λ= qi

(λ, Q

)(43) reduces to

F ′(Q)

− Qdλ

dQ− λ

(Q)

+∑i∈I

qi

(λ(Q)) dλ

dQ+∑ ∂f∗

i

∂Q= 0

or

F ′(Q)

− λ(Q)

+∑ ∂f∗

i

∂Q= 0 (44)

where, from the envelope theorem,

∂f∗i

∂Q= −∂fi

∂Q= (2P ′′)(γ − 1)qi

(λ(Q)

, Q))

Equation (44) determines the optimal Q.

Discussion. Let us consider some implications of the above analysis. Suppose forsimplicity that the cost functions can be ranked, by the following parametrization:

ci(z) = αic(z) where α1 < α2 < ... < αn

Then given λ, other things being equal, the firms with higher value αi will beinduced to produced less:

∂qi

∂αi= −∂2fi/∂qi∂αi

∂2fi/∂q2i

=−γc′ − (γ − 1)zic

′′

∂2fi/∂q2i

< 0

Similarly, given λ, for the same α, the firms with higher values of (δi + di) will beinduced to produced less.


From the first order condition of the firm’s maximization problem,

τi − δi = P + P ′qi − [(di + δi) + αic′(qi/ui)]

Let si(qi) = (di + δi) + αic′(qi/ui). Then

(τi − δi) − (τj − δj) =(P ′)

[qi − qj ] − (si(qi) − sj(qj)) (45)

Now, from (42),(−2P ′

)(γ − 1)qi = λ − γsi(qi) − (γ − 1)ziαic

′(zi) (46)

Substituting (46) into (45), we get, in the case γ = 1,

(τi − δi) − (τj − δj) =(2 − γ)(si − sj) + (γ − 1)ν(αic

′(zi) − αjc′(zj))

2(γ − 1)(47)

where we have assumed that c′(.) has a constant elasticity ν > 0. The formula (47)is a direct generalization of Proposition 2.

Consider now the case γ = 1. From (41), with γ = 1, we have

F ′(Q)

= P − (1 − β)[−P ′

]Q (48)

From (44) and (48),

P − λ = (1 − β)[−P ′

]Q (49)

The difference between the optimal per unit tax and the marginal damage isgiven by

τi − δi = P + P ′qi − [(di + δi) + c′i(qi/ui)]

But, recall that

(di + δi) + c′i(qi/ui) = λ (50)

Therefore

τi − δi =[P − λ

]+ P ′qi (51)

From (49) and (51),

∆i = τi − δi =[−P ′

]Q

[(1 − β) − qi

Q

](52)

Thus

∆i − ∆j = (τi − eiσ) − (τj − ejσ) =[−P ′

](qj − qi) (53)

From (53) we can state the following proposition:


Proposition 6 (Optimal Pigouvian Distortion under Strictly Convex Costs).Under strictly convex cost,

(i) if 1 < γ < 2, the optimal deviation from Pigouvian taxes are similar to theconstant marginal cost case.

(ii) if γ = 1, the optimal tax structure favors lower cost firms.

Remark. Under linear cost, if γ = 1, the optimal tax structure will eliminate allinefficient firms (only the lowest cost firm will survive). Under strictly convex cost,such extreme penalization does not emerge; rather, all firms will produce, but themore efficient firms are more favorably treated.

Thus, if γ = 1, the specific Pigouvian tax ti on pollution by firm i is, from (49)and (51),

ti ≡ τi

ei= δ +

1ei

S′(Q)[(1 − β) − qi

Q

](54)

We conclude that (i) ti is greater, the greater is the marginal damage cost, (ii)ti is negatively related to the weight attached to consumers’ surplus, and (iii) inequilibrium, among all firms that have the same emission coefficient ei, smallerfirms are taxed at a higher rate. This is because smaller firms are less efficient, andoptimal policy seeks to reduce their outputs.

5 Pollution standards and abatement costs

We now turn to a model in which firms can reduce emission at any given outputlevel, by incurring abatement costs (which is a function of both the output level andthe emission rate). We will focus on the use of firm-specific pollution standards.

