Maxime Charreire
email: maxime.charreire@parisnanterre.fr.y

Eric Langlais
email: eric.langlais@parisnanterre.fr.

Economix

Should environment be a concern for competition policy when …rms face environmental liability ?

Keywords: Strict liability, negligence, damages apportionment rules, market share liability, environmental liability, Cournot oligopoly, competition policy K13, K32, L13, L49

In the recent period, more and more voices have called for unconventional competition policies as a way to achieve higher environmental investments from …rms. This paper shows that this objective may come into con ‡ict with those of environmental liability laws. We introduce a basic oligopoly model where …rms produce a joint and indivisible environmental harm as a by-product of their output.

We …rst analyze the e¤ects on the equilibrium of alternative designs in environmental liability law, secondly, we discuss the rationale for "non-conventional" competition policies, i.e. more concerned with public interest such as the preservation of environment (as well as human health and so on). We study …rms decisions of care and output under various liability regimes (strict liability vs negligence) associated with alternative damages apportionment rules (per capita vs market share rule), and in some cases with damages multipliers. We …nd that basing an environmental liability law on the combination of strict liability, the per capita rule, and an "optimal" damages multiplier, is consistent with a conservative competition policy, focused on consumers surplus, since, weakening …rms' market power also increases aggregate expenditures in environment preservation and social welfare. In contrast, a shift to the market share rule, or to a negligence regime, may be consistent with a restriction of competition, since …rms' entry may instead lead to a decrease in aggregate environmental expenditures and losses of social welfare. Nevertheless the …ne tuning of the policy requires speci…c information from a Competition Authority, which we discuss as well.

Introduction

In the very recent period, competition policy has been advocated as a mean to reach environment preservation. Before Ursula von der Leyen the new President of the European Commission, announced as a priority the issue of environment sustainability (European Green Deal), the European Parliament in February 2019 asked for an adaptation of competition policies to take into account issues of public interest such as environmental sustainability, corporate social responsibility and so on. 1 The argument is that "competition law may be an obstacle to competition restraints that would nonetheless promote welfare, by ensuring more sustainable consumption and production for instance [...] When production involves a negative externality, such as a harmful by-product of output (e.g. pollution) or the decentralized use of a common resource, then a restraint of trade among competitors and the ensuing drop in output will necessarily limit the negative externality (see Cosnita-Langlais (2020) page 1)". In this paper, we contribute to this debate, analyzing possible designs for environmental liability laws, considering how …rms'decisions are constrained by these alternative environmental liability laws, and …nally discussing the consequences for (non-conventional) competition policies.

We consider the realistic situation where …rms in an oligopoly produce a joint and indivisible environmental harm as the by-product of their output. Aside of expenditures in productive inputs and the precaution cost, the existence of environmental liability law compels …rms to bear supplementary costs that re ‡ect their liability burden. The characteristics of this liability cost for environmental harm re ‡ect the speci…c design of the law: this latter may be described as the combination of a liability regime (negligence vs strict liability) that requires or not a targeted precautionary activity to avoid any liability burden, a damages apportionment rule (per capita vs market share rule) that de…nes the way total damages for harm done to the environment are allocated among responsible …rms, and a damages multiplier (loosely speaking, punitive damages) that expands the e¤ective damages paid above the observed value of the environmental damages su¤ered. Here however, we consider a basic competition policy focused on consumers surplus and the expansion of aggregate output on the market thanks to …rms' entry on the market. 2 Our concern is twofold: On the one hand, do environmental liability laws give e¢ cient incentives to …rms for the protection of environment, or at least, is there a speci…c legal design that dominates the others according to the Social Welfare criterion ? On the other hand, to what extent are environmental liability laws constraining competition policy, justifying non-conventional, more lenient actions with the purpose of increasing …rms'expenditures in environment preservation ? The analytical framework we use to tackle these issues is based on a symmetric oligopoly à la Cournot, where …rms produce a homogeneous good for a constant marginal cost. The cost of precautionary measures is modeled as a …xed cost with respect to production, and liability cost as a function of the expected environmental harm speci…c to the design of the liability law. The expected environmental harm is related both to expenditures in precaution and production, and increases with the industry output at a more-than-proportional rate.

This "cumulative e¤ect" [START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] is relevant in contexts where environmental harms are the result of non linear e¤ects, i.e. responses more than proportional to the exposition to dangerous/toxic substances, and/or because harms can only be observed after lagtimes and long latency periods, once a serious environmental deterioration is achieved. 3Regarding our …rst research question, we show that environmental liability laws generally fail in achieving e¢ cient care expenditures. Firms follow suboptimal decision rules in the setting of precautionary measures as well as in the choice of output, and thus reach equilibrium levels that may be excessive as well as insu¢ cient with regards to socially optimal ones. The introduction of optimal damages multipliers under strict liability allows …rms to adopt e¢ cient rules in care activity, but the outcome at equilibrium is nevertheless that care as well as output are lower than what is required in a Pareto e¢ ciency. Under negligence (with a standard of care) …rms face e¢ cient incentives for care activities, but choose excessive levels of output and care. The main implication of this part of the paper is that environmental liability laws, alone, cannot attain the social optimum in a context where care and output decisions are interrelated.

With regards to our second research question, we analyze how the speci…c design of environmental liability laws is constraining the objective of competition policy. A law based on strict liability associated with the per capita rule, and an optimal damages multiplier, has an appealing feature for a Competition Authority. In implementing the law, Court actions do not impinge on its domain: a standard/conservative competition policy will succeed in increasing consumer surplus thanks to the expansion of aggregate output, and this will be accompanied by an increase in aggregare care expenditures. Moreover, such a policy is welfare improving, and should market structure be closer to perfect competition then the equilibrium would coincide with the social optimum. In contrast, we show that an environmental liability law based on the market share rule under strict liability, or negligence with a standard of care, does not verify this property. Such designs translate to …rms a structure of liability cost that develops anti-competitive e¤ects, as it increases with the number of …rms on the market. Thus, …rms' entry may lead to a cut in individual output and care decisions large enough to produce a decrease of the aggregate output and care levels, despite more …rms compete on the market. Nonetheless, Social Welfare under such laws is maximized for a …nite number of …rms; in contrast, as the market structure becomes closer to perfect competition, the aggregate care and output levels fall short of their optimal values. In all, strict liability with the per capita rule and optimal damages multipliers appears as an "ideal" environmental liability law, a kind of …rst best design, and is consistent with a standard competition policy dedicated to the improvement of competition on the market. This ideal policy mix requires no speci…c coordination between Courts and the Competition Authority. In turn, alternative designs for environmental liability law put stronger constraints on …rms as well as a Competition Authority, that may justify non-conventional competition policies, limiting …rms'entry, as a kind of second best solution. However, to reach the …ne tuning of such orientation, a Competition Authority needs speci…c information on Courts' behavior as well as care technologies. To sum up, the key point for a Competition Authority is the knowledge of the impact of liability law on …rms production costs, since this cost depends on the liability regime as well as the sharing rule of damages between …rms.

The former results are obtained under the implicit assumption that …rms make positive pro…ts at equilibrium. In this perspective we also show that under the market share rule …rms make a higher pro…t than under the per capita rule (for any given number of …rms, and the optimal damages multiplier). This means that regarding the issue of …rms'entry, both rules exert di¤erent private incentives on …rms, that may potentially translate into rule speci…c consequences at the long run equilibrium : the zero pro…t condition is met with more …rms on the market under the market share rule than under the per capita rule.

However, we show that in such a long run equilibium the aggregate output, the aggregate care expenditures as well as the level of Social Welfare obtained under the per capita rule are always larger than under the market share rule.

Section 2 reviews the literature. Section 3 introduces the model and solves the social optimum.

Section 4 analyzes the equilibrium of the industry under Cournot competition, when strict liability is associated with either the per capita or the market share rule. We study whether the combination of a damages multiplier and …rms'entry have the potential to recover the social optimum. Section 5 considers the implications of a shift from strict liability to the negligence rule. Section 6 a¤ords some robustness checks, and discuss the implications for competition policy. Section 7 concludes.

Literature review

The central issue of the paper being the design of environmental liability laws, it is worth starting with some institutional and legal considerations, and existing environmental laws. Comprehensive Environmental Response, Compensation, and Liability Act -CERCLA -in USA has been adopted by the Congress in 1980 (and amended by the Superfund Amendments and Reauthorization Act of 1986) as a tool allowing to clean up uncontrolled or abandoned hazardous-waste sites existing throughout the United States, as well as accidents, spills, and other emergency releases of pollutants and contaminants into the environment. All activities targeted by the Act are subject to strict liability, and in cases where the several. 4 But CERCLA is silent on issues in relation with damages apportionment. The Environmental Liability Directive of the European Union was passed for similar reasons, stating in its preamble that: "There are currently many contaminated sites in the Community, posing signi…cant health risks, and the loss of biodiversity has dramatically accelerated over the last decades. Failure to act could result in increased site contamination and greater loss of biodiversity in the future" (Environmental Liability Directive 2004/35/CE of the European Parliament and the Council, alinea (1) page 2). Alike CERCLA, the European Directive introduces a distinction between operations and activities that are subject to strict liability all being listed in its Annex III, 5 and those (not listed) that are subject to negligence. However, it does not make a de…nite choice regarding the rule governing damages sharing among multiple party causation; instead, it states that liability apportionment should be determined in accordance with national law. Indeed, statute law (both in common law countries and civil law countries) does not provide such provision for apportioning damages among multiple tortfeasors, but case law provides traditional solutions. Courts decisions are founded on the seriousness of each defendant's misconduct to establish how the damages compensating the victims will be shared among the di¤erent injurers. Accordingly, two polar rules have emerged and are speci…cally of interest here, e.g. the no liability rule and the per capita rule. 6 Less often, Courts use the solution called the market share rule, according to which total damages are shared between tortfeasors competing in the same industry, in proportion to their market share. It …rst appeared in 1980 in the Californian case "Sindell v. Abbott Laboratories", and US Courts have limited its use up to now to toxic torts (exposure to a chemical) such as in the abestos litigation (Becker v. Baron Bros, 649 A.2d 613 (N.J. 1994).) or the MTBE (a gasoline additive) litigation (In re Methyl Tertiary Butyl Ether, 175 F. Supp. 2d 593;S.D.N.Y. 2001). In France, two cases come to the mind where the market share has been applied: the Orly Airport litigation for noise pollution (1988, Cass. 2e civ, No 86-12.543), and more recently the Distilbène litigation 7 , a case close to Sindell's one. Some 4 According to CERCLA, the liable party is the owners or operators of sites where a hazardous substance has been released, as well as the generators and transporters of hazardous substances which have been released. In cases where the Environmental Protection Agency is forced to use the Superfund, punitive damages may be imposed up to three times the cleanup costs incurred from the owner or operator of the property or from the generator of the hazardous materials. When activating the Superfund, the Environmental Protection Agency designates a Potentially Responsibile Party (ideally, the deep pocket ) to implement or …nance the cleanup of a site on which hazardous materials have been found. Because liability is joint and several, PRP is sent scrambling to identify other PRPs to whom it can look for contribution (Smith 2012). Later on, the 1990 Oil Pollution Act has been prompted after the 1988 Exxon Valdez disaster (major oil spills in US waters).

5 Indeed, Annex III lists oprations and activities that are otherwise convered by a regulation of the European Parliament or directive of the European Council.

6 Basic causation requirements imply that the contribution of each defendant, among the pool of identi…ed tortfeasors, should be proportional to its contribution to the harm of the victims. Thus, if no fault is established for any defendant, then no one is liable (no liability rule); if all the defendants committed a fault with the same intensity, then the damages are equally shared between them (per capita rule). In France, for example, Article L. 162-18 of Code de l'environnement, states that: Lorsqu'un dommage à l'environnement a plusieurs causes, le coût des mesures de prévention ou de réparation est réparti par l'autorité visée au 2 de l'article L. 165-2 entre les exploitants, à concurrence de la participation de leur activité au dommage ou à la menace imminente de dommage.

