Damien Sans
email: damien.sans@univ-amu.fr

Sonia Schwartz
email: sonia.schwartz@udamail.fr

Hubert Stahn
email: hubert.stahn@univ-amu.fr

About Polluting Eco-Industries: Optimal Provision of Abatement Goods and Pigouvian Fees

Keywords: Polluting Eco-Industry, Heterogeneous …rms, Welfare Analysis, Pigouvian Tax JEL Classi…cation: D62, Q58, L11

In this article we introduce a polluting eco-industry. Depending on the level of the damage, we …nd one of two optimal equilibria. If the damage is low, we generalize the usual results of the economic literature to the polluting eco-industry: the dirty …rm partially abates their emissions, only e¢ cient eco-industry …rms produce and the abatement level increases with the damage. However, we obtain very speci…c results if the damage is high. In this case, not all e¢ cient eco-industry …rms produce. The abatement level and the number of active eco-industry …rms both decrease as the damage increases. We …nally show that a well-designed Pigouvian tax implements these equilibria in a competitive economy.

Introduction

Pharmaceuticals and other organic wastewater contaminants are found in treated wastewater in Europe [START_REF] Ternes | Occurrence of drugs in German sewage treatment plants and rivers[END_REF][START_REF] Comoretto | Comparing pharmaceutical and pesticide loads into a small Mediterranean river[END_REF], the US, [START_REF] Boyd | Pharmaceuticals and personal care products (PPCPs) in surface and treated waters of Louisania, USA and Ontario, Canada[END_REF] and Asia (Nozaki et al., 2000). These observations suggest that wastewater treatment plants do not totally abate pollution; they are only partially e¢ cient. Moreover, their production process may also be polluting. According to [START_REF] Kim | Estimation of CO2 emission from water treatment plant -Model development and application[END_REF], wastewater treatment plants (and incineration facilities) have been reported to emit sig-ni…cant amounts of GHGs, and water treatment plants have also been categorized as one of the signi…cant public facilities emitting important amounts of CO2 by consuming immense amounts of electricity and chemicals [START_REF] Raucher | Risk and bene…ts of energy management for drinking water utilities[END_REF][START_REF] Rothausen | Greenhouse-gas emissions from energy use in the water sector[END_REF]. Along the same lines, a debate has emerged about the energy balance of the photovoltaic industry. This raises the question of grey energy. Grey energy is the hidden energy associated with a product, meaning the total energy consumed throughout the product life cycle from its production to its disposal. The issue is whether the reduction in pollution is greater than the grey energy consumed. If this is the case, we can infer that the eco-industry is e¢ cient.

Wastewater treatment, air treatment, waste treatment plants and the photovoltaic industry are all part of the eco-industry sector. This is a new industrial sector covering pollution and resource management activities, ranging from the development of clean technologies to the optimization of methods for monitoring and managing environmental impacts. 4 It appears that this sector can be partially e¢ cient and polluting. This point is crucial for policy-makers because the emergence of eco-industry …rms is often conditional on environmental policy.

The eco-industry is well-documented in the economic literature, but nevertheless mainly focuses on the fact that it is highly concentrated. The research can be divided in two main branches. The …rst branch considers innovative …rms investing in R&D to obtain a patent for a pollution-reducing new technology. The performance of taxes and tradable permits are compared in various contexts. [START_REF] Denicolo | Pollution-reducing innovations under taxes or permits[END_REF] and Requate (2005) make these comparisons under di¤erent timing and commitment regimes. A threat of imitation is introduced by Fisher et al. (2003), while Perino (2008) studies green horizontal innovation, where new technologies reduce pollution of one type while causing a new type of damage. More recently, Perino (2010a) focuses on the second-best policies for all combinations of emission intensity and marginal abatement costs.

The second branch analyses how eco-industry modi…es the usual results of the economic literature. It takes market power as a given and suggests the optimal design of environmental policy within this context. Most of these papers consider the Pigouvian tax as environmental policy tool: see for instance [START_REF] Canton | Redealing the cards: how an eco-industry modi…es the political economy of environmental taxes[END_REF] None of the studies mentioned above explores polluting eco-industry. The aim of this paper is to investigate whether the standard results in economic literature are challenged if the eco-industry is polluting. Following almost all the papers cited, we consider a vertical structure composed of a downstream polluting sector and an upstream eco-industry. Contrary to the existing literature, in this article eco-industry …rms are polluting and heterogeneous, i.e., they are more or less polluting. This situation may correspond to di¤erent generations of plants having di¤erent technologies. We also assume that they cannot reduce their emissions. To focus on this original assumption, we do not consider market power. Under these new assumptions, we …rst seek to de…ne the centralized solution. Next, we examine whether this optimal policy can be decentralized using a traditional economic instrument: the Pigouvian tax. On these points, our article is in keeping with the second branch of the economic literature described above.

We …nd that two kinds of equilibrium can emerge. The …rst equilibrium occurs if the marginal damage is not too high. In this case, we extend the usual results of the economic literature to polluting eco-industry. We …nd that the optimal level of abatement is such that the marginal social bene…t and marginal social cost of abatement are equal to the marginal damage. The dirty …rm partially abates its emissions and only e¢ cient eco-industry …rms produce. The greater the marginal damage, the less the dirty …rm produces and the higher the abatement level.