We assume that firm i’s production cost is αiqi and, in the absence of abatement,its emission is Ei = eiqi where ei is emission per unit of output. Depollution is anactivity that reduces emission per unit of output to (1 − ri)ei, where ri ∈ [0, 1].We assume that the cost of depollution is krieiqi, where k is a positive constant,it is a measure of the marginal cost of depollution. The regulator must dictate tofirm i the rate ri (the rate of depollution per unit of emission), or equivalently, therate xi ≡ riei (the rate of depollution per unit of output). In what follows, we willshow how the optimal rates xi are determined, and study the implied direction ofunequal treatments.

Firm i’s profit function is

πi = P (Q)qi − αiqi − kxiqi

Given the vector (x1, x2,..., xn) chosen by by the regulator, the Cournot equilibriumoutputs satisfy

qi =1[

−P ′(Q)] [P (Q)− αi − kxi

]


This gives the equilibrium relationship:

xi

(qi, Q

)=

1k

[P ′(Q)

qi − P(Q)

− αi

](55)

The Cournot equilibrium profit is given by (10): πi =[−P ′

(Q)]

q2i . The welfare

function is

W = βS(Q)

+n∑

i=1

πi − D(E)

= βS(Q)

+n∑

i=1

[−P ′

(Q)]

q2i − σ

n∑i=1

(ei − xi)qi

= βS(Q)

−n∑

i=1

fi

(qi, Q

)where

fi

(qi, Q

)= σ

(ei − xi

(qi, Q

))qi +

[P ′(Q)]

q2i (56)

Substituting (55) into (56), we get

fi

(qi, Q

)= −σ

k

[P(Q)

− kei − αi

]qi + (

σ

k− 1)

[−P ′

(Q)]

q2i (57)

The regulator must choose qi ≥ 0 (i = 1, 2, ..., n) and Q to maximize βS(Q)

−∑ni=1 fi

(qi, Q

)subject to

∑ni=1 qi = Q. Again, here we used the familiar two-

step procedure. In step 1, for a given Q, we choose qi, (i = 1, 2, ..., n) to solve

max βS(Q)

−n∑

i=1

fi

(qi, Q

)(58)

subject to

n∑i=1

qi = Q, and qi ≥ 0. (59)

This gives qi = qi

(Q)

.

In step 2, we choose the optimal Q.

Step 1. (for a given Q.) Here, we consider two cases. In case 1, σ < k, i.e., themarginal damage cost of pollution is smaller than the measure of marginal cost

of depollution, and hence the function fi

(qi, Q

)in (57) is strictly concave in qi,

and W is strictly convex in qi. The solution of the maximization problem (58)


subject to (59) must therefore be a corner one. Suppose kej +αj < kei +αi for alli = j. Then the optimal solution is qj = Q and qi = 0 for all i = j. Alternatively,suppose there are only two firms, with ke1 + α1 = ke2 + α2. Then there are two

optimal solutions: (q1, q2) =(Q, 0

)and (q1, q2) =

(0, Q

). We get immediately

the “unequal treatment of equals” result.In case 2, σ > k (i.e., the marginal damage cost of pollution is greater than

the measure of marginal cost of depollution), thus W is strictly concave in qi. Aninterior solution is possible, with

∂W

∂qi= −

∂fi

(qi, Q

)∂qi

+ λ = 0

which gives

qi

(λ, Q

)=

1

2(

σk − 1

) [−P ′(Q)] [λ +

(σ

k

)([−P ′

(Q)]

− kei − αi

)](60)

Summing (60) over all i, we get

nλ +(nσ

k

)([−P ′

(Q)]

− keI − αI

)= 2Q

(σ

k− 1) [

−P ′(Q)]

where eI = (1/n)∑

ei and αI = (1/n)∑

αi.Thus

λ(Q)

=2Q

n

(σ

k− 1) [

−P ′(Q)]

−(σ

k

)([−P ′

(Q)]

− keI − αI

)(61)

Substituting (61) into (60) we get

qi

(Q)

=Q

n+

(keI + αI) − (kei + αi)

2(

σk − 1

) [−P ′(Q)] (62)

Then we have several interesting results about unequal treatment of unequals:

Proposition 7. Suppose σ > k. Then, for a given total output Q,(i) The optimal standards induce firms with lower social cost (lower αi + kei)

to produce more:

qi − qj =

(σk

)2(

σk − 1

) [−P ′(Q)] [(αj + kej) − (αi + kei)]

(ii) Suppose αi = αj . Then the dirtier firms are required to achieve a higherrate of depollution. That is, xj > xi if ej > ei. Thus pollution standards favor lesspolluting firms.