7 See Tribunal de Grande Instance of Nanterre, April 10, 2014 n 12/12349 and n 12/13064. Both cases concern the diethylstilbestrol (DES), a product delivered to pregnant women and which caused years later injuries to the children exposed in utero.

scholars advocate for an extensive use of the market share rule to environmental liability or competition law [START_REF] Ferey | Pour une prise en compte des parts de marché dans la détermination de la contribution à la dette de réparation[END_REF]G'sell 2013, G'sell 2010).

As a paper focused on environmental liability, our work of course has connections with the vivid literature of the 90s focused on extended liability as a solution to injurer's insolvency and the judgmentproof problem (see for example [START_REF] Beard | Bankruptcy and care choice[END_REF][START_REF] Boyd | Noncompensatory Damages and Potential Insolvency[END_REF],1997[START_REF] Boyer | Environmental Risks and Bank Liability[END_REF][START_REF] Innes | Optimal liability with stochastic harms, judgement-proof injurers, and asymmetric information[END_REF][START_REF] Pitchford | How liable should a lender be ? The case of judgment-proof …rms and environmental risks[END_REF], Ringleb and Wiggins 1990[START_REF] Shavell | The judgment proof problem[END_REF]. [START_REF] Van 't Veld | Hazardous-industry restructuring to avoid liability for accident[END_REF] considers the e¤ects of liability on …rms'size, and discusses the welfare impacts following the restructuring of judgment-proof …rms. In the present set up with a symmetric oligopoly, the judgment proofness is of weakest interest since either all …rms are judgment-proof, or no …rm is. In contrast, our paper considers the interplay between liability and market mechanisms, taking into account the strategic interactions between …rms.

The market share rule, mainly used in personal injury cases, has motivated a debate among US scholars focusing its consistency with basic causation requirements, as well as its moral/ethical roots (see [START_REF] Dillbary | Apportioning liability behind a veil of uncertainty[END_REF][START_REF] Priest | Market Share Liability in Personal Injury and Public Nuisance Litigation: An Economic Analysis[END_REF]. The debate has known a revival in France with the 2014 Distilbène litigation. Several French scholars argue in favor of traditional solutions adopted for damages apportionment [START_REF] Molfessis | Du critère des parts de marché comme prétendu remède à l'incertitude sur l'origine d'un dommage[END_REF], Quézel-Ambrunaz 2010). Others have defended the market share rule on the grounds that market shares may be a proxy for the likelihood of individual liability at the stage of damages apportionment, in contexts of joint liability characterized by hard uncertainty and ambiguous causation [START_REF] Ferey | Pour une prise en compte des parts de marché dans la détermination de la contribution à la dette de réparation[END_REF]G'sell 2013, G'sell 2010), such as when the set of all potential o¤enders is identi…ed without any doubt, but it is impossible to establish the origin of the harmful product (evidence are missing, or destroyed). In contrast, our paper studies the case for joint liability from an ex ante perspective, a¤ording a comparative analysis of di¤erent apportionment rules (per capita vs market share rules) and their impact on the incentives to undertake precaution in richer strategic environments (oligopoly market). Moreover, we discuss di¤erent policy implications, including the choice of a liability regime (strict liability vs negligence).

The interactions between liability laws and other kinds of public interventions has motivated an important literature in Law & Economics, to begin with the mix between ex ante regulation and ex post liability in di¤erent informational contexts (Bhole and Wagner (2008), [START_REF] Friehe | On the political economy of public safety investments[END_REF], [START_REF] Innes | Enforcement costs, optimal sanctions, and the choice between ex-post liability and ex-ante regulation[END_REF], [START_REF] Kolstad | Ex post Liability form Harm vs. Ex ante Safety Regulation : Substitutes or Complements ?[END_REF], [START_REF] Schmitz | On the joint use of liability and safety regulation[END_REF], Shavell (1984a,b)). Studies focused on the mix between competition policy and liability law are scarce. [START_REF] Marino | Market Share Liability and Economic E¢ ciency[END_REF] studies product liability for joint but divisible harms, and analyses the e¤ects of …rms'entry in a oligopoly under strict liability and the market share rule. [START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] consider the case of product liability law for divisible but cumulative harms, and analyze …rms'entry under strict liability with a "modi…ed" market share rule vs negligence, while [START_REF] Friehe | Tacit collusion and liability rules[END_REF] addresses the issue of tacit collusion under liability laws.

Baumann, Cosnita-Langlais and Charreire (2020) study also the stability of cartels under liability laws in case of indivisible environmental harms, but do not introduce precautionary expenditures. Our paper considers instead the case for environmental liability law, and discuss the interplay between competition policy and di¤erent designs for environmental liability laws.

Our work is a contribution to the debate regarding the de…nition of new areas for competition policy, i.e. whether competition authorities should take into account public interests above competition objectives. Existing works focused on the issue of environmental protection assume that …rms'contributions to environmental protection are voluntary [START_REF] Hashimzade | Do Corporate Environmental Contributions Justify the Public Interest Defence[END_REF], [START_REF] Schinkel | Public Goods Provision by a Private Cartel[END_REF], [START_REF] Schinkel | Can Collusion Promote Sustainable Production[END_REF], [START_REF] Treuren | Can Collusion Promote Sustainable Consumption and Production ? Not Bene…cially Beyond Duopoly[END_REF]). In contrast, our analysis states that …rms' decisions are constrained by environmental liability, and suggests that this raises coordination issues between Courts (focused on the incentives to precautionary expenditures) and Competition Authorities (focused on consumers surplus), in order to improve environment protection measures.

Lastly, our paper is obviously connected to the vast literature on product liability law. Tracing back to the pioneering works by [START_REF] Polinsky | Strict Liability vs. Negligence in a Market Setting[END_REF], [START_REF] Polinsky | Products Liability, Consumer Misperceptions, and Market Power[END_REF], a recent stream of the Law & Economics literature has analyzed the performances of product liability in alternative competitive environments. 8 To sum up, this literature arrives at a quite optimistic conclusion regarding product liability in oligopolistic markets when consumers'expected harm is modeled as a linear function of …rms' individual output 9 : under any liability regime (strict liability, negligence, as well as no liability) …rms are induced to choose the …rst best optimal level of care. Hence, there would be "no role for the in ‡uence of market structure or strategic interaction on liability policy" [START_REF] Daughety | Economic Analysis of Products Liability: Theory[END_REF] in this case. This equivalence of liability rules, and the e¢ ciency result fall down when the expected harm to victims depends on the level of output in a more complex way (Marino 1988, Daughety and[START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF]. 10 Daughety and Reinganum (2014) show that for cumulative harms to consumers, strict liability associated with a "modi…ed" market share rule (see below) and an optimal damages multiplier, dominates the negligence rule (that degenerates into a no liability rule). In contrast our paper shows that for a joint, indivisible and cumulative environmental harm, strict liability with the per capita rule and an optimal damages multiplier, dominates strict liability with the market share rule, as well as the negligence rule. 8 To have a focus on oligopoly markets: quantity competition [START_REF] Baumann | Optimal damage multipliers in oligopolistic markets[END_REF][START_REF] Daughety | Economic Analysis of Products Liability: Theory[END_REF][START_REF] Friehe | Tacit collusion and liability rules[END_REF][START_REF] Leshem | Allocation of Liability: on the E¢ ciency of Composite Sharing Rules[END_REF], or price competition (Cournot with product di¤erentiation: Baumann, Friehe and Rasch 2018; spatial competition: Baumann, Friehe and Rasch 2016, Chen and Hua 2017).

9 [START_REF] Polinsky | Products Liability, Consumer Misperceptions, and Market Power[END_REF] assume that the harm conditional to an accident is constant, and that the probability of accident is proportional to care. [START_REF] Marino | Market Share Liability and Economic E¢ ciency[END_REF][START_REF] Marino | Product Liability and Scale E¤ects in a Long-Run Competitive Equilibrium[END_REF] and [START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] reach similar conclusions when the probability of harm is linear in the output.

1 0 [START_REF] Marino | Product Liability and Scale E¤ects in a Long-Run Competitive Equilibrium[END_REF] considers "scale e¤ects" on the probability of accident, when it depends on the level of output, and shows that comparative performances of the liability regimes depends on the one hand on the nature of these scale e¤ects (the probability of harm may increase or decreases with the output), and on the other, on the intensity of competition (pure competition vs Cournot oligopoly). [START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] deal with a case of a cumulative harm, e.g. when the expected harm is proportional to the square of the output; they …nd that for both a monopoly and an oligopoly (assuming that …rms' speci…c risks are independent) strict liability maintains e¢ cient incentives to take precaution, in contrast to no liability and negligence; however, strict liability induces an underprovision both of care and output at equilibrium as a result of the distortions due to the imperfect competition. Daughety and Reinganum show that the superiority of strict liability holds when …rms speci…c risks are interdependent, to the extent that strict liability is associated with the use of an optimal damages multiplier in order to maintain e¢ cient incentives to take precaution.

3 The framework

Assumptions

We consider a symmetric oligopoly à la Cournot where n > 2 …rms compete for a homogeneous product. 11 Both consumers and …rms are risk neutral. The quantity of goods produced by …rm i is denoted q i (i = 1; :::; n), and Q = n P i=1 q i represents the industry output. The market demand is P (Q) = a bQ, (a > 0; b > 0). We assume that a …rm i (8 = 1; :::; n) operates at a cost given by C(q i ; x i ) = x i , where x i represents the level of care of …rm i; this means that the marginal cost corresponding to the use of all productive inputs (aside from care expenditures) is constant and null, and that care expenditures do not depend on …rm's activity level. Finally following [START_REF] Tietenberg | Indivisible toxic torts: The economics of joint and several liability[END_REF], the product may cause a joint and indivisible harm to society (third party victims, not consuming the good), such that the expected harm is de…ned as H(Q; X) = h(X):Q 2 , where h(X) such that h(0) > 0 and h(X) < 1 for any X 0, is the joint probability of harm and X = n P i=1

x i is the aggregate care expenditure.

Finally, most of the central results of the paper are obtained thanks to a set of very simple assumptions, namely:

H1a: b > 2:h(0); H1b: h 0 (X) < 0 < h 00 (X) for any X 0; H1c: H(Q; X) is convex in (Q; X).
H1a is a very basic assumption regarding two parameters of the paper -requiring that the price sensibility of market demand is not too large compared to the baseline probability of accident. H1b is usual in the literature on liability rules, saying that the probability function is decreasing and convex with care expenditures. H1c has sound economic motivations, since it establishes that the expected external harm to society is supposed to be a convex function, which is a very natural and reasonable assumption for a cost function. Remark that H1a and H1b together imply that the next condition is satis…ed:

For any X > 0 and

2 : b > (h(0) h(X)) > h 0 (X): X (C1)
Several basic static comparative results require (C1) to hold. H1c in turn implies:

For any (Q; X) > 0 :

h h(X)h 00 (X) 2 (h 0 (X)) 2 i > 0 (C2)
Second order conditions are satis…ed (most of the time) under (C2), as well as the stability of Nash equilibria (see below). When necessary, we will introduce additional restrictions in order to qualify our results.

