Keywords: environmental quality, regulation, market power, …scal avoidance, incomplete information, quality choice Environmental Taxation, Monopoly, Tax Avoidance. JEL Code: D43, D82, H23, L12, Q28

Environmental quality having features of a public good is the subject of the thesis. In a theoretical framework, we study a means of environmental regulation of a polluting …rm endowed with market power and then determine the optimal decentralized choice of environmental quality. We start with examining how tax avoidance a¤ects the optimal second-best tax on polluting emissions in a monopoly setting. The …rm is owned by shareholders who di¤er in their cost of tax dodging. The optimal tax should correct two negative externalities of avoidance: the …rm's free-riding e¤ect and a tax base erosion e¤ect. This free-riding makes the regulator either impotent or unfair, depending on the severity of the environmental damage and the …rm's e¢ ciency. Next, we analyze the impact of an environmental tax on the signaling price strategy of a monopoly that communicates to consumers the unobservable information about …rm's high environmental performance. We use the intuitive and undefeated criteria of equilibrium selection.

Asymmetric information places the optimal second-best tax below the level required under complete information. In the case of undefeated equilibria selection, the tax may induce a 'migration' from separating equilibrium to pooling making the …rm prefer to conceal the private information about environmental quality. Finally, we show that market choice of environmental quality by a …rm that internalizes environmental damage from polluting emissions is yet suboptimal.

List of Tables

L'échappement …scal modi…e la composition de la taxe : elle prend en compte l'e¤et d'un comportement de passager clandestin de la …rme et l'e¤et de l'érosion de l'assiette …scale.

La taxe optimale du second rang doit équilibrer ces deux e¤ets opposés. Le comportement de passager clandestin rend le régulateur soit impuissant, soit inéquitable en fonction de la sévérité du dommage environnemental et de l'e¢ cacité de l'entreprise. Dans un second temps, on étudie l'impact de la taxe environnementale sur la politique de prix d'un monopole qui utilise le signal en prix a…n de communiquer aux consommateurs le haut niveau de qualité environnementale inobservable. On applique deux critères de sélection de l'équilibre : le critère intuitif et le critère d'invincibilité. L'asymétrie d'information place le niveau de la taxe au-dessous du niveau Pigouvien ajusté selon l'exercice du pouvoir du marché. En cas de sélection par le critère d'invincibilité la taxe peut basculer un équilibre séparateur dans un équilibre mélangeant ce qui rétablit le problème d'asymétrie d'information. En…n, on montre que le choix de qualité environnementale de la part de l'entreprise est sous-optimale même si elle internalise le dommage environnemental.

Mots-clés : qualité environnementale, régulation, pouvoir du marché, évitement …scal, information incomplète, choix de la qualité

Introduction Générale

Alors que la notion générale de qualité a évolué depuis des décennies voire même des siècles pour certains produits comme le vin et le fromage français, le phénomène de qualité environnementale est quant à lui relativement récent. Il s'est concrétisé avec la production de masse, générant la préoccupation d'une haute performance …nancière tout en négligeant l'impact sur l'environnement.

Les normes de qualité environnementale sont en train de rentrer dans les cahiers des charges pour une grande gamme des produits. Etant devenue une qualité centrale et stratégique d'un produit, la qualité environnementale concerne l'intérêt de chaque acteur du marché. Les consommateurs expriment leur intérêt (de plus en plus) pour la qualité environnementale, les entreprises cherchent à répondre aux attentes des consommateurs, quant au régulateur, il veille sur les défaillances du marché. Dans la tendance générale de mondialisation et de dérégulations, les organisations non-gouvernementales (ONGs) commencent à avoir une position particulière dans le rapport de forces en faisant le lien entre tous les acteurs du marché et en comblant les lacunes de l'intérêt public.

Ces interactions entre les acteurs du marché et la préoccupation avec des problèmes environnementaux mènent, entre autres, au changement de politique publique.

Dans un premier temps, on s'interroge donc sur l'impact d'une taxation environnementale sur le comportement de l'entreprise vis-à-vis du régulateur et/ou du …sc. On considère que la gérance de l'entreprise est con…ée aux exécutifs qui cherchent à maximiser la performance …nancière de la …rme. Cette ambition généreusement récompensée se voit menacée par la taxe permettant l'incitation aux exécutifs à réduire la charge …nancière imposée par la politique environnementale.

Le chapitre 1 étudie l'impact de l'évitement …scal sur la taxation optimale de second rang portant sur les émissions polluantes d'un monopole. L'entreprise est détenue par des actionnaires hétérogènes dans leur capacité d'esquiver le …sc. La conformité de l'entreprise à la taxation environnementale est déterminée en fonction des coûts d'évitement. Elle dépend du niveau de la taxe. Lorsque la taxe augmente, la nonconformité devient attractive pour un plus grand nombre d'actionnaires. Dans la tradition de [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF], la taxe optimale devrait corriger les deux externalités négatives qui proviennent de la pollution et du comportement en situation de monopole. La capacité à échapper le …sc introduit un nouveau paramètre de décision. L'analyse fait apparaître de deux e¤ets opposés de l'évitement …scal sur le design de la politique environnementale : l'e¤et d'un comportement de passager clandestin des actionnaires et l'e¤et de l'érosion de l'assiette …scale suite à une hausse de taxe. Le mécanisme que la taxe déclenche est le suivant : lorsque la non-conformité augmente, le niveau de la taxe doit aussi augmenter, ce qui entraîne, par conséquent, encore plus d'esquive. Etant donnée l'hétérogénéité des actionnaires face à l'évitement …scal, le régulateur doit internaliser l'externalité imposée par le comportement de passager clandestin de l'entreprise sur le reste de la société. La taxe optimale doit équilibrer ces deux e¤ets opposés. Le comportement de passager clandestin rend le régulateur soit impuissant, soit inéquitable en fonction de la sévérité du dommage environnemental et de l'e¢ cacité de l'entreprise.

Dans le cadre d'un dommage modéré où la taxation à la Buchanan aurait été instaurée, le régulateur s'abstient de la régulation car la perte des contribuables serait plus coûteuse pour la société que le gain de correction de l'externalité environnementale. Pour un dommage considérable, la taxe doit dépasser le niveau de Buchanan. En conclusion, quand la politique environnementale est mise en place, les consommateurs en supportent le fardeau …nancier. Dans le cas contraire, ils supportent la pollution. Nous démontrons par ailleurs qu'un programme d'une taxe à deux parties peut atteindre la solution du premier rang.

A part d'une réponse comportementale de l'esquive …scale, la taxation environnementale peut aussi entraîner une réponse stratégique de part de l'entreprise. Cela fait l'objet du chapitre 2. Motivée par une prime sur la haute qualité environnementale, l'entreprise propre a l'intérêt à transmettre crédiblement l'information sur la qualité de son produit a…n de béné…cier du consentement de part des consommateurs à payer plus pour une haute qualité environnementale. En même temps, l'entreprise polluante peut exploiter ces espérances des consommateurs en imitant la haute qualité. Le chapitre 2 est dédié à l'analyse des actions stratégiques de la …rme face aux problèmes d'information incomplète.

Le chapitre 2 étudie l'impact de l'asymétrie d'information sur la taxe environnementale du second rang. L'intérêt principal du chapitre est d'endogénéiser la taxe a…n d'évaluer son impact sur le bien-être social. Le cadre comprend un cas d'industrie en concurrence imparfaite et une qualité environnementale inobservable. Un monopole utilise le prix pour signaler que son produit propre (qualité haute) ne génère pas de pollution.

Pour éviter qu'un producteur de type sale (qualité faible) ne vende un produit polluant au même prix que le produit propre, le monopole doit distordre le prix du produit propre relativement au niveau d'information complète.

On distingue deux formes de régulation lorsque l'équilibre sépare les deux types : (i) la régulation faible qui préserve l'avantage en coûts du monopole sale; (ii) la régulation forte qui donne l'avantage en coûts au monopole propre.

Le niveau de la taxe optimale doit être inférieur au niveau imposé en cas de monopole polluant avec information complète. La taxe comprend trois éléments correspondant aux externalités présentes sur le marché. En partant d'un niveau Pigouvien équivalent au dommage marginal, la taxe est ajustée selon le comportement du monopole qui restreint la production a…n d'obtenir une rente non-compétitive. De plus, ce niveau est réduit encore pour prendre en compte l'impact de l'asymétrie d'information. Puisque la taxe s'in…ltre dans le prix du produit propre, que ce soit un équilibre séparateur ou mélangeant, l'espérance des consommateurs d'une haute qualité environnementale augmente l'importance que le régulateur attribue au produit propre. Cela entraîne la baisse de sévérité de la taxe environnementale relativement à la situation d'information complète.

En outre, sous un équilibre séparateur, la taxe ne peut plus égaliser la valeur sociale espérée d'un produit à son coût social espéré. Pour une régulation faible, en moyenne le prix doit être supérieur au coût social, cette relation s'inverse dans le cas d'une régulation forte. Car l'information incomplète oblige le monopole propre à distordre son prix, la réaction du prix au changement de la taxe dépend de la position relative du prix du produit propre par rapport au niveau qui maximise le pro…t.

L'analyse de bien-être donne les résultats suivants : la régulation environnementale réduit le pro…t du monopole polluant qu'il y ait ou non séparation des types. La régulation environnementale augmente le pro…t du monopole propre dans le cas d'une régulation forte et dans le cas d'un équilibre mélangeant. Or, le bien-être des consommateurs du produit propre est en général réduit suite à la régulation environnementale. Dernièrement, on s'intéresse au choix optimal de qualité environnementale de part de l'entreprise et du régulateur. Ces choix ne peuvent pas faire abstraction de question de di¤érenciation des produits. Lorsque les consommateurs valorisent à la fois la qualité environnementale mais aussi la diversité des produits vendus sur le marché, on étudie dans le chapitre 3 les moyens de provision d'un niveau de qualité socialement désirable.

Le choix de la qualité environnementale est une décision stratégique pour l'entreprise.

La littérature établit que, du point de vue d'un régulateur bienveillant, un agent privé choisit un niveau de qualité environnementale sous-optimal (la qualité ayant les attributs d'un bien public). Le chapitre 3 comprend un modèle stylisé d'un duopole caractérisé par des producteurs de la qualité verte (relativement propre) et brune (conventionnelle) verticalement di¤érenciés. Le pouvoir de marché est partagé entre les deux …rmes qui se concurrencent en quantité. Ce pouvoir augmente avec la hausse de la qualité environnementale permettant une di¤érenciation plus prononcée.

On part d'un modèle de référence dans lequel le niveau de qualité environnementale choisi est sous-optimal. Dans ces conditions, on s'interroge sur la durabilité et la stabilité de cette stratégie. L'équilibre est caractérisé par un pro…t plus faible pour la …rme brune que pour la verte. N'ayant pas une possibilité d'une déviation pro…table, le pro…t de la …rme brune est pourtant strictement croissant en fonction de la qualité environnementale.

La …rme brune aurait intérêt à inciter sa concurrente à distordre son choix de qualité environnementale vers le haut. L'entreprise verte, à son tour, en anticipant la gêne de la …rme brune concernant le rapport des pro…ts, vise à s'assurer la position dictatrice en termes du choix de qualité environnementale sur le marché. L'objectif est de dissuader l'entreprise brune de toute tentative de déviation.

On étudie donc le choix optimal de qualité environnementale dans un cadre d'une organisation des producteurs qui permet aux entreprises de communiquer a…n de se coordonner. Ce choix de long terme qui favorise l'intérêt mutuel n'assure pourtant pas un niveau de qualité environnementale socialement désirable.

A…n de tenir compte du dommage environnemental, on intègre au modèle la pression sociétale d'une ONG sur les entreprises. Menaçant d'une publicité nocive, l'ONG peut inciter l'organisation des producteurs à internaliser leur empreinte environnementale dans la décision interne. On constate que cette mesure n'atteint pas non plus le niveau de qualité environnementale socialement optimal. En e¤et, l'organisation ne prend pas en compte la valeur attribuée à la diversité des biens sur le marché par les consommateurs.

Cette défaillance nous renvoie aux questions principales suivantes : quel est le mécanisme d'incitation qui serait à la fois puissant et e¢ cace ? Quels sont les coûts associés à la tentative d'accorder un rôle social à une structure privée ? En…n, le marché correctement ajusté peut-il véritablement corriger les externalités ?

General Introduction

While the general notion of quality has evolved over the centuries for certain products, such as French wines and cheeses, the concept of environmental quality is relatively recent.

It arose together with mass production, with the ambition to ensure high …nancial gain while neglecting the impact of industrialization on the environment.

The standards of environmental quality are beginning to enter the technical speci…cations of a large range of products. Having become a central and strategic feature of a product, environmental quality concerns all the agents in a market. Consumers start to develop an authentic taste for environmental quality; …rms are looking to meet their expectations, while the regulator is concerned with market de…ciencies. The general tendency of globalization and deregulation has given non-government organizations (NGOs) a particular role to …ll in the shortcoming of public interest. They undertake initiatives to in ‡uence public opinion, inform and educate consumers'choices, try to enforce businesses to internalize their actions. The growing public awareness of environmental issues leads ultimately to changes in public policies. The imposition of environmental regulation is the subject of Chapters 1 and 2.

We examine how tax avoidance a¤ects the optimal design of a linear tax on polluting emissions in a monopoly setting. The …rm is owned by shareholders who di¤er in their cost of tax dodging. Following [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF], the optimal tax should correct two negative externalities due to pollution and the monopolist's behavior. The analysis highlights two con ‡icting e¤ects of tax avoidance on the environmental policy design: a free-riding e¤ect and a tax base erosion e¤ect. With heterogeneous tax avoidance, the regulator must also internalize the externality imposed by the free-riding of tax avoiders on the rest of the society. This free-riding makes the regulator either impotent or unfair, depending on the severity of the environmental damage and the …rm's e¢ ciency. We also show that a two-part tax schedule can achieve the …rst-best outcome.

Apart from the behavioral response in the form of tax avoidance, the environmental taxation can also evoke a strategic response at the …rm's level. This is the subject of Chapter 2. Motivated by a price premium on high environmental quality, the clean (nonpolluting) …rm has an interest to credibly transmit information about its environmental performance in order to bene…t from consumers'willingness to pay more for high quality. At the same time, the dirty (polluting) …rm can free-ride on enthusiastic consumers' expectations about environmental quality and imitate the clean type. Chapter 2 is dedicated to the analysis of strategic actions of a …rm that faces the problem of information asymmetry.

The article studies how incomplete information a¤ects the optimal second-best environmental tax in a monopoly setting.

Our contribution is to endogenize the pollution tax and to analyze its impact on social welfare. We show that for either pooling or separating equilibrium, the optimal tax must be set below the level required for a polluting monopoly under complete information. The tax is composed of three elements to correct corresponding market distortions. The …rst is the Pigouvain standard that is to internalize the pollution externality. The second adjusts the tax because of monopolistic pricing which holds back output to extract the noncompetitive rent. The third and novel element is to account for information asymmetry.

As the tax enters the price of either variety under separating and pooling equilibria, consumers'expectations of high environmental quality determine the value of the clean product, the trade of which is inevitably restrained by the pollution tax. Hence, the regulator who places a corresponding value on the clean variety scales down the optimal tax level.

We distinguish two forms of regulation: (i) a weak regulation which preserves the cost advantage of the polluting monopoly; (ii) a strong regulation which grants the cost advantage to the clean monopoly.

While under pooling, the pollution tax equalizes the expected social value to the expected social cost of the product. This is no longer attainable under separating equilibrium. The clean monopolist sets the price at a suboptimal level in the presence of information asymmetry. His reaction to price change induced by the tax depends on the relative position of the price in relation to the pro…t-maximizing level. With weak regulation, the expected product price exceeds its expected social cost and vice versa occurs with strong regulation.

A welfare analysis gives the following results: environmental regulation reduces the pro…t of the polluting monopoly, be it pooling or separating equilibrium. Environmental regulation augments the pro…t of the clean monopoly under strong regulation and under pooling equilibrium. Yet, surplus of consumers for the clean variety is generally reduced subsequent to regulation.

Finally, we study the optimal choice of environmental quality at a …rm's level relative to what a welfare-maximizing regulator would desire. This choice cannot be considered in the abstract from the issue of product di¤erentiation. When consumers value high environmental quality as well as the diversity of products sold in the market, the private quality choice is insu¢ cient. We explore in Chapter 3 the means that are necessary to attain the socially optimal level of environmental quality.

In a stylized model of vertically di¤erentiated brown-green duopoly, two …rms with market power that increases in environmental quality are competing in product quantities.

In a baseline model, we derive the privately supplied level of environmental quality and show, in line with the literature, that this level is suboptimal compared to what a benevolent regulator would desire. We argue that this equilibrium might be unstable in the long run because the …rms generate unequal pro…ts which provides an incentive to coordinate their action: for the brown …rm to bene…t from a greater quality di¤erentiation, and for the green …rm to ensure the leadership position in the choice of environmental quality.

We thus adjust the model to allow for producers' coordination, as well as with increased environmental awareness induced by the threat of NGO's negative publicity. For every case under consideration, the privately chosen environmental quality falls short of the socially desirable level. Hence, in context of a global tendency of deregulation of markets, entrusting implementation of public interests to a private entity may call for caution.

Résumé du Chapitre 1

Le chapitre 1 étudie l'impact de l'évitement …scal sur la taxation optimale de second rang portant sur les émissions polluantes d'un monopole. L'entreprise est détenue par des actionnaires hétérogènes dans leur capacité d'esquiver le …sc. La conformité de l'entreprise à la taxation environnementale est déterminée en fonction des coûts d'évitement. Elle dépend du niveau de la taxe. Lorsque la taxe augmente, la non-conformité devient attractive pour un plus grand nombre d'actionnaires. Dans la tradition de [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF], la taxe optimale devrait corriger les deux externalités négatives qui proviennent de la pollution et du comportement en situation de monopole. La capacité à échapper le …sc introduit un nouveau paramètre de décision. L'analyse fait apparaître de deux e¤ets opposés de l'évitement …scal sur le design de la politique environnementale : l'e¤et d'un comportement de passager clandestin des actionnaires et l'e¤et de l'érosion de l'assiette …scale suite à une hausse de taxe. Le mécanisme que la taxe déclenche est le suivant : lorsque la non-conformité augmente, le niveau de la taxe doit aussi augmenter, ce qui entraîne, par conséquent, encore plus d'esquive. Etant donnée l'hétérogénéité des actionnaires face à l'évitement …scal, le régulateur doit internaliser l'externalité imposée par le comportement de passager clandestin de l'entreprise sur le reste de la société. La taxe optimale doit équilibrer ces deux e¤ets opposés. Le comportement de passager clandestin rend le régulateur soit impuissant, soit inéquitable en fonction de la sévérité du dommage environnemental et de l'e¢ cacité de l'entreprise. Dans le cadre d'un dommage modéré où la taxation à la Buchanan aurait été instaurée, le régulateur s'abstient de la régulation car la perte des contribuables serait plus coûteuse pour la société que le gain de correction de l'externalité environnementale. Pour un dommage considérable, la taxe doit dépasser le niveau de Buchanan. En conclusion, quand la politique environnementale est mise en place, les consommateurs en supportent le fardeau …nancier. Dans le cas contraire, ils supportent la pollution. Nous démontrons par ailleurs qu'un programme d'une taxe à deux parties peut atteindre la solution du premier rang.

Introduction

Economists have long advocated for high Pigouvian taxes that would directly place prices on the most harmful of greenhouse gases to achieve signi…cant reductions of those. For instance, the optimal Pigouvian tax on carbon dioxide should be set in the range of $75 to $175 per ton according to the best estimates of marginal damages of emissions [START_REF] Stavins | The Problem of the Commons: Still Unsettled after 100 Years[END_REF]. Although environmental taxation has great conceptual appeal, economists also recognize the challenge of inducing tax payers to comply with taxes at such high rates.

Surprisingly enough, the concern formerly expressed by Adam Smith is rarely invoked in this respect:

"High taxes, sometimes by diminishing the consumption of the taxed commodities, and sometimes by encouraging smuggling, frequently a¤ord a smaller revenue to government than what might be drawn from more moderate taxes" (Smith 1776, Book V, Chapter II).

Long after publication of The Wealth of Nations, Arthur La¤er drew his famous curve illustrating the possibility of an inverse relationship between tax rates and government revenue [START_REF] La¤er | Statement prepared for the Joint Economic Committee[END_REF]. Understood in a broad metaphoric sense, the term "smuggling" used by Adam Smith may refer to a variety of tax dodging in today's economies, ranging from the relocation of a corporation's legal domicile to a lower-tax nation, usually known as tax inversion, to the substitution between labor and leisure discussed in general equilibrium models (see [START_REF] Fullerton | On the possibility of an inverse relationship between tax rates and government revenues[END_REF], for instance). The type of "smuggling"we examine here is tax avoidance, referring to all the transactions that take advantage of legal loopholes to reduce total tax liabilities1 .

Fiscal avoidance exerts a negative in ‡uence on the ability of governments to implement an optimal environmental regulation. This has been recognized in the incentive regulation literature for the case where the regulator cannot observe the costs of polluters'abatement alternatives. 'Foreign'shareholders of a polluting …rm -those who reside outside the regulator's jurisdiction-are identi…ed as prone to escape taxation (Baron 1985, p. 218).

Real-world examples are multinational corporations operating upstream in the supply chain of carbon-intensive goods, such as French Total, LafargeHolcim (the cement giant of merged french Lafarge and Swiss Holcim), German BASF or Russian GazProm. The presence of foreign capital in the shareholders structure of these polluting …rms varies from 30 to 70%. Hence, a regulator puts unequal weights on corporate pro…ts as on consumers'surplus. As pro…ts are partly excluded from the social welfare function, the regulator fails to fully refund the tax revenues to consumers -all of whom reside in the regulator's jurisdiction -which ultimately entails a welfare loss.

Similarly, [START_REF] Spulber | Optimal environmental regulation under asymmetric information[END_REF], [START_REF] La¤ont | The new economics of regulation ten years after[END_REF], or [START_REF] Lewis | Protecting the environment when costs and bene…ts are privately known[END_REF] acknowledge that the cost of transferring the emission tax from the polluter to the polluted limits the governance capacity of an environmental regulator. The social cost of raising public funds in these models raises a distributional problem, which amounts again to favoring consumers'over producer's interests in the regulator's objective function. The cost of public funds is traditionally justi…ed by the use of distortionary taxation for raising funds, but it might be attributable to …scal avoidance. A general insight is that, given an exogenous reduction of the tax base, the regulator must amend the environmental tax design accordingly 2 . Surprisingly enough, this literature has not considered the possibility that the environmental regulator adopts the means to in ‡uence the decision to avoid taxation. If the purpose of environmental taxation is to force individuals to consider the full set of consequences from polluting emissions, the tax should also internalize the externality caused by the trouble transferring tax.