The second kind of equilibrium occurs if the marginal damage is high: the dirty …rm abates all its emissions and not all active …rms in the …rst equilibrium produce. As pollution is very harmful for the environment, the only way to prevent even more damage is to reduce the pollution produced by eco-industry. To do this in an e¢ cient way, the regulator should not only reduce the number of active …rms but also modify the distribution of abatement in the eco-industry. We also …nd the counter-intuitive result that the number of active …rms and the level of abatement decrease with the marginal damage. The optimal abatement level is such that the marginal social bene…t is equal to the marginal social cost, but they are both lower than the marginal damage. This second equilibrium is very speci…c to polluting eco-industry.

Finally, we show that a competitive economy reaches these optimal equilibria if the regulator implements a Pigouvian tax. The rule is very simple, because it is the same whatever the level of the damage: the Pigouvian tax must be equal to the marginal damage. However, depending on the damage level, the functioning of the economy will be di¤erent, as we will see in the paper.

In Section 2, we present the model. Section 3 de…nes the social bene…ts and social costs of pollution abatement. In Section 4 we determine the e¢ cient level of abatement. Policy issues are presented in Section 5 and some concluding remarks are given in Section 6. Proofs are relegated to the Appendix

The basic assumptions

To keep the assumptions as simple as possible, we assume that the standard polluting industry is characterized by a representative …rm that produces a quantity Q at a given cost C(Q). This cost is increasing and convex (i.e., C 0 (Q) > 0 and C 00 (Q) > 0) and inaction is allowed (i.e., C(0) = 0). This activity is polluting. Emissions are given by "(Q), an increasing and convex function (i.e., " 0 (Q) > 0 and " 00 (q) > 0) with "(0) = 0. This "endof-pipe" pollution can be reduced by an abatement activity provided by the specialized external …rms which comprise the eco-industry. So if we denote by A the total abatement realized by the polluting …rm, the remaining pollution will be max f"(Q) A; 0g.

The eco-industry is composed of a continuum5 [0; 1] of …rms indexed by i. Each of them supplies a(i) pollution reduction services produced at some cost (a(i)). They share the same increasing and convex cost function and inaction is allowed (i.e., 0 (a) > 0, 00 (a) > 0 and (0) = 0) We also assume that 0 (0) = 0 in order to ensure that there is, in a competitive setting, an o¤er for each positive price 6 .

owever, we assume that this activity pollutes and that these …rms are heterogeneous with respect to their emissions. Emissions of …rm i are a proportion (i) 2 [ min ; max ] of its production and are considered as unavoidable (they cannot be abated). Since one unit of abatement good reduces the pollution of the downstream …rm in the same proportion, the coe¢ cient [1 (i)] measures the net contribution of …rm i to pollution reduction. Firms in the abatement good sector are also heterogenous: they are ranked from the least to the most polluting. We also assume that : [0; 1] ! [ min ; max ] is a continuous and di¤erentiable function, and because they are ranked, 0 (i) > 0.

Finally we assume min < 1 to ensure that at least some …rms have a net contribution to global pollution reduction. max > 1 means that at least some of these …rms contribute to pollution abatement in an ine¢ cient way since their global contribution to the emissions reduction per unit of output, (1 (i)), is negative.

The global emissions, E = max f"(Q) A; 0g + R 1 0 (i)a(i)di, are comprised of the remaining pollution from the dirty industry and the emissions generated by the abatement activity. This means that we can have situations in which the "dirty" industry is clean and some pollution remains. So, contrary to most of the literature which does not consider polluting eco-industry, it is now crucial to take into account the fact that the abatement activity becomes ine¢ cient when the pollution of the dirty industry is completely removed. We assume that these emissions create social damage, measured by

D(E) = v:E with v > 0. Hence v is the marginal damage.
Finally, to close the model, we introduce an inverse demand function for the polluting goods P (Q). This function is decreasing (i.e. P 0 (Q) < 0) and veri…es that lim Q!0 P (Q) = +1 and lim Q!+1 P (Q) = 0.

Social bene…ts and costs from pollution abatement

This section is rather traditional. We …x a production level A of the abatement good and de…ne, within our setting, the social bene…ts and costs of this abatement choice. The main di¤erence with the usual approach is that the eco-industry is polluting. This has two consequences: (i) these goods only reduce the emissions of the polluting industry and (ii) the residual pollution must be included in the social cost of the abatement production.

We then obtain the marginal social bene…t and marginal social cost of abatement.