Remark 1: Care expenditures are captured here as a …xed cost, independent of the units of output produced -care is durable, with the terminology of [START_REF] Nussim | A revised model of unilateral accidents[END_REF]. This makes sense when such expenditures correspond to the acquisition of safety technologies which are very speci…c assets adapted to a …rm's business, and as such represent inputs that are essential for the …rm's activity but that cannot be relocated to other businesses. This encompasses a large set of safety devices: it can be (smoke) alarms and detectors (for the leakage of toxic substances), a videosurveillance device, or the installation of a su¢ cient number of circuit-brakers. In the same vein, it can be also the investment in a containment system (to prevent the leakage of dangerous products outside of the plant, and limit the release in the environment), the capacity of which does not re ‡ect the normal activity on the plant on a day-to-day basis, but an abnormal event, such as the biggest accident at the plant. Alternatively, care has a durable nature when it corresponds to R&D investments dedicated to the design of "green products" and entails a substitution between polluting goods and products with lower environmental impacts. In contrast, precaution is said to be non-durable when it is related to the usual activity at the plant and the quantity supplied. It can be establishing check-lists and adopting regular check-up procedures for equipment, together with their regular cleaning to avoid and detect potential failures and repair in time; as well as the observance of resting periods for the employees in order to limit their fatigue and the occurrence of human failures. These kinds of safety measures imply expenditures that are roughly speaking proportional to the number of working hours for employees as well as the intensity with which equipments are used, and thus proportional to the …nal output of the plant. Care can also be understood as having a non-durable nature when we consider various activities dedicated to the collect of end-of-life products, and/or e¤orts focused on recycling industrial wastes and components of end of life products. 12 In a companion paper (Charreire and Langlais 2020), we compare Joint Liability vs Joint and Several

Liability for a non-durable care, under strict liability; we show that the main results of the current paper (propositions 1 to 5) still hold.

Remark 2: Indivisible harms are usually associated either with situations where several tortfeasors act in concert to produce the harm, or when the independant but successive acts of several tortfeasors have conducted to produce a single result/harm [START_REF] Kle¤ner | Successive Torts Resulting in a Single, Indivisible Injury: Plainti¤s, Prepare to Prove the Impossible[END_REF]. In both cases, the point is that it is by no mean possible to disentangle the in ‡uence of each single …rms and thus, it is impossible to assess neither individual harms, nor individual probabilities. For practicles purposes, a indisivisible harm may be the result for example of the simultaneous actions by …rms the activities of which are concentrated on the same geographical area. We can think of noise pollution (in the suburban neighborhood of Orly airport), or water pollution (rivers pollution with acids from tannery activities and dyeing plants in Indian cities; or Great Lakes pollution by mercury from the automobile industry in North America). However, indivisibility do not require the contemporaneous actions of …rms to exist. Whenever pollutions are di¤use, the harm becomes observable only after a latency periods or lag time that are quite long, such that it may be impossible to trace back the origines of the pollution and identity of the polluters, as for soil pollutions (in former mining areas in Eastern France). In famous litigation cases (see "Sindell v.

Abbott Laboratories", or the MTBE litigation in USA; in France, the Distilbène litigation ), the existence of indivisibilities has been motivated de facto because it was impossible several years after the accidental event for Courts to disentangle the individual responsibilities of each tortfeasor (evidence are missing, or destroyed), at least too costly to reach this goal (given the number of injurers and the complexity of the phenomenon) and separate individual responsibilities at a reasonable economic and social cost. By the same token, the accumulation of toxic substances produced by di¤erent polluters in a given area may also entail non linear reactions above some thresholds (du to the physical or chemical property of the polluant, at any horizon), that would not manifest should the polluter be unique. For example, in the case of the Orly Airport litigation for noise pollution (1988, Cass. 2e civ, No 86-12.543), the individual contribution of each planes and/or company cannot be easily assessed, at least at a reasonable economic or social cost -neither instantaneously, nor across periods.

Our paper is focused on indivisible harms, in contrast with the literature that deals with liability rules and market interactions pioneered by [START_REF] Polinsky | Strict Liability vs. Negligence in a Market Setting[END_REF]. Indivisibility has usually been captured in analysis of multitortfeasors cases (see [START_REF] Kornhauser | Sharing Damages among Multiple Tortfeasors[END_REF][START_REF] Tietenberg | Indivisible toxic torts: The economics of joint and several liability[END_REF] through a total expected harm function denoted as h(x 1 ; :::; x n ) D, where h(x 1 ; :::; x n ) is the overall probability of accident and D is the exogenous harm conditional to the accident; a basic property is that individual care expenditures for each tortfeasor decreases the overall probability of accident/harm ( @h @xi < 0 for any i). Our paper extends this set up taking into account care decisions together with output decisions, under symmetric Cournot competition. The functional we introduce here has two advantages: i) it maintains the separation between the probability of harm h(X), and the harm realized post-accident D = Q 2 , and ii) it enables to capture …rms symmetry with a parcimonious assumption -indeed, h(X) avoids the recourse to unavoidable technical restrictions on derivatives 13 that would be necessary in order to reach …rms's symmetry under the alternative function h(x 1 ; :::; x n ) -thus saving on notations without loss of generality as far as symmetry is a concern. As we explain above, keeping with the assumption of …rms' symmetry is useful here for the purpose of comparing our results with those issued by other papers.

The benchmark : Social Welfare maximization

Social Welfare is the sum of consumers'total utility minus the sum of …rms'operating costs (cost of care), minus the expected harm:

SW = Q R 0 P (z)dz n P i=1 c(x i ) H (Q; X).
We will directly use …rms'symmetry in order to reduce the dimension of the optimization problem, allowing the benevolent planner to focus on a symmetric optimum where q 1 = ::: = q n = q and x 1 = ::: = x n = x, with the aggregate output and function of (q; x):

SW = anq b 2 (nq) 2 n:x h(nx) (nq) 2 (1)
The …rst-order conditions for an interior solution (q sw ; x sw ) are written:

a bnq = 2h(nx)nq (2) h 0 (nx): (nq) 2 = 1 (3)
meaning that the optimal output level (see condition ( 2)) must be pushed up to the point where average market proceeds (inverse of market demand, LHS) are equal to the marginal cost associated with expected harm (RHS); similarly (condition (3)), optimal care expenditures are such that the marginal cost of care (RHS) is equal to the social marginal bene…ts associated with the decrease in the expected harm (LHS).

Second order conditions are veri…ed by convexity of the expected harm function (see Appendix 1); they also imply that output and care are strategic complements, and that the Nash equilibrium satisfying

(2)-( 3) is unique and stable.

The convexity of the expected harm function also implies that (q sw ; x sw ) decreases with the number of …rms (obvious, the proof is omitted). In contrast, the aggregate output and care at the social optimum do not depend on the number of …rms: substituting with Q = nq and X = nx, it is easy to see that

(2)-(3) give Q sw = a b+2h(X sw ) and h 0 (X sw ) (Q sw ) 2 = 1 which do not depend on n. As a consequence, the optimal expected harm h 0 (X sw ) (Q sw ) 2 and Social Welfare at optimum SW (Q sw ; X sw ) do not depend on n. Hence, an increase in n has also no e¤ect on Social Welfare (as long as the cost of care is the unique source of …xed costs for …rms, which is consistent with our interpretation that care is durable).

Oligopoly equilibrium under strict liability

We assume in this section that the liability regime is based on strict liability according to which Courts do not rely on a negligence test, and only need to establish causation to prove the responsibility of …rms.

We discuss the case for negligence in section 5. In our context of joint and indivisible harms to the environment, the legal doctrine that allows the Plainti¤ to sue several defendants in a single trial is Joint Liability. Given Joint Liability, the main issue when Defendants are strictly liable is the way total damages will be apportioned between them.

Remark 3: Other legal doctrines of large application exist, for cases where multiparties are involved in the harm a victim is su¤ering from [START_REF] Kornhauser | Sharing Damages among Multiple Tortfeasors[END_REF]. One is Several Liability, according to which the Plainti¤ is entitled with the right to sue each one of the …rms in separate trials. Several Liability is not relevant here because in cases with indivisibility, establishing facts on an individual basis and separate causations for each defendant is not possible -or will leave the Plainti¤ with the impossibility to obtain the payment of damages and be compensated for the harms borne. An alternative doctrine is Joint and Several Liability which makes each defendant responsible in solidum, and provides the Plainti¤ with the opportunity to obtain the payment of total damages from any one of the defendants. We discuss the case with Joint and Several Liability in a companion paper.

Remark 4: We do not provide speci…c details regarding procedural rules, nor with settlement issues (existence of pre-trial negotiations), and generally speaking we ignore the existence of litigation costs, as well as strategic aspects of settlements and litigations and the associated impacts on litigation costs.

Similarly, we are not speci…c with Plainti¤ type/identity. It may be a (public) environmental agency -in this case, the trial will be heard by an administrative jurisdiction/Court; or it may be a private association involved in the protection of environment, or generally speaking a third-party victim (the Plainti¤ is an individual who has neither economic nor contractual relationships with the Defendants)hence, the case will be settled in front of a civil Court. Finally, we assume here that in the implementation of the law, Courts choose the way total damages are shared between multidefendants. This corresponds to reality for some national jurisdictions (see the Distilbène litigation in France), whereas for other jurisdictions or under alternative doctrines, the sharing of total damages is a decision of the Plainti¤ (see the implementation of CERCLA by Environmental Agencies in USA) who may strategically consider the solvability of the di¤erent defendants to secure the recovering of damages. Indeed, this is neutral here since we assume complete information, we rule out the insolvability problem, and do not consider strategic aspects of litigations. The rationale is that the paper is rather focused on liability costs and the way some speci…c designs of liability rules may shape this liability cost, with the ensuing consequences for …rms.

Joint liability and damages sharing

Let us denote L JL i (q i ; x i ) = s i H(X; Q) the amount of compensation accruing to …rm i; s i is …rm's i liability share, and > 0 is a damages multiplier, exogenously set to 4.2, that Courts may use to in ‡ate total damages. We will investigate the e¤ects of two di¤erent damages sharing rules.

The per capita rule. Courts may divide total expected damages equally between all …rms pertaining to the industry, i.e. s i = 1 n , 8i = 1; :::; n. (C2) guarantees the convexity of the individual liability cost function L pc i (q i ; x i ) = n H(Q; X) with respect to (q i ; x i ). In this case, each …rm i(8i = 1; :::; n) chooses a level of output and a level of care in order to maximize its pro…t:

pc (q i ; x i ) = (a bQ)q i x i n h(X)Q 2 (4)
Using the …rst order conditions (see Appendix 1; second order conditions are met under (C2)), the symmetric Cournot-Nash equilibrium where q 1 = ::: = q n = q pc and x 1 = ::: = x n = x pc solves the system:

a b(1 + n)q = 2 h(nx)q (5) h 0 (nx): nq 2 = 1 (6)
meaning that the output level (see condition ( 5)) must be pushed up to the point where marginal market proceeds are equal to the marginal cost of liability (RHS); similarly (condition ( 6)), care expenditures are such that the marginal cost of care (RHS) is equal to their marginal bene…ts, de…ned as the decrease in the expected liability cost (LHS). Remark more speci…cally that the LHS in (5) exhibits the standard distortion due to imperfect competition according to which the marginal market proceeds are smaller than the market demand price (RHS in ( 5)). In Appendix 1, we verify that output and care levels de…ned by ( 5)-( 6) are strategic complements, and that the associated Nash equilibrium is stable and unique (under (C2)).

The market share rule. Courts may assess individual contributions according to individual market shares, i.e. s i = qi Q , 8i = 1; :::; n. Let us assume that the individual liability cost function

L ms i (q i ; x i ) = q i Q H(Q; X) is convex in (q i ; x i ), which implies that 2q i Q:h(X):h 00 (X) (Q + q i ) 2 (h 0 (X)) 2 > 0.
This requirement is stronger than (C2), but as shown in Appendix 1, it allows that second order conditions for pro…t maximization under the market share rule are satis…ed. Firm i chooses now a level of output and care that maximize the pro…t:

ms (q i ; x i ) = (a bQ)q i x i h(X)q i Q (7)
Once more, using the …rst order conditions and focusing on the symmetric Cournot-Nash equilibrium, the solution q 1 = ::: = q n = q ms and x 1 = ::: = x n = x ms solves the system:

a b(1 + n)q = (1 + n) h(nx)q (8) h 0 (nx): nq 2 = 1 (9)
with a meaning equivalent to ( 5)-( 6). The LHS in (8) also exhibits the standard distortion due to imperfect competition according to which the marginal market proceeds are smaller than the market demand price.