In this paper we assume that the environmental regulator is able to deal with tax avoidance, thereby controlling tax payment by the …rm's shareholders instead of undergoing it. There is evidence of a substantial heterogeneity in tax avoidance [START_REF] Christian | Voluntary compliance with the individual income tax: results from the 1988 tcmp study[END_REF]). Among the shareholders of a polluting …rm, there are some who comply with taxation, and some who do not. The model in [START_REF] Baron | Regulation of prices and pollution under incomplete information[END_REF] suggests that tax avoidance arises from di¤erences in the shareholders' residential location, which involves various cost-bene…t calculations. In [START_REF] Slemrod | Tax avoidance, evasion, and administration[END_REF], tax avoidance mostly involves high-income individuals and depends on regulatory expenditures on enforcement. The heterogeneity in tax avoidance can also be explained by di¤erences in intrinsic motivation such as civic virtue, as argued by [START_REF] Frey | A constitution for knaves crowds out civic virtues[END_REF]. In our setting, …rm's shareholders di¤er in their opportunity to avoid taxation. Our model departs from [START_REF] Slemrod | Fixing the leak in Okun's bucket optimal tax progressivity when avoidance can be controlled[END_REF] in that the regulator controls the extent of tax avoidance through the design of the environmental tax rather than through an enforcement mechanism.

When the polluting …rm has no information rent, [START_REF] La¤ont | The new economics of regulation ten years after[END_REF] shows that the Pigouvian tax leads to internalize correctly the environmental externality and a lumpsum tax on pro…t solves the distributional problem. This result suggests that, if the regulator could observe the cost of abating pollution, tax avoidance would not be an issue provided that the combination of two instruments, namely the tax per unit of emissions and the lump-sum taxation on pro…t, is available. We adopt a di¤erent perspective and consider that the regulator is armed only with a tax on polluting emissions. Unlike [START_REF] La¤ont | The new economics of regulation ten years after[END_REF], the regulator in our model is able to modify the amount of transfers that must be handed back to consumers. Fiscal avoidance undermines the regulator's commitment to redistribute the tax yields, leading to an endogenous disturbance in tax transfer. Building on [START_REF] Baron | Regulation of prices and pollution under incomplete information[END_REF], we assume that the regulator's objective excludes part of the monopoly pro…ts belonging to the shareholders who avoid taxation. However, unlike Baron, the regulator in ‡uences the changes in the tax base by setting the secondbest optimal tax rate. Our analysis di¤ers further in that it abstracts from information asymmetry and assumes environmentally conscious consumers, implying that consumer disutility of pollution internalizes, to some extent, the environmental damage.

We shed light on adverse e¤ects from the free-riding of tax avoiders on the e¢ ciency of environmental regulation. In a partial equilibrium setting where a regulator levies an environmental tax to correct for the polluting behavior of a monopolistic …rm, we characterize optimal taxation, assuming that …rms' shareholders can choose between paying and avoiding the tax. That is, we consider the second-best problem of choosing the welfare-maximizing tax for which the regulator endogenizes the free-riding behaviors of tax avoiders.

In a recent work based on the "World Induced Technical Change Hybrid" model, [START_REF] Carraro | Şinvestments and public …nance in a green, low carbon, economy µ T[END_REF] predict that carbon taxes might sometimes generate …scal revenues which …rst increase, then achieve the highest levels of revenue, and …nally decrease, thereby forming a "carbon La¤er" curve (p. 25). Their analysis questions the political and economic feasibility of large taxation schemes. We raise similar concerns regarding a polluting monopolist, following the tradition of [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF]. As has been thoroughly formalized by [START_REF] Lee | E¢ ciency of pollution taxation and market structure[END_REF] and [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF], the regulator must scale down the environmental tax below the Pigouvian level to correct for the monopolist's tendency to underproduce. Therefore, compared to markets with perfect competition, the emissions control of imperfectly competitive …rms requires a less stringent taxation.

We take a fresh look at this second-best policy and examine how the environmental regulator copes with the possibility that the …rm's shareholders bypass taxation. Clearly, this is an issue of theoretical importance. If the purpose of environmental taxation is to force individuals to consider the full set of consequences from polluting emissions, the optimal tax should also internalize the negative externality caused by the free-riding of tax avoiders. Furthermore, tax avoidance has emerged as a systemic problem with the globalization of the economy. The deregulation of the …nancial system in industrialized countries, which took place in the 1980s, together with the technological progress, has made the mobility of capital far greater than that of labor. While labor mobility is largely limited to national jurisdiction, the capital can almost instantaneously move around the globe.

As a result, almost every multinational …rm is to some extent involved in avoidance activities (Christensen and Kapoor 2004, p. 9). Indeed, many multinational companies such as Apple, Amazon and Microsoft [START_REF] Young | The interview: John Christensen, director, tax justice network[END_REF] or Starbucks, Total and Colgate [START_REF] Harel | Evasion …scale: le hold-up du siècle [The robbery of the century : Tax evasion[END_REF]) have drawn public attention for remarkably low taxes paid on pro…t. More importantly for our purposes, tax avoidance is closely related to companies operating in greenhouse gas producing sectors, including the extractive industries, aviation, shipping, pharmaceuticals, traded commodities and weapons industry (Christensen and Kapoor 2004, p. 3).

As pointed out by the optimal taxation literature, avoidance or sheltering is the behavioral response to taxation where an individual searches a legal opportunity to reduce his tax liability [START_REF] Cowell | Tax sheltering and the cost of evasion[END_REF]. Besides this issue, the mainstream literature on avoidance is concerned with optimal enforcement expenditure [START_REF] Mayshar | Taxation with costly administration[END_REF] and [START_REF] Slemrod | Fixing the leak in Okun's bucket optimal tax progressivity when avoidance can be controlled[END_REF]).

In the present paper, we ignore income e¤ects and abstract from enforcement e¤orts to focus on the impact of tax avoidance on environmental regulation; only the e¤ect of diminishing the tax base caused by tax sheltering is under consideration. To our knowledge, the concern of tax base erosion has not yet been addressed in the …eld of environmental economic. The only study of environmental tax evasion belongs to [START_REF] Liu | Tax evasion and optimal environmental taxes[END_REF]) who questions its impact on environmental taxation. He shows in a generalized equilibrium model with perfect competition that an environmental tax may enhance welfare provided that environmental tax is more di¢ cult to evade than other 'regular' taxes.

In our framework, the …rm's pro…t is distributed to shareholders who di¤er in their opportunity cost of avoiding taxation. They will gradually bypass taxation as the tax rate increases. Hence, the regulator's objective function excludes part of the monopoly pro…ts attributable to tax avoidance. The environmental tax is designed so that the regulator optimally chooses the level of tax avoidance compatible with the dual task of internalizing the polluting externality and mitigating the monopolist's overpricing. The analysis highlights two con ‡icting e¤ects of tax avoidance on the environmental policy design: a free-riding e¤ect and a tax base erosion e¤ect. On the one hand, tax avoiders free-ride on tax payers as well as consumers by passing on to them the burden of the tax. The regulator internalizes this externality by taxing the complying shareholders more heavily than it would do under full compliance. On the other hand, the regulator must take into consideration that the tax increase will induce more shareholders to avoid taxation. As shrinking tax base reduces tax revenues, the regulator has an opposite incentive to soften taxation in order to compensate for the shareholders' mobility. As a result, depending on which e¤ect of tax avoidance prevails, the second-best optimal taxation proves more or less severe relative to full compliance.

We …nd that the tax base erosion e¤ect dominates when the monopolist is rather clean and the polluting good is relatively inexpensive to produce, so that environment has a low rank in the agenda of economic priorities. The regulator gives up a positive tax that would otherwise internalize the monopolist's environmental externality under full tax compliance, for the sake of maintaining the tax base. Anticipating that positive taxation would induce some shareholders with low costs of avoidance to escape, the regulator refrains from taxation. This may somewhat explain the sluggishness in introducing carbon taxes when the focus on economic growth takes priority over the environmental concern. If, on the contrary, the monopolist is rather dirty and/or moderately e¢ cient, the environmental damage becomes the major concern and the free-riding e¤ect of tax avoidance dominates. To cope with the dire need of internalizing the polluting externality, the regulator can no longer a¤ord not to tax emissions. Instead, the regulator taxes pollution at the cost of losing shareholders. The revenue lost as a result of tax avoidance creates a further negative externality on those who bear the burden of environmental taxation. These are the shareholders who remain under the regulator's authority, as well as the consumers who pay the monopoly price in ‡ated by the tax. To correct for the tax avoiders'free-riding, the regulator must raise the environmental tax above the level recommended by [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF]. Consequently, the remaining taxpayers must bear a greater tax than that under no tax avoidance.

We …nally allow the regulator to use the combination of a unit tax on emissions and a lump-sum tax. We show that this two-part tax schedule successfully achieves the …rst-best outcome in our setting.

The present article is organized as follows. In Section 1.2, we set up the formal model. Section 1.3 characterizes the second-best optimal tax under the assumption of tax avoidance. Section 1.4 examines the use of the two-part tax schedule. In Section 1.5, we present a brief history intended to illustrate the reluctance of an environmental regulator to correct for the environmental externality because of tax avoidance. Section 1.7 concludes.

The model

The industry consists of a single polluting …rm, the …rm's shareholders, consumers and an environmental regulator.

The …rm produces quantity q of a good that releases the amount (1 e) q of polluting emissions, where parameter e 2 [0; 1] represents the state of the abatement technology.

If, for instance, polluting emissions are greenhouse gases and the good is electricity, e measures the use of non-emitting methods such as hydroelectric, nuclear, or geothermal, and 1 e the use of emitting methods through the combustion of coal, natural gas, or petroleum distillates. We restrict attention to short-run decisions, so that the level e is not a decision variable. If the abatement is at a maximum, the emissions are zero. If no abatement is undertaken, the emissions are equal to the output. The overall marginal cost of producing the good is c(e), with c 0 (e) > 0, meaning that devoting more resources to abatement raises the overall cost of production. Industry-speci…c emissions cause environmental damage d(q; e), which is assumed to be strictly proportional to the amount of polluting emissions: d(q; e) = (1 e) q, where > 0 measures the marginal damage from emissions.

The regulator is facing two market distortions due, respectively, to monopoly power and environmental externality. In that event, we know from [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF][START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF]) that a second-best optimal policy strikes the balance between the need to correct for the monopolistic behavior and the Pigouvian task of internalizing the marginal social damage. To allow comparison with this literature, we assume that the regulator is benevolent and the only regulatory instrument available to correct for both distortions is a tax per unit of emissions. Hence, throughout the article, the environmental tax can be thought of as the application of a broader regulatory policy: besides being concerned with the environment, the regulator is also responsible for controlling the market price.

In other terms, the regulator is wearing two hats, that of an environmental protection agency and a public utility commission. A tax will take negative values if it turns out to be a subsidy. Indeed, it is conceivable that the tax cut resulting from the need to correct for the …rms'market power becomes so sharp that the optimal policy imposes to subsidize the product.

The …rm is owned by a continuum of shareholders, holding one unit of pro…t each.

We normalize the mass of shareholders to be one. For simplicity, we will assume, …rst, that a shareholder has valuation 1 for holding one unit of pro…t, and second, that a shareholder incurs a cost 2 [0; 1] of …scal optimization. Hence, a shareholder with a cost gets the surplus 1 from tax dodging. Paying the cost may help the shareholder economize on the amount (1 e) of the due tax. For instance, is the cost of sourcing out for the best way of being tax exempt. In practice, there exists a market for tax optimization services. Banks o¤er their corporate and private clients services such as wealth management, which contain "recommendations" for organizing their …scal obligations in the most bene…cial way [START_REF] Gravelle | Tax havens: International tax avoidance and evasion[END_REF]. Alternatively, an internal department of the …rm may specialize in tax issues, screening for countries where a subsidiary would bene…t from the most favorable tax regimes. From [START_REF] Christian | Voluntary compliance with the individual income tax: results from the 1988 tcmp study[END_REF],

there is evidence of a substantial heterogeneity in tax avoidance. This will be captured by assuming that shareholders di¤er according to , which is uniformly distributed on [0; 1]. Now consider the behavior of a shareholder whose objective is to maximize surplus.

The shareholder b ( ) who is indi¤erent between paying and avoiding the environmental tax is determined by

1 (1 e) = 1 b ( ); (1.1)
which implies that the fraction of shareholders who comply with taxation is given by 1 b ( ), provided that 0 < < 1 1 e . If 0, the environmental policy takes the form of a subsidy, and there is full compliance. Moreover, if

1 1 e , then all the shareholders bypass taxation. Let ( ) denote the compliance rate for taxation:

( ) = 8 > > > < > > > : 1 if 0; 1 (1 e) if 0 1 (1 e) ; 0 if 1 (1 e) v c(e)
1 e :

(1.2)

The monopolist sells the polluting good to a continuum of risk-neutral consumers with heterogenous "green" preferences. Each consumer purchases at most one unit of the good, which thus indirectly generates the amount 1 e of polluting emissions via consumption. Although consumers have a common reservation price v, they di¤er in their tastes for the good due to their personal dislike of pollution. For a given consumer X, the dislike of pollution is measured by the monetary loss (1 e)X, which, besides psychic discomfort, may represent health care expenditure and all the adaptation costs to the polluted environment. The heterogeneity of preferences may also re ‡ect various degrees of social environmental conscience among consumers. If, for instance, the good is fossil energy, consumers may di¤er in their aversion to the negative impact on global warming, and if it is nuclear energy, they may di¤er in their dislike of the potential risks imposed on future generations by nuclear repositories. To simplify the analysis, we make the assumption that X is uniformly distributed along a segment of unit length, which is convenient to generate a linear demand function. Hence, consumers distinct from 0 partly internalize the polluting externality, and none of them fully internalizes it as long as 1 < . The total number of consumers is normalized to unity. Assuming that consumers receive zero surplus from consuming outside goods, consumer X derives a surplus v (1 e)X p from purchasing the good at price p, which yields the demand function

D(p; e) = 8 > > > < > > > : 0 if p v; v p 1 e if v (1 e) p v; 1 if p v (1 e) :
(1.3)

Socially optimal allocation of the good.-The welfare function is the conventional one of gross bene…ts to consumers less production and pollution costs. The welfare function is

W (X) = X Z 0 [v c(e) (1 e)x (1 e)]dx (1.4) = [v c(e) (1 e)]X (1 e) X 2 2 :
The expression (1 e)X is the actual damage caused to the environment, while (1 e) X 2 2 represents the monetary equivalent of the consumer's dislike for pollution3 .

At the socially optimal solution, the marginal consumer X solves equation v (1 e)X = c(e) + (1 e) so that the marginal social value of the good v (1 e)X must exactly o¤set the total social marginal cost c(e) + (1 e). Thus from the social standpoint, the market size should be

X = v c(e) 1 e : (1.5)
To ensure that the good is socially desirable, we will restrict the parameters of the model to satisfy the following assumption:

(1 e) v c(e):

(1.6)

We will abbreviate the marginal social cost of pollution where "(p e ) = pe v pe . Substituting this expression into the right-hand side of (1.8), we obtain the monopoly price pe = v+c(e) 2 , and the resulting demand is D(p e ; e) = v c(e) 2(1 e) . Comparing this outcome to the socially optimal solution, two separate cases emerge within the parameter con…guration de…ned by (1:6). In one case where (1 e) v c(e) 2 , the MSC is low relative to the MSV-say, because the monopolist is signi…cantly clean and e¢ cient -, and we have D(p e ; e) X : the monopolist's production is lower than that implied by the socially optimal solution. Following the policy recommendations from [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF] or [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF], the regulator should rely on the monopolist's tendency to underproduce and scale down the environmental tax below the Pigouvian level (possibly until the tax turns into a subsidy paid to the …rm for fairly low values of the MSC). In the other case where the MSC (1 e) exceeds v c(e) 2 -the monopolist is signi…cantly dirty or moderately e¢ cient -, we have D(p e ; e) > X , so that the monopolist by itself produces too much of the polluting good. In that case, the monopolist is no longer 'the environmentalist's best friend'; the second-best optimal policy calls for a regulation more stringent than in the previous case.

Lemma 1 summarizes this discussion.

Lemma 1 Under Assumption (1:6), in the absence of regulation, the monopolist underproduces relative to what would be socially desirable when (1 e) < v c(e) 2 , and otherwise, the monopolist overproduces.

Second-best optimal tax

We now consider the second-best problem faced by a welfare-maximizing regulator who can control the pollution emitted by the …rm but not its monopolistic behavior. Hence, the regulator cannot ensure that the …rm will behave as a price-taker. We further assume that the regulator is a Stackelberg leader who commits to a policy. Thus, we model the policy implementation as a two-stage game. In the …rst stage, the regulator speci…es a tax on each unit of emissions that maximizes the social welfare W ( ) and generates the revenue R( ). We allow to be negative, in which case it will be a subsidy paid to the …rm instead of a tax. The regulator is also committed to transferring all tax yields to consumers. Stage two is the production period. In this stage, the monopolist sets the price of the good, the consumers decide whether to purchase the good, and the shareholders decide whether to comply with taxation. Should a shareholder decide to avoid taxation, it must incur its own cost of …scal optimization. This sequence of play amounts to assuming that the regulatory policy has a commitment value, so that, once made, the choice of cannot be reversed in stage two, and the regulator cannot renege on his commitment to transfer the tax yields to consumers.

Denoting by p e ( ) the monopoly price under regulation, the tax base is

(1 e)D (p e ( ) ; e) = v p e ( ), and the expected emissions payments R( ) correspond to the …scal revenue corrected by the compliance rate R( ) = ( ) (v p e ( )):

(1.9)

As will be seen later, R( ) plots the …scal revenue as a function of the environmental tax, yielding a new variant of the La¤er curve. Furthermore, tax avoidance drives a wedge between R( ) and the total amount of tax yields to be transferred to consumers, that is,

(1 e)D (p e ( ) ; e) = (v p e ( )).

Anticipating tax avoidance, the regulator recognizes that part of the corporate pro…t escapes the regulator's jurisdiction, which distorts the refund of the tax proceeds to ;

(1.11) or, equivalently,

p e ( ) = v + c(e) + (1 e) 2 :
(1.12)

The regulator will take this monopoly price into consideration to implement the environmental policy. Clearly, we can see from (1:11) that, besides internalizing the environmental externality, the regulator will employ the environmental tax to correct for the externality exerted on the society by the monopolistic behavior. Moreover, the comparison between the monopoly prices pe given by (1:8) and p e ( ) given above shows that the environmental tax will be partly passed on to consumers in a higher price for their polluting good purchases.

Substituting (1:12) and (1:2) into (1:9) leads to the following relationship between tax rates and …scal revenue:

R( ) = 8 < : 2 (1 (1 e)) (v c (e) (1 e)) if 0 1 (1 e) ; 0 if 1 (1 e) v c(e)
1 e :

(1.13) Some calculations given in Appendix 1.8.1 show that R( ) reaches a maximum at b , falls to zero at = 1

(1 e) , and remains zero on the interval

h 1 1 e ; v c(e) 1 e
i , where tax rates are so high that all the shareholders bypass taxation (see

(1:2)). Figure 1 represents R( ) with
respect to as a solid line. This curve is always lower than the dotted curve R n ( ) that depicts the …scal revenue in the absence of tax avoidance, that is, when ( ) = 1 for all (the expression of R n ( ) is relegated to Appendix 1.8.1). Interestingly enough, …scal revenues in both regimes of avoidance and compliance are consistent with the statement by [START_REF] Dupuit | On the measurement of the utility of public works[END_REF]:

"By thus gradually increasing the tax it will reach a level at which the yield is at a maximum . . . Beyond, the yield of tax diminishes . . Lastly a tax will yield nothing".

This early insight into the existence of a revenue maximizing tax was later illustrated by the La¤er curve.

Like the La¤er curve, the two curves in Figure 1-1 exhibit a downward-sloping segment for high tax rates,known as the "prohibitive"range because the same revenues can

τ  τ  n = v-c (e) 2 1-e 1 1-e v-c (e)
1-e

Tax rate, τ

R (τ), R n (τ)
Tax revenue with compliance and avoidance

Revenue with full compliance

Revenue with avoidance

Figure 1-1: La¤er Curve be collected at lower tax rates [START_REF] Fullerton | On the possibility of an inverse relationship between tax rates and government revenues[END_REF]. General equilibrium models such as that used by [START_REF] Fullerton | On the possibility of an inverse relationship between tax rates and government revenues[END_REF]) in a perfectly competitive regime, have highlighted the importance of income e¤ects and substitution e¤ects between labor and leisure in explaining the emergence of a negative relationship between tax rates and …scal revenue. The reasons for this are clearly di¤erent in the present setting since our partial equilibrium approach focuses exclusively on the market for the polluting good while ignoring income e¤ects. In the absence of tax avoidance, tax rates above b n lower the revenue R n ( ) because they raise the monopoly price and, …nally, shrink the sales volume, thereby reducing the tax base down toward zero at = v c(e) 1 e . When shareholders have the possibility to avoid taxation, tax rates above the maximum b depress the revenue R( ) even further because a growing number of shareholders with su¢ ciently low costs of …scal optimization prefer to bypass rather than pay the tax, as its burden increases. Therefore, R( ) falls gradually to zero, from b to = 1

(1 e) , where all the shareholders …nd it less costly to bypass taxation. If the regulator raises the tax above this threshold, but below v c(e) 1 e , so that at least one consumer purchases the good, environmental taxes have no bene…t and the burden of environmental taxation is fully passed on to consumers through the price set by the monopolist. Finally, the comparison between R n ( ) and R( ) shows that tax avoidance extends the prohibitive range of the "environmental La¤er curve".

Let us now examine how consideration of tax avoidance shapes the social welfare function. For this, we plug (1:12) and(1:2) into (1:10). After some calculations given in Appendix 1.8.2, we obtain that W ( ) is made of three parts due to the shareholders' responses to taxation. When takes the form of a subsidy, no shareholder avoids taxation and W ( ) coincides with the function W n ( ) on this range. For non-negative tax rates inside the interval 0; 1 1 e , the welfare W ( ) has a di¤erent functional form given by W p ( ), due to the emergence of partial tax avoidance: some shareholders with su¢ ciently low costs of …scal optimization prefer to bypass rather than pay the tax, and the foregone revenue entails a loss in social welfare. Finally, the welfare W ( ) turns into the function W f ( ) for higher tax rates inside the interval

h 1 1 e ; v c(e) 1 e i
, where there is full avoidance: in this range, the regulator has zero revenue from the tax since all the shareholders prefer to bypass it, and the burden of environmental taxation is fully passed on to consumers through the monopoly price.