The social bene…t of abatement The social bene…t from a level A of pollution abatement is obtained by choosing the production of the dirty industry. This production level maximizes the welfare of consumers net of the production costs and of the pollution induced by this activity. This function is given by:

SB(A; v) = max Q 0 Z Q 0 P (q)dq C(Q) v max f"(Q) A; 0g (1)
This de…nition of the social bene…t is very conventional, especially for "end-of-pipe" pollution. But in most treatments of this problem, the condition stating that the emission of the dirty industry must be non-negative (i.e., max f"(Q) A; 0g in our article) is quickly forgotten simply because this corner solution in which no pollution occurs is not really interesting. However, this is far from being the case when the eco-industry also pollutes, because there is now a possible arbitrage between upstream and downstream pollution, i.e., between the emissions of the abaters and those of the …nal goods producers. This is why we have to solve this non-smooth optimization problem globally. The method (see the proof of Lemma 1 in the appendix) essentially makes use of the sub-di¤erential introduced by Rockafellar (1979). In any event, non-smooth optimization involves case studies and thresholds. In this article, if we solve this program for all levels of pollution abatement A 2 R + , there are three possible outcomes.

The …rst situation, the usual one, is characterized by partial abatement: "(Q) A > 0. In this case, the …rst order condition is given by P

(Q) C 0 (Q) v:" 0 (Q) = 0. If we set (Q) = P (Q) C 0 (Q) " 0 (Q)
, the optimal level of production Q = 1 (v); that solves this condition is simply a decreasing function of the marginal damage v. But this solution only occurs if "(Q) A > 0; which requires that 1 (v) > " 1 (A) or that the …xed level of abatement good veri…es A < " ( 1 (v)).

The second situation corresponds to full abatement of the emissions of the downstream industry:. Q = " 1 (A). This requires that the previous condition is not met i.e., 1 (v) " 1 (A) or A " ( 1 (v)). But if we now bear in mind that this full abatement condition means that max f"(Q) A; 0g = 0, this production level is optimal as long as we do not reach the production level Q max which is e¢ cient when there is any damage. In other words, this also requires that (P (Q) C 0 (Q))j Q=" 1 (A) 0 or that A < " (Q max ).

Finally if A " (Q max ), the optimal production level that solves program (1) will be equal to Q max . As we will see later, this last case never occurs, simply because pollution is not taken into account. It is given here for the sake of completeness.

From all these observations, we can construct the social bene…t SB(A; v) of pollution abatement. It is a piecewise continuous function depending on the …xed level of abatement A. However what really matters is the marginal social bene…t:. @SB(A;v) @A :

if there is only partial reduction, this marginal bene…t will be, as usual, equal to the marginal damage v;

if there is full abatement with A > "(Q max ), additional abatement is fully ine¢ cient since the optimal production level does not depend on A. The marginal social bene…t is clearly 0;

-if there is full abatement with A 2 [" (Q(v)) ; "(Q max ))
, the optimal production level is positively correlated with A. The social marginal bene…t is then given by

P (" 1 (A)) C 0 (" 1 (A)) " 0 (" 1 (A)) = (" 1 (A)) (2)
We know that this case occurs if 1 (v) " 1 (A), which is equivalent to v > (" 1 (A)). In this last case the marginal social bene…t is therefore smaller than the marginal damage.

More formally, we can state that:

Lemma 1. If (Q) = P (Q) C 0 (Q) " 0 (Q)
; inspection of program 1 shows that: (i) The optimal production level is given by:

Q(v; A) = max f 1 (v); " 1 (A)g if A < " (Q max ) Q max if A " (Q max ) (3)
(ii) The marginal social bene…t is given by:

@SB(v; A) @A = max min v; (" 1 (A)) ; 0 (4)
The social cost of abatement The social cost induced by the production of abatement goods in quantity A is obtained, as usual, by choosing an optimal distribution of the production between the di¤erent plants which comprise the eco-industry. But in our case, this process involves not only the cost structure of these …rms but also their pollution structure. This cost is de…ned by:

SC(A) = min a( ) 0 R 1 0 (a(i)) di + v R 1 0 (i)a(i)di (5) subject to R 1 0 a(i)di = A (6)
If we denote by the Lagrangian multiplier associated with the constraint, the …rst order conditions of this convex minimization problem are given by:

8i 2 [0; 1] , 0 (a(i)) + v (i) 0 (with equality if a i > 0) and R 1 0 a(i)di = A (7)
From these FOC we see that a given …rm i is active if v (i) > 0 and in this case its production level is of ( 0 ) 1 ( v (i)). Since the emissions of these …rms are increasing with their index i, his also means that there exists a pivotal …rm7 i 0 = 1 min v ; max which is the …rst for which it is optimal to stop the production. If we now keep bear in mind that the total production level A of production of abatement goods is given, the index of this …rm can be obtained by making sure that the total level of production of …rms i i 0 is equal to A.

In other words, even if these …rms share the same cost function, the optimal distribution of the global production is not symmetrical because they are heterogeneous in their contribution to pollution. We can therefore expect that not all …rms will be selected at the optimal allocation. In order to de…ne this allocation, we also need some information about the marginal social cost of the production of an additional unit of abatement goods. This quantity is given by @SC(v; A) @A . Since the constraint to the problem is R 1 0 a(i)di = A, the envelop theorem immediately tells us that the social marginal cost is equal to the Lagrangian multiplier associated with this program.