In Appendix 1, we verify that output and care levels de…ned by ( 8)-( 9) are strategic complements, and that the associated Nash equilibrium is stable and unique (under (C2)).

Before going into the analysis of the performances of both rules and the comparison with the social optimum, remark how the di¤erent sharing rules shape the relationship between output decisions and the individual cost of liability, and the way it translates to the marginal cost of production. More precisely, it turns out that the marginal cost of production/liability under the market share is proportional to the number of Defendants (…rms). Throughout the paper, this di¤erence with the per capita rule will drive the results. Remark also that despite the damages multiplier is exogenously set, we assume until the next section that it cannot include values that are too large (see below). This quali…cation will be relaxed later on, when we will discuss the case for optimal damages multipliers. We have the following result:

Proposition 1 Assume 1 < n. i) Strict liability under the per capita rule yields a level of output and a level of care larger than under the market share rule (x pc > x ms and q pc > q ms ). ii) Under strict liability with the per capita rule, the equilibrium output and care levels may be larger as well as smaller than their socially optimal levels. iii) Under strict liability with the market share rule, the equilibrium output and care levels may be larger as well as smaller than their socially optimal levels if

2n 1+n ; if > 2n
1+n , the equilibrium output and care levels are smaller than their socially optimal levels.

Proof: Let us denote q sw (x) the output level that solves (2), for any given value of care; while x sw (q)

is the care level that solves (3), for any given value of the output. Similarly, let us denote q pc (x) the output level that solves (5), for any given value of care; while x pc (q) is the care level that solves (6), for any given value of the output; and let q ms (x) be the output level that solves (8), for any given value of care; while x ms (q) is the care level that solves (9), for any given value of the output. i) According to conditions (6) and ( 9): 8q > 0; x pc (q) = x ms (q). In contrast, according to ( 5) and ( 8):

2qh(nx) < (1 + n)qh(nx)
implying that: 8x > 0; q pc (x) > q ms (x). Hence: x pc > x ms and q pc > q ms since care and output are strategic complements under both rules.

ii) and iii) From ( 3)-( 6)-( 9), it comes that: h 0 (nx): nq 2 < h 0 (nx): (nq) 2 ; hence: 8q > 0; x sw (q) >

x pc (q) = x ms (q). In turn, (2)-( 5)-( 8) show there are two opposite e¤ects on output. On the one hand (LHS): a (1 + n)bq < a nbq; this reduces the incentives to produce under strict liability at any level of care, compared to the social optimum. On the other hand (RHS): if 1 2n 1+n , then 2nh(nx)q > (1 + n) h(nx)q > 2 h(nx)q; this increases now the incentives to produce at any level of care, compared with the social optimum. Thus, q sw (x) is not generally comparable with q pc (x) or q ms (x).

But if > 2n

1+n , then 2nh(nx)q < (1 + n) h(nx)q and thus q pc (x) > q ms (x), implying that x sw > x ms and q sw > q ms . Hence the result.

For both damages apportionment rules, …rms have the same best response in terms of care to any feasible activity level. However, …rms'private marginal bene…ts associated with care under strict liability are smaller than their socially optimal value, implying that …rms have ine¢ cient incentives to take care at any output level under strict liability.

In contrast, for any given feasible level of care, …rms face a marginal cost for liability under the per capita rule that is smaller than under the market share rule (for the same marginal market proceeds); thus at any level of care, …rms'best response function in terms of output is larger under the per capita than under the market share rule. The intuition is simple to get: under both liability sharing rules, …rms obtain obviously the same market share at equilibrium, 1=n. However, output decisions are driven by the ex ante individual liability cost which is rule-speci…c. Under the market share rule, the cost of liability for a …rm is related to its individual market power, the larger its market share, the larger its liability burden. Thus, aside from the standard negative competition externality, the market share rule adds a liability cost externality that drives down the output decisions of …rms at any level of care. As a result, the market share rule provides …rms with more incentives to reduce output and care than the per capita rule.

This said, strict liability (whatever the damage sharing rule) leads to an ine¢ cient rule for output choice since the furniture of the output is driven by two opposite e¤ects. On the one hand, …rms' marginal market proceeds under strict liability (for both rules) fall short of their socially optimal value: this is nothing else but the standard output distortion of output that re ‡ects …rms'market power under imperfect competition. On the other hand, the marginal cost of liability accruing to a …rm under the per capita rule falls short of its optimal value; this is also true under the market share rule if the damages multiplier is small enough ( 2n 1+n ). As a result, strict liability (whatever the damage sharing rule) also exerts ine¢ cient incentives to produce at any level of care, but this is the result of two countervailing in ‡uences and the net e¤ect is ambiguous: …rms may produce either too much or not enough at any care level. Hence the comparison between the equilibrium values for output and cares under strict liability (for any damages sharing rule) and their optimal values is also generally ambiguous. 14 When the damages multiplier trespasses a threshold ( > 2n 1+n ), the marginal cost of liability under the market share rule increases above its social value, thus shifting the equilibrium below the social optimum.

In the next section, we will discuss the combination of a optimal damages multipliers (used by Courts) and …rms'entry (forstered by a Competition Authority). As a matter of comparison, we brie ‡y review here what is the impact of …rms' entry on the equilibrium under strict liability for exogenous values of the damages multiplier.

Proposition 2 (…rms' entry under strict liability with a exogenous damages multiplier) i) Under both rules, …rms' entry yields a decrease in individual output and care levels. ii) Under the per 1 4 It can be understood as the result of the comparison between the slope of two curves: the slope of the marginal market proceeds (mainly driven by the value of b), and the slope of the marginal cost of liability; the proof is omitted. capita rule, the aggregate output increases with the number of …rms, while the aggregate care decreases with the number of …rms. iii) Under the market share rule, the aggregate output increase (decrease) with the number of …rms when b is large (small) enough, while the aggregate care always decreases with the number of …rms.

Proof. See Appendix 2. i) We show that under the capita rule, the result holds for n 1;

although the market share rule does not require such a restriction. ii) In contrast to the social optimum, it is obvious that the equilibrium levels of aggregate output and care under both rules depend on the number of …rms. Using ( 5)-( 6), the aggregate output and care levels under the per capita rule satisfy:

a b 1 + 1 n Q = 2 n h(X)Q (5bis) h 0 (X): n Q 2 = 1 (6bis)
It is easy to see that …rms'entry entails opposite e¤ects on aggregate output and care expenditures.

The LHS in (5bis) is increasing with n, while the RHS is decreasing with n: hence the larger n, the larger the aggregate output Q pc (X) at any level of X. In turn, the LHS in (6bis) is decreasing with n, while the RHS does not depend on n: hence the larger n, the smaller the aggregate care X pc (Q) at any level of Q. However, the conditions b > 2h(0) and 1 < n 1 are su¢ cient to sign the e¤ect at equilibrium.

iii) Similar e¤ects arise under the market share rule. Using (8)-( 9), the aggregate output and care levels satisfy now:

a b 1 + 1 n Q = 1 + 1 n h(X)Q (8bis) h 0 (X): n Q 2 = 1 (9bis)
The LHS in (8bis) is increasing with n, while the RHS is decreasing with n: hence the larger n, the larger the aggregate output Q ms (X) at any level of X. In turn, since (9bis) is identical to (6bis), the larger n, the smaller the aggregate care X ms (Q) at any level of Q. Once again, the net e¤ect at equilibrium is ambiguous. Explicit comparative statics show that H1a is not enough, and a su¢ cient condition for the aggregate output to increase (decrease) is that b be large (small) enough with respect to a new theshold value de…ned in Appendix 2, larger than 2h(0).

Proposition 2 establishes that following …rms'entry in oligopoly, the e¤ect on output and care at …rm level under both rules is the usual/intuitive one. Less market power yields less production at …rms level, and given the strategic complementarity between care and output, this also implies less individual care.

The impact on the aggregate care level, identical under both rules, is particularly interesting. The intuition of the result is related to the situation where …rms share a joint harm they have produced, and the public good characteristics of care activity: as the number of …rms increases, the equilibrium individual share (1=n) in total harm also decreases under both rules, hence diminishing the expected bene…ts of care more than what would be justi…ed by the decrease in the output level. As a result, …rms' entry has such a large a negative impact on individual care levels, that the aggregate care level also decreases: the additional number of …rms that invest in care does not compensate the decrease in the individual care (there are more …rms, but poorly investing in care).

In contrast, the impact on the aggregate output is rule-speci…c. Under the per capita rule, we obtain the usual e¤ect according to which the aggregate supply expands following …rms'entry (the individual output decreases, but this e¤ect is compensated by more …rms producing the output). Inspection of the RHS (5bis) shows that the feedback in ‡uence of care on output decision is neglictable as n becomes large under the per capita rule. Thus the e¤ect on the LHS is dominating. In contrast, the e¤ect is ambiguous with the market share rule: inspection of the RHS in (8bis) suggests that the explanation is in the importance of the feedback in ‡uence of care on output (the RHS is proportional to 1 + 1 n h(X) and thus tends to 1 as n becomes large) compared to the impact on marginal market proceeds (the LHS is also proportional to 1 + 1 n :b), the net e¤ect depending on the price sensibility of the market demand (see Appendix 2 for an explicit proof). Depending on the relative size of the marginal cost of production and marginal market proceeds, the decrease in the individual output may be so large that it results in a decrease of the output at the industry level -the entry of new …rms does not compensate the large cut in individual production levels.

Obviously, these distortions in care and output decisions imply a loss of welfare. Speci…cally as regards with the discussion below, the di¤erentiated responses of aggregate output and care suggest that there may exist a trade-o¤ between consumers surplus and environmental harm. Let us assess how Social Welfare evolves as the number of …rms increases, starting with the per capita rule. Evaluating (1) at (Q pc ; X pc ) and di¤erentiating in n, we have (using (5bis)-(6bis)):

dSW dn (Q pc ; X pc ) = [b 2h(X pc )(n )] : Q pc n : dQ pc dn (n ) (Q pc ) 2 n h 0 (X pc ): dX pc dn
where h 0 (X pc ): dX pc dn > 0. Still considering a damages multiplier small enough (1 n), it comes that the impact of …rms' entry on Social Welfare is generally ambiguous. When the slope of the (inverse) market demand is small enough (b < 2h(X pc )(n )), indeed Social Welfare decreases -the intuition is that aggregate output is too large, and aggregate care expenditures are too low, implying losses of welfare related to an excessive expected environmental harm that is not compensated by consumers surplus. However, as the damage multiplier increases (such that b > 2h(X pc )(n )), there is a balance between the welfare gains provided by aggregate output expansion and the losses that result from the increase in expected harm (decrease in aggregate care expenditures).