(iii) Suppose ei = ej . Then firms with higher production costs will be requiredto achieve a higher rate of depollution (that is, αj > αi implies xj > xi) if σ < 2k,but a lower rate of depollution (that is, αj > αi implies xj < xi) if σ > 2k.


Proof. (i) use (60).(ii) and (iii) use (55) and (60) to get

k(xi − xj) =1

2(

σk − 1

) [(αj − αi)(σ

k− 2)

− σ(ej − ei)]

Step 2. We now determine the optimal industry output. We focus on the caseσ/k > 1, because the other case is trivial. In this step, we maximize

W = βS(Q)

−∑

fi

(qi

(Q)

, Q)

(63)

where qi

(Q)

is given by (62). From (63) and (57),

W = βS(Q)

+σ

kP(Q)

Q − σ

k

n∑i=1

(kei + αi)qi

(Q)

−(σ

k− 1) [

−P ′(Q)] n∑

i=1

[qi

(Q)]2

(64)

To illustrate, consider the linear demand case, P = A−Q. Assume A > keI −αI .The objective function W is concave in Q if

( 2σk − β

)+ 2

n

(σk − 1

)> 0. In this

case, maximizing (64) yields

Q =

(σk

)[A − keI − αI ]( 2σ

k − β)

+ 2n

(σk − 1

)Thus, the greater is β, the greater is the optimal industry output. The greater is σ,the smaller is the industry output. These are intuitively plausible results.

6 Concluding remarks

We have characterized the optimal structure of penalties for polluting firms in anoligopoly with heterogenous costs. We have shown that there is a bias in favor ofefficient firms. In achieving efficiency, a structure of systematic biases emerges.

Our paper goes beyond the existing result of unequal treatment to ex ante iden-tical firms. Several important insights emerge. It is shown that optimal firm-specificregulations are partly driven by the motive to increase the industry concentration,because increased concentration can enhance productive efficiency. However, taxrevenue can be an important consideration, and any increase in the marginal costof public funds will lead to an increased tax rate on the more efficient firms. Thedegree of industry concentration is increased by the structure of efficient taxes.

Our analysis can be extended to study the role of strategic trade policy in thepresence of a polluting international oligopoly. There are a number of insightfulpapers that deal with this topic (Conrad (1993), Barrett (1994), Kennedy (1994),Ulph and Ulph (1996), Ulph (1996a,b), Neary (1999)). However, the optimal struc-ture of taxes was not explored in these papers, because the models did not allow forasymmetry within the domestic industry, and firm-specific taxes or standards wereruled out.


Appendix A1

The benchmark case: perfect competition

Perfect competition means that each firm thinks that its output has no effect on theprice, i.e., the term P ′(Q) does not appear (i.e., is assigned the value zero) in thefirm’s first order condition. Therefore (15) reduces to

τi = P − di − c′(qi) (65)

With γ = β = 1 social welfare (16) becomes

W = S(Q)

+ P Q −∑i∈I

[(di + δi)qi + ci(qi)] (66)

Writing Q =∑

i∈I qi and maximizing (66) with respect to the qi’s, we obtain

P − di − δi − c′i(qi) = 0 (67)

which says that marginal social cost of firm i’s output must be equated to price, astandard result. From (67) and (65), we get τi = δi, and hence, using (1) and (14),

ti =σ

γ≡ δ for all i ∈ I (68)

(where γ = 1) that is, the tax per unit of pollutant discharged by firm i is equal tothe marginal damage cost.