More precisely, we can say: Lemma 2. If A = 0 then a(i) = 0 for almost all i and (A) 2 ( 1; v: min ] and for A > 0 :

(i) The productions of abatement goods are given by:

a (i; A; v) = ( 0 ) 1 ( (A; v) v: (i)) if i 1 v 0 else (8)
(ii) (A; v) is implicitly de…ned by: R 1 (minf v ; maxg);

0 ( 0 ) 1 ( v: (i)) di = A (9)
(iii) The marginal social cost is given by:

@SC(v; A) @A = (A; v) (10)
As we have de…ned the marginal social cost and bene…t, it remains for us to …nd the optimal level of abatement.

Optimal outcome

With regard to our previous results, let us …rst identify the optimal provision of abatement goods. This level results from a trade-o¤ between the marginal social bene…t and the marginal social cost. It is given by:

A opt (v) = arg max A 0 SB(A; v) SC(A; v) (11)
As expected, this program shows that the optimal production of abatement goods never exceed " 1 (Q max ), the level of abatement which maximizes the social bene…t without damage. If the last case occurs, the social bene…t is constant while the costs are increasing with the abatement e¤ort (see (ii) of Lemma 1), which contradicts optimality. We can also observe that the optimal provision of abatement goods is always positive, i.e., A opt (v) > 0.

Otherwise, the marginal cost of abatement is, by Lemma 2, smaller than v: min < v while the marginal bene…t of an additional unit of abatement when there is no abatement at all is of v since lim A!0 (" 1 (A)) = +1 (see Lemma 1). Following these observations, we can a¢ rm that the FOC associated with program (11) is:

@SB(A; v) @A @SC(A; v) @A = min v; " 1 (A) (A; v) = 0 (12)
This condition clearly suggests that two kinds of e¢ cient outcomes occur, depending on the level of the marginal damage. The …rst situation is rather classical: the dirty …rm partially abates its emissions and the marginal damage of pollution is equal to both the marginal bene…t and the marginal cost of abating pollution. The second case occurs if the pollution of the dirty industry is totally removed but some pollution persists due to the activity of the eco-industry. In this case, the marginal bene…t remains equal to the marginal cost, but lower than the marginal damage induced by pollution. Intuition suggests that the marginal damage v admits a threshold for which we switch from one situation to the other. To get this intuition, let us start with a case in which there is partial abatement in the dirty industry or, more formally from Equation [START_REF]Facts and …gures: the link between EU's economy and environment[END_REF], in which (" 1 (A)) > v = (A; v). By equation ( 9), we can compute the optimal provision A opt (v) of the abatement good simply by replacing (A; v) by v. This quantity, given by:

A opt (v) = R 1 (1) 0 ( 0 ) 1 (v (1 (i))) di (13)
is obviously increasing with the marginal damage. But this case only holds seeing that (" 1 (A opt (v))) > v. So if on has in mind that is a decreasing function, the left hand side of the previous condition is decreasing in v while the right hand side in increasing. This rather intuitive argument therefore suggests that the case of partial abatement disappears for su¢ ciently high marginal damage levels.

Lemma 3. There exists a unique threshold v given by:

" 1 ( v) R 1 (1) 0 ( 0 ) 1 ( v (1 (i))) di = 0 ( 14
)
with the property that if v < v there is only partial pollution reduction in the downstream industry, while in the other case there is full abatement of the pollution emitted by the dirty industry. The pollution of the eco-industry nevertheless remains in the last case.

If the marginal damage of pollution is lower than v, we are in the standard case described by the literature. There is, at the optimal allocation, partial abatement of the pollution emitted by the dirty industry. The only di¤erence is that there are now some additional emissions due to the eco-industry. The aggregated level of abatement is a usually chosen such that the marginal cost and bene…t are both equal to the marginal damage of pollution and the aggregated level of production of the …nal industry is commonly decreasing with the marginal damage. In other words, there is a traditional arbitrage between the reduction of the …nal production and the increase in production abatement: both quantities are negatively correlated when the marginal damage v changes.

However, the remaining emissions of the eco-industry contribute to a selection of which …rms in this sector should produce. This selection is not based solely on private costminimizing considerations, but also takes into account the emissions of the eco-industry. Since the aggregated marginal social cost is equal to the marginal damage, only the …rms which have a positive net contribution to pollution abatement (i.e., [1 (i) > 0] produce, and because these …rms are heterogeneous in their emissions, the less polluting …rms contribute more.

Moreover, since one unit of the abatement good removes one unit of pollution, we observe that at the e¢ cient allocation, the marginal cost and bene…t from the aggregate level of abatement are both equal to the marginal damage of pollution. Finally, we have the usual arbitrage between the reduction of the …nal production and the increase in pollution abatement, which depends on the level of the marginal damage, since these quantities are negatively correlated when v changes. More precisely, we can say that: Proposition 1. If the marginal damage is not too high, i.e, v v,only partial abatement of the pollution of the dirty …rm occurs, and e¢ cient allocation has the usual properties: (i) the marginal bene…t and marginal cost of pollution abatement are both equal to the marginal damage of pollution, i.e. @SB(A opt ;v) @A = @SC(A opt ;v) @A = v, (ii) the optimal level of production Q opt = 1 (v) is decreasing with the marginal damage of pollution while the total production of abatement good

A opt (v) = R 1 (1) 0 ( 0 ) 1 (v (1 (i))) di (15)
and the individual production are increasing with the level of the marginal damage.