Lemma 2 The social welfare function can be decomposed into the following functions:

W ( ) = 8 > > > < > > > : W n ( ) if 0 (no tax avoidance); W p ( ) if 0 < 1 (1 e) (partial tax avoidance); W f ( ) if 1 (1 e) < v c(e)
1 e (full tax avoidance):

(1.14) would be socially desirable. Hence, a positive tax would have the undesirable e¤ect of in ‡ating the price of the good, leading to the aforementioned e¤ects on consumption and monopoly power, both detrimental to welfare. Rather, a subsidy has a desirable e¤ect on price. The dual bene…t of boosting consumption on the demand side and mitigating the upward pressure that the monopolists puts on price on the supply side, balances the adverse e¤ect of increasing pollution. In addition, with a subsidy, the regulator does not need to worry about tax avoidance.

Regarding the three other welfare functions displayed in Figure 1-2, we suspect that the MSC of pollution is now high enough so that (1 e) v c(e) 2 holds, namely the monopolist is moderately clean or moderately e¢ cient. For this parameter con…guration, we know from Lemma 1 that the monopolist produces too much of the polluting good from the social standpoint, unless the MSC of pollution coincides with the MSV of the good. The regulator should now consider taxing the monopolist rather than subsidizing the polluting output. If, by chance, there is a coincidence between (1 e) and v c(e) 2 , Lemma 1 suggests that regulation is pointless since the monopolist is behaving in a socially e¢ cient way on its own. It happens, however, that the three welfare functions other than that in the upper left of the …gure depict situations in which there is a local maximum at zero tax, although the MSC of pollution exceeds the MSV of the good.

What is intriguing in the upper right of Figure 1-2 is that the local maximum at zero tax clearly dominates the local maximum resulting from positive taxation. This suggests that tax avoidance disrupts the previous balance between the environmental bene…t of taxation and the detrimental upward pressure it puts on the monopoly price. We can guess that the fear of a novel adverse impact on welfare refrains the regulator from taxing the monopolist. From Proposition 3, we know that a positive tax, however light it is, induces the shareholders with the lowest costs of avoidance to bypass taxation. This immediately creates a welfare loss that can be seen at = 0, where the slope of every welfare function turns from positive to negative. To o¤set this loss, the regulator must resort to a sharp increase in taxation, leading to the local maximum on the right-hand side of the welfare functions. The tax increase pays the bill for the free-riding of the tax avoiders, thereby enhancing welfare. However, this bene…cial e¤ect may not be su¢ cient to reach a welfare level higher than that obtained with the zero tax, as illustrated by the welfare function in the upper right of Figure 1-2. It is su¢ cient in the two remaining cases (the lower left and right of the …gure), which suggests that the MSC of pollution is much greater than the MSV of the good, relative to the other cases. Although the positive tax fosters tax avoidance, it internalizes the externalities due to the tax avoiders' free-riding and the pollution as well.

Observe …nally that, in the lower right of Figure 1-2, the monopolist is signi…cantly dirty and the good is expensive to produce. This situation calls for such a severe taxation that it induces every shareholder to dodge paying tax.

We now analyze the second-best optimal tax on a case-by-case basis.

Low tax rates: the range of partial tax avoidance

The most interesting case of our analysis occurs when some, but not all, shareholders avoid taxation, that is, W ( ) = W p ( ). Assuming a solution inside the interval 0; 1 1 e , the …rst order-condition for welfare maximization is A detailed analysis of welfare maximization is relegated to Appendix 1.8.3. Proposition 3 shows the extent to which tax avoidance a¤ects the optimal tax design in the case where some shareholders still comply with the taxation.

Proposition 3 The second-best optimal tax with partial avoidance satis…es = + p e ( ) "(p e ( ))

1 1 e + 2 (1 ( )) 1 e + 2d d p e ( ) c(e) (1 e) (1 e) 2
The tax departs from the …rst-best Pigouvian level (the …rst term in the right-hand side of the equation in Proposition 3) only if the producer possesses some market power, that is, pe( ) "(pe( )) > 0. Hence, the deviation from the Pigouvian level is higher in markets where consumers are less sensitive to price changes. The second term in the right-hand side of equation in Proposition 3 captures Buchanan's correction: the regulator reduces the tax to mitigate the monopolist's tendency to overprice.

The remaining two terms represent two opposite e¤ects of tax avoidance on social welfare: a free-riding e¤ect and a tax base erosion e¤ect. The shareholders with avoidance costs lower than (1 e) free-ride on those with higher costs, who thus …nd it worthwhile to comply with taxation. The regulator must strengthen the tax to internalize this externality. Therefore, the tax is increased by an amount exactly equal to the term pe( ) " 2 1 ( ) 1 e , which is positive when ( ) falls below 1. On the other hand, the regulator knows that every tax increase makes …scal avoidance attractive for some more shareholders, and every lost taxpayer worsens the …scal imbalance between consumers and shareholders. To o¤set this welfare loss, the regulator decreases the tax by an amount corresponding to the fourth term in the right hand side of equation in Proposition 3, i.e., pe( )"

2(1 ( )) 1 e
. Like the correction for the monopolist's overpricing, the overall correction for tax avoidance, whether positive or negative, is greater when price demand is less elastic.

Subsidies: the range of no avoidance

We now turn to the case where is a subsidy, so that social welfare is given by the function W n ( ). The shape of this function depends upon three forces appearing in the following derivative:

dW n ( ) d = @D( ) @p dp e d
(1 e)

(1 e) + p e " ( ) :

(1.16)

Raising the tax tends to push the monopoly price up, which both curtails consumption and reinforces the monopolist's tendency to overprice. These two adverse e¤ects on welfare are re ‡ected in, respectively, the …rst and the third terms in the right-hand side of (1:16), which are negative since @D( ) @p dpe d < 0. It turns out, however, that increasing the emission tax to combat pollution lowers aggregate emissions. This ameliorating e¤ect on welfare is captured by the second positive term in the right-hand side of (1:16). The …nal outcome ultimately depends on the balance of these con ‡icting forces. It clearly involves a subsidy when the shape of the welfare function is that depicted in the upper left of Figure 1-2.

High tax rates: the range of full avoidance

Finally, we consider the case where 1

(1 e) v c(e) 1 e , so that no shareholder complies with taxation, i.e., ( ) = 0. In that case, the environmental tax is fully passed on to the consumers in a very high price for the polluting good. Social welfare reduces to consumer surplus less the environmental damage. One can check that

dW f ( ) d = @D( ) @p dp e d
(1 e) ( ) p e " ( ) :

(1.17)

Here again, there is no welfare impact of taxation through tax avoidance because the tax base has completely vanished. Taxing polluting emissions boils down to manipulating the consumer price in order to achieve the desired allocation of the good.

In Appendix 1.8.2, we explicitly de…ne n , 2 and f as the tax levels that respectively maximize W n ( ), W p ( ), and W f ( ).

Note that n would be the optimal choice for the regulator if there were no tax avoidance in the economy. Hence, n is the traditional tax on emissions under monopoly, as stated by [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF]) and later computed by [START_REF] Lee | E¢ ciency of pollution taxation and market structure[END_REF]) and [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF].

Equating (1:16) to zero yields n = p e (1 e) " ( ) :

(1.18)

Without tax avoidance, the standard result holds in the present setting: the regulator reduces the tax below the Pigouvian level to o¤set the welfare loss due to the monopolistic behavior, which corresponds to pe

(1 e)"( ) . By substituting (1.18) into (1.11), we immediately see that n leads to the price that achieves the …rst-best outcome. This is due to the assumption that the amount of polluting emissions is strictly proportional to the output. As the regulator can mechanically control the output through the tax applied on emissions, this tool is su¢ cient by itself to correct both the environmental distortion and the monopoly distortion.

We now equate (1:17) to zero and get the following result:

f = + p e (1 e) " ( ) ; (1.19)
which clearly exceeds n . The second term in the right-hand side of the equation above suggests that when all the shareholders avoid taxation, the regulator is facing externalities other than pollution. We provide a detailed analysis of these externalities in the remainder of the article. For the moment, one synthetic interpretation is that pe (1 e)"(:) re ‡ects the net cost of the tax avoiders'free-riding, borne by consumers.

To provide an explicit solution for the second-best optimal tax under tax avoidance, we need to divide the parameter con…guration into the four regions de…ned in Table 1.

Region Boundaries

I v c(e) 2(1 e) II v c(e) (1 e) 1 4 and 2 h v c(e) 2(1 e) ; min n 24(v c(e)) 1 32(1 e) ; ( p 6 1)(v c(e)) 2(1 e) oi or v c(e)
(1 e) 1 4 and 2 ; v c(e) 1 e

h v c(e) 2(1 e) ; v c(e)+(v c(e)) 2 2(1 e) i III v c(e) ( 1 e
Table 1.1: Parameter con…guration

The four regions are depicted in Figure 1-3. This …gure can be read as a map of the regulator's priorities. Region I represents a situation where the environment has a low rank in the agenda of economic priorities because the monopolist is su¢ ciently clean and e¢ cient. Furthermore, Lemma 1 shows that, in Region I, the unregulated industry is not producing a su¢ cient amount of the good, from the social standpoint. In contrast, in Region IV, the regulator is very concerned about the environmental damage and, besides the environmental issue, the monopolist has a tendency to produce too much output given consumer preference for the polluting good. The next lemma compares the welfare levels reached at the di¤erent local maxima on a case by case basis.

Lemma 4 In Region I, n 0 and

W n ( n ) max fW p ( 2 ); W f ( f )g. In Region II, W n (0) max fW p ( 2 ); W f ( f )g. In Region III, W p ( 2 ) max fW f ( f )); W n (0)g. In Region IV, W f ( f ) max fW p ( 2 ); W n (0)g.
From this lemma, it is straightforward to characterize explicitly the optimal tax e under the threat of …scal avoidance, and compare it to Buchanan's tax n .

Proposition 5 The optimal solution for the regulator is to set e such that:

(i) e coincides with the subsidy n in Region I;

(ii) e = 0 < n in Region II;

(iii) e = 2 > n > 0 in Region III;

(iv) e = f > n > 0 in Region IV.

In Region I, the monopolist is so clean and e¢ cient that the regulator is mainly concerned to correct the monopolist's tendency to overprice and produce too little output from the social standpoint. The optimal policy is to decrease the consumer price through a subsidy in order to boost consumption and mitigate the monopolist's market power.

The subsidy is designed in accordance with Buchanan's rule. Clearly, tax avoidance is not an issue in this case. However, the idea that a …rm gets paid to pollute may be politically unacceptable [START_REF] Lee | E¢ ciency of pollution taxation and market structure[END_REF].

In Region II, the environmental issue is more serious and/or the good is more costly to produce than in Region I. Left to itself, the monopolist would produce a too large amount of the polluting good. Observe that this holds true for all parameter values outside Region I: the imperfect competition distortion fails to o¤set the externality distortion. To combat pollution and correct the monopolist's behavior as well, the regulator, in the absence of tax avoidance, should set the positive tax n on emissions consistent with Buchanan's rule. Under the threat of tax avoidance, however, the regulator prefers to withhold tax in Region II, worrying about the following chain reaction: a positive tax would have a knock-on e¤ect on tax avoidance, creating a shortfall in tax revenue that would, in turn, call for a further increase in the tax. This chain reaction would create a welfare loss relative to the zero-tax decision. To some extent, the optimal policy in Region II boils down to letting the monopolist regulate the market for the regulator. The internalization of the environmental externality is passed on to consumers through the monopoly price premium they have to pay for the polluting good. Tax avoidance is an issue in that it is a deterring threat directed against taxation.

In Regions III and IV, the severity of the environmental damage as well as signi…cant production costs require a positive tax e , which exceeds Buchanan's tax. In Region III, the tax e = 2 induces a number of shareholders with low costs of tax avoidance to avoid taxation. Not all of the shareholders avoid taxation, but those who do exert a negative externality on the rest of society in terms of lost …scal revenue. To o¤set this welfare loss, the regulator can only hit the remaining taxpayers harder than if there were no tax avoidance, through a tax increase that also in ‡ates the consumer price.

The optimal solution strikes the balance between the need to collect tax yields and the dual correction for the pollution and the monopolistic behavior. In Region IV, taxation is so severe that the regulator completely loses the tax base, thereby losing the …scal revenue to correct for distortions in the economy. The regulator leaves the whole burden of the tax on the consumers through the price at which they purchase the good. All the regulatory corrections rely on the manipulation of the monopoly price through taxation.

The regulator chooses the tax f that raises the monopoly price up to the level at which the consumers'purchase decisions fully internalize the environmental externality. at level zero, below n , re ‡ecting that, to prevent tax avoidance, the regulator refrains from taxing emissions. At some critical level of the MSC, however, e jumps above n to correct for the free-riding of the tax avoiders.

The two-part tax schedule

We now assume that the regulator is not restricted to linear taxation, as is traditionally the case. Instead, we let the regulator use the a¢ ne (but nonlinear) tax schedule q + T , where is a unit tax for the …rm's emission and T is a lump-sum tax T -an "entryfee" into the market -that the …rm must pay regardless of the number of units of the polluting emissions.

Social welfare is now The participation constraint in the regulator's optimization program requires that the …rm gets no rent. This constraint is binding at the optimal solution since the regulator wants T to be as large as possible:

W ( ; T ) = Z D(pe;e) 0 [v (1 e)x ( 1
(p e ( ) c(e)

(1 e))D(p e ( ); e) = T (1.22)

We substitute the expression for T into (1:21) and use (1:12). This yields the following reduced form The …rst-order condition for welfare maximization is

W ( ) = [v c(e) ( 1
@D ( ) @p dp e d (1 e) (1 e) + p e " ( ) = 0 (1.24)
From (1:18), we can see that n solves (1:24).

Proposition 6 The optimal two-part tax schedule entails As a result, the two-part tax schedule can solve the problem of tax avoidance. The lump-sum tax enables the regulator to capture the whole pro…t of the polluting …rm, thereby eliminating any incentive on the part of the shareholders to avoid taxation. The regulator can then set the unit tax in accordance with Buchanan's rule to correct the two remaining distortions caused by pollution and market power. In the present setting, this combination helps the regulator achieve the …rst-best outcome.

The example of BHPP

The following example is one among others 4 that may illustrate the reluctance of the environmental regulator to correct for the environmental externality presumably because of tax avoidance. In the Far Eastern region of Krasnoyarsk, the Russian government has allowed Boges Limited to operate a hydro power station called the Boguchany Hydroelectric Power Plant (BHPP). This large business enjoys a natural monopoly position.

The construction of the dam reservoir has been largely deplored as a threat to the environment. The reservoir ‡oods villages, forests, agricultural land, meadows and pastures, bringing about signi…cant changes in the geographical landscape. According to [START_REF] Jagus | Environmental consequences of the construction of the boguchany dam-present change and projections[END_REF], the main environmental hazards stem from rising water levels which alter the landform, raising the ground water table, and compromising the water quality. The ‡ooding of lands and forests a¤ects the local climate, which in turn leads to a loss of natural ecosystems and habitats. On the other hand, the government aims to develop and urbanize the region, increasing living standards. The commission of BHPP should collect 2.4 billion rubles (84 million USD) of tax revenues in the public budget and provide at least ten thousand jobs in the region (Khotuleva and al. 2006, p. 16).

However, more than 95% of the shareholders of Boges Ltd. are registered in Cyprus-a well-known tax haven [START_REF] Baranova | Siberia: Sold down the river[END_REF]. In what follows, we provide a detailed analysis of this real-world situation in the light of our …ndings. 2009), [START_REF] Baranova | Siberia: Sold down the river[END_REF] claims that "the damage caused by the power station to the area in terms of forest, land, water and …shing resources, as well as birds and animals, will amount to 4.5 billion rubles (£ 90 million)-that is assuming an out ‡ow head of only 185 metres". Besides, the damage from the project also encompasses the loss of local cultural amenities such as archeological monuments, wooden architecture, nonmaterial heritage like local dialects, songs, and customs (Khotuleva and al. 2007, p. 428). In this context, the monopolist's tendency to underproduce can hardly o¤set the distortion due to the various externalities caused by the dam, so that a sound environmental regulation calls for a positive tax along Buchanan's line. However, with no device to ensure tax compliance, the regulator has no leeway: a positive tax would exacerbate the free-riding of the tax avoiders without yielding enough revenue to outweigh the further welfare loss.

The table in

As a result, the regulator …nds it optimal to set a zero tax on pollution.

Discussion on the modelling choice of avoidance function

In the present version, we have chosen to model tax avoidance subsequent to an imposition of environmental taxation as an internal decision of the …rm which is exogenous to our analysis. There is no or little speci…cation for the mechanism that the avoidance decision may take. Indeed, di¤erent avoidance schemes may come to mind: it can be pro…t shifting, relocation of production facilities to jurisdictions with a laxer regulation; it can be green washing, or political lobbying against the regulation among others.

It is common knowledge that taxation in general gives rise to tax optimization and may entail non-compliance. Environmental taxation is certainly no exception. Asym-metric information being an inherent feature of environmental performance, in which quality is unobservable and hardly veri…able, renders the enforcement of environmental policy and its monitoring problematic. In this analysis, we have proposed the simplest way to frame the idea that higher tax rates give taxpayers greater incentives to avoid liability. Therefore we let the …rm be immediately responsive to the level of tax, adjusting accordingly her compliance levels. Hence the assumption that …rm's shareholders are heterogenous in the cost of tax avoidance.

Certainly one may argue that dividends are paid on the after-tax pro…t, and shareholders have little interest in or grasp of how to in ‡uence a …rm's compliance behavior. Yet, a …rm has a complex hierarchy for decision-making and …rm's management is entrusted to executives who are generously rewarded to maintain high …nancial performance. Hence, we abstract from the mechanism of avoidance decision assuming it comes from a black box.

Ultimately, the incidents of successful global environmental policy are few. The EU emission trading system struggles to maintain a 'meaningful' carbon price that would re ‡ect its social cost. Yet, it still varies around 7 e per ton6 . During the past decade the allowances were in oversupply and nevertheless often allocated freely7 . An attempt to introduce an eco-tax in France under the Grenelle law has turned out to be a complete failure. It has cost taxpayers the construction of the infrastructure that has never been in use, plus there are penalties for violation of contract terms to the …rm commissioned for the project.

Thus we believe that a …rm with market power and an international infrastructure such as a multinational can and will …nd ways to reduce its environmental liabilities. The micro foundation of avoidance decision is, however, beyond the scope of this paper and hence is left for a future research.

Conclusion

'Free-riding on environmental taxation'implies that the monopolist takes advantage of consumers' willingness to pay for higher environmental quality. Yet, the …rm doesn't entirely carry out her …scal duties.

If the purpose of environmental taxation is to induce a …rm to consider the full set of consequences of polluting emissions, the tax should also internalize the externality caused by those shareholders who avoid taxation. The novelty of our approach is to allow the environmental regulator to endogenize tax avoidance and examine how this framework a¤ects traditional policy recommendations. In a monopoly setting without allowing for tax avoidance, the standard environmental taxation follows Buchanan's rule: as the monopoly distortion partly outweighs the externality distortion, the tax on polluting emissions must be set below the Pigouvian level; the tax cut resulting from the need to correct for the …rms'market power may be so sharp that the optimal policy imposes to subsidize the product when the monopolist is signi…cantly clean and the good is cheap to produce.

Outside these circumstances, we …nd that tax avoidance biases the second-best environmental tax away from Buchanan's tax in two opposite directions.

First, in situations where Buchanan's tax would be positive, it may happen that the fear of triggering tax avoidance compels the regulator to refrain from taxing the polluting emissions. This occurs when the monopolist is moderately clean and/or moderately e¢ cient. In that case, the revenue raised from a positive tax would not fully o¤set the welfare loss produced by those shareholders who avoid taxation. Then, tax avoidance proves to be a deterrent threat against taxation.

Second, when the monopolist is signi…cantly dirty and moderately e¢ cient, the regulator must internalize the free-riding of tax avoiders by taxing emissions more severely than if there were no tax avoidance. The shareholders who comply with taxation pay the bill for the free-riding of those who bypass taxation. In the extreme case where all the shareholders avoid taxation, the tax increase in ‡ates the monopoly price so that the whole burden of internalizing the environmental externality is passed on to the consumers.

Ultimately, we show that the problem of tax avoidance can be solved by the combination of a lump-sum tax that captures the whole pro…t of the polluting monopolist and Buchanan's tax on emissions.

The policy implications of our …ndings are rather grim. When the environmental regulator is restricted to linear taxation, tax avoidance is a serious stone in the regulator's shoe, forcing the regulator to be either impotent or unfair. The regulator proves impotent when the threat of tax avoidance deters the regulator from taxing emissions. In this case, the best policy is to let the monopolist charge the consumers a price that underinternalizes the environmental externality. On the other hand, the regulator proves unfair when tax avoidance forces the regulator either to overtax those who comply with taxation, or to drastically in ‡ate the monopoly price with a tax that everyone avoids. In both cases, one objective is to internalize the externality caused by the free-riding of the tax avoiders.

However, unlike the environmental externality, those who pay the bill for the free-riding are not those who generate the externality. This may be seen as politically unacceptable both by taxpayers and consumers.

The two-part tax schedule seems, on paper, an attractive solution to the problem of tax avoidance. Nevertheless, the literature on environmental taxation usually considers that lump-sum taxes and transfers are not available in practice [START_REF] Bovenberg | Environmental policy, public …nance and the labour market in a second-best world[END_REF]. We have also demonstrated how tax avoidance makes it di¢ cult to design a linear tax on polluting emissions, which …nally argues in favour of implementing speci…c enforcement measures against tax avoidance.