Turning to the market rule now, evaluating (1) at (Q ms ; X ms ) and di¤erentiating in n yields (using (8bis)-( 9bis)):

dSW dn (Q ms ; X ms ) = h b 2h(X ms ) n 2 (1 + n) i : Q ms n : dQ ms dn (n ) (Q ms ) 2 n h 0 (X ms ):
dX ms dn with once more h 0 (X ms ): dX ms dn > 0. The same comments are relevant here, with one noticeable di¤erence. Although the impact on Social Welfare resulting from the decrease in aggregate care expenditures is still conditioned by the sign of n , the impact of the increase in the aggregate output is now depending on the sign of 2n 1+n

. As seen before (in the proof of proposition 1), when > 2n 1+n the marginal cost of liability is above its socially optimal value; thus the equilibrium corresponds to output and care levels that are below their optimal values. Hence in this situation, increasing n provides welfare gains thanks to the increase in aggregate output, despite the decrease in aggregate expenditures -the intuition is that in this situation the market equilibrium is associated with output distortions that large, that an increase in the number of …rms will provide a gain for consumers, whatever the price sensibility of market demand, which will do more than compensate the loss associated with environment deprivation. This e¤ect disappears when < 2n 1+n , and we obtain e¤ects on Social Welfare similar to the per capita rule. These considerations suggest that there may exist a …nite number of …rms that maximizes Social Welfare, at least as long as the damage multiplier is low enough, under strict liability. However, in any case, the distortion in care decisions is an issue. We now discuss the potential for improving …rms care decisions, and the consequences that arise.

Optimal damages multipliers

We have found for both rules that …rms choose an ine¢ cient rule of care (this is not an e¢ cient response at any level of output). Thus, …rms'incentives in care activities may be corrected, with the means of a speci…c value for the damages multiplier > 1, at the disposal of Courts. We focus here on this optimal value of this damages multiplier.

Proposition 3 (impact of an optimal damages multiplier) i) For both rules, the optimal damages multiplier is = n. ii) In a regime of strict liability (the damages sharing rule being either the per capita or the market share rule) with an optimal damage multiplier, the industry provides insu¢ cient levels of output and care, compared with the social optimum (q sw > q pc > q ms and x sw > x pc > x ms ).

Proof: i) Comparing the LHS in ( 6) or ( 9) and (3), it comes that = n ) h 0 (nx): nq 2 = h 0 (nx): (nq) 2 , in which case x sw (q) = x pc (q) = x ms (q), 8q > 0. ii) According to the RHS of ( 2) and

(5), when = n ) 2 h(nx)q = 2nh(nx)q implying that q sw (x) > q pc (x), 8x > 0. Given i), we obtain that: q sw > q pc and x sw > x pc . In turn considering the RHS in ( 8) and ( 2), = n ) (1 + n) h(nx)q > 2nh(nx)q, implying that q sw (x) > q ms (x), 8x > 0; as a result given i): q sw > q ms and x sw > x ms .

Part i) of Proposition 3 suggests that an optimal damages multiplier may be quite easy to assess for Courts, since it corresponds to industry size = n. Part ii) illustrates that when the per capita rule is combined with a damages multiplier optimally chosen, …rms face a marginal cost of liability equal to its social value. Hence an optimal damages multiplier solves the issue of care incentives (…rms face e¢ cient incentives to take care at any level of output), and also part of the distortion on production. Of course, this is not su¢ cient to reach optimal levels of care and output at equilibrium, since the distortion coming from imperfect competition still holds. A single instrument such as a damages multiplier is obviously not enough to solve two distortions; nevertheless, under the capita rule, the optimal damages multiplier allows to improve both the incentives to take care and the incentives to produce. This property does not hold under the market share rule. Under the market share rule, the optimal damages multiplier will exert perverse incentives on output choice now, since it puts an excessive liability cost on …rms. This aggravates the problem of output underprovision created by the distortion due to imperfect competition. 15 The issue we consider now is whether fostering …rms'entry will allow that the equilibrium be closer to the social optimum given the constraint of the environmental liability law considered here. We proceed in two stages. First, we focus on a situation where as the Competition Authority forsters …rms' entry, …rms still make a positive pro…t (short run equilibrium). In this case, we study the (potential) convergence of the equilibrium towards the optimum as the number of …rms increases, and compare the equilibrium pro…ts and social welfare between both rules. Second, we consider the situation were the zero pro…t condition is met, and compare the properties of the long run equilibrium and the social welfare level obtained under both rules.

As it will appear, we may need a very simple and intuitive assumption in order to qualify some e¤ects:

H2: h 0 (nx): (nq) 2 is decreasing with n.
H2 means that the marginal bene…t of care must be decreasing with the number of …rms; it implies that the next condition holds in this case:

1 5 One may argue that under the market share rule, the multiplier could be designed according to output distortions (equation ( 8)), i.e. in order to correct the marginal liability cost associated with the output decision. It can be shown that such a multiplier has the same e¤ects as those described in proposition 5 below. The results are available on request. In the text, we consider the usual de…nition of the optimal damages multiplier, relying on care incentives.

For any x; n : 2h 0 (nx) + nxh 00 (nx) > 0

(C3)

Let us …rst consider equilibria where …rms make a positive pro…t. The next proposition collects the results for the per capita rule.

Proposition 4 Assume environmental liability law relies on the combination (strict liability, the per capita rule, an optimal damages multiplier); then: i) The individual output decreases with the number of …rms; individual care decreases with the number of …rms under (C3). ii) The aggregate output and care levels increase with the number of …rms. iii) As n ! 1, the equilibrium industry (Q pc 1 ; X pc 1 ) converges to the social optimum (Q sw ; X sw ). iv) fostering …rms' entry is always welfare improving.

Proof. i) See Appendix 2. Regarding the ambiguous impact on care, it is shown that (C3) is a su¢ cient condition for x pc to decrease with n; thus, should the inequality (C3) not hold, it may be that

x pc still decreases with n. ii) To illustrate, let us write (5bis)-(6bis) substituting with = n; it comes:

a b 1 + 1 n Q = 2h(X)Q (5bis, ) h 0 (X)Q 2 = 1 (6bis, )
Condition (6bis, ) does not depend on n. In turn, (5bis) shows that an increase in n shifts upward the marginal market proceeds (LHS). Thus, an increase in n yields an increase in Q pc and X pc . iii) As n ! 1, the LHS in (5bis, ) is equal to the demand price; thus as n ! 1, the equilibrium industry (Q pc 1 ; X pc 1 )

converges to the social optimum (Q sw ; X sw ). iv) Evaluating ( 1) at (Q pc ; X pc ) and di¤erentiating in n, we have (using (5bis, )-(6bis, ) now):

dSW dn (Q pc ; X pc ) = b n :Q pc : dQ pc dn > 0 since dQ pc dn > 0.
The optimal damages multiplier correct …rms incentives regarding care decisions; as the public good e¤ects is removed, …rms are bound to follow e¢ cient rule of care at any level of output, on the pathway that leads to the social optimum. Part i) and ii) of Proposition 4 illustrates a nice property of an optimal damages multiplier under the per capita rule. Speci…cally, the aggregate output and the aggregate care both increase now as more …rms enter the market. Part iii) of Proposition 4 shows that under the per capita rule, fostering …rms'entry on the market increases Social Welfare, and when competition between …rms turns out to be perfect, the …rst best optimum is recovered. In this perspective the per capita rule appears as a quite ‡exible damages apportioning arrangement: once the incentives to invest in care and the incentives to produce are aligned with the socially optimal ones, there is a clear-cut separation between

Courts'action and Competition Authorities policies. The decisions of the former do not impinge on the domain of the latter, and as long as Courts commit to use optimal damages multipliers, Competition

Authorities have room to pursue their traditional objectives moving the equilibrium closer to the social optimum.

In turn, the next proposition collects the results for the market share rule:

Proposition 5 Assume environmental liability law is designed according to the combination (strict liability, the market share rule, an optimal damages multiplier); then: i) The individual output decreases with the number of …rms; (C3) is su¢ cient for that individual care decreases with the number of …rms. ii)

If b is large (small) enough, then the aggregate output and care levels increase (decrease) with the number of …rms. iii) As n ! 1, the industry vanishes, i.e. (Q ms 1 = 0; X ms 1 = 0). iv) If dQ ms dn > 0, fostering …rms' entry is Social Welfare improving. If dQ ms dn < 0, fostering …rms' entry reduces Social Welfare. v) Social Welfare is maximized for a …nite number of …rms, n ms = q b h(X ms ) .

Proof. i) See Appendix 2. Once more we …nd that (C3) is a su¢ cient condition for x pc to decrease with n. ii) To illustrate, let us substitute = n in conditions ( 8bis)-(9bis):

a b 1 + 1 n Q = (1 + n)h(X)Q (8bis, ) h 0 (X):Q 2 = 1 (9bis, )
Condition (9bis, ) is similar to (6bis, ). According to (8bis, ), the marginal cost of liability increases with n, which drives downward the output; to the opposite, the marginal market proceeds also increases with n, driving upward the output. Thus, Q ms (X) may increase or decrease at any level of X, depending on whether the impact of n on the market proceeds is larger or smaller than on the cost of liability. In Appendix 2, we show that b > n 2 h(X) implies dQ ms dn > 0 and dX ms dn > 0; to the converse, b < n 2 h(X) implies dQ ms dn < 0 and dX ms dn < 0. iii) According to the RHS in (8bis), the marginal cost of liability goes to in…nity as n ! 1. Thus the aggregate output and care levels become smaller and smaller with n (Q ms 1 ! 0 and X ms 1 ! 0). iv) Evaluating ( 1) at (Q ms ; X ms ) and di¤erentiating in n (using (8bis, )-(9bis, )), we obtain:

dSW dn (Q ms ; X ms ) = b n + (n 1)h(X ms ) :Q ms : dQ ms dn
The result iv) is straightforward. v) In Appendix 2, we show that Social Welfare is maximized neither with perfect competition (n ! 1) nor with a monopoly (n = 1); there exists a …nite number of …rms n ms > 1, for which SW (Q ms ; X ms ) is maximized, satisfying the condition:

dSW dn (Q ms ; X ms ) = 0 ) dQ ms dn = 0 (10)
Given that b n + (n 1)h(X ms ) > 0, it must be that dQ ms dn = 0 , b n 2 h(X) = 0; solving yields:

n ms = q b h(X ms ) .
Proposition 5 (Part i) and ii)) shows that if at …rms level, the behavior of the individual output and care levels under the market share rule are qualitatively very similar to those obtained for the per capita rule, at the industry/aggregate level, the implied adjustments following …rms' entry are very di¤erent (rule-speci…c). In short, optimal damages multipliers do not remove the public good e¤ect for sure.

Proposition ii) highlights that under the market share rule, the net e¤ect at the aggregate level is driven in a complex way by the relative size of the slopes of market demand and the marginal cost of liability.

When n increases (see (8bis, )), both the aggregate marginal proceeds (LHS) and aggregate marginal cost of liability (RHS) increase. If the price-elasticity of market demand is low (b is large) enough, the …rst e¤ect (LHS) dominates, entailing an increase in the aggregate output and care levels -the rationale is that despite the increase in the individual marginal liability cost at a …rm level, the cut in the individual output level is of limited scale, such that the aggregate output increases with …rms'entry. In contrast as the price-elasticity of market demand becomes low (b is large) enough, the second e¤ect (RHS) is large compared to the …rst one -the associated cut in the output level at a …rm level is that large that it is not be compensated at the aggregate level by the entry of new …rms; as a result, there is a contraction in the market supply, associated with a decrease in aggregate care expenditures. On the other hand, Proposition 5 Parts iii) and i) illustrate that the implications for Social Welfare analysis are also very contrasted between both rules; part iii) may be also understood as follows. Starting with a given number of …rms, and fostering …rms'entry may allow in a …rst stage to improve both the output and care levelthis is the "traditional" e¤ect: as n increases, both individual output and care expenditures decrease at …rm level, but given that more …rms operate on the market, the aggregate output and care expenditures both increase with n. However, as n becomes great enough, fostering …rms' entry a little further may have countervailing e¤ects, in the sense that this will induce a decrease of individual output and care expenditures at …rm level that large, that the aggregate output and Social Welfare also decrease. This explains Part v): restricting …rms'entry is socially worth in this case.