(iii) all …rms in the eco-industry that e¢ ciently reduce pollution, i.e, (i) 1,contribute to the abatement but to di¤erent extents depending on their own emissions, i.e., 8i

1 (1), a opt (i) = ( 0 ) 1 (v (1 (i))
) and this quantity is decreasing with (i),

The second case in which the marginal damage is higher than the threshold, v, is less usual. Its interpretation largely depends on the assumption that the eco-industry pollutes. In this situation, it becomes optimal to remove all the emissions of the upstream industry, even if some pollution persists due to the activity of the eco-industry. As this irreducible pollution is harmful, this may not be su¢ cient to improve the environment. This is why it is also optimal (i) to reallocate the production of abatement goods toward the less polluting …rms in the eco-industry and (ii) to slow down the production of abatement goods and therefore also the production of the …nal good, since the emissions from this activity are totally abated thanks to the eco-industry production. So it is not really surprising that (i) the number of active …rms decreases with the marginal damage, contrary to the previous case in which all e¢ cient …rms produce, and (ii) the level of …nal production is now positively correlated with the level of abatement, simply because the maximal abatement level is reached and both are decreasing with the level of marginal damage.

What is perhaps more surprising is that the marginal cost and bene…t from abatement are now lower than the marginal damage of pollution. In order to understand this property, let us start with a level of production in the eco-industry corresponding to the total abatement, for which the marginal social bene…t of pollution reduction is equal to the marginal damage. If the damage is high, this often requires a large reduction in the …nal output. So it is possible that the marginal social cost of producing enough abatement goods to totally remove downstream emissions remains lower than the marginal damage of the pollutants. This provides an incentive to produce more abatement goods and to expand the production of the …nal good in a way that ensures full upstream emission abatement. More precisely, we observe that: Proposition 2. If the marginal damage is high, i.e. v > v, the pollution of the dirty …rm is totally abated. The e¢ cient allocation has the following less usual properties: (i) the marginal bene…t remains equal to the marginal cost of abatement, but this value is now smaller than the marginal damage:

@SB(A opt ; v) @A = @SC(A opt ; v) @A = opt (v) < v (16)
(ii) the optimal level of production Q opt = " 1 (A opt ) bis now positively correlated with the optimal level of abatement. The total production of abatement goods is now decreasing with the marginal damage, since the pollution of the eco-industry can only be reduced by reducing the production of these goods, i.e., dQ opt (v) dv 0 and dA opt (v) dv 0: (iii) some …rms that can reduce pollution e¢ ciently are not active at the optimal allocation. The …rm i is active if

(i) < opt0 (v) v < 1.
Moreover, the number of active …rms decreases with the marginal damage:

i opt (v) = 1 opt0 (v) v veri…es di opt (v)
dv < 0: However, their contribution to the production of abatement good a opt (i) = ( 0 ) 1 ( opt (v) v (i)), remains decreasing with their emission rate.

(iv) the Lagrangian multiplier opt (v) 2 (v min ; v) is the unique solution to:

" 1 ( ) = Z 1 ( v ) 0 ( 0 ) 1 ( v: (i)) di (17)
It remains for us to analyze how to decentralize these two optimal equilibria.

The policy issues

We have shown that the model exhibits two kinds of e¢ cient allocation, depending on the level of the marginal damage (v) and that these allocations have rather di¤erent properties. In this section, we investigate whether a standard instrument like the Pigouvian tax can implement each of these equilibria in a competitive setting. This last point will be veri…ed in two steps. We …rst assume that there is a price signal representing the emission tax and we compute the competitive allocation for each value of . In the second step, we determine the level of the Pigouvian tax that implements the e¢ cient allocation in each case.

The competitive behaviors

We …rst analyze the competitive behavior of the dirty …rm, and then that of the eco-industry …rms. Lastly, we expose the abatement market equilibrium.

The dirty …rm So let us start with the dirty …rm. If there is a price signal associated with the emission of pollution, this …rm will choose its production supply and its demand for the abatement good by solving the pro…t equation:

(Q; A) = max Q 0 8 > > > < > > > : p:Q C(Q) min A 0 fp a :A + : max f"(Q) A; 0gg | {z } =C A (pa; ;Q) 9 > > > = > > > ; (18)
We see that the objective function is linear in A on [0; "(Q)]. This implies that the optimal conditional demand for abatement goods will be 0 if p a > , "(Q) if p a < , and any quantity within [0; "(Q)] if p a = . It follows that the abatement cost is given by C A (p a ; ; Q) = min fp a ; g :"(Q) and that the FOC characterizing the product supply is:

p C 0 (Q) min fp a ; g " 0 (Q) 0 (with equality if Q > 0) (19)
Since we know that the inverse demand is given by P (Q), the quantity which clears the commodity market is obtained by:

Q(p a ; ) = 1 (min fp a ; g) (20)
In the opposite case, i.e. > v, the equilibrium price of the abatement good market is lower than but nevertheless higher than min : because there is no supply of abatement at any price lower then min : (see equation [START_REF] Rockafellar | Convex Analysis Princeton landmarks in mathematics[END_REF] ). In fact this price solves:

" 1 (p a ) = Z 1 ( pa ) 0 ( 0 ) 1 (p a : (i)) di (26)
This also implies (i) that not all e¢ cient …rms in terms of pollution reduction are active at equilibrium since pa < 1 and (ii) the pollution of the dirty downstream …rm is completely abated. More precisely, we can say:

Lemma 5. If > v, the price of the abatement goods p c a ( ) is the unique solution to equation [START_REF] Schwartz | Competitive permit markets and vertical structures: the relevance of imperfectly competitive eco-industries[END_REF] and is lower than the implicit price for emissions. The quantities traded on the markets are given by:

( Q c ( ) = 1 (p c a ( )) = " 1 (A c ( )) A c ( ) = R 1 ( pa(t) ) 0 ( 0 ) 1 (p a ( ) : (i)) di (27)
The number of active …rms in the eco-industry is given by i

c ( ) = 1 p c a ( ) < 1 each of them producing a c (i; ) = ( 0 ) 1 (p c a ( ) : (i)).
Finally the equilibrium of the commodity market is p c ( ) = P (Q c ( ))

The level of the Pigouvian tax

In the previous section we analyzed competitive behaviors following the implementation of a Pigouvian tax. In this section we ask if there is a tax rate which implements the e¢ cient allocation obtained either in Proposition 1 or Proposition 2.

If the level of the damage is such that it is optimal to partially abate the pollution of the dirty downstream …rm (v v), we are back to the traditional case largely covered by the literature: the Pigouvian tax has to be equal to the marginal damage. If = v v the quantities traded at the competitive equilibrium (Q c (v), A c (v) and a c (i; v) in Lemma 4) are exactly the same as the optimal quantities obtained in proposition 1. In this case the price of the abatement good also re ‡ects the marginal damage since p c a (v) = v. If the marginal damage is higher than the threshold identi…ed in Lemma 3 (v > v) the value of is less obvious. Assuming that the policy maker keeps the same rule (i.e.

= v). Eq (26) tells us that the equilibrium price of the abatement good has to bebe equated with the optimal Lagrangian multiplier representing the marginal social cost (see Eq [START_REF] Nawaz | Embodied energy analysis of photovoltaic (PV) system based on macro-and micro-level[END_REF] in proposition 2). Hence, the quantities traded at the competitive equilibrium are exactly the same as the e¢ cient quantities (see lemma 5). In other words, the policy rule remains the same but the mechanism leading to the e¢ cient allocation is totally di¤erent. By setting the Pigouvian tax at the level of the marginal damage, the adjustment of the abatement good market results in a price corresponding to the social cost of abatement. Hence, the ratio between this equilibrium price and the Pigouvian tax selects the number of active …rms in an optimal way. Proposition 3. As usual, the e¢ cient allocation is reached if the Pigouvian tax is set at the level of the marginal damage. However, if the marginal damage is high (v > v) this tax is higher than the marginal bene…t from abatement, which is given by the equilibrium price of the abatement good.

Concluding remarks

In this article, we have investigated whether the hypothesis of a polluting eco-industry challenges the usual results in economic literature. To this purpose, we considered a vertical structure composed of a polluting downstream …rm and an upstream eco-industry. We assumed that eco-industry …rms are heterogeneous and that they cannot reduce their pollution level. Under these assumptions, we obtained two kinds of equilibrium. The …rst equilibrium, with a lower level of damage, extended the standard results of economic literature to the case of a polluting eco-industry, but our results are very di¤erent when the damage is high. In this case, the dirty …rm must totally abate its emissions. To reduce the remaining pollution, produced by the eco-industry, not all e¢ cient eco-industry …rms produce and the level of production among these …rms is di¤erent to that of the …rst equilibrium. We found that the greater the damage, the lower the abatement level and the smaller the number of producing eco-industry …rms.

We …nally show that both equilibria can be decentralized in a competitive economy by means of a Pigouvian tax. Whatever the equilibrium, the regulator can follow a very simple rule, because the Pigouvian tax should always be equal to the marginal damage. However, this rule plays a di¤erent role in reaching each equilibrium.

Our results suggest that a polluting eco-industry is not a problem for the regulator, because the competitive equilibrium selects the right …rms to be in production, provided that the regulator sets the correct level of the Pigouvian tax. However, this optimistic conclusion depends on the crucial assumption of perfect information that we implicitly make in our model. In the real world, the regulator cannot de…ne this tax so well, and our results may not hold. Moreover, eco-industry is characterized by the fact that it is highly concentrated. In this respect, one may wonder what level of the Pigouvian tax would decentralize the optimum. Finally, this article takes as given the pollution features of each …rm. Taking into account the innovative process would make it possible to endogenize the pollution distribution among …rms. Further research is needed to investigate these di¤erent questions.

  , Canton et al. (2007), Canton et al. 2012, David and Sinclair-Desgagné (2005, 2010), David et al. (2011), Nimubona (2012) and Nimubona and Sinclair-Desgagné (2005, 2010). Greaker (2006) and Greaker and Rosendahl (2008) introduce non-tradable quotas, while Schwartz and Stahn (2014) study a pollution permit market

See[START_REF] Ernst | Eco-industry, its size, employment, perspectives and barriers to growth in an enlarged EU" Final Report to the DG environment of the European Commission[END_REF], Sinclair-Desgagné (2008), European Commission Report(1999, 2007, 2009) and OECD (1999) for an analysis and description of eco-industry.

This continuum assumption is essentially introduced to simplify the treatment of an industry composed of heterogeneous agents. The same arguments hold with a …nite number of …rms.

[START_REF] David | Environmental regulation and the ecoindustry[END_REF] A discussion about the emergence of an eco-industry related to the fact that 0 (0) > 0 can be found inCanton et al. (2007).

As long as i 0 is smaller than one. This is why we introduce the min v ; max in the de…nition of this pivotal …rm.

while the demand for abatement goods is:

The eco-industry …rms Let us now study the supply of the abatement good. Each …rm i 2 [0; 1] in the eco-industry maximizes its pro…t:

the …rst-order condition of which is given by: p a 0 (a(i)) : (i) 0 (with equality if a(i) > 0)

So, if p a < min : , no abatement good is supplied while if p a max ; each …rm produces and its production level is given by a(i) = ( 0 ) 1 (p a : (i)). Finally if p a 2 [ min : ; max : ), only the …rms with an index i 1 pa produce. Hence, the aggregated supply of abatement goods is:

The abatement good market It now remains for us to study the equilibrium of the abatement good market for any given price of pollution. So let us denote by z(p a ; ) = A d (p a ; ) A s (p a ; ) the excess demand correspondence. A …rst look at this correspondence tells us that for any p a > there is always an excess supply: when the price of the abatement good is higher than , nobody is willing to buy abatement goods and therefore no equilibrium can be reached. We can now investigate whether p a = and min : < p a can each be an equilibrium. We begin by analyzing if p a = clears the market. This requires that the upper bound of the demand " ( 1 ( )) at price is higher than the supply at this price, i.e.:

This conditions is similar to condition [START_REF] Greaker | Spillovers in the development of new pollution abatement technology: a new look at the Porter-hypothesis[END_REF]. This means that there exists a threshold = v, with the property that for all implicit pollution prices v , p a = is the market clearing price of the abatement good market. With this observation we can a¢ rm that: Lemma 4. If t v, the equilibrium production levels of the aggregated market are given by Q

The equilibrium prices are p c ( ) = P ( 1 ( )) and p c a ( ) = . Moreover each e¢ cient …rm in the eco-industry is active and its production is given by a c (i; ) = ( 0 ) 1 ( (1 (i))).

APPENDIX A. Proof of Lemma 1

Step 0: Some notations.

This quantity exists and is unique since (P (Q) C 0 (Q)) is, under our assumptions, a continuous and decreasing function with the property that lim

is invertible and

and lim Q!0 (Q) = +1 and (Q max ) = 0.

Step 1: The existence of a solution Q(v; A) to program (1).

To prove this point, let us verify that we maximize (i) a strictly concave function on (ii) a domain which can be reduced to the compact convex set [0; Q max ].

(i) Let us …rst observe that R Q 0 P (q)dq C(Q) is a strictly concave function since its second derivative is given by (P 0 (Q) C"(Q)) < 0. Now, note that (" (Q) A) is convex in Q while v max fx; 0g is convex and increasing in x (for v > 0), hence their combination v max f(" (Q) A) ; 0g is convex. We therefore conclude that:

is strictly concave. (ii) By (i) of step 0, and since v max f(" (Q) A) ; 0g is non decreasing in Q, 1 (Q; A; v) decreases after Q max . We can therefore reduce the maximization domain to [0;

Step 2: The characterization of the solution Q(v; A): Even if this problem is non-smooth but nevertheless concave, we can always de…ne the subdi¤erential (see Rockafellar 1979 part V) of 1 (Q; A; v) (see Eq (29)) with respect to Q. This quantity is given by:

this one is given by:

(31) Now note that:

Thus, we deduce that:

Step 3: The computation of @SB(A;v) @A : If we replace Q by Q(v; A) in 1 (Q; A; v) (see Eq (29)) and remember, by step 0, that 8v > 0, 1 (v) < Q max , we obtain:

Moreover if we di¤erentiate this function piecewise with respect to A, we can see that:

is a continuous function (remember step 0) which can be summarized by @SB(A;v) @A = max min v; (" 1 (A)) ; 0 .

B. Proof of Lemma 2

Step 1: The solution to program [START_REF] Comoretto | Comparing pharmaceutical and pesticide loads into a small Mediterranean river[END_REF]. Let us remember that the FOCs of program (5) are given by:

It is a matter of fact to observe that if A = 0, almost all a(i) = 0, and Eq (35a) requires that v min since 0 (0) = 0 and (i) increasing. So let us us concentrate on the situations in which A > 0 and > v min . From Eq (35a), we observe (i) that for < v max , only the …rms i 2 0; 1 v produce while, for v max , each …rms is active, and (ii) that their individual production is given by ( 0 ) 1 ( v (i)). It remains to use Eq (35b) to get . This quantity is implicitly de…ned by:

Let us now note that 8(A; v) 0, (i) lim !v min 2 = A > 0, (ii) lim !+1 2 = 1 since lim a!+1 0 (a) = +1 and (iii) this function is continuous and decreasing in since:

for 6 = v: max (remember that 0 (0) = 0). It therefore exists a unique (A; v) which solves Eq (36) for each (A; v) and the optimal solution to program (5) is given by:

Step 2: The computation of @SC(A;v) @A .

is the pivotal agent so that a (i; A; v) = 0 and since (0) = 0, the second term vanishes. Moreover by Eq (35a):

Step 3: Additional results for latter use. Let us observe that for < v max we have:

It follows that:

C. Proof of Lemma 3

Step 1: There exists a unique threshold v Let us verify that:

admits a unique solution v. This result is rather immediate since:

3 (v) is continuous and decreasing since:

Moreover we know that (i), by assumption, " 0 (Q) > 0, (ii), by step 0 of the proof of lemma 1, 0 (Q) < 0, and (iii), by the range of the integral,

1 (v) since 0 (0) = 0. Using again step 0, we know that 1 (0) = Q max > 0. The result follows. lim v!+1 3 (v) < 0. By step 0 of the proof of lemma 1, we know that lim Q!0 (Q) = +1. It remains to remember that "(0) = 0 in order to conclude that :

Step 2: If v < v then the optimal abatement provision only partially reduces the emissions of the dirty …rm By contraposition, assume that the e¢ cient solution requires full pollution abatement of the dirty industry. At this optimal allocation opt and A opt veri…es:

From the …rst equation, we get that opt v and A opt " 1 (v) . It follows, from step 3 of the proof of lemma 2, that:

Now remember by Step 2 that 3 ( v) = 0. Since 0 3 (v) < 0, this implies that v v.

Step 3: If v v then the optimal abatement provision requires full pollution abatement of the dirty industry By contraposition, assume now that the e¢ cient solution requires partial abatement. At this optimal allocation opt and A opt veri…es:

Since now A opt < " 1 (v) , we can say by using the second condition and step 3 of the proof of lemma 2 that:

Now remember by Step 2 that 3 ( v) = 0. Since 0 3 (v) < 0, this implies that v < v.

D. Proof of proposition 1

Point (i): This result follows from the de…nition of the case.

Point (ii): By step 0 of the proof of lemma 1, we can say that dQ opt dv = 0 1 (v) 1 < 0 and by computation:

)

since 8i 2 0; 1 (1) we have (1 (i)) > 0 and " > 0:

Point (iii): This follows from the proof of step 1 of lemma 2 for = v

E. Proof of proposition 2

Point (i): This result follows from the de…nition of the case.

dv 0. To verify this point, let us remember that an optimal allocation is in this case de…ned by: 2 ( opt ; A opt ; v) = 0 and opt = " 1 A opt (48)

By di¤erentiation, this implies that:

and the result follows from the proof of Lemma 1 (step 0) and Lemma 2 (step 3).

Point (iii): Since opt (v) < v, the proof of the …rst part of the result directly follows from equation (38).

It remains to verify that di opt (v)

50) By point (ii) of this proof and Eq (39), we know that the …rst term of the previous equation is negative. Now let us note, by Eq (39), that the second term can be written as:

If we now replace the derivatives of 2 by its value (see step 3 of the proof of Lemma 2), we obtain:

Point (iv): In this case, the marginal social bene…t is @SB(

We can therefore say that opt (v) is implicitly de…ned by:

and this equation admits a unique solution 2 (v min ; v) since:

1 di < 0 (remember that 0 < 0, " 0 > 0 and " > 0) lim !v min 4 ( ) = " 1 (v: min ) > 0 because there is no production in the eco-industry (see the proof of Lemma 2)

In fact by Eq (46) we know that 0 2 (v; "

v) ; v) < 0 . It remains for us to observe that lim !v 4 ( ) = 2 (v; " 1 (v) ; v).

F. Proof of Lemma 4

This result directly follows from our discussion.

21

G. Proof of Lemma 5

We simply have to make sure that for all > v, there exists a unique price p c a ( ) 2 [ min : ; ] which solves 5 (p a ; ) = " 1 (p a ) A s (p a ; ) = 0. To verify this point let us observe that: (i) 8 > v, lim pa! min:t 5 (p a ; ) = " 1 ( min ) > 0 since A s ( min ; ) = 0 (see Eq [START_REF] Rockafellar | Convex Analysis Princeton landmarks in mathematics[END_REF].

(ii) 8 > v, lim pa! 5 (p a ; ) = " 1 ( ) R 1 (1) 0 ( 0 ) 1 ( (1 (i))) di < 0 by the de…nition of this case. (iii)8p a 2 ( min ; ) , @ pa 5 (p a ; ) < 0 since: @ 5 (p a ; ) @ = " 0 1 (p a ) 0 ( 1 (p a ))

(remember (0) = 0), " > 0, " 0 > 0 by assumptions and 0 < 0 by step 0 of the proof of Lemma 1.

H. Proof of Proposition 3

This result directly follows from our discussion.