1.8 Appendix

La¤er curve

Consider …rst the case where all the shareholders comply with taxation, that is, ( ) = 1 for all . Using (1:3) and (1:12), the …scal revenue with respect to is given by

R n ( ) = (1 e) D (p e ( )) (1.25) = 2 (v c (e) (1 e))
This function is an inversely u-shaped parabola in , which takes zero value at = 0 and

= v c(e)
1 e , and which reaches a maximum at b n = v c(e) 2(1 e) . We now take tax avoidance into account and obtain the revenue function R( ) given by (1:13). For all 2

h 0; 1 (1 e) i , the …rst derivative is @R( ) @ = 1 2 3 2 (1 e) 2 2 (v c (e) + 1) (1 e) + v c (e) :
(1.26)

The …rst-order condition yields only one admissible extremum inside . One can check that the upper solution of equation

@R( ) @ = 0 is v c(e)+1+ p (v c(e)) 2 (v c(e))+1 3(1 e)
, which exceeds 1

(1 e) . The second derivative is

@ 2 R( ) @ 2 = 3 (1 e) (v c (e)) 1; (1.27)
which takes the following value at 2

@ 2 R( ) @ 2 b = q (v c (e)) 2 (v c (e)) + 1 < 0: (1.28) Thus, b is a maximum. Straightforward calculations show that the maximum of R( ) is always lower than that of R n ( ), that is, b < v c(e)
2(1 e) .

Proof of Lemma 2

The proportion of shareholders who comply with taxation depends on the tax level. As a result, the social welfare function can be decomposed as follows:

If 0, no shareholder avoids taxation, hence ( ) = 1. From (1:10), W ( ) is then given by 1 e , so that

W n ( ) = ( 1
W n ( n ) = (v c(e) (1 e)) 2
2(1 e)

.

If 0

1

(1 e) , the proportion of shareholders who comply with taxation is ( ) = 1

(1 e). From (1:10), W ( ) is then given by 2(1 e) . It follows that n is a local maximum for this parameter con…guration. Furthermore, two cases can be distinguished, depending on the sign of p . First, if p < 0, then W p ( ) is decreasing on the interval 0; 1 1 e , and f is also a local maximum provided that f > 1 1 e , which amounts to > 3 (v c(e)) 2(1 e) . Nevertheless, it is straightforward that W n ( n ) > W f ( f ) always holds, hence n is always more bene…cial than f from the social standpoint. The second case corresponds to parameter values such that p 0, or equivalently,

W p ( ) = (1 e) 3
4(v c(e)) 2 +20(v c(e)) 1 24(1 e)
.

Then, 2 may be another local maximum because 2 1 1 e when v c(e)

2(1 e) . Indeed, for all parameter values in Region I,

(v c(e)) 2 3(v c(e))+4 2(1 e)
, which is equivalent to 2 1 1 e and less restrictive than 2. We now turn to Region II. If

v c(e)
2(1 e) , then W n ( ) is increasing for negative values of up to zero, which makes = 0 a local maximum provided that W p ( ) decreases at this value, which amounts to 1 0. In addition, 2 may also be a local maximum provided that, …rst, p 0 (which guarantees the existence of 2 ), second,

v c(e)+(v c(e)) 2 2(1 e)
or v c(e)

(1 e) 1 4 (in which case 2 0), and third, 2 1 1 e (or, equivalently,

(v c(e)) 2 3(v c(e))+4
2(1 e)

) as previously mentioned. Furthermore, f happens to be a local maximum too when f 1 1 e (i.e., for parameter values such that

3 (v c(e)) 2(1 e) ).
Let us …rst assume that v c(e) ].

In this parameter con…guration, < 3 (v c(e)) 2(1 e) , and so W f ( ) is a decreasing function of on the interval , hence W p ( ) is also decreasing on 0; 1 1 e . Thus, = 0 is a global maximum in this parameter con…guration.

We can conclude that W n (0) max fW p ( 2 ); W f ( f )g for parameter values inside Region II.

3. Consider now Region III. One can check that p > 0 for these parameter values, because 4(v c(e)) 2 +20(v c(e)) 1

24(1 e)

< 24(v c(e)) 1

32(1 e)

< v c(e)+(v c(e)) 2

2(1 e)

. It follows that 2 exists and is positive when either v c(e)

(1 e)

1 4 or v c(e)
(1 e) .

Moreover, it turns out that 2 < 1 1 e because, …rst, when v c(e)

(1 e)

1 4 , we have

5+ p (3 2(v c(e)))(2(v c(e)) 3) 2 6(1 e) < (v c(e)) 2 3(v c(e))+4
2(1 e)

, and second, when v c(e)

(1 e)

1 4 , we have v c(e)

(1 e) < (v c(e)) 2 3(v c(e))+4

2(1 e)

. Hence, 2 is a local maximum.

Furthermore, = 0 may be a local maximum too, because v c(e) 2(1 e) < holds, and so n > 0. Indeed, one can check, …rst, that v c(e) 2(1 e) < 24(v c(e)) 1

32(1 e)

when v c(e)

(1 e) 1 4 , and second, that v c(e)

2(1 e) < v c(e)+(v c(e)) 2

2(1 e)

when v c(e)

(1 e) 1 4 , thereby implying v c(e) 2(1 e) < . However, in the case where v c(e)

(1 e)

1 4 or v c(e)+(v c(e)) 2
2(1 e)

, we know that 1 0, meaning that W p ( ) is increasing at = 0, so this tax is not a local maximum.

In addition, f happens to be a local maximum when f

1 1 e ( 3 (v c(e))
2(1 e) ).

Assuming v c(e)

(1 e) > 1 4 and < v c(e)+(v c(e)) 2

2(1 e)

, so that = 0 is actually a local maximum, routine calculations show that W n (0) W p ( 2 ) for all inside h for which W p ( 2 ) W f ( f ).

We conclude that W p ( 2 ) max fW f ( f )); W n (0)g in Region III. Moreover, straightforward calculations yield that 2 > n for all inside

h 2(v c(e))
3(1 e) ; v c(e)

(1 e) i , which includes all the parameter values inside Region III unless v c (e) 2 0; 1 3 .

4. We …nally turn to Region IV. All > max{

5+ p (3 2(v c(e)))(2(v c(e)) 3) 2 6(1 e) ; ( p 6 1)(v c(e)) 2(1 e)
} is also greater than 3 (v c(e)) 2(1 e) . This inequality guarantees that f > 1 1 e , mean-58 ing that f is a local maximum. In the same parameter con…guration, one can check that p 0, hence 2 exists, and moreover 2 is positive since v c(e)

(1 e) > 1 4 . It follows that 2 is a local maximum too, unless 2 > 1 1 e , which amounts to > (v c(e)) 2 3(v c(e))+4

2(1 e)

. Suppose this is the case. Then = 0 is a local maximum because, …rst, v c(e) 2(1 e) ( 1 0), and second, v c(e)

(1 e) > 1 4 and < v c(e)+(v c(e)) 2

2(1 e)

( 1 > 0) for parameter values inside Region IV. In this case,

( p 6 1)(v c(e)) 2(1 e) implies that W f ( f ) W n (0). Otherwise, 2
1 1 e , hence 2 is a local maximum and, for all inside

5+ p (3 2(v c(e)))(2(v c(e)) 3) 2 6(1 e) ; v c(e)
1 e

, we have

W f ( f ) W p ( 2 ).
To sum up, W f ( f ) max fW p ( 2 ); W n (0)g in Region IV. (v -c(e))

1 6 Environmental regulation with price signaling

(5 + -( -3 + 2 (v -c (e))) 3 )  6 -1 (v -c (e)) 2
The article studies how incomplete information a¤ects the optimal second-best environmental tax in a monopoly setting. We endogenize the pollution tax to analyze its impact on social welfare. We show that the optimal tax must be set below the level required for a polluting monopoly under complete information. The tax is composed of three elements to correct corresponding market distortions: the Pigouvain standard to internalize the pollution externality, an adjustment of monopolistic restraint of output and an element to account for information asymmetry. As the tax enters the product's price, consumers'expectations of high environmental quality determine the value of the clean (non-polluting) product, the trade of which is inevitably reduced by the pollution tax.

Hence, the regulator who places a corresponding value on the clean variety, scales down the optimal tax level. While under pooling, the pollution tax equalizes the expected social value to the expected social cost of the product, it is no longer attainable under separating equilibrium. With weak tax, the expected product price exceeds its expected social cost and vise versa with severe tax.

Keywords: Signaling, Environmental Regulation, Information and Environmental Product Quality JEL Code: D82, L15, Q5

Introduction

The regulation of polluting emissions plays an important role in preserving the natural environment. Mechanisms of pollution control and incentive schemes reduce environmental damage by constraining polluting emissions. Understanding how environmental regulations a¤ect producers' strategic choices of price and quantity is thus of primary importance. The goal of the present paper is to analyze in an economy with imperfectly competitive markets and incomplete information about producers'environmental quality, how environmental regulations a¤ect price signaling decisions.

The problem is relevant in markets with information asymmetry between producers and consumers where information disclosure is either not feasible or problematic. There are many examples of such. For instance, a producer in transition to organic agriculture1 is unable to disclose his environmental type. The Volkswagen emissions scandal shows that certi…cation or compliance with environmental quality standards does not necessarily ensure perfect availability of information on actual pollution. Sometimes there are grounds to dismiss labeling. An excessive use of organic fertilizes creates a high concentration of nitrates which degrades the virtue of organic quality. Cross pollination between organic and GM crops, possible under the current EU regulation, may confound environmental quality [START_REF] Tickell | EuroParl votes in new era of GMO farming[END_REF]. Some wine producers deliberately avoid organic certi…cation as it entails heavy bureaucratic procedures, in ‡exible certi…cation costs and a long term commitment; while price adjustments are much less rigid. Indeed, there is a notion in France of reasonable agriculture which implies a reasonable use of chemicals.

It is common knowledge that wine growers also consume their own product and logically don't have an incentive to ingest pesticides. Finally, the epidemic of ‡avescence dorée in France in 2012 may well illustrate how environmental labeling can be compromised. The ‡avescence dorée is a highly contagious epidemic that attacks vines threatening them in rapid extinction. The only e¢ cient treatment is synthetic and does not conform to organic principles. However, given the severity of the threat, the authorities and the certi…cation entity have decided to enforce treatment in the a¤ected and neighboring regions. In order to avoid noncompliance, the organic label is nevertheless maintained.

As a result, the majority of producers have given way to synthetic treatment while few have opposed such [START_REF] Berdah | Savoie : la ‡avescence dorée, dilemme des producteurs de vin bio [Savoie: the fralvescence dorée, dilemma for producers of organic wine[END_REF].

To overcome information issue, a high quality producer could use price as an instrument of communication with consumers, thus signaling their own type2 .

Alas, environmental regulation, being put in force to constrain the pollution, may have a twisted impact on social welfare. Indeed, price signaling is based on the cost di¤erence between the types allowing the high type to distinguish himself. The greater the cost di¤erence, the cheaper is the signal3 . In this light, the environmental regulation which increases the e¤ective production cost of the polluting type directly reduces the cost di¤erence. Thus, the price signal grows more expensive because it requires the high type to distort the price more. Ultimately, it produces a loss in consumer surplus.

However, one can argue that severe environmental regulation which inverts the order of cost e¢ ciency between types could increase the welfare. When the high type is more cost e¢ cient, then the price is distorted downwards and the signal increases consumer surplus.

The aim of the paper is to analyze the impact of environmental regulation on the social welfare, taking into account distortions of the economy.

Literature review.

In this subsection we provide a brief overview of previous studies on communicating quality and how they relate to our research.

In the context of the second-best optimal policy with monopoly and pollution externalities we compare our results to those derived by [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF]) and [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF]. The former questions Pigouvian approach for the case where organization cost -in the tradition of Coase, bribery between polluter and victims-is prohibitively high. He demonstrates than equalizing the marginal social to marginal private cost can degrade welfare since Pigouvian tax unduly restricts output that is already below the socially optimal level because of imperfect competition. The latter formalizes the model of market power in hands of polluting monopolist. [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF] derives the second-best solution

showing that optimal tax may fall short of the marginal damage and that the Pigouvian level of taxation may not be desirable when price elasticity of demand is not elastic. [START_REF] Bagwell | High and declining prices signal product quality[END_REF] analyze the impact of incomplete information on monopoly pricing strategies. In a two-type model with informed and uninformed consumers and a costly quality provision, they demonstrate that the high type can …nd it pro…table to distort the price of the high quality product, raising it above the full information monopoly level, to thus reveal its true type to consumers. By such deviation the high type gains the demand of uninformed consumers, who correctly infer the true type, at the cost of losing some pro…t margin due to suboptimal pricing. Given that the low type is more cost e¢ cient, signaling price distortion must be just high enough to make mimicry of the low type unpro…table.

In a discrete type monopoly model, [START_REF] Mahenc | Are green products over-priced?[END_REF] analyses pricing strategies for a …rm of a superior environmental quality. Consumers, unable to observe or verify …rm's environmental performance, have green preferences and are willing to reward high quality with price premium. He demonstrates that although signaling quality is a valid strategy for the clean type, it has a signi…cant opportunity cost which increases with consumers' optimistic expectations about the …rm's environmental performance, lower market power of the conventional variety or a lower gap between marginal costs of two types. These factors decrease pro…tability of the signaling strategy. A high level of optimism increases willingness to pay of uninformed consumers thus increasing pooling pro…ts and the opportunity cost of price signal. The lower market power of a dirty producer corresponds to a lower price margin of the conventional variety implying that price deviation from a full information level becomes less expensive in view of the low type. Equivalently, the less the cost di¤erential, the lower the dirty type's opportunity cost, as its per unit loss from price distortion decreases. We continue the logic of plausible equilibria selection to analyze the impact of environmental regulation.

We adopt the market model similar to that of [START_REF] Daughety | Communicating quality: A uni…ed model of disclosure and signalling[END_REF] in which signaling high quality and costly disclosure are uni…ed in the monopoly setting with a continuum of types. Quality is costly, unveri…able and requires extra willingness to pay.

The authors show that in signaling equilibria, with no possibility of disclosure, pro…ts are decreasing in quality due to the need to distort price This makes a …x cost disclosure pro…table for highest quality types explaining the incentive for voluntary disclosure and high quality provision.

In terms of pollution control, we complement the approach of (Sengupta 2012). In a two-type asymmetric information monopoly with green consumers, a clean …rm signals its environmental performance. Pollution control is enforced through a market of pollution permits. The regulatory instrumentt, the price of the pollution permit, is exogenously given. Hence, the …rm adapts its pricing strategy to the regulation which a¤ects the relative cost e¢ ciency between types. The author de…nes 'strong' regulation as such that inverts the cost order making the clean type more e¢ cient. Thus, strong regulation induces a downward price signal and provides an incentive to invest in clean technology.

Strong regulation is viewed as a means of overcoming the distortion of the incomplete information. We endogenize the instrument, derive the second-best optimal tax and perform a welfare analysis of environmental regulation taking into account the social cost of signaling price distortion under the condition of plausible equilibria. [START_REF] Antelo | Asymmetric information, signaling and environmental taxes in oligopoly[END_REF] analyze the e¤ect of quantity signalling and asymmetric information on environmental taxation. In a two-period Cournot oligopoly model where the regulator acts as a von Stackelberg leader, they show that the optimal second-best tax must be set below the level of the polluting monopoly, i.e., below the Buchanan benchmark. They …nd two additional e¤ects induced by information asymmetry. The …rst information e¤ect, deprives the regulator of an ability to distinguish between types, requiring him to expand environmental regulation on clean types. The second signaling e¤ect, makes both types of producers distort output from the pro…t-maximizing level.

The attain separation the clean type must suppress and the dirty type must expand their output. Therefore, pollution coming exclusively from dirty types will inevitable grow with asymmetric information. The environmental tax must target not only the trade-o¤ between output and pollution but also signaling distortion. Overall they show that the gab between marginal damage and the second-best optimal tax increases with incomplete information. Qualitatively, this is in line with our …nding of a less stringent regulation, relative to the Buchanan benchmark, that we obtain for environmental regulation in a monopoly market with price signaling.

The paper is organized as follows. Section 2.2 presents the model. Section 2.3 derives equilibrium prices under incomplete information. Section 2.4 introduces the optimal environmental regulation. The implications for social welfare are discussed in Section 2.5. Section 2.6 concludes.

The model

Quality. Assume that a monopoly produces a product of environmental quality e which is unobservable to consumers before and after purchase, as for credence goods. Environmental quality can be high, e, or low, e; and is supplied by the clean or dirty producer respectively. The environmental quality implies a certain mode of production. The high environmental quality leaves an insigni…cant footprint as it uses clean electricity, recyclable materials, organic fertilizers etc. In contrast, the conventional type uses cheaper and dirtier electricity, creates waste and applies chemical fertilizers Hence e describes the production technology or abatement intensity. We make a standard assumption that the environmental damage, d, is proportional to emissions4 . The damage function is thus: d (q; e) = (e e) q (2.1) denotes the intensity of environmental damage per unit of output. It is a scienti…c estimate of the harm caused to the natural environmental, to health and living quality, to air or water quality etc. q denotes the quantity of purchased units. e captures the actual environmental quality. The higher it is, the less damage is done per unit of output. We assume that the highest environmental quality is clean and free from pollution, while the conventional quality is dirty and its production ejects harmful emissions. The distribution of environmental quality is thus:

e = fe; eg (2.2)
with e < e:

The following assumption states that the conventional variety is socially desirable and that consumers underestimate the damage to the environment caused by the dirty variety.

< < (A.3)

When disutility from pollution is less than the e¤ective damage, then green consumers internalize environmental damage only partially 5 .

Consumers. Any type of product provides positive utility to consumers but the one with a higher environmental quality generates a higher utility. Consumers have green preferences (for the same price they favor the clean variety) as they experience disutility from insu¢ cient (with respect to their individual ideal level of) environmental quality.

The bene…ts derived from environmental quality are public and private simultaneously.

On the one hand, pleasure procured from a clean environment is non-rival and nonexcludable. On the other hand, consumers get a direct and personal health bene…t from the lack of or low content of nitrates, sul…tes and pesticides. Consumers value these semi-public components of environmental quality and are eager to pay a premium for them.

The consumer's utility function is quasi-linear in all other goods6 :

U (p; q; b e) = ( (e b e)) q 1 2 q 2 + I pq (2.3)
where denotes gross willingness to pay for one unit of product, measures the disutility from lacking environmental quality, > 0, b e stands for perceived environmental quality determined as part of perfect Bayesian equilibrium where the …rm's strategy is its price, I is income and p is price. Consumers may purchase multiple units of good, however sampling doesn't improve chances of future consumption satisfaction as with restaurant visits, medical consultations etc.

The consumers'demand for the good of perceived quality b e is given by: q (p; b e) = (e b e) p (2.4)

A representative consumer buys multiple units of a product; sampling succeeds or fails independently but with the same probability. To check the robustness, one can construct a market with heterogeneous consumers of mass M with unit demand. The taste or the reservation price for environmental quality, r; is distributed on the interval To exclude trivial solutions, the following parameter restriction is necessary:

> c (e) (A.1)
Assumption (A.1) implies that demand for each type is positive when the product is sold at marginal cost and with certainty perceived as dirty. (A.1) guarantees that even in the worst case a …rm generates a positive pro…t7 .

The …rm. There is a single …rm of type e. Its realization is determined by nature 8 .

The …rm produces the good at an e¤ective marginal production cost c (e)9 . As a baseline setting we assume that:

c 0 e > 0 (A.2)
where the subscript indicates the variable with respect to which the derivative is taken.

The baseline assumption, (A.2), implies that the provision of environmental quality is costly. It is a conventional assumption, see for instance [START_REF] Kurtyka | The switching e¤ect of environmental taxation within bertrand di¤erentiated duopoly[END_REF].

Indeed, in agriculture engaging more manpower can substitute an extensive use of synthetic chemicals. Manual weeding that preserves soil quality has a much greater cost than the application of herbicides.

The …rm may face a …xed cost of investment in high quality F (e) with F (e) = 0.

However, the purpose of the investment is unobservable to consumers and therefore they are not able to infer quality level by observing F (e)10 . An initial payment for the installation of clean-up …lters or other abatement materials may illustrate this case.

The gross pro…t for the …rm depends on its true price-cost margin and on the perceived environmental quality which a¤ects the demand for the product. Hence, the generalized form of pro…t is: The pro…t function is strictly concave in p, it attains its maximum at p (e) =

(e e( 0 ))+c(e) 2

. Henceforth, the 'star' superscript indicates a quantity obtained by monopoly maximization calculus.

The maximized pro…t is ( (e e( 0 )) c(e)) 2

4

F (e). Lemma (7) summarizes the full information benchmark. Note also that the monopolist will optimally set the price in the price-elastic interval.

Indeed, the monopolist chooses a price to solve the following maximization problem: This is a standard result of monopoly pricing in which the price mark-up is equal to the inverse price elasticity of demand. It can be veri…ed that (p (e)) is increasing in e when evaluated at a pro…t-maximizing price. This implies that higher environmental quality has higher price elasticity compared to conventional quality11 .

The regulator. A benevolent regulator is aware of the environmental damage from the production of a conventional variety. He acts as a von Stackelberg leader wherein he chooses an optimal instrument that maximizes the standard utilitarian welfare.

The game. The sequence of the game is as follows: at stage 1 the regulator commits to an environmental policy. At stage 2 nature draws the …rm's type, the …rm learns its realization and sets the price. At stage 3 consumers purchase.

Asymmetric information

In this section we state the separating conditions and derive the equilibria under information asymmetry.

When consumers have an incomplete knowledge of the producer's environmental type, the …rm's strategy is to choose a price that allows consumers to correctly identify its type/ environmental quality. Denote 0 to indicate consumers'prior belief of high environmental quality. It embraces all the information about environmental quality publicly available to consumers. Then, the perceived quality, b e; is the expected probability that the product is clean, that is b e e ( 0 ) = 0 e + (1 0 ) e. Rearranging (2.4) we obtain the demand as a function of the perceived quality: q (p; 0 ) = (e e ( 0 )) p (2.7)

Note that (2.7) increases in 0 implying that optimistic expectations of environmental quality raises demand. Since every producer type prefers to be taken for the clean regardless of its true type, the …rm of type e can bene…t from revealing its true type when it distinguishes itself from the dirty type thus reaping the green consumption. The The …rst condition states that for each type the price strategy must be pro…t-maximizing

given consumers'beliefs. The second condition imposes the Bayes rule for belief updating. When the price is informative, consumers correctly identify the producers' type.

When the price is uninformative, prior and posterior beliefs equal; and beliefs updating has no e¤ect.

Separating equilibrium.

A producer of true environmental quality e who charges price p and who is perceived as type (e)12 , obtains a pro…t of: We use the necessary and su¢ cient conditions derived in [START_REF] Mailath | Incentive compatibility in signaling games with a continuum of types[END_REF] to de…ne the separating equilibrium:

1. @ @ 0 = (p c (e)) > 0 (belief monotonicity) 2. @ 2 @e@p 00 Condition 1 states that the …rm's pro…t is always increasing in perceived environmental quality. Thus, the worst inference consumers'may draw about a …rm's type is that = 0. In that case, the dirty type would yet obtain the full information pro…t; the clean type, in contrast, would be worse o¤ because it loses the entire demand for the clean variety. Hence, the clean …rm always has an incentive to reveal its type. Condition 2

ep = e +
shows that the pro…tability of changing the price increases in e when c 0 e > 0. Condition 3, known as single crossing property, indicates that the marginal rate of substitution between price and perceived environmental quality is a strictly monotone function of e.