More generally, the implication of proposition 5 is that the market share rule appears as less ‡exible, in the sense that Courts'action impinges on the domain of Competition policy. The use of the market share rule yields an additional distortion in output decisions at …rms'level (the higher the market share, the higher the (marginal) cost of liability), and thus it is not granted that fostering …rms' entry under the market share rule be socially welfare improving. Moreover recovering perfect competition under the market share would entail a severe contraction in the aggregate output supply and aggregate care expenditures, and Social Welfare would be reduced compared to the oligopoly.

Note once more that this analysis relies on the implicit assumption that …rms always make a positive pro…t at equilibrium. In this perspective it appears that …rms'interest is opposite to society' s interest, as we show now :

Proposition 6 (short run equilibrium, …rms' entry and damages sharing) Assume Courts use an optimal damages multiplier. For any n exogenously given, …rms' equilibrium pro…t under the market share rule is larger than under the per capita rule; in contrast, social welfare under the market share rule is smaller than under the per capita rule.

Proof: For more details, see Appendix 3. The comparison of pro…ts and social welfare level under both rules is easier using a parametrization of the damages sharing rule, such that the individual liability cost to any …rm is de…ned as:

L k i (q i ; x i ) = k 1 n Q 2 + (1 k)q i Q ,
i.e. a mix of the per capita and market share rules, with respective proportions k; 1 k. The symmetric Cournot-Nash equilibrium is thus parametrized by k 2 [0; 1], such that the equilibrium levels of the aggregate output and care expenditures, Q(k); X(k), are two increasing

functions of k, with Q(0) = Q ms ; X(0) = X ms and Q(1) = Q pc ; X(1) = X pc .
As a result, the equilibrium pro…t may be written as (with

= n) : k (Q(k); X(k)) = (a bQ(k)) Q(k) n X(k) n h(X(k)) (Q(k))
2 , meaning that it depends on k only through Q(k); X(k), with:

d k dk (Q(k); X(k)) = 1 n n qk Q(k) 3 b n h 00 (X(k)) + (1 + k)h(X(k)):h 00 (X(k)) 2(h 0 (X(k)) 2 )
where > 0 by second conditions, qk > 0 (see in Appendix 3), and using (C2) :

(1 + k)h(X):h 00 (X) 2(h 0 (X) 2 ) > 0 for any k 0. Thus d dk (Q(k); X(k))
< 0 : increasing k (moving from the market share rule k = 0, to the per capita rule k = 1) yields a decrease in …rms' individual pro…t for any given n (i.e. (Q ms ; X ms ) > (Q pc ; X pc )).

In turn, evaluating (1) at Q(k); X(k), and di¤erentiating in k gives (after substituting with the …rst order conditions):

dSW dk (Q(k); X(k)) = b n + (n 1)(1 k)h(X(k)) Q(k) dQ(k) dk
with dQ(k) dk > 0. Hence, we obtain dSW dk (Q(k); X(k)) > 0 for any k 0 : increasing k (moving from the market share rule: k = 0, to the per capita rule: k = 1) increases social welfare for any given n (i.e. SW (Q ms ; X ms ) < SW (Q pc ; X pc )).

The …nding in proposition 6 calls for the analysis of the long run equilibrium reached under each rule. 16 As more …rms enter the market, the equilibrium pro…t decreases, and it becomes harder for a Competition Authority to ignore …rms'private incentives to enter the market. The rational is that as the pro…t is higher under the market share rule all else equal (for any n given), then the zero pro…t condition will be met under the per capita rule for a number of …rms smaller than under the market share rule. Thus in the long run, more …rms are in the industry under the market share rule; hence it could happen that this latter is associated with higher levels of aggregate output and care expenditures and …nally, a higher level of social welfare, than the per capita rule. Indeed, the next proposition shows that this is not the case.

Proposition 7 (long run equilibrium, …rms' entry, and damages sharing) Assume Courts use an optimal damages multiplier, and that …rms make zero pro…t; then, the levels of aggregate output, aggregate care expenditures, and social welfare reached under the market share rule are larger than under the per capita rule.

Proof: For more details, see Appendix 3. Using

L k i (q i ; x i ) = k 1 n Q 2 + (1 k)q i Q, we …nd that the associated symmetric long run equilibrium Q(k); X(k); n(k), is such that dQ(k) dk > 0; dX(k) dk > 0 and dn(k) dk < 0 (hence Q(0) = Q ms < Q(1) = Q pc ; X(0) = X ms < X(1) = X pc but n(0) = n ms > n(1) = n pc ). The zero pro…t condition gives (with = n(k)) : k (Q(k); X(k); n(k)) = 0 , (a bQ(k))Q(k) X(k) = n(k):h(X(k)) (Q(k)) 2
Now, substituting in (1), we obtain the next expression for social welfare :

SW (Q(k); X(k); n(k)) = b 2 + (n(k) 1):h(X(k)) (Q(k)) 2
Di¤erentiating in k, it comes that :

dSW dk (Q(k); X(k); n(k)) = k qk ( ) h(X(k))(Q(k)) 5 :h 00 (X(k)) b (n(k)) 2 + (1 k) n(k) 1 n(k) h(X(k))
where < 0 by second conditions, k qk > 0 (see in Appendix 3). Thus dSW dk (Q(k); X(k); n(k)) > 0 for any k 0 : with the zero pro…t condition, increasing k (moving from the market share rule: k = 0, to the per capita rule: k = 1) leads to a higher social welfare level (i.e. SW (Q ms ; X ms ; n ms ) < SW (Q pc ; X pc ; n pc )).

1 6 We are thankfull to the referee for having suggested a detailed analysis of this issue.

Proposition 7 implies that the advantages (in terms of aggregate output, care expenditures, and social welfare level) of the per capita rule over the market share rule still survive in a long run equilibrium (zero pro…t), despite this latter allows that more …rms enter the market compared with the former.

At that point, two issues are worth of consideration before tuning to the analysis of environmental liability regimes based on a negligence test.

Remark 5 (No liability regime): Let us consider the impact of a no liability regime. In this case, each …rm i chooses a level of output and a level of care in order to maximize its pro…t: nl (q i ; x i ) = P (Q):q i x i . Thus, …rms do not invest in care x nl = 0, and we obtain the standard symmetric Cournot-Nash equilibrium where …rms choose a level of output equal to q nl = a (1+n)b , associated with an aggregate output Q nl = na (1+n)b . It is straightforward that compared with the equilibrium with strict liability (whatever the damages rule), the no liability regime yields the highest level of output q nl > q pc > q ms and the lowest level of care expenditures x nl = 0 < x pc < x ms . However, the comparison with the social optimum yields an ambiguous result; it can be veri…ed that b > (<)2nh(x sw ) ) q sw > (<)q nl . Moreover, we obtain dSW dn (Q nl ; 0)) = b n 2h(0) :Q nl : dQ nl dn where dQ nl dn > 0. On the one hand:

lim n!1 Q nl = a b and lim n!1 dQ nl dn = 0, implying that lim n!1 dSW dn (Q nl ; 0)) ! 0 ; on the other hand, Q nl (n = 1) = a 2b
and dQ nl dn n=1

= a 4b , implying that dSW dn (Q nl ; 0)) n=1 > 0 under H1a. As a result the number of …rms that maximizes Social Welfare under no liability is n nl = b 2h(0) .

Remark 6: Our analysis of the market share rule (propositions 2, 5) reaches conclusions that are close to Marino (1989) although with a very di¤erent framework. Marino uses the standard "proportional harm model" where the expected harm is divisible (the equivalent with our notations being

H = ( P n i=1 h(x i )q i ))
, and …nds that under the market share rule (without an optimal damages multiplier) …rms undertake ine¢ cient decisions of care (both in terms of response to any output level, and in terms of equilibrium level); moreover, he shows that the equilibrium care level decreases with …rms' entry, and that Social Welfare is maximized for a …nite number of …rms. In contrast, recent papers a¤ord formal arguments according to which the market share rule allows to reach an e¢ cient goal. The papers by [START_REF] Dehez | How to share joint liability: a cooperative game approach[END_REF] and [START_REF] Ferey | Multiple causation, apportionment and the Shapley value[END_REF] rely on a cooperative game-theoretic frameworks to analyze apportionment rules that satisfy both e¢ ciency and fairness criterions; however, they consider the case for a exogenous harm, and this way eliminate the issue of care incentives and the interdependence with endogenous decisions of production, as well as the backward in ‡uence of market interactions. [START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] 17 discuss a case with correlated and cumulative individual harms (the equivalent with our notations is H = ( P n i=1 h(x i )q i ) 2 ); the resulting aggregate expected harm being divisible, this enable them to introduce a "modi…ed market share" rule whose equivalent with our notations here is h(x i )q i P n i=1 h(x i )q i . Daughety and Reinganum show under this modi…ed market share rule and an optimal damages multiplier, that the oligopoly equilibrium has several properties that are close to those described for the per capita rule in our framework (see proposition 4), with the noticeable exception that the equilibrium level of individual care in their set up increases with the number of …rms (whereas, it is decreasing in our set up). Remark that this modi…ed market share rule is not relevant for our case since the environmental harms is indivisible, 18 and going back to the "simple" market share rule qi Q makes sense in this context. However, we …nd instead that the use of the basic per capita rule for apportioning indivisible environmental damages between …rms has nice properties for care incentives and production decisions, since it introduces no distortion in competition above those due to strategic market interactions. In contrast, the market share rule will exert perverse incentives on …rms regarding their choice of output, since the liability cost increases with their market power, thus aggravating the issue of output underprovision due to imperfect competition. Interestingly enough this result mirrors an old debate about the comparison between the per capita rule and the market share rule in personal injury case (see [START_REF] Kornhauser | Sharing Damages among Multiple Tortfeasors[END_REF], although we reach the exact opposite conclusion, as far as imperfect quantity competition is concerned.

Oligopoly equilibrium under negligence

CERCLA as well as The Environmental Liability Directive of the European Union are focused on speci…c polluting or dangerous activities and/or operations, thus being explicitly recognized as subject to strict liability. By default, those not listed are subject to negligence. Having provided the analysis of strict liability, we now investigate here the case with the negligence regime.

Assume Courts rely on a negligence test considering a ‡exible standard of care: 19 a …rm will be considered as not negligent to the extent that its care expenditures proove to be an e¢ cient behavioral response, in the sense of …rm's best care expenditure it may have chosen considering any relevant foreseeable instances, that Courts consider as being adapted to the situation -in which case the …rm will again avoid any liability cost (i.e. s i = 0). It is natural to consider in that case that Courts promote a standard of care x = x i (q i ; q i ; x i ) de…ned as:

1 8 Remark also to have a complete picture that in the situation investigated by Marino (1989), the "modi…ed market share" rule indeed provides …rms with e¢ cient incentives in care decision, without requiring any damage multiplier to achieve this result. The proof is obvious: with joint but divisible and non cumulative harms, the individual liability cost borne by each …rm under strict liability with this modi…ed market share is equal to individual harm:

L i (q i ; x i ) = h(x i )q i P n i=1 h(x i )q i
P n i=1 h(x i )q i = h(x i )q i Thus, each …rm faces the social cost it imposes to the society, and the e¢ cient goal is attained. 1 9 A classical distinctions in the Law & Economics literature [START_REF] Kaplow | Rules versus standards: An economic analysis[END_REF][START_REF] Sullivan | The justices of rules and standards[END_REF] is made between a rule and a standard of care. When Courts use a rule of care, a …rm will be considered as not negligent once its provides at least a predetermined …xed level of care (x, i.e. the due care level). A natural candidate for such a due care level, usually considered in the literature, is the socially optimal level of care x = x sw (see Charreire and Langlais 2020).

h 0 (x i + x i ):(q i + q i ) 2 = 1 (11) for any positive (q i ; q i ; x i ). The rationale for Courts is that an individual …rm will not be seen as liable as long as its contribution to the safety have been designed in order to minimize the expected cost of the accident, given its own output and its competitors decisions (including output and care), i.e.

x i (q i ; q i ; x i ) = min xi (h(x i + x i ):(q i + q i ) 2 + x i ) whatever (q i ; q i ; x i ).