Appendix A2

Oligopoly with constant marginal costs

Subcase (a): γ = 1 :Without loss of generality, assume s1 < s2 < s3 < ... < sn. Then, if

∂W/∂q1 = F ′(Q)

− s1 = 0 it must be true that, for all j > 1, ∂W/∂qj =

F ′(Q)

− sj = s1 − sj < 0, implying that qj = 0. It follows that at the social

optimum, only firm 1 produces.An intuitive explanation of this result is as follows. Suppose that at an equi-

librium both firms 1 and 2 produce positive outputs, and they satisfy the Cournotequilibrium conditions

P(Q)

+ P ′(Q)

q1 = d1 + α1 + τ1

and

P(Q)

+ P ′(Q)

q2 = d2 + α2 + τ2


Then, social welfare can be increased by raising τ2/[−P ′(Q)] by ∆ > 0 and

reducing τ1/[−P ′

(Q)]

by ∆, so that firm 1’s output will increase by ∆ and

firm 2’s output will fall by ∆, leaving industry output and price unchanged. Socialwelfare increases because the total cost of producing the given output Q is nowlower. Tax revenue will change, but industry profit, defined as sales revenue, minusproduction cost, minus tax payment) will change by the same amount, therefore,given that γ = 1, the tax revenue change is completely offset by the change inindustry profit.

Subcase (b) γ > 1

The two-step procedure. In step 1, we fix an arbitrary level of industry output, Q,and maximize welfare by choosing the qi’s subject to the constraint that

∑i∈I qi =

Q. This gives the optimal value of qi, conditional on the given Q. In the secondstep, we determine the optimal industry output.

Step 1. Given Q, we write the Lagrangian as

L = F(Q)

−∑i∈I

fi

(qi, Q

)+ λ

[∑i∈I

qi − Q

](69)

From this we obtain the conditions

−γsi − 2b(Q)

qi + λ = 0, for all i ∈ I (70)

where qi denotes the optimal value of qi, conditional on the given Q, and wheresi ≡ (di + δi + αi), b(Q) ≡ (γ − 1) [−P ′(Q)] > 0.From (70),

qi =λ − γsi


Summing (71) over all i, we obtain an expression showing that λ is uniquelydetermined by Q

λ = λ(Q)

= γsI +2n

b(Q)

Q (72)

where sI ≡ (1/n)∑

i∈I si. Substituting (72) into (71), and letting qI ≡ Q/n, weget

qi

(Q)

= qI − γ

2b(Q) [si − sI ] for all i ∈ I (73)

This equation gives us:


Lemma A1. The optimal deviation of the output of firm i from average industryoutput is a linear function of the deviation of its marginal social cost from theindustry average.

Remark. To illustrate, consider a pair of firms (1, 2) with marginal social costss1 < s2. Then (73) gives

q1

(Q)

− q2

(Q)

=γ

2(γ − 1)[−P ′

] [s2 − s1] (74)

That is, the solution of the optimization problem has the property that the firmwith higher marginal social cost produces less than the firm with lower marginalsocial cost. Note that, for a given Q, a greater γ implies a smaller gap between q1

and q2, but this gap is always positive and greater than [s2 − s1]/2[−P ′

]. This

may be explained as follows: a greater γ implies that a greater weight is givento tax revenue. Thus, for any given Q, a marginal increase in γ would increasethe government’s desire to increase tax revenue at the cost of reduced productiveefficiency (here, productive efficiency includes not only private cost considerations,but also environmental cost). The government would therefore raise τ1 by somesmall amount ε > 0 and at the same time reduce τ2 by ε , thus leaving total outputQ constant. The increase in tax revenue is approximately (q1 − q2)ε and this mustbe balanced against the marginal loss in productive efficiency associated with theincrease in the output of the high-cost firm and the reduced output of the low-costfirm.

Step 2. Using the results in step 1, we are now ready to find the optimal industryoutput. We make use of the fact that the optimal value of the Lagrangian, given Q,is equal to the maximized W , given Q. Thus

L(Q)

= F(Q)

− λ(Q)

Q +∑i∈I

f∗i

[λ(Q)

, Q]

where

f∗i

(λ, Q

)≡ sup

qi

λqi − fi

(qi, Q

)(f∗

i

(λ, Q

)is called the conjugate function of fi

(qi, Q

), see Rockafellar, 1970,

Sect. 12.) In the present case,

f∗i

(λ, Q

)= [γsI + 2bqI ] qi − [γsiqi + bq2

i

]= −γ(si − sI)qi − b [qi − qI ]

2 + bq2I

Thus, using (73)

f∗i

(λ, Q

)= −γ(si − sI)qI + bq2

I +γ2

4b(Q) (si − sI)2


and, using (72), the maximized welfare, for given Q, is

W(Q)

= F(Q)

−[γsI + 2b

(Q) Q

n

]Q

+nb(Q)( Q

n

)2

− γ2

4b(Q)∑

i∈I

(si − sI)2

Hence

W(Q)

= F(Q)

− γsIQ − 1n

b(Q)

Q2 − γ2

4b(Q)∑

i∈I

(si − sI)2 (75)

Maximizing (75) with respect to Q, we get the necessary condition

F ′(Q)

− γsI − 1n

b′(Q)

Q2 − 2n

b(Q)

Q +b′(Q)

γ2

4b2(Q) ∑

i∈I

(si − sI)2 = 0

(76)

This equation determines the optimal value of Q, which we denote by Q.

Remark. To illustrate, consider the case of linear demand, P = A − BQ. ThenF ′(Q) = −(γ − β)BQ + γ(A − BQ), and b(Q) = (γ − 1)B and b′(Q) = 0.Equation (76) yields the optimal Q:

−(γ − β)BQ + γ(A − BQ) − 2(γ − 1)BQ

n= γsI (77)

If β = 0 (for example, the whole industry output is exported, and social welfaredoes not include foreign consumers’ surplus), and firms are identical,then, as γtends to 1, we obtain the optimal exporting monopoly result, marginal revenueequals marginal social cost.

The optimal output for firm i is

q∗i =

λ(Q)

− γsi


From this, we derive the optimal firm-specific tax τi, using (15):

τi = q∗i P ′

(Q)

+(P(Q)

− di

)− c′(q∗

i ). (79)

where q∗i is given by (78).


Appendix A3

The duality approach

The following is the outline of the duality approach contained in Rockafellar (1970).Consider the problem

maxxi

J = F (X) −∑i∈I

fi(xi, X)

subject to ∑i∈I

xi = X , xi ≥ 0, i ∈ I

where X is given, fi(xi, X) convex with respect to xi and differentiable withrespect to (xi, X), and proper ( fi(xi, X) is never −∞, and is not identically +∞).

To solve this problem, define the extended functions gi(xi, X) = fi(xi, X) ifxi ≥ 0 and gi(xi, X) = +∞ if xi < 0. Then we have the program

maxxi

J = F (X) −∑i∈I

gi(xi, X)

subject to ∑i∈I

xi = X , i ∈ I

For agiven X, the Lagrangian of this problem is

L(x, λ, X) = [F (X) − λX] +∑i∈I

[λxi − gi(xi, X)]

where x = (x1, ..., xn).The saddlepoint duality theorem (see Rockafellar, 1970, pp 284–285) states

that x = (x1, ..., xn) is an optimal solution of the program if and only if (i) given λ,x maximixes the function L(x, λ, X), and (ii) λ minimizes L(x(λ, X), λ, X) withrespect to λ, where x(λ, X) achieves the minimum of L(x, λ, X) for each given λ.

The determination of xi(λ, X) is given by the first order condition of the pro-gram

supxi

[λxi − gi(xi, X)] = g∗i (λ, X)

g∗i (λ, X) is called the conjugate function of gi(xi, X). We have

∂g∗i

∂λ= xi(λ, X) +

[λ − ∂gi

∂xi

]∂xi

∂λ= xi(λ, X)


We also have

L = L [x(λ, X), λ, X] = [F (X) − λX] +∑i∈I

g∗i (λ, X)

and λ(X) = λ achieves the minimum of L with respect to λ. The first ordercondition for that is ∑

i∈I

∂g∗i

∂λ(λ, X) = X

that is, ∑i∈I

xi(λ, X) = X

This equation gives λ(X). The optimal xi follows: xi(X) = xi

[λ(X), X

], i ∈ I.

Appendix A4

A theorem on existence and uniqueness of Cournot equilibrium with strictly positiveoutputs under linear costs

The following theorem shows that under suitable assumptions on the range ofthe non-identical constant marginal costs, we can ensure both (i) existence anduniqueness of Cournot equilibrium, and (ii) strictly positive outputs for all firms.

Let the constant marginal costs be ranked as follows: 0 < cmin = c1 ≤c2 ≤ ... ≤ cn = cmax < P (0). Define Q# by P (Q#) = cmax, and letcM = (1/n)

∑i∈I ci.In addition to Assumptions 1 to 5 in Section 2.2, we add:

Assumption 6. Each firm’s output cannot exceed a capacity output level q whereq is sufficiently great in the sense that

q >P (0) − cmin

[−P ′(0)]

Assumption 7. n(cmax − cM ) <[−P ′(Q#)

]Q#, i.e., the dispersion of marginal

costs is not too great.

Theorem. Under Assumptions 1 to 7, there exists a unique Cournot equilibrium,with q∗

i ∈ (0, q).

Proof. Let πi = P (Q)qi − ciqi denote the profit of firm i. Define the functiongi(qi, Q) by gi(qi, Q) = P ′(Q)qi + P (Q) − ci = ∂πi/∂qi

The second order condition for an interior maximum is satisfied because ∂2

πi/∂q2i = P ′′ (Q)qi + 2P ′ (Q) < 0, from Assumption 5. The output q∗

i is strictlyinterior, iff

q∗i =

P (Q∗) − ci

[−P ′(Q∗)]∈ (0, q)


Let us define the functions

φi(Q) =P (Q) − ci

[−P ′(Q)]

and the function

Φ(Q) =n∑

i=1

φi(Q)

Now, if we can show that there exists a unique Q∗ ∈ (0, Q#)

such thati) gi(q∗

i , Q∗) = 0, i ∈ I,i.e., q∗i = [P (Q∗) − ci] / [−P ′(Q∗)] = φi(Q∗)

ii) Q∗ =∑

i∈I q∗i =

∑i∈I φi(Q∗) = Φ(Q∗),

then we have a proof that there exists a unique Cournot equilibrium where allfirms produce a positive quantity.

Step 1: Proof that for all Q ∈ [0, Q#), there exists for each firm i a unique

corresponding value qi, 0 < qi < q, such that gi(qi, Q) = 0, i ∈ I .Since (a) gi(qi, Q) is continuous in qi, (b) gi(0, Q) = P (Q) − ci > 0 ,for all

Q ∈ [0, Q#), and (c) gi(q, Q) < 0, i ∈ I for all Q ∈ [0, Q#

),the mean value

theorem tells us that for each Q ∈ (0, Q#

)there exists a qi ∈ (0, q) such that

gi(qi, Q) = 0. This value qi is unique because gi(qi, Q) is strictly decreasing in qi.

Observation on condition (b). gi(0, Q) = P (Q) − ci > 0 for all Q ∈ [0, Q#)

from the definition of Q#.

Observation on condition (c). g1(q, 0) < 0 because of Assumption 6. Nowg1(q, 0) < 0 implies g1(q, Q) < 0 for all Q ∈ [0, Q#

), and gi(q, Q) < 0, i ∈ I,

for all Q ∈ [0, Q#), because gi(q, Q) is strictly decreasing in Q,for all i ∈ I, by

Assumption 5.Note that gi(q, Q) ≤ g1(q, Q),for all i ∈ I,from P ′(Q)q + P (Q) − ci ≤

P ′(Q)q + P (Q) − cmin,where c1 = cmin.

Step 2: The existence of Q∗

We have Φ(0) =∑

i∈I φi(0) =∑

i∈I [P (0) − ci] / [−P ′(0)] > 0, from As-sumption 6.

We have Φ(Q#) < Q# iff∑

i∈I

[P (Q#) − ci

]/[−P ′(Q#)

]< Q#

Thus n[P (Q#) − cM

]/[−P ′(Q#)

]< Q#, which is Assumption 7.

Note that Φ(Q) is continuous (differentiable) and strictly decreasing (becauseall φi(Q) are strictly decreasing in Q for all Q ∈ [0, Q#

), from φ′

i(Q) = −1 +[P (Q) − ci] /(P ′(Q)) < 0). Since Φ(0) > 0 and Φ(Qmax) < Qmax by the meanvalue theorem there exists Q∗, 0 < Q∗ < Qmax such that Φ(Q∗) = Q∗.Theuniqueness of Q∗ comes from the fact that Φ(Q) is strictly decreasing

This completes the proof of the theorem.


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selective penalization of polluters: an inf-convolution approach

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