Indeed, when the product becomes cleaner, the …rm is willing to charge more to bene…t from green consumers' demand. Note that generally, the sign of S ( ) in condition 3 depends on the p location relative to p . For an upward price distortion p < 0 and S ( ) > 0, for a downward price signal the signs reverse.

Recall, however, that c (e) denotes the e¤ective marginal cost of production. It implies that in the case of considerable environmental damage, a severe taxation regime on polluting emissions is capable of reverting this relation. This would entail a turnover in terms of cost e¢ ciency, i.e. c 0 e < 0. Then, Condition 1 holds with no change. As for Conditions 2 and 3, the signs of their expressions invert. Indeed, d ( (e)) de = @ @ @ @e < 0; since @ @e @p @e p=p > 0 and @ @ < 0 when c 0 e < 0: As stated earlier, the price increases with environmental quality. When the clean type is less costly than the dirty, a downward price distortion ensures separation because by that action the clean type with certainty gains the green demand and loses less pro…t than the dirty type if the latter attempts to replicate the clean type's price. Since environmental regulation is common knowledge and consumers can observe its stringency, and beliefs in equilibrium are consistent, i.e.

( ) = e for e = fe; eg, consumers correctly infer the clean type by its action -downward price distortion. As the dirty type has a greater e¤ective marginal cost, a downward price signal, to mimic the clean type, is unpro…table. Condition (IR C ) requires that the clean type's pro…t at separation is greater than any other pro…t it may achieve when it is not correctly identi…ed. Condition (IC D ) doesn't allow the dirty type to obtain a higher pro…t by deception than by revealing its true type.

Denote % IR and % IC the set of prices that verify conditions (IR C ) and (IC D ) respectively (see 2.7.1 for details).

Proposition 9 There is a set of separating equilibrium prices such that (e) = +c(e) 2

and (e) 2 % IR \ % IC .

The set of prices that sustain separation (and verify 9) contains an in…nite number of prices. The equilibrium re…nement criterion of [START_REF] Cho | Signaling games and stable equilibria[END_REF] requires that price distortion is minimal. This gives the unique signaling price for high environmental quality.

Corollary 10 The unique price that signals high environmental quality is (i) the lower bound p s 2 % IR \ % IC when c 0 e > 0 and (ii) the upper bound p s 2 % IR \ % IC when c 0 e < 0. The explicit formula for signaling price is:

p s = 1 2 h + c (e) p (2 2c (e))
i for c 0 e ? 0 respectively (2.12)

To reveal the true environmental quality, the clean type must deviate from the full information pro…t maximizing level. In equilibrium, this allows consumers to correctly infer …rm's environmental performance. When the clean type has greater marginal production costs, the signaling price exceeds the full information benchmark. In the opposite case when the clean type is more cost e¢ cient, i.e. c 0 e < 0, the price deviation falls below p .

It is valuable to examine how the change in the marginal production cost of the dirty type a¤ect the signaling price:

@p s @c (e) = 1 2 " 1 p (2 2c (e))
# for c 0 e ? 0 respectively (2.13)

It is straightforward that the sign of (2.13) is positive.

Corollary 11 The signaling price always increases in the marginal cost of the dirty type.

Proof. See 2.7.1.

An increase in the marginal production cost of the dirty variety reduces its pro…t and makes the deception strategy of price deviation from the full information level more attractive. To e¤ectively separate itself from the conventional type, the clean …rm must increase signaling distortion accordingly to discourage the dirty type from mimicking.

The pro…t the clean type earns when it signals its type is:

@ s @c (e) = @p s @c (e) q s [ ] (p s c ( e)) @q s @p s = @p s @c (e) s0 p
(2.15)

As it has been shown, the …rst term is always positive. The second term in (2.15)

represents the change of pro…t in the price: As pro…t has a normal concave form, it attains maximum at p ; hence 8p > p 0 p < 0 and 8p < p 0 p > 0. Than is, pro…t is decreasing in price when the price is distorted upward and it is increasing when the price is distorted downward.

Corollary 12 The variation of s in c (e) has the same sign as 0 p : 8 < :

@ s @c(e) < 0 if p s > p @ s @c(e) > 0 if p s < p (2.16)
Signaling pro…t varies in the opposite direction of the price signal.

The intuition of this result is as follows. When the marginal cost of the dirty type increases, the signaling price rise. With upward distortion it implies an increase in the gap between the signaling and the pro…t-maximizing prices. With downward distortion, an increase of the signaling price brings it closer to the pro…t-maximizing level.

Pooling equilibrium.

Despite the existence of separating equilibria, there also exist pooling equilibria, some of which possibly dominate the least-costly separating price p s ; if the clean type's pro…t under pooling exceeds that under separating.

When prices are uninformative, consumers are unable to distinguish between the types 13 . Yet, given the assumption A.1 each product variety gets a positive demand (even with lowest expectations, i.e., 0 = 0). When pooling price p p is an element in a set of undefeated pooling equilibria prices, then a …rm of type e obtains a pooling pro…t (p p ; e; 0 ) : As the pooling price is arbitrary, we assume that it is set by a producers' organization where interests of clean and dirty types are represented by their respective probability of occurrence. Since each e type producer would prefer to set its pro…tmaximizing monopoly price p p (e) = 1 2 ( ( e 0 ) + c (e)), the interval of undefeated pooling prices can be represented by the following:

p p 2 [p p (e) ; p p ( e)]
(2.17) p up denotes the set of undefeated pooling prices in (2.17). It is clear that p p (e) < p p < p p ( e) 8 0 2 (0; 1) :

The explicit form of pooling price chosen by the producers'organization is thus:

p p = 0 p p ( e) + (1 0 ) p p (e) = 1 2 [ ( e 0 ) + 0 c ( e) + (1 0 ) c (e)] (2.18)
which yields a pooling pro…t of:

p (p p ; e; 0 ) = 1 2 [ ( e 0 ) + 0 c ( e) + (1 0 ) c (e) 2c (e)] (2.19) q (p p ) F (e)
Now, consumers are not 'selective': each variety faces the same demand level for a given price. Obviously, with prior beliefs about environmental quality 8 2 (0; 1), the dirty type pooling pro…t is higher than with complete information. The exact opposite is true for the clean type.

For our purpose, it is of interest to see how a change in the e¤ective marginal cost a¤ects the pooling pro…t for each variety:

@ p (e) @c (e) = @p p @c (e) q ( ) + p p @q @p p @p p @c (e) (2.20)

It is obvious that @p p @c(e)

> 0 for either type. Rearranging (2.20) we obtain: @p p @c(e) h q ( ) + p p @q @p p i = @p p @c(e) p 0

p (e). Hence, the sign of expression (2.20) depends on the location of the chosen pooling price with respect to that which maximizes each type's pro…t. Since for the dirty variety p p > p p (e) ; the increase in its own marginal cost reduces the pooling pro…t, and vise versa for the clean variety.

Note further that @ p ( e) @c(e) > 0; i.e., an increase in the dirty product's marginal cost increases the clean type's pooling pro…t. An increase in c (e) reduces the set p up ; allowing the chosen pooling price come closer to the clean variety's pro…t-maximizing level.

Equilibrium selection.

The intuitive criterion [START_REF] Cho | Signaling games and stable equilibria[END_REF] eliminates all the pooling equilibria and singles out the separating price p s : Among all the equilibria (pooling and separating), one can rank the equilibria according to the Pareto dominance criterion. This is the spirit of the undefeated equilibria concept developed by [START_REF] Mailath | Belief-based re…nements in signalling games[END_REF]. This criterion selects p s over pooling equilibria when 0 is su¢ ciently low. For high enough values of 0 , some pooling equilibria, characterized by p p ; defeat p s [START_REF] Mahenc | Are green products over-priced?[END_REF].

According to the undefeated equilibria criterion, at Stage 2, the producer having learned its environmental type e; selects the price strategy that generates the highest pro…t. Given a parameter vector ( ; ; ; F ), the pricing strategy is: When the pooling equilibrium pricing strategy Pareto dominates separating, then pooling is more pro…table than separating for each type. The plausible choice of the producer of type e is to take a pooling price from the set p up : Hence, following the logic of the undefeated criterion, the …rm prefers concealing information. In the opposite case, when separating is less costly than pooling, the clean type sets the signaling price and the dirty type sets the pro…t-maximizing full information price.

p (e) =
Figure (2-1) 14 illustrates precisely this process. On the upper panel, continuous curves cover the loss of conventional consumers due to this price increase.

In Figure (2-1) from left to right, with an increasing c; …rst pooling equilibrium defeats the separating price, then the relation inverts. Note that all along the c axis; the pro…t of the clean type is steadily decreasing.

Environmental regulation

The misallocation of resources comes from three sources in this economy; from external damage, from imperfect competition, and from incomplete information. Ideally, three corrective instruments would be needed to restore e¢ ciency: one to increase production of the …nal good, a tax to control external diseconomies and an information revelation device. It is assumed, however, that product market distortion cannot be directly corrected. The unobservability is inherent in environmental quality and can only call for the search for e¢ cient revealing mechanisms 16 . So the pollution tax is the optimal second best trade-o¤ among the diseconomies.

Assuming that the society is indi¤erent to redistribution e¤ects (all funds collected through the environmental regulation are transferred back to consumers) and abstracting from the shadow price of public funds, the sum of consumers'and producer's surpluses, and the taxes levies net of environmental damage can be used as welfare measures 17 . For a given realization of environmental quality e, social welfare is thus expressed:

W (e) =
Z q(e) 0 P (x) dx c (e) q (e) ( e e) q (e) F (e) (2.22)

= ( e e) 1 2 q (e) c (e) ( e e) q (e) F (e)

It measures the social value of traded goods, net production costs and aggregate damage to the environment.

In the following subsection, we separately examine the properties of the optimal second-best environmental tax and its impact on producer's pricing decisions when equilibria are selected according to intuitive and undefeated criteria.

Environmental taxation

The regulator has an objective to maximize the aggregate social welfare by imposing a tax on the polluting emissions. At the time when the regulation is set, a realization of the production type has not yet taken place, so the regulator commits to a policy in anticipation of a probability 0 ; that the producer is of the clean type. Hence, the regulator's objective function is W (t):

max t 0 W ( e; t) + (1 0 ) W (e; t) (2.23)
The probabilistic welfare is similar to that of [START_REF] Freixas | Planning under incomplete information and the ratchet e¤ect[END_REF], in our context, alternatively to consumers'prior optimism about the …rm's environmental performance,. 0 can represent the percentage of clean markets in the economy as a whole.

Optimal second-best tax with equilibrium selection by intuitive criterion.

The unique equilibrium, which survives the intuitive criterion, is separating with the price pair (p s ; p (e)), and is such that the clean type sets the least distortionary signaling price while the dirty type sets a full information monopoly price. Then, using (3.30) we obtain the explicit form of the regulator's objective function:

W (t) = 0 1 2 q (p s ( e; t)) c q (p s ( e; t)) F ( e) + (2.24) (1 0 ) ( e e) 1 2
q (p (e; t)) c (e) ( e e) q (p (e; t))

With no loss of generality, set e e 1: The …rst order condition (FOC) of (2.24) requires that:

0 @q @p s @p s ( e) @t (p s ( e; t) c ( e)) = (1 0 ) @q @p @p (e) @t ( [p (e; t) c (e)]) (2.25)
To ensure that welfare attains in t its maximum, the second order condition (SOC) must be negative, the demonstration of which is relegated to Appendix 2.7.3.

Condition 2.25 states that the weighed sum of the expected value of the 'social'price margins of both types must equal zero. The term 'social'emphasizes the total production cost incurred by society, including environmental damage. As the clean variety is pollution free by assumption, its social price margin is the same as its (standard) price margin which measures the di¤erence between the signaling price and the marginal production cost. For the dirty type, in contrast, the social price margin, accounting for its marginal damage, is the gab between the full information monopoly price and the total social cost.

These price margins are weighed by the probability of the occurrence of the corresponding environmental type, and by the demand sensitivity to the price change induced by the tax. The social price margin of the clean variety is always positive because the signaling strategy ensure strictly positive pro…ts. Consequently, the social price margin of the dirty variety must be strictly negative at the optimum. This implies that the dirty type's unregulated monopoly price is below its social optimum, and, under complete information, the optimal second-best tax on the dirty variety would be positive. Hence, the optimality of environmental regulation is conditional on su¢ ciently high damage from production of the dirty variety.

De…nition 13 Denote t wk : c (e; t) < c ( e) weak environmental tax associated with weak environmental regulation; and t st : c (e; t) > c ( e) strong environmental tax associated with strong environmental regulation.

Recall that since the level of environmental tax, t; depends on the intensity of polluting emissions, which are in turn a function of e, we count the tax a part of the e¤ective marginal production cost.

All tax rates that preserve the cost advantage of the dirty variety relative to the clean variety correspond to weak environmental regulation. All tax rates that turn over the cost e¢ ciency between the types represent strong regulation.

Recall that @p s @t = 1 2 1 p

(2 2t)
> 0 for upward and downward price distortion respectively Then simplifying (2.25), the FOC reduces to:

0 @p s @t (p s ( e; t) c ( e)) = (1 0 ) ( + c (e) p (e; t)) (2.26)
Since with upward price distortion 0 < @p s @t < 1 and with downward price distortion 1 < @p s @t , it allows the following.

Proposition 14 With weak regulation the expected price is above the expected social cost of the two varieties, with strong regulation the expected price is below the expected social cost:

8 < : 0 p s ( e; t) + (1 0 ) p (e; t) > 0 c ( e) + (1 0 ) ( + c (e)) if c e > 0 0 p s ( e; t) + (1 0 ) p (e; t) < 0 c ( e) + (1 0 ) ( + c (e)) if c e < 0 (2.27)
The condition speci…ed in Proposition 14 states that under weak regulation consumers pay for the product on the average more that it costs to society, while under strong regulation they, in expectation, underpay. This result is driven by the inertia of the clean type's signaling price. The response of the clean producer to an increase in c (e; t), induced by t, under weak regulation pushes the clean type to 'reluctantly'distort the signaling price further away from the pro…t-maximizing level. Under strong regulation however, an increase in tax directly increases the clean type's pro…t, so the clean type is willing to further distort its signaling price toward its optimal monopolistic price.

Therefore, when it come to comparison with the optimal solution under full information, then standard Buchanan result requires the second-best optimal tax to equalize the price of the polluting monopoly to its social cost: p Buch (t; e) = ( e e) + c (e) : This achieves optimal resource allocation and the (…rst-best) optimal output level (apart from redistributional issues since the monopoly exerts market power). Under incomplete information, optimal resource allocation is no longer attainable.

Claim 15 Strong environmental regulation can pertain and be optimal if and only if the probability of the clean variety is low, i.e., for 0 < 1 2 :

Proof. See 2.7.3.

Claim ( 15) is intuitive because the expected environmental damage must exceed the expected value of clean variety.

Since the pollution tax is set proportionally to emissions, it enters linearly into the e¤ective marginal cost function, i.e., c (e; t) c (e) + t ( e e) ; the full information monopoly price of the dirty …rm rises at a +c(e)+t( e e)

2

. The private FOC of pro…t maximization states @ @q = @p(q) @q q + p (q; e) c (e) t ( e e). This makes it possible to present p (q; e) c (e) = t ( e e) @p(q) @q q or in terms of price elasticity p(q;e) c(e) p(q;e)

= t( e e) p(q;e) p(q;e) (p;e) . Substituting for the price margin of the conventional environmental quality gives the general form of the optimal second-best environmental tax:

t = 1 e e
p (e; t) (p (e; t) ; e)

+ 0 1 0 q 0 t (p s ) q 0 t (p ) [p s ( e; t) c ( e)]
(2.28)

For expositional reasons replace e e = 1 and q t (p s ) = @q @p p 0 t ; then the tax becomes:

t = p (e; t) (p ; e) 0 1 0 p s0 t p 0 t [p s ( e; t) c ( e)]
(2.29)

Or explicitly:

t = ( c (e) ) 0 1 0 p s0 t p 0 t [p s (e; t) c ( e)]
(2.30)

Clearly, expression (2.29) is not the explicit solution for t because t is on both sides of the equation. It allows nevertheless for economic interpretation. The optimal secondbest environmental tax is composed of three elements that account for three distortions in the market: marginal environmental damage, , monopoly pricing and incomplete information. Each distortion, having its proper impact on the welfare, …nds its re ‡ection in the formula of the optimal environmental tax.

The Pigouvian taxation level serves the ultimate benchmark of regulation, fully internalizing pollution damage. However, in the presence of market power as it has been shown by [START_REF] Buchanan | External diseconomies, corrective taxes, and market structure[END_REF]) and [START_REF] Barnett | The pigouvian tax rule under monopoly[END_REF], the severity of environmental regulation must be set below the Pigouvian level. The monopolist restraining output to obtain monopoly rent thus restrains polluting emissions as well. Let us adopt the term 'Buchanan tax'to designate the optimal second-best tax on the polluting monopoly. A …nite price elasticity of demand, ; is crucial to support Buchanan tax level. Otherwise, market power vanishes and optimal tax regains Pigouvain level since the third term in (2.29) disappears as the price margin falls to zero.

The last element in (2.29) is in the center of our interest. This term measures the impact of incomplete information on the optimal second-best tax. This term contains three elements: the relative probability of occurrence of the clean type, the relative sensitivity of the price to a change in the tax and the signaling price margin of the clean variety. All three components are always positive implying that the externality of imperfect information further reduces the severity of environmental regulation. This happens because the signaling price is dependent on the marginal production cost of the dirty variety. The imposition of a tax increases the dirty type's cost making deception more attractive. It forces the clean type to additionally adjust its signaling price to discourage imitation. Hence, green consumers carry the burden of environmental taxation as they must pay a higher signaling price. The clean type's pro…t varies in the opposite direction of the signaling distortion. Hence, when the strong regulation is enforced, the clean type bene…ts because its signaling price moves closer to the full information pro…tmaximizing level. However, the producer's gain doesn't make up for the reduction in the surplus of consumers. The price increase induced by the emission tax generates a deadweight loss for the clean variety with no bene…t of pollution control.

Following [START_REF] Mahenc | Are green products over-priced?[END_REF]), let us examine the impact of consumers' prior beliefs on the level of an optimal environmental tax.

Corollary 16 Under separating equilibrium, consumers' greater expectations with respect to environmental quality alleviate the severity of regulation:

@t @ 0 < 0 (2.31)
Proof. See (2.7.4).

Corollary 16 gives the relation between tax stringency and the consumers'prior beliefs about environmental performance. As consumers grow more enthusiastic, the tax level must decrease. This result is in line with Claim 15. As consumers place greater expectations on high environmental quality, the relative importance of pollution control diminishes.

Let us examine the extreme values of consumers' expectations. When 0 ! 0, the optimal second-best tax in (2.29) shrinks to the Buchanan tax on a polluting monopoly.

When, on the contrary, 0 ! 1, the FOC in (2.25) is reduced to: @p s @t [p s (e; t) c ( e)] : With no pollution externality , the tax aims at eliminating the positive price margin of the clean variety. Yet, given @p s @t > 0, any positive tax would only aggravate the existing distortion. Thus, intuitively, a pollution tax is unable to correct pricing distortions stemming from the signaling behavior of the non-polluting variety.

Optimal second-best tax with equilibrium selection by undefeated criterion.

As stated earlier, the Mailath's undefeated equilibrium criterion may select pooling equilibria over separating. This is the case for su¢ ciently high 0 [START_REF] Mahenc | Are green products over-priced?[END_REF]. We argue that a su¢ ciently low c; that is 8c < b c : p (p p (b c) ; e; 0 ) = s ( (b c) ; e; ( (b c))), may as well invoke similar results.

To see how environmental regulation may a¤ect plausible equilibria selection by the undefeated criterion consider the situation depicted in Figure 23). On the upper panel, the before-tax separating equilibrium is Pareto dominant. Yet, tax implementation, reducing relative cost di¤erence between the types, changes the equilibria ranking: the after-tax pooling equilibrium18 Pareto dominate separating, depicted on the lower panel.

Thus, for all marginal separating equilibria where the clean type is only slightly better o¤ revealing information than concealing it, environmental taxation will necessarily reverse the choice of pricing strategy.

Given distribution of types, production costs and consumers'disutility from pollution, the regulator can anticipate which equilibrium survives the selection by the undefeated criterion. For all undefeated pooling equilibria and all marginal separating equilibria, the regulator will maximize the probabilistic welfare (2.23) given uninformative pricing .

Then, the regulator's objective function is:

max t W (t) = 0 ( e 0 ) 1 2 q (p p (t; 0 )) c ( e) q (p p (t; 0 )) + (1 0 ) ( e 0 ) 1 2 q (p p (t; 0 )) c (e) q (p p (t; 0 ))(2.32)
Rearranging social welfare gives:

W (t) = ( e 0 ) 1 2 q (p p ( )) q ( ) (2.33) [ 0 c ( e) + (1 0 ) (c (e) + )] q ( )
the objective function, with the optimal second-best tax, must be to equalize the product's social value given the expected environmental quality with the expected social production cost of the clean and dirty varieties.

Under pooling equilibrium, the FOC requires that the optimal second-best environmental tax is set at:

t p = ( c (e) ) 0 1 0 [ c ( e)]
(2.34)

As in (2.29), under pooling equilibrium the optimal second-best environmental tax is composed of three elements responsible for the correction of corresponding diseconomies.

First, it is the correction of pollution externality at the level of marginal environmental damage. Second, it is the correction of monopoly pricing given expectations of environmental quality, and lastly, the correction for incomplete information. Also in this tax version, the presence of information asymmetry requires the scaling down of tax stringency: 8 0 > 0 the optimal tax must be below the Buchanan level. However, now it is the net social value of the clean variety (and not the price margin) by which the tax is adjusted. The pooling price is the same for each variety but proportional to pollution tax. With greater social value of the clean variety, the regulator must take into account the restrictive e¤ect of the tax on the pooling price.

Corollary 17 Under pooling equilibrium, greater consumers'expectation with respect to environmental quality alleviate the severity of regulation:

@t p @ 0 < 0 (2.35) Proof. Stems directly from: @t p @ 0 = c( e)
(1 0 ) 2 : Under pooling equilibrium, an increase in 0 increases the value the clean variety receives in the eyes of the regulator. Therefore when setting an optimal tax, higher consumers'expectations make the regulator favor the clean variety, and because of the negative e¤ect of the tax on social welfare that results from the trade of the clean variety, the tax level must be reduced accordingly.

Special case: c (e) = c (e)

By construction the clean …rm di¤ers from the dirty one in either its marginal production cost and/or in its …xed cost. Complementary to our previous discussion, in this subsection we examine the case where only technology investment is at the origin of high environmental performance (holding marginal production costs equal). With no loss of generality, set c (e) = 0 and assume that the …xed cost is su¢ ciently small not to interfere with monopolist's incentive to produce output, i.e., F ( e) ( ) 2 4 . Before pollution control is enforced, Condition (2) for a separating equilibrium does not hold. When both types receive equal price margins, the clean type can no longer attract green demand and at the same time e¤ectively discourage mimicry. Hence, in an unregulated market the only possible equilibria type is pooling.

It is obvious that implementation of environmental regulation would immediately satisfy Condition (2) enabling separating equilibria. Since after tax the dirty …rm becomes less e¢ cient with c < 0, the clean type is better o¤ with downward price signal.

Welfare implications

In this section we evaluate the impact of environmental taxation on social welfare.

Separating equilibrium. When weak regulation is required, it partially internalizes the externality from the polluting production of the dirty product. The clean product also grows more expensive making consumers'and producer's surpluses shrink. It represents a mere deadweight loss because the reduction of output doesn't serve to control or limit pollution.

When strong regulation is required, the clean variety signals its quality by downward price distortion. Compared to upward signaling, consumers are better o¤ as they bene…t from the lower price of the clean variety. In terms of surplus, the clean producer is indi¤erent to upward and downward price distortion because in the former case a high price margin compensates for thinner demand and vise versa in the latter case. Compared to socially optimal resource allocation, strong regulation with downward price distortion performs relatively better than the weak because both price and output of the clean product are closer to the competitive level. It is clear however, that the optimality of such a sever environmental policy is conditional on relatively high environmental damage, low consumers'expectations, and a su¢ ciently high di¤erence in marginal costs. Figure

(2-4) illustrates a parameter map for which all the conditions are veri…ed. As either constraint is relaxed, the strong tax ceases to be optimal because of its high social cost.

It is noteworthy that for both types of regulation, optimistic expectations about environmental quality lessen the severity of the environmental tax. With upward signaling, there is a deadweight loss from the reduced amounts of trade of the clean variety. With downward signaling, it is rather the exercise of market power by the clean type, which enables the signaling price to approach monopoly level.

Pooling equilibrium. When prices are uninformative, the pollution tax a¤ects the pooling price. The optimal tax corresponds to the Buchanan level, softened by the probability of occurrence of the clean variety. The optimal tax equalizes the product's expected social value to its expected social cost. For the dirty variety, the tax doesn't entirely corrects pollution externality. For the clean variety, the tax reduces the amount of trade. In addition, private information about environmental quality remains concealed.

Equilibrium selection by undefeated criterion. As it has been shown for equilibria selection according to Mailath's undefeated criterion, environmental taxation may a¤ect a producer's pricing strategy. Thus, the marginal producer who is ex-ante better of with a separating price strategy, would ex-post 'migrate', subsequent to the imposition of weak environmental regulation, to a pooling price strategy. Similarly, to have a downward price signal induced by strong environmental regulation, environmental damage must be su¢ ciently close to the dirty variety's social value and the tax level should more than double the di¤erence in marginal production costs, i.e., t > 2c. The …rst c is necessary to reverse the order of cost e¢ ciency, the second -to induce a signaling strategy. Because even being more cost e¢ cient, the clean type's pooling pro…t would Pareto dominate its separating pro…t, if it costs almost the same to produce a high or a low environmental quality. Thus, the tax may make the clean type prefer the pooling strategy over the separating strategy.

The 'equilibrium migration'induced by environmental regulation has a negative e¤ect on welfare because consumers lose the ability to distinguish between environmental types when environmental regulation renders pooling equilibrium Pareto dominant.

E¢ ciency of environmental regulation. Given the complexity of consequences of environmental taxation on imperfectly competitive markets plagued by incomplete information about environmental quality, one may wonder if such a policy can in fact be e¢ cient. The result of uninformative pricing is deemed suboptimal while the regulation under separating equilibrium is associated with high social costs. What might seem appealing in environmental policy -apart from the control over pollution externalityis that strong regulation has the ability to induce a downward price signal so bene…cial to consumers. However, it can be shown that a welfare neutral strong subsidy from consumers to the clean producer (such that would not only buy o¤ clean type's marginal production cost but also make separating equilibrium Pareto dominant) would attain the same result but at a lower social cost, casting doubts on the desirability of strong environmental taxation.

Conclusion & discussion

In this paper we studied the impact of price signaling under asymmetric information on the optimal design of the second-best environmental tax.

The main contribution to existing literature is that we endogenize environmental regulation to emphasize its impact on social welfare. In the framework of imperfectly competitive industry with incomplete information about environmental quality, the …rm may communicate this quality to consumers by distorting the product's price from its pro…t-maximizing level. This price strategy allows the …rm to attract green consumers who can infer a …rm's true type by observing price deviation.

We …rst examine equilibrium pricing for two quality types according to intuitive and undefeated equilibria selection criteria. Then we derive the optimal second-best environmental tax. We show that the tax rate must be scaled down below the level optimal for a polluting monopoly under complete information. This result is driven by two effects. First, positive consumers' expectations of clean production lower the stringency of the tax, because the clean type does not emit pollution. Second, the mechanics of price signaling make the clean variety more expensive for no environmentally grounded reason. This constitutes a social cost of regulation for which incomplete information is responsible. While an environmental tax a¤ects optimal pricing of both clean and dirty varieties, for the dirty type, the tax serves to correct pollution externality, for the clean type it represents a mere undesirable by-product. The reduction of trade of the clean variety induced by environmental taxation represents a direct loss to society.

Our …nding with respect to the optimal second-best tax level is consistent with the literature. Indeed, [START_REF] Antelo | Asymmetric information, signaling and environmental taxes in oligopoly[END_REF] come to a similar conclusion that informational de…ciency on part of the regulator makes the regulation less severe compared to the level required for a polluting monopoly.

We demonstrate that …rst-best resource allocation, still attainable in the absence of information asymmetry, cannot be achieved under incomplete information. As price signaling has an unequal sensitivity to policy stringency, it must be on the average either unduly lax or unduly stern at the expense of the consumers'welfare.

Finally, we show that environmental policy may produce a perverted e¤ect on market equilibrium. When considering the criterion of undefeated equilibria, the implementation of an environmental tax may remove the incentive for the clean producer to reveal its true type to uninformed consumers. Thus, subsequent to environmental taxation, pooling may turn Pareto dominant over separating equilibria.

The complex structure of environmental taxation, its high social cost and the possibility of undesirable equilibria migration, all call for great caution when contemplating the implementation of a market-based instrument to correct pollution externality when the market in question is characterized by multiple distortions. and CI LEQ :

Appendix

8 < : (p c) ( p) ( c) 2 4 ( ) 2 4 p ( p)
(2.43)

Note that the two conditions can be rewritten as a function:

f (p; c) = (p c) ( p) ( c) 2 4
(2.44)

To …nd the separating equilibrium the system must hold

f (p; c) 0 f (p; 0) (2.45)
The 2.43 represents two inequalities for polynomials, whose solution must the overlap of intervals de…ned by the two conditions. The price solving the equation

f (p; c) = 0 is p (c) 1;2 = 1 2 + c p (2 2c) (2.46)
To have a solution to the polynomial function f ( ), the discriminant has to be positive:

(2 2c) 0 (2.47)

Rewriting 2.47 we obtain 2 c (2.48)

This inequality holds always under Assumption (A.1).

To verify the conditions in 2.43, signalling price p s must be element of sets of intervals, i.e.

p s 2 [p (c) 1 ; p (c) 2 ] \ (( 1; p (0) 1 ] [ [p (0) 2 ; +1)).
Because of the cost di¤erence between the varieties, the separating price must belong to the interval [p (0) 2 ; p (c) 2 ],

which corresponds to

h 1 2 + p (2 ) ; 1 2 + c + p (2 ( c) ) i .
At the lower bound, p s = 1 2 + p (2 ) , the pro…t distortion of the HEQ from signalling is minimal, which is the lowest upward distortion of the price necessary .Then, we can rewrite (2.23) to get:

W (t) = 1 2 q s (t) c q s (t) + (1 ) 1 2 q f d (t) q f d (t) (2.51)
The corresponding …rst order condition, when replacing q f d (t) by t 2 and q f 0 d (t) by 1 2 , is given by q s0 (t) (p s (t) c) 1 4 (1 ) q f 0 d (t) ( + t) = 0. If the regulator knows that the negative externality from pollution is internalized, he must use permit market to correct the monopoly distortion and the informational distortion. Substituting

1 2 1 q 2 2t
for q s0 (t) for weak and strong regulation respectively and rewriting the above …rst order condition we obtain the implicit form of the optimal permit price:

t e = ( ) 2 1 1 r 2 2t (p s (t) c) (2.52)
The expression in (2.52) is negative for weak and strong regulation 19 . When = , the regulator would need to subsidy the producer to correct the distortions in the economy.

It is almost intuitive that when polluting externality has been internalized there is no room for tis correction. The environmental regulation is incapable of improving welfare.

The choice of optimal pricing

Comparison between pooling and separating pro…ts

The pooling pro…t for a producer's type e is:

p (p; e; 0 ) = (p c (e)) ( ( e 0 ) p) F (e) (2.53) 19 The expressiong 1 2 1 q 2 2t < 0 given Assumption (A.3).

103

The price that maximizes p for each type e is: For the exposition purpose, some simpli…cation of notation is worthwhile. As …xed costs have solely a level e¤ect, we set F (e) = 0: The e¤ective di¤erence in cost e¢ ciency is crucial/ substantial for separation, thus we set c (e) = 0 and c (e) = c which measures now on the di¤erence between e¤ective production marginal costs of the dirty and the clean types.

p p =
To obtain the parameters under which the separating pricing is optimal, we solve for s ( ) > p ( ) given the assumptions of the model. The solution yields:

2 6 6 6 6 6 6 6 6 6 4 0 < c 1 + p 2 ^0 < < ^ ^c + < 8 > > > < > > > : 1 + p 2 < c < 4 2 4 0 < < ^ ^c + < ^ < < 1 ^ < c 4 ^0 < < 1 ^ < (2.58)
with ^ being a critical value for which and equal (their intersection doesn't belong to the area of product desirability);

c 2 (2 ) 2c (1 ) +2c p 1 jc j+ 2 (2 ) 2 2 2 and c 2 (2 ) 2c (1 ) 2c p 1 jc j+ 2 (2 ) 2 2 2
: For parameters given by (2.58) the signaling pro…t exceeds that of pooling.

Parameter conditions for full information price signaling Under separation, that is when the conditions of the individual rationality, IR C , and incentives compatibility, IC D , are veri…ed and s > p , if p 2 % IR \ % IC ; the full information price is informative and no distortion is required. Thus, we solve for p = p s conditional on (2.58) to obtain:

c > 1 + p 2 ^ c 2 + 2 2 (2.59)
which determines the area of full information separating prices.

Proof of Claim 15: < 1 2 :

Proof. For dirty variety with monopoly position the socially optimal environmental tax (Buchanan tax) requires that p = c (e) + : This result in t Buch = 2 ( ). For a positive tax must hold that > 2 :

With probabilistic welfare and 1 2 , environmental damage must outweigh the social value of the clean variety. Taking the lowest bound of = 1 2 implies that which violates assumption A.3 for social desirability of conventional variety.

Demonstration: @ 2 W @t 2 < 0
To ensure that t maximizes welfare, the second order condition must verify:

@ 2 W (t) @t 2 = @q @p s " @ 2 p s @t 2 (p s c ( e)) + @p s @t 2 # + (1 ) @q @p @p @t 2 < 0 (2.60)
Given that @q @p < 0 for both environmental types, it must hold that the second derivative of clean type's price margins is positive. It is so for downward price distortion as 22t)] 3=2 > 0: For upward price distortion, the second derivative of price in t, @ 2 p s @t 2 = 2[ (2 2t)] 3=2 ; is negative. Therefore must hold: This condition places the lower bound on : It can be shown that for separating equilibrium with upward price distortion, it is not constraining (see Figure 234). On the plot, the bound restricts some area for the low values of . This is because we set t ! c , which is for the weak regulation over constraining. It has been shown that for all tax values su¢ ciently close to c, the pooling equilibrium prevail. Reducing accordingly the slope of the line removes the constraining e¤ect. Hence, for all separating equilibria with upward price distortion, the second order condition holds.

@ 2 p s @t 2 = 2[ (
1 p (2 2t) ! 2 (p s c ( e)) 2 [ (2 2t)] 3=2 (
2.7.4 Proof of Corollary 16: @t @ < 0

Proof. Recall that expression (2.25) is an implicit function:

@W (t) @t F (t; ; ; ; ; c; e) = 0 (2.66) which is de…ned in the neighborhood of t . Using implicit-function rule, dt d = F Ft , note that F and F t correspond to W t and W tt : In (2.7.3) we've shown that W tt < 0; to demonstrate negative relation of dt d it is enough to have F < 0: Taking derivative of (2.25) with respect to we obtain: F = @q @p @p s @t (p s c ( e)) @q @p @p @t (p c (e) )

(2.67)

Since @q @p < 0, the …rst term of the right hand side is negative. In the second term, the 'social'price margin is negative when FOC holds. Hence, F < 0: We thus adjust the model to allow for producers'organization, also with environmental awareness induced by the threat of NGO's negative publicity. For every case under consideration, the privately chosen environmental quality falls short of the socially desirable level. Hence, in context of a global tendency of deregulation of markets, entrusting implementation of public interests to a private entity may call for caution.

Literature review Among most closely related literature, in the framework of a di¤erentiated Bertrand duopoly [START_REF] Amacher | Environmental quality competition and eco-labeling[END_REF]) study the optimal market choice of green technology investment. The …rms'decision is contrasted to the socially optimal level of investment. Two versions of social optimum are considered: a standard utilitarian and one with environmental externality. The externality is positive and represents consumers' perception of an average environmental quality (EQ). The externality however does not imply any damage to the natural environment. The model has three stages with the investment, quality and price decisions being made successively and resolved by backward induction. The technology investment is assumed to reduce marginal cost. The …xed cost of the investment can be higher or lower for the high quality …rm than for its low quality competitor. Thus all the strategic interactions are linked to the relative e¢ ciency of the high quality …rm. The resulting quality choices a¤ect the severity of competition. If the high quality …rm is more e¢ cient, she invests more than socially optimal in the environmental quality to relax price competition. However, since it is assumed that the investment also increases the willingness to pay for the product the resulting market quality is highest when low quality …rm is e¢ cient in investment because it takes more e¤ort to relax price competition. [START_REF] Conrad | Price competition and product di¤erentiation when consumers care for the environment[END_REF] analyzes the optimal EQ choice made by …rms in the market of a horizontally di¤erentiated duopoly with price competition. The author views EQ as a horizontal attribute. It is rather a matter of consumers'tastes than a sign of superior product quality. Indeed, EQ is often negatively related to intrinsic quality: polluting cars are more powerful, bleached or recycled paper is brighter and is seen as of a higher quality, organic vegetable or fruit have less visual ‡aws than conventional ones. This particular assumption renders the utility function (and the intrinsic reservation price) concave in EQ. The model is resolved by a two-step game. Depending on the relation between the dislike of pollution and the (private) cost of production , it can result in 3 quality spaces. When an extreme environmental concern is present, the duopolists choose the highest EQ and make zero pro…ts. Lowering environmental concern, makes quality space and pro…ts gradually increase. It turns out that for intermediate environmental concern, the quality space is always a corner solution. [START_REF] Mankiw | Free entry and social ine¢ ciency[END_REF] study the equilibrium number of …rms in a free-entry oligopolistic market with a …xed cost investment at entry. In the baseline model -a

Cournot oligopoly with homogenous goods -the market equilibrium is characterized by an excessive entry than is socially optimal. Each new entry increases welfare by the pro…t the new entrant obtains and reduces aggregate pro…ts of competitors since they have to reduce their output (business stealing e¤ect). Although the excessive entry is welfare decreasing, it doesn't have to be the correct measure of welfare loss. The welfare loss due to free entry however declines as the socially optimal number of …rms increases. The result of excessive entry doesn't hold speci…cally for an oligopolistic market, it holds also for cartels where …rms continue to entry until "all of the collusive monopoly pro…ts are dissipated into set-up costs". However, if consumers appreciate diversity of homogenous products, then each new entry has a positive e¤ect on welfare since it increases the variety. Depending on which e¤ect -business stealing or variety -dominates, the market equilibrium can result in excessive, insu¢ cient or optimal number of …rms. Ultimately, when the number of …rms grows larger and …rms come to act as price takers and if the set-up cost approaches zero, in that case the market equilibrium number of …rms approaches the social optimum. [START_REF] Dixit | A model of duopoly suggesting a theory of entry barriers[END_REF]) develops a theory of entry barriers. In a Cournot duopoly, he points out that there are discontinuities in response function if market entry requires a …xed cost investment. The analysis of an incumbent …rm reveals that threatened by an entry of a competitor, depending on the magnitude of the …xed costs, the incumbent may …nd it optimal to either accommodate or to deter entry. If the competitors …xed costs are so high than there is no entry. If they are low, the incumbent is better o¤ accommodating entry because the deterrence strategy becomes excessively costly. If entrant's …xed costs are intermediate, the incumbent is better o¤ deterring entry by producing limit quantity.

Further, the author analyzes the circumstances to determine the facility of entry. He demonstrates that the increase in net private value (net advantage) of the incumbent variety hinders entry while a decrease in substitutability between goods facilitates entry.

The model

We wish to compare the level of environmental quality (EQ) that a …rm chooses on the basis of pro…t maximization with the level that a benevolent welfare-maximizing regulator would desire.

We suppose that the market is composed of two producers that can supply brown and/or green EQ. The brown quality signi…es a conventional production which emits polluting particles. The producer of brown variety has a conventional production technology that exerts polluting emissions. The producer of green EQ undertakes some abatement e¤ort and thus the production of a green product is characterized by less intense emissions. So the output of the two …rms di¤ers in the level of environmental cleanliness. As more abatement is undertaken, the more di¤erentiated the products are, and thus less substitutable one for another.

We assume that despite pollution, both varieties are socially desirable. Let e denote the level of EQ chosen by the green producer, e 2 [0; 1] : When e = 0; there is no abatement e¤ort and emissions are proportional to the output, and the two products are equal, the competition is a duopoly of homogenous goods, when e = 1; the emissions are zero, the green and brown product attain the maximal di¤erentiation which divides the market in two independent niches for each type of EQ. Thus, the level of abatement allows relaxing the competition intensity

The order of the baseline game is as follows: at Stage 1 the green producer chooses the pro…t-maximizing level of EQ e g . At Stage 2, both producers participate in Cournot competition with di¤erentiated varieties choosing pro…t-maximizing output. At Stage 3, consumers make their purchasing decisions.

We solve the game by backward induction.

The consumers'utility function is quasi-linear in all other goods in the sense/style of [START_REF] Dixit | A model of duopoly suggesting a theory of entry barriers[END_REF]:

U (p i ; p j ; q i ; q j ; e i ; e j ) (a (1 e i ) d) q i + (a (1 e j ) d) q j (3.1) 1 2 q 2 i + q 2 j + 2 (e i ; e j ) q i q j + I with (e i ; e j ) being a function that measures substitution between the goods q i and q j .

To capture the idea that the more distinct the products are the less substitutable they are, a most 'neutral'function for is (e i ; e j ) = 1 je i e j j. Indeed, if both …rms choose zero abatement, their output is homogeneous in the eyes of consumers, then = 1. From the IO literature, we know that the producers would tend to -in the simplest and most basic model -maximally di¤erentiate, this would result in, say, e i choosing maximal abatement and e j choosing minimal abatement; in that case = 0 and both producers enjoy monopoly power. One could have thought of a more general substitution function of , as for example, (e i ; e j ) = 1 je i e j j x . Then for x 2 (0; 1), the substitution e¤ect is ampli…ed and for x > 1, the substitution e¤ect is diminished. However, for our present purpose, let us stick to the most neutral version in terms of substitution e¤ect and further let us suppose the …rm i always chooses a higher abatement level compared to …rm j (the results would be symmetric anyway). Since the tendency to di¤erentiate products is an established fact in the IO literature, let us suppose that …rm j is brown and she doesn't clean up the a-uents from hers manufacturing/industrial activities while …rm i chooses her level of abatement e i , then:

U (p i ; p j ; q i ; q j ; e i ; 0) (a (1 e i ) d) q i + (a d) q j (3.2) 1 2 q 2 i + q 2 j + 2 [1 e i ] q i q j + I Let a measure the gross reservation price for each product type, indexes i; j = fb; gg : i 6 = j indicate the level of EQ for each product: brown or green; d describes consumers'taste for green EQ, d > 0;, q i indicates the number of units of i type's product that consumers purchase at price p i with some income I: The last term, (1 e g ) q i q j , speci…es the degree of di¤erentiation between the two products, that is the green producer optimally chooses the level of di¤erentiation (or substitutability) with the rival product. This term can be viewed also as consumers'taste for variety, that is, the provision of similar varieties reduce consumers'utility.