Let us show that in this regime of negligence, there exists a Nash equilibrium where all …rms comply with the standard of care.

Assume that one …rm is negligent, while its n 1 competitors do comply with the standard required.

The pro…t of the non compliant …rm is: ^ neg (q i ; x i ) = (a bQ)q i x i Q 2 h(X), since it bears the full external cost. As a result, it faces the liability cost of strict liability with the per capita rule when = n, implying that the …rm chooses a level of care that satisfy (11) -this amounts to say that the deviant …rm has inconsistent belief and should have not considered to be liable because Courts shouldn't have conclude for its liability.

As a result, the equilibrium now is such that any …rm abides the standard of care x i (q i ; q i ; x i ), and the individual output is maximizing the pro…t:

^ nl (q i ; x i (q i ; q i ; x i )) = (a bQ)q i x i (q i ; q i ; x i ) (12) under x i (q i ; q i ; x i ) de…ned by the constraint (11). The …rst and second order conditions are given in Appendix 4, such that the symmetric Cournot-Nash equilibrium where q 1 = ::: = q n = q neg and x 1 = ::: = x n = x neg solves the system:

a b(1 + n)q = 2 (h 0 (nx)) 2 h 00 (nx) ! nq (13) h 0 (nx): (nq) 2 = 1 (3)
The RHS in condition (13) shows that the marginal cost of liability accruing to each …rm has now a more complex expression compared with (2); it still depends on the expected harm, although in a more elaborate way (captured through the two …rst derivatives of h(X)). This implies that negligence associated with a standard of care outperforms strict liability as we show now:

Proposition 8 (negligence with a standard of care vs strict liability, and social optimum) i)

Negligence with a standard of care leads to a level of output and a level of care larger than under strict liability associated with an optimal multiplier = n (either with the per capita or the market share:

q neg > q pc > q ms and x neg > x pc > x ms ). ii) If b is large (small) enough, then negligence with a standard of care yields equilibrium levels of output and care smaller (respectively, larger) than their optimal values (b large enough ) q neg < q sw and x neg < x sw ; b small enough ) q neg > q sw and x neg > x sw ).

Proof. i) Under (C2) we have h(nx) > 2 (h 0 (nx)) 2 h 00 (nx) and thus h(nx) > (h 0 (nx)) 2 h 00 (nx) ; as a result (comparing the RHS in ( 5), ( 8) when

= n with ( 13)): q ms (x) < q pc (x) < q neg (x) 8x. On the other hand,

x ms (q) = x pc (q) = x neg (q) 8q if = n. Hence: q ms < q pc < q neg and x ms < x pc < x neg . q neg (x) 8x.

ii) See also Appendix 4. By construction, x sw (q) = x neg (q) for any feasible q > 0. From the comparison of the LSH in ( 2) and ( 13), it comes that a (1 + n)bq < a nbq; in contrast, the comparison of the RHS shows that: 2nqh(nx) > 2nq (h 0 (nx)) 2 h 00 (nx) . Hence the comparison of q sw (x) and q neg (x) at any x > 0 is ambiguous. So it is at equilibrium. In Appendix 5, it is shown that:

i) if b > 2n h(nx) (h 0 (nx)) 2 h 00 (nx)
, it comes q sw (x) > q neg (x) for any feasible x > 0; thus we obtain x sw > x neg and q sw > q neg ; but ii) if

b < 2n h(nx) (h 0 (nx)) 2 h 00 (nx)
, then q sw (x) < q neg (x) for any feasible x > 0; and thus we have x sw < x neg and q sw < q neg . Proposition 8 Part i) re ‡ects that the marginal cost of liability under negligence with a standard of care is smaller than under strict liability associated with the per capita rule, at any level of care. The intuition is that the incentives constraint (11) a¤ord …rms with a strategic advantage: anticipating its care activity as being an e¢ cient response to the output decision, this materializes through a smaller (marginal) cost of liability, at any level of output. As a consequence, the equilibrium output and care levels under negligence are larger than under strict liability (whatever the damages rule).

Part ii) shows that under negligence with a standard of care, …rms face once more two opposite incentives regarding the furniture of the output. On the one hand, …rms'marginal market bene…ts under negligence with a standard of care are smaller than their socially optimal value (this is the standard distortion due to imperfect competition) -this reduces the incentives to produce compared to the social optimum, at any level of care. On the other hand, the marginal cost of liability accruing to …rms under negligence with a standard of care is lower than its socially optimal value (according to the convexity of the expected harm function) -this increases now the incentives to produce under strict liability, compared to the optimum, at any level of care. The net e¤ect at equilibrium is ambiguous, and depends on the relative size of the slopes of the two marginal market proceeds (mainly, the value of b), and the slopes of the two marginal cost of liability (convexity of the expected harm function).

We show now that the equilibrium attained in this regime is ine¢ cient from a social point of view, with either too much or in contrast not enough of both output and care.

Regarding the impact of …rms'entry, the consequences are described in the next proposition.

and solving yields

n neg = b 2(h(X neg ) V neg ) = b 2 h 00 (X neg ) h(X neg ):h 00 (X neg ) (h 0 (X neg )) 2 .
Proposition 9 Part i) re ‡ects that negligence with a standard of care a¤ords …rms a strategic advantage with regard to their output and care decisions, and this leaves them with a degree of freedom that materialize through the adaptation to market conditions and changes in their market power. On the one hand, depending on the price sensibility of market demand, …rms may choose to increase or decrease their output following …rms' entry; on the other hand, depending on whether the marginal bene…ts of care activities decrease or increase, …rms may decrease or increase their investments in care. Although the public good e¤ect is removed by the incentives constraint (11), this liability regime introduces a direct link between equilibrium care expenditures at …rms level, and the variation of the marginal bene…ts of care with n. Part ii) illustrates that once the public good e¤ect is neutralized thanks to the incentives constraint (11), …rms' entry yields the normal, intuitive e¤ect on the aggregate output and agggregate care. Part iii) may be understood as follows: as n becomes great enough, the marginal market proceeds increase, while the average marginal liability cost for the industry (RHS in (13bis) is constant: thus the aggregate output and aggregate care expenditures reach level above their optimal values. Part iv) and v) follow: an excessive aggregate output may entail loss of welfare; this restricting …rms'entry may be socially worth.

6 Environmental law and competition policy: tentative of interpretation

Our results illustrate that when environmental harms are characterized by invisibility, strict liability fails to induce an e¢ cient rule of care from …rms, unless when Courts consider optimal damages multipliers.

When Courts do so, then the speci…c sharing rule they use for apportioning total harms is not innocuous:

if the per capita rule does not distort …rms output decision, in contrast, the market share rule adds to the standard competitive externality (that drives the marginal market proceeds downwards) because it makes …rms bearing an excessive marginal cost of liability, which aggravates the underprovision of output.

In turn negligence associated with a standard of care proves to be an e¤ective rule in obtaining that …rms abide with an e¢ cient rule of care, but provides …rms with a strategic advantage, they use in setting their output and care according to market conditions above or below the socially optimal ones. In all, any environmental liability law alone fails in reaching the social optimum: strict liability has a tendency to reach an equilibrium with too little care and output, while negligence may provide both outcomes at equilibrium, i.e. too much or not enough of both care and output. Thus, there is room for other kinds of public interventions. The point, we now turn to, is to discuss the constraints environmental laws are putting on a Competition Authority, and when competition policy may be oriented with respect to environmental goals.

Without going into details (see above), let us simply remind here that the results of the paper re ‡ect nothing but the way the speci…c design of the environmental law shapes the liability cost borne by a …rm, and the way this individual liability cost will vary with the number of …rms (because of damages sharing rules, and/or because of return to scales in care activity). The debate in the public arena, although not explicitly articulated, seems to focus on protective measures at …rms/individual level. The paper shows it is worth distinguishing between behaviors at …rms/individual level, behaviors at industry/aggregate level, and …nally the general move towards the social optimum.

In the di¤erent scenarios we have analyzed, the individual output and individual care levels are both decreasing with the number of …rms on the market, these results being obtained under very simple assumptions (H1a, b and c). The additional assumption H2 is important to consider only when optimal damages multipliers are introduced; moreover, relaxing H2 in this case does not necessarily revert the comparative statics for the individual care level. 20 Given the strategic complementarity between output and care in our model, this is hardly a surprise: both individual output and care levels decrease with …rms' entry. Does it support the view according to which less competition (the preservation of their market power) allows insider …rms at individual level to invest more in the preservation of environment? Indeed in the present set up, this does re ‡ect anything else but that …rms adapt their care expenditures to their size and to the cost of liability -given that less competition would increase market power and a larger individual output, individual care expenditures would be also larger to avoid an excessive cost of liability.

Most important, our analysis suggests at the same time that the aggregate level of care may have erratic reactions following (restrictions to) …rms entry. It depends on the speci…c design of the environmental liability law, but also on the properties of the technology of care (including the cost of care, and the environmental impact of care activities and productions). Saying di¤erently, the environmental law put some constraints on a Competition Authority, and for achieving better decisions this later needs generally two di¤erent sets of information. On the one hand, it requires the understanding of the legal and the institutional design regarding to the way Courts implement the environmental liability law, which is a practical issue; as we remind before, CERCLA and the European Directive operate at …rst glance a clear separation between activities that are subject to strict liability and those driven by negligence.

However, the interventions of a Competition Authority are designed to improve competition on markets in the sense of business. Thus identifying whether the product on the market is subject to strict liability rather than negligence may be uneasy (when it combines activities that are targeted by strict liability and others by negligence). Moreover, although the distinction between strict liability and negligence is 2 0 To be a little more speci…c: relaxing H2: 1) may not necessarily change dx pc dn < 0 or dx ms dn < 0 for a positive sign, as long as the convexity of the expected harm is large enough (i.e. the term in (C2) is large enough); 2) and change dx neg dn < 0 in dx neg dn > 0 only when b is low enough. a matter of statute law, it is not the case with damages sharing rule; it may be a matter of precedents (Courts may apportion damages between multidefendants according to one rule or the other, following some general guidelines, or conventional use, or the evolution of the doctrinal debate), or it may a right entitled to Plainti¤s. On the other hand, the Competition Authority needs to know the cost structures and bene…ts curve associated with safety activities in di¤erent business and markets at scrutiny, and the assessment of existing returns to scale -this is an empirical issue, but also a technical point since a high level of engineering expertise is required.