From (3.2) we derive demand for k th good/variety with k = i; j. Then, the quantity of good k consumed depend on/the demand is a function of k th variety's price, k th variety's output weighed by the abatement intensity of …rm i: and k th own level level of EQ:

q k (p k ; e k ; q k ; e i ) = a (1 e k ) d p k (1 e i ) q k (3.3)
condition (FOC) for the green …rm: @ g @g = @P g @g g + P g = 0 (3.8)

g (b) = P g P 0 g (g) = P g ( ) (3.9)
The FOC in (3.9) allows to state explicitly the reaction function of the green product to the brown …rm's output:: We assume that @b @e > 0 imposing restrictions between parameters a and d: @b @e = a(e + 1) 2 4d(1 e) (3 e) 2 (1 + e) 2 > 0 (A.2)

g (b) = 1 2 [a (1 e) d ( 1
Note that the second derivative is strictly positive for all admissible values: 0 8e 2 [0;1] (3.17)

@ 2 b @e 2 = d (e + 1) 3 + 2a + d (3 e) 3 >
Thus, b (e) is increasing and convex in e:

The equilibrium pro…t of the brown variety given a level of e is:

b (e) = (1 + e) a (1 + 2e e 2 ) d 3 + 2e e 2 2 (3.18) = b (e) 2
Clearly, b is strictly increasing in the level of EQ that the green producer determines as the sign of @ b @e corresponds to that of @b @e ; hence:

@ b @e = 2b (e) @b @e > 0 (3.19)
The equilibrium demand for the green variety given a quality choice e is:

g The equilibrium pro…t of the green type is given by: If the …rms chose not to diversify their products, the equilibrium pro…t of undi¤erentiated duopoly for 8e 2 [0; 1] is: max q i g (q i ; q i ; e) = (a d (1 e)) 2 9 F (e) (3.25)

To exclude monopoly solution, assume that for 8e 2 [0; 1] 9e ê :

(a (1 ê) d) 2 9 F (ê) = (a d) 2 9 (A.3) Assumption (A.3) states that there is an intermediate level of EQ at which the pro…ts of undi¤erentiated green duopoly fall short of the pro…ts of the undi¤erentiated brown duopoly. This implies that the …xed cost necessary to attain the split of the duopoly in two monopolies is too high.

As discussed earlier, both the green and the brown …rms emit pollution. Let D denote the corresponding environmental damage function from i th variety, then: D (q i ) = (1 e i ) q i (A.4)

where denotes marginal damage per unit of output. It is clear that for the brown variety the emissions are equal to the output of brown units, while for the green variety, the emissions are reduced by the abatement e¤ort de…ned by e g = e:

Assumption (A.5) implies that disutility from pollution together with environmental damage outweighs the cost of EQ provision: [START_REF] Par Exemple | Action pour le climat[END_REF] hence it is socially e¢ cient to produce high EQ.

d + > c 0 e (A.

Short-term choice of EQ

In this section, we examine the optimal choice of EQ by the green …rm.

In line with a standard approach in the literature, the green type will autonomously choose such level of e that maximizes its own pro…t in anticipation of subsequent quantity competition on the product market against the brown variety, thus: e = arg max e g (g (e) ; e) (3.26)

The FOC for (3.26) in e must solve the equation:

@ g @e = 0
2g (e) @g @e F 0 (e) = 0 (3.27)

With the general form of (3.27) being: @P g @e + @P g @b @b @e + @P g @g @g @e g (e) + P g ( ) @g @e F 0 (e) = 0 @g @e @P g @g g (e) + P g ( ) + @P g @e + @P g @b @b @e g (e) F 0 (e) = 0 (3.28)

The …rst term in (3.28) is zero since it is the FOC in quantity for the green product, i.e., g0 (g) : Hence, the FOC in e reduces to: @P g @e + @P g @b @b @e g (e) = F 0 (e) (3.29) Equation (3.29) speci…es the condition that maximized the pro…t of the green …rm in anticipation of a Cournot competition against the quality di¤erentiated brown variety.

This private choice of optimal EQ takes into account the impact the variation of e on prices of the green and the brown varieties. This re ‡ects consumers'taste for EQ and the diversity of the green product.

Social choice of environmental quality

In this section, we examine what is the optimal choice of EQ in the eyes of a welfaremaximizing benevolent regulator. His objective is to maximize the standard utilitarian welfare function composed of consumers'and producers'surpluses and the environmental damage stemming from …rm's output.

Hence, the regulator keeping in mind the di¤erence in pollution intensity of the …rms, Let e F denote the socially optimal level of EQ that solves (3.32), i.e., e F = arg max e W (e).

Then, it is straightforward that e F must break-even the gains and the costs of the environmental quality provision at the social level.

An increase in e enhances the output of both varieties, hence the pollution; relaxes the competition and thus allows both producers to raise prices. Moreover, it satis…es consumers taste for high EQ and the diversity of products sold in the market. In line with standard results, the private solution of EQ provision is below of what is socially desirable. The …rm makes a myopic quality choice because it considers solely its private bene…t from an increase of EQ. Note that the equilibrium choice e always gives the green …rm a greater pro…t compared to the brown …rm (this holds always provided that the green …rm's pro…t is concave in e, while b is convex). Although there is no pro…table deviation for the brown …rm, its pro…t is strictly increasing in e: It is clear that the ideal EQ in the eyes of the brown …rm is e = 1. However, Assumption (A.3) excludes this possibility, rendering this option too costly.

Thus, the di¤erence in pro…ts of the two …rms suggests certain instability of the equilibrium at e = e : If only the …rms could communicate with each other facilitating a transfer that would equalize their pro…ts, it would result in a long run stable equilibrium.

In this setting, the brown …rm has no longer any incentive to deviate, while the green …rm ensures itself the leadership in the strategic choice of EQ.

Suppose that the producers can unite within a producers'organization that represents their mutual interests with the goal to maximize joint pro…t (under condition of equal pro…ts). Denote T the amount that the green …rm transfers to the brown …rm. Then the joint objective function is: (1 ) @P b @e + @P b @b @b @e + @P b @g @g @e b + P b @b @e + (3.34) @P g @e + @P g @b @b @e + @P g @g @g @e g + P g @g @e = 0

Substitute the FOCs from pro…t maximization, @ i @i for i = fb; gg for P i , the condition () reduces to:

(1 ) @P b @e + @P b @g @g @e b + @P g @e + @P g @b @b @e g = 0 (3.35) Condition (3.35) speci…es the jointly chosen e F serving mutual interests.

Figure illustrates the jointly chosen level of EQ.

Proposition 22 Producers'mutual choice of EQ is below of what is socially desirable.

When the producers chose the EQ determined by their joint interests, they take into account how the variation in e a¤ects the prices as well as the equilibrium output of each variety.

e * e jnt The following passage captures the essential role of the NGOs when challenging large businesses:

"Aided by advances in information and communications technology, NGOs have helped to focus attention on the social and environmental externalities of business activity. Rewriting also the FOC for (3.36) gives:

(1 ) @P b @e + @P b @g @g @e b + @P g @e + @P g @b @b @e g (3.37)

[(1 e) g 0 (e) g (e) + b 0 (e)] = 0

Right: "It's time to act -What can we do? ", 2017, https://www.greenpeace.org.uk/what-wedo/oceans/coke/.

Denote e

z the level that solves for (3.37) and thus satisfy the joint …rms'e¤ort under the societal pressure of the NGO.

Figure (3)(4)[START_REF] Par Exemple | Action pour le climat[END_REF] illustrates this equilibrium EQ choice relative to the social optimum. The reason why the joint …rms'EQ choice falls short of the socially optimal level is that the producers' organization doesn't consider the surplus that consumers get from an enhanced EQ.

Conclusion

The article is to analyze the optimal choice of environmental quality when consumers value product diversity as well as high EQ.

We have derived the autonomous level of EQ chosen as a result of an environmental leadership by the green …rm. Further, we've shown that a private entity has a tendency to fall short of the EQ level necessary from the point of view of the benevolent regulator.

These …ndings call for caution when a private entity is granted power to undertake the imposition of public interests. This initiative/ambition may be doomed to failure.

Appendix

3.8.1 The sequence of the game

At stage 1 both …rms decide whether to invest in green production technology or not.

At Stage 2 the green …rm chooses its optimal level of EQ. At Stage 3, the …rms compete in quantities in the product market.

Let us resolve the game by backwards induction.

At Stage 1 each …rm must decide whether to make the green investment. Let us represent the pro…t in the tabular form: Given that consumers have green preferences, the green-green undi¤erentiated duopoly (Pareto) dominates the brown-brown undi¤erentiated duopoly.

At Stage 2, given the optimal EQ e set by the clean type, the two producers compete in quantities. This allows to write down pro…t and reaction functions.

Conclusion Générale

Cette thèse est structurée en trois chapitres analysant le rôle de la qualité environnementale comme une caractéristique intrinsèque d'un bien. Throughout the dissertation, the environmental regulation is mostly conducted with the help of a sole instrument -an environmental tax on polluting emissions. Indeed, when considering pollution, an imperfect competition and an additional market distortion, the general result is that the optimal second-best tax must be set below the Pigouvian level.

It turns out that a number of market failures corrected by a means of an environmental taxation weakens both the stringency and the e¢ ciency of the regulation. Clearly, a set of regulatory instruments is capable to improve the e¢ ciency in terms of the cost and pollution control. One can imagine a separation within the regulatory entity as such a set. The separation can be achieved when, for example, an environmental agency or an environmental NGO is concerned with product's environmental quality and …scal authority is in charge of the imposition and the enforcement of the tax. However, an inevitably problem that will arise is coordination. This suggests possible extensions to allow a comparison between these regulatory strategies and the analysis of their limits and advantages.

Another direction the extensions could take is to compare in the same framework the e¢ ciency of a market-based environmental regulation against the command and control instruments as minimal quality standards, mandatory quality certi…cation etc. I believe that the matter of environmental tax avoidance in conjunction with market power opens up an entire fruitful and vital …eld of research. Most natural extension is to consider the micro foundation of the avoidance mechanism that would allow to formulate the enforcement scheme against 'environmental'…scal avoidance.

Further developments that follow from the …rst chapter are to consider environmental regulation of a single powerful …rm (a multinational) by a number of jurisdictions or states. Despite a common environmental goal the regulation may be subject to con ‡icting interests for the reasons of tax competition, di¤erent monetary policies or diverging objectives for employment and competitiveness. The court hearing between Apple, Ire-

1. 1

 1 Parameter con…guration . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Résumés de la thèse Résumé en français La qualité environnementale ayant des caractéristiques d'un bien public fait objet de cette thèse. Dans un cadre théorique, on étudie d'abord les moyens de régulation d'une entreprise polluante et ensuite le choix décentralisé de la qualité environnementale. Pour une entreprise capable d'esquiver le …sc, la conformité à la taxation est déterminée en fonction du niveau de la taxe. Lorsque elle augmente, la non-conformité devient plus attractive.

  (1 e) by MSC, and v c(e), the marginal social value of the good, net of its production cost, absent any environmental consideration, by MSV. Monopolist's behavior in the absence of regulation.-The monopoly pro…t is e (p; e) = (p c(e))D(p; e) and the …rst-order condition for the monopolist's optimization problem is given by (p c(e)) @D(p; e) @p + D(p; e) = 0: (1.7)One can easily check that the second-order conditions are satis…ed. Let "(p) = ;e) denote the price elasticity of demand for the good. We further denote by pe the price set by the unregulated monopolist. The …rst-order conditions can be rewritten in the usual way to show that the Lerner index is equal to the inverse of the price elasticity of demand, which implies that market power is a decreasing function of the price elasticity of demand:

  consumers. Hence, the regulator only takes into consideration the complying part of producer surplus in the welfare function. The regulator's problem is to choose the tax that maximizes the social welfare function the …rm's noncompetitive behavior. As will be shown, the regulator mobilizes the force of the monopoly power to generate a socially bene…cial allocation of the good.We concentrate …rst on the …rm's monopolistic behavior. The pro…t is e (p e ( )) = (p e ( ) c(e)(1 e))D(p e ( ); e), and the …rst-order condition for pro…t maximization can be rearranged to compute the Lerner index of the regulated monopolist p e ( )

Figure 1 -Figure 1 - 2 :

 112 Figure1-2 displays the various forms that W ( ) can take, depending on the relative values of the MSV of the good and the MSC of pollution. As can be seen, the welfare functions exhibit two local maxima.When the shape of the welfare function is that depicted in the upper left of Figure1-2, the global maximum is a subsidy. Intuitively, this situation occurs when the MSC of pollution is low relative to the MSV of the good, meaning that the monopolist is signi…cantly clean and the good is cheap to produce. If (1 e) falls short of v c(e) 2 , Lemma 1

  ) p e ( ) (1 e) D (p e ( ) ; e) + (1 e)) @D (p e ( ) ( ) c(e) (1 e)) D (p e ( ) ; e) = 0:

Figure 1 -

 1 Figure 1-4 illustrates how the optimal tax regimes change with the severity of the environmental damage and the …rms'e¢ ciency. The bold curves plot e as a function of (1 e) , whereas the dashed line depicts n as a function of (1 e) . The three cases are drawn for low, intermediate and high MSVs of the good. The function e remains ‡at

  e) p e ( ) + (1 e)]dx + T + ( ) e (p e ( ) ; e); (1.20) where the monopoly pro…t is e (p e ( )) = (p e ( ) c(e) (1 e))D(p e ( ); e) T . Substituting this expression into (1:20), we have W ( ; T ) = [v p e ( ) + ( ) (1 e) + ( )(p e ( ) c(e) (1 e))] D (p e ( ) ; e) (1 e) D (p e ( ) ; e) 2 2 + (1 ( )) T (1.21) The welfare maximization problem of the regulator is max ;T W ( ; T ) s. t. e (p e ( )) 0

  e)] D (p e ( ) ; e) (1 e)D (p e ( ) ; e) 2 2 (1.23)

  (i) the unit tax n , (ii) and the lump-sum tax T = (p e ( n ) c(e) n (1 e))D(p e ( n ); e).

  Figure 1-5 displays the annual report of Boges Ltd. for the year 2012 5 . It represents the main indicators of economic activity of the …rm. The highlighted row gives a piece of evidence that although the Russian law on water use, water and air pollution, land-retirement and such, provides for the existence of enforceable taxes, Boges Ltd. has paid no taxes, whether revenues be positive in 2011 or negative in 2010 and 2012. This is the actual outcome corresponding to Region II of Figure 1-3.Given the large number of Boges Ltd. shareholders registered in Cyprus, the regulator is highly exposed to …scal avoidance. Compared to the overwhelming task of fostering economic growth in the region of Krasnoyarsk, environmental issues are at the bottom of the regulator's agenda. The government has still not achieved the construction of an aluminum plant which is expected to become the major client of the dam and one of the main employers in the region. In the meantime, Boges Ltd. is producing in excess of the environmental requirement in the region-this situation …ts the parameter con…guration delimited by (1 e) > v c(e) 2 in Figure1-3. On the basis of(Shishikin and Sirotsky

  e) 3D (p e ( ) ; e) 2 =2 + ( ) D (p e ( ) ; e) ; (1.29) which reaches a maximum at n = 2 v c(e)

  2 (1 e) D (p e ( ) ; e) 2 + ( ) D (p e ( ) ; e) ; c(e)) 2 20(v c(e))+1+24 (1 e) 6(1 e), provided that the discriminant p = 4 (v c(e)) 2 20 (v c(e)) + 1 + 24 (1 e) 0; otherwise, W p ( ) no shareholder complies with taxation, hence ( ) = 0. From (1:10), W ( ) is then given byW f ( ) = (1 e) maximum at f = 2 3 + v c(e)3(1 e) , so that W f ( f ) = (v c(e) (1 e)) -order condition for welfare maximization Using the fact that (1 e)D (p e ( ) ; e) = v p e ( ), Equation (1:15) )D(p e ( ); e) + d ( ) d [p e ( ) c(e) (1 e)]D(p e ( ) ; e) = 0: Substituting " (p e ( )) = @D(pe( );e) @pe( ) pe( ) D(pe( );e) into the left-hand side above, we have + p

  some calculations, we …nd that W n ( n ) W p ( 2 ) if and only if v c(e)))(1 6(v c(e))) 2 . It turns out that these parameter values are outside Region I, hence W n ( n ) > W p ( 2 ) holds in this parameter con…guration.

  , = 0 is a local maximum too, and routine calculations show that W n (0) W f ( 2 ) when24(v c(e)) 1 32(1 e) (< v c(e) 1 e ). Further calculations yield that W n (0) W f ( f ) when ( p 6 1)(v c(e)) 2(1 e) (< v c(e) 1 e). As a result, W n (0) max fW p ( 2 ); W f ( f )g for all parameter values in this con…guration.Examine now the part of Region II where v c(e)

  e) ; (v c(e)) 2 +(v c(e)) v c(e)))(2(v c(e)) 3) 2 6(1 e)

  Figure 1-3: Optimal taxation by zones

Figure 1 - 5 :

 15 Figure 1-5: Boges Ltd. pays zero tax on current income

[ 0 ;

 0 R] : A consumer with a reservation price r buys one unit if r ( e ê) > p and zero otherwise. The aggregate demand for the good is thus q(p; ê) = M R (R ( e ê) p)which has the same form as the demand derived for the representative consumer.Denote price elasticity of demand: (p; b e) ) p) 2 < 0. Higher expectations of envi-ronmental quality for a given price make consumers more captive from the monopolist's point of view.

Lemma 7

 7 Under full information about environmental quality the producer charges the price p (e) = (e e) + c (ee) provided that (p (e)) > 1 (for an interior solution).

  max p (p c (e)) D (p; ) : The FOC requires that D (p; ) + (p c (e)) Since the right hand side is clearly negative, it must hold that > 1:

  producer's objective is to maximize pro…t with respect to p given consumers beliefs about environmental quality. The …rm's strategy must form perfect Bayesian equilibrium where consumers infer environmental quality by observing the price. Let ( ) : R + ! [0; 1] denote a posterior belief function that relates the …rm's price to consumers' perception of environmental quality. Let (e) and (e) be the equilibrium prices for clean and dirty producers respectively. Thus, when a …rm charges price (e), then it is inferred to have quality ( ).Then, ( (e) ; (e) ; (p)) is the equilibrium strategy given the conditions: 1. For e = e; e (e) = max p (p; e; ) : 2. If (e) 6 = (e), then ( (e)) = 0 and ( (e)) = 1. If (e) = (e), then ( (e)) = 0 .

  c e is strictly monotone in e (type monotonicity) 3. S (p; e; ) dp d = 0 (p;e; ) 0 p (p;e; ) is strictly monotone in e (single crossing) The explicit form of the marginal rate of substitution between the price and perceived environmental quality is: q (p; (e)) + (p c (e)) e ] (2.11) It is straight forward that S e > 0 because and c e are positive by assumption, demand and the price margin are positive numbers. Finally, when c e > 0, consumers infer that high environmental quality is communicated by an upward price distortion, hence e > 0:

Lemma 8 :F

 8 Under conditions 1-3, separating equilibrium prices are such that (e) = p (e) and (i) (e) p (e) when c 0 e > 0 and (ii) (e) p (e) when c 0 e < 0: Lemma 8 summarizes equilibrium pricing strategies. The dirty type charges the full information price, while the clean type distorts its price thus communicating to consumers superior environmental quality. The di¤erence in e¤ective production marginal costs determines the direction of price distortion. Separating prices. Separating prices reveal producers' types. The dirty …rm sets the full information monopoly price: p (e) = +c(e) 2 According to its price strategy, (e) = p (e) ; it obtains the full information monopoly pro…t ( (e) ; e; 0) = ( c(e)) 2 4 . Yet, if the dirty type were to be taken for the clean through setting p = ( e), it would have reached the green consumers gaining the 'imitation' pro…t (p; e; 1) = (p c (e)) ( p). If, however, the clean type were to set up an uninformative price, it in turn would be taken for the dirty type. This would lead to 'confusion' pro…t (p; e; 0) = (p c (e)) ( p) F ( e) which achieves maximum p = ( e).To sustain separation, the clean type must set price (e) that veri…es:

s

  (p s (c (e)) ; e; 1; c ( e)) = [p s (c (e)) c ( e)] q [p s (c (e)) ; e] (2.14) Note that the signaling pro…t depends on both dirty and clean marginal production cost, c ( e) and c (e). It is clear that s decreases in c ( e). It is instructive to see how an increase in c (e) a¤ects the signaling pro…t:

  p (p p ; e; 0 ) > s ( ; e; ( )) for e 2 fe; eg (e) when p (p p ; e; 0 ) s ( ; e; ( )) for e 2 fe; eg (2.21)

Proposition 18

 18 When c ( e) = c (e), environmental taxation with t > 0 (i) induces a downward price signal and (ii) represents strong regulation.Proof. For (i) see the discussion following separating equilibrium conditions (3), (ii) by de…nition.Thus, environmental regulation may have a major positive impact on the informational issue proposing an alternative to uninformative pricing. The intuitive criterion selects the unique separating equilibrium with a downward distortion of the clean variety's price: ( e) (e) = p (e) ; the undefeated criterion selects separating equilibria once t is large enough to Pareto dominate the pooling equilibria.

Figure

  Figure 2-1: Equilibrium selection

  e) b c (e)] (3.10) The brown …rm's pro…t is: b (b; g; e) = P b (b; g; e) b(functions allows obtaining equilibrium quantities of green and brown products. The equilibrium output for the brown variety is given by:

  e) = (1 + e) a (1 e) d 3 + 2e e 2 (3.20) = a d + (a + d) e (3 e) (1 + e) Equilibrium pro…t of the green …rm can be stated as: g (g (e) ; e) = P g (g (e) ; b (e) ; e) g (e) F (e) (3.21) 123Note that the equilibrium output of the green variety is strictly increasing in the EQ: is increasing and concave in e 2 [0; 1] :

  = g (e) 2 F (e)

Figure ( 3

 3 Figure (3-1) depicts the pro…ts of green (green color) and brown (red) …rms as functions of EQ e:

  Figure 3-1: The choice of EQ by the green producer

Figure ( 3 Figure 3 - 2 :

 332 Figure (3-2) illustrates the position of e F relative to e to emphasize the gap between socially and privately chosen levels of EQ.

3. 5

 5 Long-term private choice of environmental quality Let us return to Figure (3-1) to examine the 'durability' of the equilibrium in e = e derived in Section (3.3).

  : g ( ) T = b ( ) Where measures the weight of the green …rm's interests within the producers' organization. The transfer cancels out in the objective function. Then, the FOC requires the following:

Figure 3 - 3 :

 33 Figure 3-3: Choice of EQ by producers'organization

Figure 3

 3 Figure 3-4: NGO's campagne against Coca-Cola. Reprinted with permission from Greenpeace UK.

Figure 3 - 5 :

 35 Figure 3-5: NGO's enduced choice of EQ

  represent an input of the type t ij that indicates the corresponding pro…ts for …rm i = f1; 2g of environmental type j = fb; gg ; i.e. brown or green. The upper superscript indicates the type t of the resulting duopoly, that is t = fu; dg that is for undi¤erentiated or di¤erentiated duoply.

  La qualité environnementale a¤ecte les préférences des consommateurs, les stratégies des producteurs et le degré de sévérité de la réglementation environnementale. Cette thèse met en avant di¤érents aspects du coût social de la pollution. La dégradation de l'environnement sensibilise les consommateurs à la qualité environnementale des produits, suscitant ainsi une certaine aversion envers les produits polluants. Une meilleure qualité environnementale est donc privilégiée. D'autre part, les producteurs considèrent la dimension environnementale d'un produit comme un moyen de di¤érenciation, ce qui leur permet d'attirer des consommateurs supplémentaires. Finalement, la réglementation environnementale permet d'induire un niveau de pollution socialement optimal. Bien que j'ai principalement examiné une taxe incitative sur les émissions polluantes, j'envisage également la possibilité d'une intervention décentralisée, alternative à la régulation, par le biais des ONGs. Ainsi, cette thèse théorique se situe au croisement des domaines de l'économie publique et de l'économie industrielle au travers l'environnement. Le premier chapitre traite de l'évasion …scale du monopole polluant inhérent à une régulation environnementale. Dans une deuxième partie, la taxe sur les émissions d'un monopole potentiellement polluant et dont l'engagement environnemental est inobservable pour les consommateurs est endogénéisée. Le troisième article étudie la concurrence entre une entreprise propre et une entreprise polluante. Le choix décentralisé de la qualité environnementale est déterminé en fonction des intérêts d'une organisation de producteurs ou / et d'une ONG environnementale. Généralement, l'aversion des consommateurs pour la pollution et leur préférence pour un produit d'une meilleure qualité environnementale internalise une partie du dommage environnemental. En même temps, elle (l'aversion) donne aux producteurs une incitation socialement indésirable à extraire le surplus dite vert des consommateurs. Dans competition.

Table 15 (

 15 in thousands of rubles)

	Number	Indicator	2010	2011	2012	The growth
						rate, (4/5)%
	(1)	(2)	(3)	(4)	(5)	(6)
	1	Revenues from sales	2 148 547	2 029 271	1 412 899	70%
	2	Prime cost	-2 069 294	-1 938 693	-1 573 157	81%
	3	Gain / loss from sales	79 253	90 578	-160 258	-277%
	4	Other income	38 353	678 756	117 412	17%
	5	Other expenses	-145 788	-496 684	-485 919	98%
	6	Profit before taxation	-28 182	272 650	-528 765	-194%
	7	Current income tax	0	0	0	0%
	8	Change in deferred tax liabilities	-181 428	-115 496	-305 165	264%
	9	Change in deferred tax assets	105 372	117 348	382 143	326%
	10	Other / miscellaneous	37 786	-1 711	-121	7%
	11	Net profit	-66 452	272 791	-451 908	-266%

  The clean variety will signal the high environmental quality (HEQ) it must hold that the pro…t under the signaling strategy is greater the HEQ is not correctly identi…ed. The individual rationality (IR) for the clean variety, RI HEQ , is thus: Hence, 2.38 is the only bounding condition among the RI.The incentive compatibility constraints ensures that the pro…t generated from truthtelling is higher than mimicking the other type. The IC HEQ is for either type is thus:

			(p; e; 1)			max p	(p; e; 0)	(2.36)
	(p c) (	p)			max p	(p; e; 0)	(2.37)
	Note that in the worst case when consumers fail to identify HEQ, the highest pro…t
	the clean variety can obtain solves: max p (p c) (	p) :The optimal pro…t at the
	level of pro…t-maximizing price is thus (	4	c) 2	.
	Rewriting the RI HEQ we obtain the condition:
	(p c) (	p)	(	4	c) 2	(2.38)
	Similarly, for the dirty type, RI LEQ is p (	p)	(	4	) 2	which is an identity.
	(p ; e; 1)			(p; e; 0)	(2.39)
	(	4	c) 2		(p c) (	p)	(2.40)
	This constraint is veri…ed if the RI HEQ holds.
	2.7.1 Derivation of signalling price Lastly, the compatibility constraint for the low environmental quality (LEQ) type,
	Case 19 1: signalling high environmental quality without regulation CI LEQ , is:
		(p ; e; 0)			(p; e; 1)	(2.41)
		(	4	) 2			p (	p)	(2.42)
	Note that p = max p p (		p) = ( ) 2

More precisely, throughout the article, we will use the term "tax avoidance"in the sense of "abusive tax avoidance transactions" (GAO

2011), that is, all the practices that don't contradict the law but diverge from its spirit. For instance, a company that uses arti…cial non-productive transactions through o¤shore entities with complex but legal pro…t-shifting techniques is involved in abusive avoidance, as it uses loopholes in law to optimize tax obligations in the ways not intended by the legislation.[START_REF] Slemrod | Cheating ourselves: The economics of tax evasion[END_REF] distinguishes tax avoidance from illegal tax evasion, although he highlights the di¢culty of distinguishing illegal from legal intent on the part of a taxpayer.

In[START_REF] Baron | Regulation of prices and pollution under incomplete information[END_REF], the foregone tax revenue combined with the information problem leads the environmental regulator to use an emission tax as a penalty aimed at diminishing the incentive to misreport information. In[START_REF] La¤ont | The new economics of regulation ten years after[END_REF]) and[START_REF] Lewis | Protecting the environment when costs and bene…ts are privately known[END_REF], …scal avoidance imposes on the environmental regulator through changes in the social cost of public funds. It turns out that …scal avoidance under asymmetric information modi…es the Pigouvian taxation because the need of an incentive correction hinges on the …scal transfer between the polluting …rm and consumers. The higher the cost of raising funds, the higher the information rent captured by the truthtelling …rm, the greater the deviation from the Pigouvian taxation.

The model in[START_REF] Kurtyka | The switching e¤ect of environmental taxation within bertrand di¤erentiated duopoly[END_REF] with heterogeneous preferences for the environment.results in the same distinction between the actual damage to the environment and consumer adaptation costs.

See also[START_REF] Mambondiyani | The pollution fallout from ZimbabweŠs blood diamonds[END_REF]) and (The Economist 2013).

The original Russian version can be consulted at: http://www.boges.ru/…les/…n_otchet/2012/ Otchet_2012_2.pdf, page 33.

"EU Market: EU carbon prices again recover from sub-e6.80 levels to post small weekly loss", https://carbon-pulse.com/category/eu-ets/.

"EU Emissions Trading System", http://carbonmarketwatch.org/category/eu-climate-policy/euets/.

The transition takes french vegetable farmers

to

years (Chambre d'Agriculture Rhône-Alpes 2015, p. 7).

High environmental quality, organic quality are used as synonyms.

I assume that quality is costly.

See, for example,[START_REF] Kurtyka | The switching e¤ect of environmental taxation within bertrand di¤erentiated duopoly[END_REF].

This assumption is necessary to ensure that there is a pollution externality. If consumers were to have a correct estimate of environmental damage, i.e.= , they would have internalized entirely pollution externality by their purchasing decisions. The demonstration is relegated to the appendix, see (2.7.2).

We adopted the structure of utility function from[START_REF] Daughety | Communicating quality: A uni…ed model of disclosure and signalling[END_REF].

Note that Assumption A.1 implies > c e : the cost increment from enhanced environmental quality is lower than the disutility from insu¢ cient environmental quality. This ensures that the environmental quality provision is socially e¢ cient.

We abstract from producer's technology choice.

c (e) accounts for all per unit expences the …rm incurs during production, that is an after-tax -when relevant -per unit cost to the producer for a given environmental quality.

In our model we adopt a similar cost structure as in[START_REF] Andrè | Strategic quality competition and the porter hypothesis[END_REF].

Price elasticity increases with price @ @p > 0 and under complete informaiton, the price increases with environmental quality @p @e > 0 . Hence, the direct and indirect e¤ects move in the same direction. This implies that price elasticity is increasing with environmental quality, that is, for the monopolist consumers become less captive.

Since the posterior belief at equilibirum is the function of e; i.e., the true environmental quality, we can replace ( ) with (e) :

A full scale analysis of pooling equilibrium is beyond the scope of this paper. Therefore the pooling equilibrium serves solely to give an intuition of our argument.

The graph represents a general case for arbitrary c for expositional reasons. However, according to Assumption (A.1), only the part for which c < is relevant to ensure socially e¢ cient supply of

In our model, incomplete information of a product's environmental quality concerns exclusively the product market, as the regulator can observe the …rm's type. An alternative model of incomplete information between the regulator and the producer would require a menu of contracts capable of revealing producers'true types, in the spirit of[START_REF] Freixas | Planning under incomplete information and the ratchet e¤ect[END_REF]. Although in practice, lumpsum transfers are met with strong political resistance, it would be of interest to examine the e¢ ciency of those contract menus in terms of their capacity to reveal information. However, it is beyond the scope of this

paper.17 It is the exact measure of welfare for in our model preferences are quasi-linear coving a large range of consumer goods plagued by incomplete information (wines, for example).

For technical reasons, we focus on pooling equilibria at p p ( e) which produces qualitatevely the same results.

.The signalling price of the HEQ must verify the system of bounding conditions RI HEQ

Acknowledgements

I would like to good-heartedly thank my thesis supervisor, Philippe Mahenc, for his generous advice, attentive guidance, all-around help, wit, endless patience, and inexhaustible con…dence! I could have never thought of a better thesis director. Without his expertise, sense of humor, enthusiastic attitude to work and authentic interest and curiosity in research, the years spent on the dissertation would have been less exciting.

I would also like to thank the members of the LAMETA laboratory and the University of Montpellier for all their help, support, and openness. In particular, a special thank for Sabine de Béchevel, Guillaume Cheikbossian, Thésy Pothet, Francesco Ricci, Isabelle Romestan, Julie Rozas, Mabel Tidball. I thank cordially Edmond Baranes and the LabEx Entreprendre for their grand generosity and support who enabled the participation in many national and international conferences.

I would also like to thank my peer doctoral students who have been always ready to share moments of hardship as well as instances joy. Especially, Bocar Samba Ba, Marc Dubois, Sylvain Hours, -they've been of great help to advance my knowledge of the French language and the French society.

I would like to thank my family and friends who are always hoping for the better. Maria Chistiakova, Nikolay Chistyakov, Ilya Chistyakov, Alexandre Shafranov, Nadezhda Shamrova, Anne Boquet, Soline Caillé, Oksana Dovzhenko, George Egorov, René Fouilleul, Xia Guang, Sara Harris, Pierre Peylhard, Marina Soloveva, thank you all for being such a support! I would like to thank my teachers who have always inspired my studies, curiosity and taught to have consistency and will: Vilia Gelbras, Ludmila Fadeeva, Grigory Korbukov, Natalia Miroshnikova, Nigora Nacimova, Irina Volkova and Chantal Delaplace.

Résumé du chapitre 2

Le chapitre 2 étudie l'impact de l'asymétrie d'information sur la taxe environnementale du second rang.

L'intérêt principal du chapitre est d'endogénéiser la taxe a…n d'évaluer son impact sur le bien-être social. Le cadre comprend un cas d'industrie en concurrence imparfaite et une qualité environnementale inobservable. Un monopole utilise le prix pour signaler que son produit propre (qualité haute) ne génère pas de pollution. Pour éviter qu'un producteur de type sale (qualité faible) ne vende un produit polluant au même prix que le produit propre, le monopole doit distordre le prix du produit propre relativement au niveau d'information complète.

On distingue deux formes de régulation lorsque l'équilibre sépare les deux types : (i) la régulation faible qui préserve l'avantage en coûts du monopole sale; (ii) la régulation forte qui donne l'avantage en coûts au monopole propre.

Le niveau de la taxe optimale doit être inférieur au niveau imposé en cas de monopole polluant avec information complète. La taxe comprend trois éléments correspondant aux externalités présentes sur le marché. En partant d'un niveau Pigouvien équivalent au dommage marginal, la taxe est ajustée selon le comportement du monopole qui restreint la production a…n d'obtenir une rente non-compétitive. De plus, ce niveau est réduit encore pour prendre en compte l'impact de l'asymétrie d'information. Puisque la taxe s'in…ltre dans le prix du produit propre, que ce soit un équilibre séparateur ou mélangeant, l'espérance des consommateurs d'une haute qualité environnementale augmente l'importance que le régulateur attribue au produit propre. Cela entraîne la baisse de sévérité de la taxe environnementale relativement à la situation represent both types' pro…t functions when correctly identi…ed by consumers. Dashed lines depict pro…ts when each type is taken for its counterpart. The thick horizontal lines correspond to incentive constrains for dirty and clean producers, the overlapping of which gives a set of signaling prices that allows e¤ective separation between the types.

As mentioned earlier, the pair of prices (p s ; p (e)) survives selection by the intuitive criterion. This speci…es the unique separating equilibrium.

On the lower panel, dot-dashed curves depict pooling pro…ts for each variety as an expected quality characterized by prior beliefs 0 :

Obviously, for an in…nite number of su¢ ciently large 0 ; the set p up will always defeat separating if applying Mailath's criterion. This is precisely the e¤ect identi…ed in [START_REF] Mahenc | Are green products over-priced?[END_REF] of overly optimistic consumers'expectations. For our purpose, notice that the change in c (which measures the di¤erence in e¤ective marginal production costs, i.e., c c ( e) c (e)) will have a similar e¤ect on the equilibria selection.

Converting the same concept into the (c; ; ) space makes it possible to map each (c; ) pair to an equilibrium selected by the undefeated criterion 15 . In Figure 2-2, plausible equilibria in terms of are projected on the (c; ) plane. Hence, on the horizontal axes c denotes the di¤erence in the e¤ective marginal production cost between the clean and dirty types. On the vertical axes denotes the gross product valuation. The colored upper left triangle corresponds to the region where Assumption A.1 is veri…ed, i.e., the manufacturing of either variety is desirable. Here, 0 is …xed at an arbitrary level and c is set as the variable. Intuitively, for a given , in the limit, for c ! 0 (with no di¤erence in marginal costs), the set of undefeated pooling prices is reduced to a unique point which yields the maximal monopoly pro…t for expected environmental quality. For c ! 1, the full information monopoly price allows for separation with no price distortion because the dirty type can no longer pro…tably imitate the clean variety. Indeed, for su¢ ciently large c; the gain from additional green consumers and the higher price margin doesn't environmental quality.

15 See Appendix 2.7.3 for its derivation.

to discourage the LEQ from mimicking the clean variety. Note that the signalling price of the clean variety is linked to the dirty variety's marginal cost.

Case 20 2: signalling high environmental quality under the environmental regulation Same as in case 1 except that in the interval

) taking into account the emission permit price as entering the marginal cost function of the dirty variety.

Proof of Corollary 12: the sign of @p s @c(e) :

Proof. Let c e < 0; then @p s @c(e) = 1 2 1 + p

> 0 for all parameter values. Let

: Suppose its sign is negative, then:

As both sides of inequality (2.49) are positive, taking square and rearranging gives:

(2.50) violates Assumptions (A.1).

Consumers entirely internalize the pollution externality

Let us brie ‡y examine the outcome of optimal environmental regulation when consumers entirely internalize the pollution externality. Suppose that consumers' disutility from pollution correspond exactly to the objective (scienti…c) level of damage from pollution, i.e. = . Hence, the regulator doesn't need to take into account the negative externality from pollution because consumers by their behavior have already accounted for it.

Otherwise it would lead to a double/redundant according of the environmental damage

Chapter 3

Taste for diversity, pollution aversion and optimal choice of environmental quality

In a stylized model of vertically di¤erentiated brown-green duopoly, two …rms compete in quantities. The baseline model derives the level of environmental quality determined solely by the green …rm. We argue that this equilibrium might be unstable in the long run because the …rms generate unequal pro…ts. The possibility to coordinate their action allows the brown …rm to bene…t from a greater di¤erentiation, and the green …rm to preserve its leadership position in quality choice. We thus adjust the model to allow for producers' organization, also with environmental awareness. For every case under where k denotes the competitor of k.

Hence the level of EQ a¤ects the demand in two ways. First, it measures consumers' willingness to pay for i th variety. Second, the green variety's choice of EQ determines the level of demand: the more di¤erentiation there is, the greater the demand is.

Since the dirty type doesn't abate pollution by assumption, e b is set to zero and e g = e. Replacing q i by fb; gg and q j by fg; bg respectively; where b and g represent quantities of brown and green products, we can obtain from (3.1) the inverse demand for the brown product:

we emphasize that e is the level of EQ that the green producer optimally sets. The inverse demand for the green variety is given by:

For the sake of simplicity we assume that marginal production costs of either green or brown are equal, and thus can be set to zero with no loss of generality. Assume further that the provision of high EQ requires a …xed investment, F (e). F (e) is continuous, twice di¤erentiable and increasing and convex in EQ: F (0) = 0; F 0 (e) > 0; F 00 (e) > 0 (A.1) Suppose that per unit cost is independent of environmental level.

Then, the pro…t the green …rm earns is: Maximizing pro…t with respect to green variety's output, g, that gives the …rst order

Multinational brands have been acutely susceptible to pressure from activists and from NGOs eager to challenge a company's labour, environmental or human rights record.

Even those businesses that do not specialize in highly visible branded goods are feeling the pressure, as campaigners develop techniques to target downstream customers and shareholders.

In response to such pressures, many businesses are abandoning their narrow Milton

Friedmanite shareholder theory of value in favour of a broader, stakeholder approach which not only seeks increased share value, but cares about how this increased value is to be attained.

Such a stakeholder approach takes into account the e¤ects of business activity -not just on shareholders, but on customers, employees, communities and other interested groups.

There are many visible manifestations of this shift. One has been the devotion of 1 "The rise and role of NGOs in sustainable development", http://www.iisd.org/business/ngo/roles.aspx 2 "A Transformative Partnership to Conserve Water: The Coca-Cola Company and WWF 2016 Annual Report", https://www.worldwildlife.org/publications/a-transformative-partnership-to-conserve-water-the-cocacola-company-and-wwf-2016-annual-report 3 Image credits. Left: "5 Reasons Why We're Outside Coca-Cola's HQ", 2017, https://www.greenpeace.org.uk/5-reasons-why-greenpeace-are-outside-coke-hq/. le cadre d'une concurrence imparfaite, les entreprises pro…teront donc des préférences des consommateurs pour un environnement propre en exerçant leur pouvoir de marché.

Cela rend la mise en oeuvre de la réglementation environnementale plus coûteuse et plus problématique que dans une situation de concurrence parfaite.

Tout au long de cette thèse, la réglementation environnementale est basé sur seul dispositif -la taxe optimale du second degré sur les émissions polluantes. En e¤et, lorsque l'on considère plusieurs défaillances de marché (la pollution, la concurrence imparfaite, etc.), le résultat général prévoit que la taxe optimale doit être …xée en dessous du niveau Pigouvien. Il s'avère que si l'ensemble des défaillances du marché est corrigé par un seul dispositif, cela a¤aiblit à la fois le degré de sévérité et l'e¢ cacité de la régulation environnementale. Il est clair que si l'on envisage un ensemble de dispositifs réglementaires pour confronter l'ensemble des défaillances du marché, il est, alors, possible d'atteindre un meilleur résultat en termes d'e¢ cacité de coût et de qualité environnementale. On peut imaginer une séparation entre les buts environnementaux et ceux de mise en conformité en tant qu'un tel ensemble de dispositifs règlementaires. Lorsque, par exemple, une agence ou une ONG environnementale s'occupe de la qualité environnementale, le …sc est seulement responsable de l'imposition et de l'exécution de la taxation. Cependant, cette approche poserait inévitablement le problème de la coordination entre les organismes.

Ceci suggère des perspectives possibles à cette thèse pour permettre une comparaison entre ces stratégies réglementaires et l'analyse de leurs limites et avantages.

Une autre perspective envisageable pourrait être une comparaison de l'e¢ cacité d'une taxe environnementale et d'un dispositif de commande et de contrôle, comme des normes de qualité minimale ou des certi…cations de qualité obligatoire.

Je crois que la question de l'évasion …scale environnementale en conjonction avec le pouvoir du marché ouvre tout un champ de recherche intéressant et d'actualité. La problématique serait alors de considérer le micro fondement du mécanisme d'évitement qui permettrait d'assurer la conformité à la régulation environnementale.

General Conclusion

The thesis is composed of three chapters that analyze the role of environmental quality viewed as an intrinsic feature of a product.

Environmental quality a¤ects consumers' preferences, producer's strategies and the stringency of environmental regulation. The analysis sheds light on di¤erent aspects of the social cost of pollution. Degrading natural environment pollution alters consumers' awareness of environmental quality forming certain aversion of environmentally dirty product, making consumers willing to spend a premium on environmentally superior quality. Producers view the environmental feature of a product as a means of product di¤erentiation allowing to extract more of consumers'surplus. Environmental regulation is to incite socially optimal level of pollution. Although I mostly consider a tax on polluting emissions as an incentive instrument, I also explore a possibility of an alternative decentralized regulation/intervention by semi-public entities such as an NGO. Hence, the dissertation is theoretical and it reunites the domains of public and environmental economics with industrial organization.

The …rst article treats …scal avoidance on the part of a polluting monopoly. The second endogenizes environmental taxation on emissions of a potentially polluting monopoly whose environmental engagement is unobservable to consumers. The third article allows competition between a clean and a polluting …rms. The choice of environmental quality is decentralized and determined in accordance with the interests of a producers'organization or/and an environmental ONG.

Generally, consumers'aversion to pollution, their preferences for environmentally superior product endogenize by itself a part of pollution damage. At the same time it gives producers a socially undesirable incentive to extract the 'green' rent from consumers.

Under imperfect competition, …rms will take advantage from consumers'preferences for clean environment and will exercise market power. This renders the implementation of environmental regulation more costly and more problematic than it appears with perfect Doctorat de l'Université de Montpellier Three problems in the market of a green product: …scal avoidance, signal and vertical di¤erentiation

Abstract: Environmental quality having features of a public good is the subject of the thesis. In a theoretical framework, we study a means of environmental regulation of a polluting …rm endowed with market power and then determine the optimal decentralized choice of environmental quality. We start with examining how tax avoidance a¤ects the optimal second-best tax on polluting emissions in a monopoly setting. The …rm is owned by shareholders who di¤er in their cost of tax dodging. The optimal tax should correct two negative externalities of avoidance: the …rm's free-riding e¤ect and a tax base erosion e¤ect. This free-riding makes the regulator either impotent or unfair, depending on the severity of the environmental damage and the …rm's e¢ ciency. Next, we analyze the impact of an environmental tax on the signaling price strategy of a monopoly that communicates to consumers the unobservable information about …rm's high environmental performance. We use the intuitive and undefeated criteria of equilibrium selection. Asymmetric information places the optimal second-best tax below the level required under complete information. In the case of undefeated equilibria selection, the tax may induce a 'migration' from separating equilibrium to pooling making the …rm prefer to conceal the private information about environmental quality. Finally, we show that market choice of environmental quality by a …rm that internalizes environmental damage from polluting emissions is yet suboptimal.

Discipline : Sciences Economiques (Section