For the purpose of this discussion, tables 1 to 3 summarize our central results for a constant marginal cost of care, and add those for a more general cost for care (assuming that c(0) > 0; c 0 (x) > 0; c 00 (x) > 0; and c 000 (x) 0 8x; cf in the Appendix): Skipping to the most clearcut result, we …nd that when the marginal cost of care is increasing (and associated with a large degree of convexity), it is very likely that aggregate care expenditures increase with …rms'entry (thus, decrease with restriction to …rms'entry). This holds under strict liability -whatever the damage sharing rule, and whether an optimal damages multiplier is used or not -as well as under negligence (with a standard of care). In turn when the marginal cost of care is constant (or with low returns to scale, i.e. c 00 (x) ! 0), the impact of (restrictions to) …rms'entry on aggregate care expenditures may depend in a complex manner on the speci…c design of the liability law, as well as on the elasticity of market demand. Once again going to the essence, aggregate care increase for sure with …rms'entry: 1) under strict liability associated with the per capita rule and an optimal damages multiplier, or 2) under the negligence rule with a standard of care. Apart of such designs/cases, things are less predictable with strict liability, and as a consequence, the …ne tuning of competition policy requires more information, some being quite commonplace and standard for a Competition Authority (value of elasticity of demand, number of …rms). But others that are less usual, such as those regarding the way Courts implement the law (use of sharing rules). Our results suggest that the opportunity to use damages multipliers is of great importance here; on the one hand, damages multiplier design the incentives to take care at individual level; on the other hand, their impact on aggregate care expenditures may dramatically change with the regime liability to which they are associated. Under strict liability, aggregate care expenditures always increase with entry restrictions when suboptimal (no) damages multiplier are used; but they decrease (may increase or decrease) with entry restrictions if an optimal damages multiplier is combined with the per capita rule (market share rule). It is worth to remind that if CERCLA considers the use of "punitive damages", the use of "punitive damages" is forbidden up to now in the European context outside of competition law. The point is that suboptimal (no) damages multipliers lead to distortions in care and output decisions at individual level, and restrictions to …rms' entry produce rule-speci…c distortions in aggregate care levels, and thus welfare losses.

n 1 = n c 00 (x) = 0 dQ pc dn > 0 dX pc dn < 0 dQ pc dn > 0 dX pc dn > 0 c 00 (x) > 0 dQ pc dn > 0 dX pc dn ? 0 (c 00 (x) large ) dX pc dn > 0) dQ pc dn > 0 dX pc dn > 0 TABLE 1 -Aggregate equilibrium,
The …nal challenge is whether exist liability rules ‡exible enough, in combination with competition policy, in order to improve Social Welfare? Our results suggest as a …rst best policy mix achieving higher aggregate output and care expenditures, as well as higher Social Welfare, the association of an ("ideal") environmental liability law based on three pillars -a regime of strict liability, a damages sharing arrangement given by the per capita rule, an optimal damages multiplier -on the one hand, and on the other hand, any tools at the disposal of a competition policy focused on the maximization of consumers surplus (here, an increase in the aggregate output thanks to …rms'entry). Each of these tools is in the hands of an independent body (Courts enforce environmental liability law; the Competition Authority implements the competition law), but this policy mix requires a low degree of coordination: as long as each body commits to its targeted objective, …rms undertake e¢ cient decisions at the individual level both (in terms of care and in terms of output), aggregate expenditures for environment preservation increase with the degree of competition on the market (towards the socially optimal level), and …rms' entry is Social Welfare improving.

To the converse, an unusual lenient competition policy (preserving …rms markets power, through restrictions to …rms'entry) may be justi…ed on the grounds that the existing design of the environmental liability law departs from the ideal one, and is constraining the Competition Authority action. The rationale is that any design for environmental liability laws (other than our "ideal" law) will fail to increase aggregate care expenditures and improve Social Welfare when exists an excessive number of …rms on the market -thus restricting market access may be a kind of second best policy for a Competition Authority. But as explained above, this needs a close scrutiny at the situation, case by case, in order to reach a …ne tuning -otherwise, the aggregate care expenditures may decrease with restrictions to …rms' entry. When Courts depart from the per capita rule to adopt the market share rule, the knowledge of the price sensibility of market demand as, well as the speci…c characteristics of the technology of safety (cost structure for care, up to c 00 ; and productivity of care h 00 ) is determinant to establish that restricting …rms' entry should improve aggregate care and Social Welfare. Instead, when the assessment of liability is set according to a negligence test, …rms will always adhere to a ( ‡exible) standard of care, but reaching the …ne tuning of …rms' entry is even more demanding for a Competition Authority: this means collecting even more speci…c information on the technology of safety (up to c 000 ), not only on the price-elasticity of market demand.

Last but not the least, our analysis shows that the our so-called "ideal" environmental liability laws enables a traditional competition policy (focused on the improvement of competition in the industry) to move the equilibrium closer to the social optimum (as n ! 1); in contrast for any other liability regime, there exists a …nite number of …rms that maximizes Social Welfare. The consequence is that adopting restrictions to …rms' entry cannot be seen as a second best solution given the institutional design of the environmental liability law, unless the current number of …rms is above this threshold. This is an empirical/practical issue, but maybe an uneasy job to establish. 21

Concluding remarks

Our analysis rests on a simple framework (linear demand, durable care), but relaxing such assumptions will not change the general story (see for example [START_REF] Charreire | Apportioning indivisible environmental harms and the Joint and Several Liability Doctrine[END_REF]). We are also very focused here on a rough instrument (…rms' entry) for a Competition Authority; additional works in di¤erent competitive environments will be useful to complete the picture. The case for cartels and various forms of collusion will be of speci…c interest. In this perspective Baumann, Charreire and Cosnita-Langlais (2020) develop the analysis of cartels stability under an environmental liability law based on the market share rule.

Remark also that the di¤erent points addressed here, as well as the conclusions of the paper focused on the environmental liability law, extend very generally to any situation with "third-party" victims (that is, having no economic nor contractual relationships with the injurers), e.g. injurers competing on the same legal market, producing a good with potential harmful consequences for human health, society and so on (which are not the consumers of the good produced by …rms). 22 Most of the conclusions will also easily extend to consumers harms as the result of competition distorsions. Nonetheless, this needs more scrutiny. [START_REF] Friehe | Product liability when harm is incurred by both consumers and third parties[END_REF] provide the analysis of harms to both consumers and third-parties victims for a monopolistic market, and …nd signi…cant departures related to the comparison of strict liability and negligence for example.

Finally, remark that (optimal) damages multipliers appear as an important pillar of the …rst best as well as second best policies in our paper. These so-called "punitive damages" are still a matter of debate in Europe. With punitive damages, e¤ective damages paid by each …rm represent a multiple of e¤ective harms to the environment that fall down public pockets, or provide fundings for a Compensation Fund for victims of environmental harms. Such high levels for e¤ective damages raise, and at the same time may help in solving, the judgment-proofness problem or the disappearing defendants issues. These two important problems are beyond the scope of the present paper which consider symmetric …rms. The judgment-proofness issue requires a more detailed analysis of liability sharing in an asymmetric environment, whether this is understood as considering an asymmetric oligopoly, 23 or vertically di¤erentiated markets; the disappearing defendant problem calls for a dynamic approach. This will be the topic of future researches.

0 b

 0 …rms'entry, and the per capita rulen 1 = n c 00 (x) = 0 b high (low) ) dQ ms dn > (<)0 dX ms dn < 0 b high (low) ) dQ ms dn > (<)0 b high (low) ) dX ms dn > (<)0 c 00 (x) > 0 dQ ms dn ? 0 (c 00 (x) large ) dQ ms dn > 0)dX ms dn ? 0 (c 00 (x) large ) dX ms dn > 0)dQ ms dn ? 0 (b or c 00 (x) large ) dQ ms dn > 0)dX ms dn ? 0 (b or c 00 (x) large ) dX ms dn > 0)TABLE 2 -Aggregate equilibrium, …rms'entry, and the market share rule c 00 (x) 0; c 000 (xx) > 0; c 000 (x) > 0 b or c 00 (x) large ) dQ neg dn > or c 00 (x) large ) dX neg dn > 0)

TABLE 3 -

 3 Aggregate equilibrium, …rms'entry and negligence with a standard of care

https://www.competitionpolicyinternational.com/eu-parliament-demands-fundamental-overhaul-of-competitionpolicy/.

2 One of us has analyzed the stability of cartels under the market share rule[START_REF] Baumann | Market collusion with joint harm and liability sharing[END_REF].

[START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] give several examples in the area of environment, food, and health.[START_REF] Friehe | Prevention and cleanup of dynamic harm under environmental liability[END_REF] deal explicitly with cases where the cumulative e¤ect arises because of repeated accidents, and they analyze the issue of dynamic incentives under tort law.

1 Firms' symmetry is also considered in the literature dedicated to (product) liability and market structures, as well as the one about unconventional competition policies. We deal with the asymmetric case inCharreire and Langlais (2017).

2 Considering the di¤erent precautionary measures undertaken at a plant, one may argue obviously that some will have a durable nature and the others will be of a non-durable nature. We contrast the two "pure" cases for pedagogical reasons, whereas the literature about product liability assumes usually that care is non-durable.

3 Such restrictions are just a pure technical point, adding no intuition.

7 Although[START_REF] Daughety | Cumulative Harm and Resilient Liability Rules for Product Markets[END_REF] paper addresses the issue of product liability, their results regarding strict liability are still relevant for environment liability. In turn, what is speci…c to a product liability context is their comparative analysis of no liability, negligence vs strict liability -as well as our comparative analysis of these di¤erent liability regimes is speci…c to the case with indivisible environmental harms we focus on; see above Remark 1.

1 As matter of comparison, see the analysis of the consequence of fusions in European markets for mobile communications[START_REF] Genakos | Evaluating market consolidation in mobile communications[END_REF], or the debate about the introduction of a fourth operator on the French market[START_REF] Thesmar | L'impact macro-économique de l'attribution de la quatrième licence mobile[END_REF].

2 Authorized by Title III of the Superfund Amendments and Reauthorization Act (SARA), the Emergency Planning & Community Right-to-Know Act (EPCRA) was enacted by Congress as the national legislation on community safety. This law is designed to help local communities protect public health, safety, and the environment from chemical hazards.2

[START_REF] Charreire | Compensation of third party victims, and liability sharing rules in oligopolistic markets[END_REF] we provide the comparative analysis of the market share and the per capita rules for an asymmetric oligopoly. We show that the market share rule induces more distortions in output decisions than the per capita rule. On the one hand, high-cost …rms obtain a strategic competitive advantage when aggregate damages are shared according to the market share rule compared with the capita rule, allowing them to reduce their output less than low-cost …rms. On the other hand, the market share rule yields a distribution of equilibrium markets shares that is less spread than the per capita rule, away from the socially optimal one.

Proposition 9 Assume environmental liability law is designed according to negligence with a standard of care; then: i) The individual output and care may increase as well as decrease with the number of …rms.

ii) The aggregate output and care levels increase with the number of …rms. iii) As n ! 1, the equilibrium industry converges to levels (Q neg 1 ; X neg 1 ) above the social optimum (Q sw ; X sw ). iv) If b is large (small) enough, fostering …rms'entry increases (decreases) Social Welfare. v) Social Welfare is maximized for a …nite number of …rms,

Proof. i) See Appendix 6. Speci…cally we show that if b is large (small) enough, then the individual output decreases (increases) with n. The e¤ect on individual care requires more quali…cations: we show that if b is large enough and (C3) holds, then the individual care decreases with n; in contrast, if b is small enough and (C3) does not hold (2h 0 (nx) + nxh 00 (nx) < 0 for any n; x), then the individual care increases with n. ii) Using ( 13)-( 3), the aggregate output and aggregate care levels satisfy:

We will denote as Q neg (X) the aggregate output level that solves (13bis), for any given value of aggregate care; while X ms (Q) is the aggregate care level that solves (3bis), for any given value of the aggregate output. According to (3bis), the aggregate care expenditures X neg (Q) doe not depend on n, but increase with Q. According to (13bis), the marginal market proceeds increases with n, hence Q neg (X) increases with n at any X. Hence ii). iii) In the limit case where n ! 1, then according to the LHS in (13bis), the marginal market proceeds tends to the market price; given that the marginal cost of liability still satis…es (h 0 (X)) 2 h 00 (X) < h(X) 8X > 0, the aggregate output and care levels do not reach their socially optimal values (Q sw 1 ; X sw 1 ) as n ! 1, but in contrast take some positive values larger than (Q sw 1 ; X sw 1 ) that solve (13bis)-(3bis), or: 1) at (Q neg ; X neg ) and di¤erentiating in n (using (17bis)-( 3bis)), we obtain, denoting

In Appendix 6, we show that Social Welfare is maximized neither with perfect competition (n ! 1) nor with a monopoly (n = 1); there exists a …nite number of …rms, n neg > 1, that maximizes SW (Q neg ; X neg ), which is the solution to: