Anicet B Kabre
email: kabre.anicet@parisnanterre.fr

Cobb-Douglas preferences and pollution in a bilateral oligopoly market *

Keywords: Bilateral oligopoly, Pollution, Cobb-Douglas preferences. JEL Classification: D43, D51, Q52

In this note, we introduce pollution and examine its effects in a finite bilateral oligopoly model where agents have asymmetric Cobb-Douglas preferences. We define two strategic equilibria: the Stackelberg-Cournot equilibrium with pollution (SCEP) and the Cournot equilibrium with pollution (CEP). While the supplied quantities of the polluting and the non-polluting good depend on the preferences of all economic agents in the case of symmetric preferences, we show that when preferences are asymmetric, i) at both equilibria, each polluter's equilibrium supply depends only on the non-polluters' preferences for the non-polluting good; ii) at the CEP and the SCEP, the elasticity of the polluters emissions is greater when nonpolluters preferences for the non-polluting good increase, compared to an increase in their own preferences for this good; iii) firm's emissions'elasticity decreases with the market power if their marginal cost is lower than their competitor.

Introduction

In this paper, we investigate the consequences of asymmetric Cobb-Douglas preferences on equilibrium strategies and pollution emissions level, in a bilateral oligopoly model with Stackelberg competition. To this end, we extend [START_REF] Julien | Market price mechanisms and stackelberg general equilibria: An example[END_REF]'s bilateral oligopoly model based on [START_REF] Gabszewicz | Oligopoly Equilibria in Exchange Economies[END_REF] by introducing a polluting good, and assuming asymmetric preferences. In this framework, we study two strategic equilibria in an exchange economy with production and pollution emissions. In the Stackelberg-Cournot model, we assume that a Stackelberg leader and one follower produce a non-polluting good, while other followers supply a polluting good. By contrast, in the Cournot equilibrium model, all firms set their decisions simultaneously.

Studies dealing with pollution mostly focus on partial equilibrium and are devoted to pollution permit markets. In a seminal paper, [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF] pioneered the analysis of strategic interactions in pollution permit markets. He considers two different scenarios.

In the first, it is assumed that firms sell their products on perfectly competitive markets. In the second scenario, one dominant firm is assumed to face a competitive fringe. In this case, it is shown that the permit market is cost-effective if the dominant firm's initial endowment of permits is such that he chooses not to trade. [START_REF] Westskog | Market Power in a System of Tradeable C02 Quotas[END_REF] extends [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF]'s model by considering several dominant firms and a competitive fringe in the permit market. In line with [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF], he finds that the permit market is costeffective if the dominant firms endowments of permits are such that they don't need to be exchanged. All the above mentioned studies share the assumption that the dominant firm behaves non-strategically in the final product market, and the economy embodies competitive fringe. Some firms are therefore sentenced to be price-takers.

When the input market is strategic and the final good market is competitive, [START_REF] Salop | Raising Rivals' Costs[END_REF], [START_REF] Misiolek | Exclusionary Manipulation of Markets for Pollution Rights[END_REF] show that the dominant firm would manipulate the price of the input in order to increase the production costs of its rival. Thus, these studies do not provide an answer to the strategic markets of the final goods and, consequently, do not integrate the fact that a firm use its market power on the permit market to increase its efficiency on the final goods market. [START_REF] Montero | Market power in pollution permit markets[END_REF] relax that assumption by allowing firms to compete on both the permit markets and the product market. [START_REF] Kolstad | Using Environmental Emissions Permit Prices to Raise Electricity Prices : Evidence from the California Electricity Market by[END_REF] show that electric utilities in California used the N O x market to enhance their ability to exercise (unilateral) market power in the electricity market. More recently, [START_REF] Dickson | Strategic trade in pollution permits[END_REF] study strategic trade in pollution permit market where firms decide endogenously to be buyers or sellers. They investigate the interplay between market power in the product market and the permit market equilibrium, and examine the effect of increased demand in the product market by assuming that firms have symmetric market power. They show that there is a unique equilibrium in which trade in permits takes place. Their assumption is not always appropriate because there are economies where agents display heterogeneous market powers

The oligopoly models with a finite number of traders were introduced by [START_REF] Gabszewicz | Oligopoly Equilibria in Exchange Economies[END_REF] and pursued by [START_REF] Bloch | Stable Trading Structures in Bilateral Oligopolies[END_REF], [START_REF] Bloch | Strategic complements and substitutes in bilateral oligopolies[END_REF], [START_REF] Dickson | The strategic Marshallian cross[END_REF], [START_REF] Julien | Market price mechanisms and stackelberg general equilibria: An example[END_REF]. In these models, both sides of the market are linked by a price mechanism. This mechanism was developed by [START_REF] Shapley | Trade Using One Commodity as a Means of Payment[END_REF] and refined by [START_REF] Sahi | The non-cooperative equilibria of a trading economy with complete markets and consistent prices[END_REF] and [START_REF] Amir | A strategic market game with complete markets[END_REF]. Our model is closely related to [START_REF] Julien | Market price mechanisms and stackelberg general equilibria: An example[END_REF]'s who assumes that all traders have the same preferences which can be represented by the same log linear utility function. We relax this assumption by considering that agents located on both sides of the market exhibit different preferences which are adequately captured by Cobb-Douglas utility functions. Morever, our economy enables agents to have different market power. The simple model we develop here allows for the investigation of the role of preferences on production strategies and pollution emissions in two scenarios that differ in terms of symmetric market power (Cournot equilibrium with pollution, namely CEP) and asymmetric market power (Stackelberg-Cournot equilibrium with pollution, namely SCEP).1 To the best of our current knowledge, this paper is the first to examine the role of asymmetric preferences on emissions level, in bilateral oligopoly model when firms have asymmetric market power.

This market structure has several applications in ecological economics.2

We show that: 1) at the SCEP and the CEP, the strategic supply of each polluter only depends on non-polluters preferences while the emissions level depends on the preferences of all agents; 2) at the CEP and the SCEP, the elasticity of polluters emissions is more responsive to non-polluters preferences for the non-polluting good compared to their own preferences for this good; 3) at the CEP, the marginal variation of the polluters emissions is more elevated when non-polluters preferences for the non-polluting good rises compared to an increase in their own preferences for this good; 4) at the SCEP, when polluters have a higher preference for the non-polluting good than non-polluters', the marginal variation of the leader's emissions (the follower's) is more sensitive to non-polluters (polluters) preferences for the non-polluting good compared to those of the non-polluters'. 5) when the marginal cost of the firm is low (high) compared to its competitor, the polluters emissions flexibility when their preferences for the non-polluting good raise, decreases (increases) with the market power.

The article is structured as follows. The next section outlines the model. In sections 3 and 4, we present and analyze the SCEP and CEP respectively. Section 5 provides a comparison between the SCEP and the CEP. We conclude in Section 6.

Example 2: Imagine two agri-food companies that produce a final good (good 1) with cereals, and several (n) other companies that own the cereals (good 2). It is assumed that producers are also consumers and the utility of each agent of the economy depends on the consumption of both goods. In addition, all the economic agent have Cobb-Douglas preferences that differ in the value of the preferences. The production of good 1 requires good 2 whose combustion generates pollution. We know that acting on preferences could reduce pollution. From the social planner point of view, the main questions are: in which cases should we act on producer-consumer preferences rather than pure-consumer preferences to better reduce pollution? Conversely, in which cases would an action on pure-consumers preferences reduce more pollution compared to an action on producer-consumers'preferences ?

The model

Let us consider an exchange economy with a productive sector. It consists in n + 2 traders of two types (indexed respectively by i = 1, 2 and j = 1, ....n) and two divisible commodities (A and B). Good B which is not produced, is used as an input to produce good A. p A denotes the price of good A in terms of good B so that good B is assumed to be a numeraire; i.e p B = 1. Pollution results from the processing of good B. Firms i are consumer-producers and firms j are pure consumers. The preferences of each agent are captured by the following utility functions:

U (x i A , x i B ) = α ln x i A + (1 -α) ln x i B , α ∈ (0, 1) ∀i = 1, 2 (1)
U (x j A , x j B ) = Ω ln x j A + (1 -Ω) ln x j B , Ω ∈ (0, 1) ∀j = 1, ..n (2)
Following [START_REF] Gabszewicz | Oligopoly Equilibria in Exchange Economies[END_REF], [START_REF] Julien | Market price mechanisms and stackelberg general equilibria: An example[END_REF], the initial endowments in good A and B for both types of agents are respectively given by:

w i = (0, 0), ∀i = 1, 2 (3)
w j = 0, 1 n , ∀j = 1, ..., n (4)
As in [START_REF] Gabszewicz | Oligopoly Equilibria in Exchange Economies[END_REF], an oligopolist must produce to consume. By using z i quantity of good B, an oligopolist produces a quantity y i of good A according to the linear technology:

y i = 1 β i z i , β i > 0, ∀i = 1, 2 (5)
Following [START_REF] Crettez | Tax Policy in a Simple General Oligopoly Equilibrium Model with Pollution Permits[END_REF], the use of an amount z i of the polluting input generates a quantity of emissions:

e i = 1 γ i z i , γ i > 0, ∀i = 1, 2 (6)
where γ i measures the magnitude of the pollution. From the last two equations, we express the production y i of good 1 in terms of the emissions e i and obtain:

y i = γ i β i e i , ∀i = 1, 2 (7)
Traders try to manipulate the market price through their strategic supply. Let q i denote the strategy of agent i; e i their emissions level and b j the strategy of agents j, the strategy sets for the supply of both oligopolists are:

s i = (q i , e i ) ∈ R 2 + |0 ≤ q i ≤ γ i β i e i , ∀i = 1, 2 s j = b j ∈ R + |0 ≤ b j ≤ 1 n , ∀j = 1, ...., n
Market price is then given by:

p A = n j=1 b j 2 i=1 q i = B Q (8)
Individual allocations are given by:

(x i A , x i B ) = y i -q i , B q i + q -i q i -γ i e i , i = 1, 2 (9)
(x j A , x j B ) = Q b j + B -j b j , 1 n -b j , j = 1, ...., n (10)
and yield the following indirect utility levels:

V i (q i , q -i , B) = α ln y i -q i + (1 -α) ln B q i + q -i q i -γ i e i , i = 1, 2 (11)
V j (Q, b j , B -j ) = Ω ln Q b j + B -j b j + (1 -Ω) ln 1 n -b j , j = 1, ...., n, (12)
where b = (b 1 , b 2 , ....., b n ) is the vector of equilibrium strategies of traders j and q = (q 1 , q 2 ) is the vector of equilibrium strategies of traders i.

The Stackelberg-Cournot equilibrium with pollution

In this game, polluting firms compete "à la Stackelberg"; agent 1 behaves as a Stackelberg leader with respect to the (n + 1) remaining agents. The game consists in two stages. In the first, the leader solves the following program:

(q 1 , e 1 ) ∈ arg max α ln γ 1 e 1 β 1 -q 1 + (1 -α) ln n j=1 g j (q 1 ) q 1 + f (q 1 ) q 1 -γ 1 e 1 (13)

At the second stage, the n + 1 followers simultaneously solve the following problems:

(q 2 , e 2 ) ∈ arg max α ln γ 2 e 2 β 2 -q 2 + (1 -α) ln B q 1 + q 2 q 2 -γ 2 e 2 , ∀q 1 (14) b j ∈ arg max Ω ln q 1 + q 2 b j + B -j b j + (1 -Ω) ln 1 n -b j , ∀q 1 j = 1, ..., n, . (15)
Proposition 1 : The solution of the SCEP is given by the following equilibrium strategy profiles (q 1 , q2 , b) and emissions level (ẽ 1 , ẽ2 ):

q1 = Ωβ 2 4(β 1 ) 2 φ (16) q2 = Ωφ 4(β 1 ) 2 2β 1 -β 2 (17) b j = Ω(n -1) n(n -Ω) ∀j = 1, ..., n (18)
ẽ1 = Ω(1 + α) 4γ 1 β 2 β 1 φ (19) ẽ2 = Ωαφ 2γ 2 2 - β 2 β 1 1 2 + 1 -α 2α β 2 β 1 2 - β 2 β 1 , (20)
where φ = n-1 n-Ω .

From these equilibrium strategies given in proposition 1, the market price is:

pA = 2β 1 (21)
The individual allocations are:

(x 1 A , x1 B ) = αΩβ 2 4(β 1 ) 2 φ, Ω(1 -α)β 2 4β 1 φ (22) (x 2 A , x2 B ) = x2 A , x2 B ( 23
)
x2 A = Ωφ 2 α β 2 2 - β 2 β 1 + 1 -α 2(β 1 ) 2 2 -α β 2 β 1 (24) x2 B = Ω(1 -α) 2 φ 2 - β 2 β 1 - β 2 2β 1 2 - β 2 β 1 (25) (x j A , xj B ) = Ωφ 2nβ 1 , (1 -Ω)φ n ∀j = 1, ..., n. (26)
Finally, equilibrium utility levels write as:

Ũ 1 = α ln α β 1 (1 -α) + ln Ωβ 2 φ 4β 1 + ln(1 -α) (27)
Ũ 2 = α ln x2 A + (1 -α) ln x2 B ( 28
)
Ũ j = Ω ln Ωφ 2nβ 1 + (1 -Ω) ln 1 -Ωφ n , j = 1, ..., n. (29)
We remark that:

∂ẽ 1 ∂Ω = n n-Ω α+1 4γ 1 β 2 β 1 φ; ∂ẽ 1 ∂α = Ω 4γ 1 β 2 β 1 φ; ∂ẽ 2 ∂α = Ωφ 2γ 2 2 -β 2 β 1 1 -β 2 2β 1 2 -β 2 β 1 ; ∂ẽ 2 ∂Ω = n n-Ω αφ 2γ 2 2 -β 2 β 1 1 + 1-α α β 2 2β 1 2 -β 2 β 1 .
While partial equilibrium models with pollution usually assume an exogenous (often linear) market demand function [START_REF] Hahn | Market Power and Transferable Property Rights[END_REF], [START_REF] Montero | Market power in pollution permit markets[END_REF], [START_REF] Chen | An Oligopolistic Power Market Model with Tradable NOx Permits[END_REF], [START_REF] Sanin | A Note on Clean Technology Adoption and its Influence on Tradeable Emission Permits Prices[END_REF], [START_REF] Sanin | Clean Technology Adoption Under Cournot Competition[END_REF]), the market demand in our model is endogenous and depends on agents' preferences.

Proposition 2 : In the SCEP, marginal variations of the leader and the follower emission levels are greater when non-polluters preferences for the non-polluting good (Ω) increase, compared to an increase in their own preferences (α).

The emissions level of polluters increase with their preferences for the product good, but this increase remains low compared with what we would have obtained if non-polluters preferences varied in the same proportions.

Proof: The differences of the marginal variations are

∂ẽ 1 ∂Ω -∂ẽ 1 ∂α = ẽ1 Ω(n-Ω)(1+α) [n+n(α- Ω) + Ω 2 ] > 0 and ∂ẽ 2 ∂α -∂ẽ 2 ∂Ω = φ 2γ 2 2 -β 2 β 1 (Ω -α n n-Ω ) -β 2 Ω 2β 1 2 -β 2 β 1 [Ω + n n-Ω 1-α 2 ] < 0, if Ω < α.
However, when α is high, the value obtained from ∂ẽ 2 ∂α -∂ẽ 2 ∂Ω is greater than that obtained with a small value of α, and the follower marginal variation of emissions become more and more sensitive to polluters preferences compared to those of the nonpolluters

. If α > Ω, α -Ω > 0, (α -Ω) < 1. This implies n + n(α -Ω) + Ω 2 > 0, so that ∂ẽ 1 ∂Ω -∂ẽ 1 ∂α > 0.
Morever, the emission elasticities resulting from a variation of preferences yield the following expressions:

ε ẽ1 /α = α 1 + α < 1 2 ε ẽ2 /α = 1 -β 2 2β 1 2 -β 2 β 1 1 2 1 + 1-α α β 2 2β 1 2 -β 2 β 1 1 2 < 1 ε ẽ1 /Ω = ε ẽ2 /Ω = n n -Ω > 1
The elasticity of emissions resulting from a variation of non-polluters' preferences is the same for the leader and the follower: this elasticity does not depend on market power and is greater than what recorded if non-polluters' preferences increase. Then, to reduce pollution when producers are also consumers, it is better to act on pureconsumers'preferences. However, the emissions elasticity resulting from a variation of polluters' preferences is different for the leader and the follower. Indeed,

ε ẽ2 /α < (>) 1 2 ⇒ 1-X 1+ 1-α α X < (>) 1 2 with X = β 2 2β 1 2 -β 2 β 1 1 2
. By solving this inequality, we get α 1+α < (>)X, so ε ẽ2 /α < (>) 1 2 if ε ẽ1 /α < (>)X. This result show that to reduce pollution, acting on the follower give a better result than acting on leader, if the elasticity of the leader's emission is < to a certain threshold (X).

The Cournot equilibrium with pollution

Here, polluters compete "à la Cournot". The Cournot equilibrium is obtained as the solution of the following system of simultaneous optimization programs:

(q 1 , e 1 ) ∈ arg max α ln γ 1 e 1 β 1 -q 1 + (1 -α) ln B q 1 + q 2 q 1 -γ 1 e 1 , i = 1 (30) (q 2 , e 2 ) ∈ arg max α ln γ 2 e 2 β 2 -q 2 + (1 -α) ln B q 1 + q 2 q 2 -γ 2 e 2 , i = 2 (31) b j ∈ arg max Ω ln q 1 + q 2 b j + B -j b j + (1 -Ω) ln 1 n -b j , j = 1, ..., n. ( 32
)
Proposition 3 : The solution of the CEP is given by the following strategy profiles ( q 1 , q 2 , b) and emissions level ( e 1 , e 2 ):

q 1 = Ωβ 2 (β 1 + β 2 ) 2 φ (33) q 2 = Ωβ 1 (β 1 + β 2 ) 2 φ (34) b j = Ω(n -1) n(n -Ω) = Ω n φ ∀j = 1, ..., n (35)
e 1 = Ω γ 1 β 2 (αβ 2 + β 1 ) (β 1 + β 2 ) 2 φ (36) e 2 = Ω γ 2 β 1 (αβ 1 + β 2 ) (β 1 + β 2 ) 2 φ, (37)
where φ = n-1 n-Ω . From those strategies, we deduce the market price p A = n j=1 b j 2 i=1 q i which is:

p A = β 1 + β 2 (38)
Individual allocations are

x 1 = ( x 1 A , x 1 B ) = αΩ β 1 β 2 β 1 + β 2 2 φ, Ω(1 -α) β 2 β 1 + β 2 2 φ (39) x 2 = ( x 2 A , x 2 B ) = αΩ β 2 β 1 β 1 + β 2 2 φ, Ω(1 -α) β 1 β 1 + β 2 2 φ
(40)

x j = ( x j A , x j B ) = Ωφ n(β 1 + β 2 ) , (1 -Ω)φ n . ( 41
)
and yield the following utility levels

U 1 = 2 ln β 2 β 1 + β 2 + (1 -α) ln(1 -α) + ln αΩφ -α ln β 1 (42) U 2 = 2 ln β 1 β 1 + β 2 + (1 -α) ln(1 -α) + ln αΩφ -α ln β 2 (43) U j = Ω ln Ωφ (β 1 + β 2 ) n + (1 -Ω) ln(1 -Ωφ) + (Ω -1) ln n. (44)
The elasticity are given by:

ε e 1 /α = αβ 2 αβ 2 + β 1 < 1 ε e 1 /Ω = ε e 2 /Ω = n n -Ω > 1
Polluters react more to an increase in non-polluters' preferences than to an increase in their own preferences.

Comparison of equilibrium outcomes

We now proceed with the comparison of equilibrium outcomes in the SCEP and CEP. The strategic supplies of goods A and B exclusively depend on non-polluters preferences while emissions level depend on the preferences of all agents (polluters and non-polluters).

Proposition 4 : In the SCEP and CEP, when preferences are asymmetric, agents' supplies only depend on the non-polluters preferences. But, when preferences are symmetric, equilibrium supplies depend on the preferences of all agents.

Proof: If preferences are asymmetric,

∂ q i ∂Ω = n n-Ω β -i (β i +β -i ) 2 > 0; ∂ q1 ∂Ω = n n-Ω β 2 4(β 1 ) 2 φ > 0; ∂ q2 ∂Ω = n n-Ω φ 4(β 1 ) 2 [2β 1 -β 2 ] > 0; ∂ q i ∂α = ∂ q1 ∂α = ∂ q2 ∂α = 0; ∂b j ∂α = 0.
However, when all agents have the same utility function U

(x h 1 , x h 2 ) = α ln x h 1 + (1 -α) ln x h 2 , α ∈ (0, 1) ∀i = 1, 2, ∀j = 1, 2, ....n, the strategic supplies are given by q 1 = αβ 2 (β 1 +β 2 ) 2 φ; q 2 = αβ 1 (β 1 +β 2 ) 2 φ; b j = α(n-1) n(n-α) j = 1, ..., n.
We now focus on the effect of a change in agent preferences on the level of emissions. Indeed we have:

∂ e i ∂Ω = n n-Ω β -i (αβ -i +β i ) γ i (β -i +β i ) 2 φ > 0 and ∂ e i ∂α = 1 γ i Ω(β -i ) 2 (β -i +β i ) 2 φ > 0.
The emissions level of a relatively inefficient firm (with a high value for β) is less sensitive to a variation in preferences compared to that of a relatively efficient firm: indeed, if

β -i < β i , 1 γ i Ω(β -i ) 2 (β -i +β i ) 2 φ < 1 γ -i Ω(β i ) 2 (β -i +β i ) 2 φ and 1 γ -i β -i (αβ -i +β i ) (β -i +β i ) 2 φ < 1 γ i β i (αβ i +β -i ) (β -i +β i ) 2 φ.
Then, the difference in the marginal variation of emissions level resulting from a variation of the preferences is represented by:

∂ e i ∂α - ∂ e i ∂Ω = β -i φ γ i (n -Ω)(β i + β -i ) 2 β -i (n(Ω -α) -Ω 2 ) -nβ i (45)
Proposition 5 : At both equilibria, if α > Ω, polluters' marginal variations of emissions remain more elevated when non-polluters preferences increase compared with an increase in their own preferences. However, at the CEP, the polluter is more sensitive to their own preferences rather than non-polluters preferences if β -i β i > n n(Ω-α)-Ω 2 . At both equilibria, the polluters' emissions elasticity is greater when non-polluters' preferences increase, compared to an increase recorded when polluters' preferences increase.

Proof: ∀β i , if α > Ω, ∂ e i ∂α -∂ e i ∂Ω < 0. In addition, solving ∂ e i ∂α -∂ e i ∂Ω > 0, needs

β -i (n(Ω -α) -Ω 2 ) -nβ i > 0 ie β -i β i > n n(Ω-α)
-Ω 2 ; the ratio of marginal costs remaining positive and

1 n(Ω-α)-Ω 2 being negative (α > Ω).
Remark 1: When α > Ω, this condition is sufficient to yield ∂ e i ∂α -∂ e i ∂Ω < 0. This indicates that a higher preference of polluters for their own good, their level of emissions increases more with non-polluters preferences compared to the increase observed if their own preferences varied.

Proposition 6 : An increase in non-polluters preferences for the non-polluting good affects more the polluter's marginal variation of emissions if their market power is sufficiently high. However, when their market power decreases, polluters preferences have no impact.

Proof: At the SCEP, if α > Ω, we have ∂ẽ 1 ∂Ω -∂ẽ 1 ∂α = ẽ1 Ω(n-Ω)(1+α) [n+n(α-Ω)+Ω 2 ] > 0 and ∂ẽ 2 ∂α -∂ẽ 2 ∂Ω = φ 2γ 2 2 -β 2 β 1 (Ω -α n n-Ω ) -β 2 Ω 2β 1 2 -β 2 β 1 [Ω + n n-Ω 1-α 2 ] < 0. However with α > Ω, at the CEP we get ∂ e 2 ∂α -∂ e 2 ∂Ω = β 1 φ γ 2 (n-Ω)(β 2 +β 1 ) 2 [β 1 (n(Ω -α) -Ω 2 ) -nβ 2 ] > 0 because β 1
β 2 is always positive.

Remark 2: If polluters market power is below a certain threshold, their emissions' marginal variation is higher when their own preferences vary compared to a variation in that of non-polluters. From this threshold, their emissions becomes more responsive to non-polluters preferences. Moreover, at the CEP, the impact on polluters' emissions level resulting from a variation of their preferences may be identical to that which would be recorded if the preferences of non-polluters varied in the same proportion. However, if the same firm competed at the SCEP as a leader, this coincidence could not be observed. Indeed,

∂ e i ∂α -∂ e i ∂Ω = 0 if β i β -i = n(Ω-α)-Ω 2 n while ∂ẽ 1 ∂Ω -∂ẽ 1 ∂α = 0 because the value recom- mended for α is greater than 1, i.e α = 1 + Ω -Ω 2 n < 0.
The ratio of marginal costs remaining positive, ∂ e i ∂α -∂ e i ∂Ω = 0 if and only if Ω > α.

Remark 3: The emission elasticity does not depend on firms' marginal costs (β i and β -i ) when Ω increases while it depends on it when α increases. At the SCEP as at the CEP, when firms have the same marginal cost, they have also the same vulnerability in their emission. At the SCEP as at the CEP, polluting firms react in the same way to a change in the preferences of non-polluting agents (ε e i /Ω = ε ẽi /Ω ), and this value is greater than what recorded with a variation in their own preferences. 3 However, at the CEP, ε e 1 /α > 1 2 if and only if β i β -i > 1 while at the SCEP, ε ẽ1 /α is always < 1 2 , ∀β i , β -i .

Proposition 7 : If β i < β -i , firm's emission elasticity decreases when their market power increase. However, if β i > β -i , firm's emission elasticity increases when their market power increases.

Proof : Indeed, ε e 1 /α = α β 1 β 2 +α and ε ẽ1 /α = α 1+α ⇒ ε e 1 /α > ε ẽ1 /α if β 1 < β 2 .
Also, ε ẽ2 /α < ε e 2 /α if and only if

β 1 β 1 +β 2 < β 2 2β 1 2 -β 2 β 1 1 2
. This is equivalent to solve

β 2 2β 1 β 1 +β 2 (β 1 ) 2 2 -β 2 β 1 1 2 > 1. • If β 1 > β 2 , ie β 2 β 1 < 1, we get β 2 2β 1 < 1 2 , 2 -β 2 β 1 1 2 < √ 2 : this yields β 2 2β 1 β 1 +β 2 (β 1 ) 2 2 -β 2 β 1 1 2 < 1 ⇒ ε e 2 /α < ε ẽ2 /α .
• If 1 < β i β -i < 2, ie β i > β -i , then we get ε e 2 /α > ε ẽ2 /α i = 2 ε ẽ1 /α > ε e 1 /α i = 1

• When β 1 = β 2 , we get ε ẽ1 /α = ε ẽ2 /α = ε e i /α = α 1+α .

Finally, we determine the conditions under which the two equilibria coincide.

Proposition 8 : When β 1 = β 2 = β and γ i = γ, the SCEP coincides with the CEP.

Remark 4: Morever, ∂ẽ 1 ∂Ω -∂ẽ 1 ∂α = ∂ẽ 2 ∂Ω -∂ẽ 2 ∂α = ∂ e i ∂Ω -∂ e i ∂α = φ 4γ(n-Ω) [n+n(α-Ω)+Ω 2 ] > 0.

Proof: If β i = β and γ i = γ, we get q1 = q2 = q i = Ω 4β φ and ẽ1 = ẽ2 = e i = Ω(α+1) 4γ φ.

Conclusion

In this note, we pionneered the analysis of asymmetric preferences on pollution in bilateral oligopoly market by extending the model of [START_REF] Julien | Market price mechanisms and stackelberg general equilibria: An example[END_REF]. We set out to examine the consequences of asymmetric preferences on emissions when polluters have asymmetric and symmetric market power. Three results can be emphasized. First, the supply of each polluter depends on the non-polluters' preferences. Second, when the firm's market power increases, their emission increases or decreases according to the relationship between its marginal cost and that of its competitor. Third, when producers are also consumers, an action on pure-consumers'preferences reduce more the pollution, and this regardless of the producers' market power. From the point of view of the social planner, to reduce pollution it is more benefit to act on firms with big market power rather than small firms. This study can be extended to a dynamic context and in particular to the dynamics of government health expenditures.

Environmental externalities are not studied in this paper. Also, issues related to the existence and uniqueness of oligopolistic equilibrium in general equilibrium models are beyond the scope of this paper (due to problems raised in[START_REF] Gabszewicz | Strategic Multilateral Exchange: General Equilibrium with Imperfect Competition[END_REF]).

Example 1: Consider a polluting company (for example McDonald's, a fish processing company) owned by several shareholders who have different market (decisional) powers. These shareholders consume another good (meat or fish) used as an input to produce the final good (burgers or canned fish). All agents of the economy consume the two goods. In such a context, we describe how the agents' preferences for the non-polluting good would affect strategies and, discuss climate change mitigation by asking how to act effectively on these preferences to alleviate pollution, and thus preserve the environment.

This result is not obvious when producers consume the good they produce.


Tim Friehe
email: tim.friehe@uni-marburg.de

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

Elisabeth Schulte
email: elisabeth.schulte@uni-marburg.de.

On consumer preferences for (partial) products liability

Keywords:

Traditional law and economics analyses of products liability …nd that di¤erent liability regimes lead to the same market outcome, implying that risk-neutral consumers are indi¤erent between products liability and no products liability. We present a setup in which a group of consumers supports the implementation of products liability although its enforcement is costly. All consumers may prefer the same level of (partial) products liability.

Both the size and the scope of the potential liability have increased over the past …fty years (e.g., [START_REF] Rubin | Public Choice and Tort Reform[END_REF]. Potential explanations are an optimal adaptation of the legal system to changing circumstances (e.g., [START_REF] Landes | A Positive Economic Analysis of Products Liability[END_REF] or rent seeking. Traditional law and economics analyses of products liability …nd that di¤erent liability regimes lead to the same market outcome in terms of product safety and output (e.g., [START_REF] Daughety | Market Structure, Liability, and Product Safety[END_REF]. This irrelevance result practically excludes rent seeking on part of the consumers as an explanation for the above observation. The very small literature on the political economy of products liability so far focuses particularly on lawyers'e¤orts in changing the structure of the law [START_REF] Epstein | The Political Economy of Product Liability Reform[END_REF][START_REF] Rubin | The Role of Lawyers in Changing the Law[END_REF][START_REF] Rubin | Public Choice and Tort Reform[END_REF]).

We present a setting in which products liability is costly and show that, nevertheless, some consumers strictly bene…t from products liability, and in some constellations all consumers do. Our formal analysis can support the informal discussion in [START_REF] Osborne | What's yours is mine: Rent-seeking and the common law[END_REF] that recent developments of products liability may be largely due to e¤orts by consumers.

The model

Consumers Consumers may purchase a product that generates gross consumption ben-e…t v. Consumers may su¤er harm h after an accident due to the consumption of the product. The probability of an accident is 1 x, where x denotes (observable) product safety. Consumers may sue the …rm under strict liability for damages amounting to h, thereby incurring litigation costs `where `is distributed according to G(`) on [0; L]. We assume L < h, 2 [0; 1], and g 0 (`) 0.

Firm A monopolist chooses the product safety x and the price p. Producing the product with a safety level x implies a per-unit cost of c(x), with c 0 ; c 00 > 0. In case the …rm is sued by a consumer after an accident, it incurs litigation costs y and pays damages h.

Timing

In Stage 1, the …rm chooses x and p. In Stage 2, consumers make their purchasing decisions and accidents may occur. Consumers who had an accident decide whether or not to sue the …rm in Stage 3.

E¢ ciency

We assume v > h + y + c(0) which guarantees that the …rm always wants to serve the market. The allocation is e¢ cient if the entire market is served and if the level of product safety maximizes the joint surplus. Products liability exhibits a built-in ine¢ ciency because its enforcement is costly.

The analysis

Stage 3 After an accident, consumers with ` h sue the …rm for compensation h. We will refer to consumers who (do not) sue the …rm as type S (N ) consumers.

Stage 2 Consumers S obtain an expected net bene…t

U S (x; p; ) = v p (1 x)h + (1 x)[ h `] (1)
from consuming the product. Consumers N obtain an expected net bene…t

U N (x; p; ) = v p (1 x)h = U S (x; p; ) (1 x)[ h `]: (2)
Consumers purchase the product only if U J 0, J = N; S.

Stage 1 The …rm maximizes expected pro…ts using p and x. The …rm may select to serve either consumers of type S and N (Scenario 1 ) or only consumers of type S (Scenario 2 ).

Scenario 1: If the …rm seeks to attract all consumers, it can charge a maximum price of P max (x) = v (1 x)h, which is increasing with product safety at the rate h. Charging P max , the …rm's pro…ts are

The …rst-order condition for the pro…t-maximizing level of product safety x in this scenario reads

@ 1 @x = h c 0 (x ) + G( h)( h + y) = 0: (4)
For (>) = 0, x is (in)e¢ cient. The …rm e¤ectively considers own litigation costs and that of the marginal consumer just indi¤erent between suing and not suing (i.e., h), where the latter exceeds expected litigation costs incurred by suing consumers.

Scenario 2: If the …rm chooses p > P max , it attracts only consumers of type S. For any combination of price and product safety, the indi¤erent consumer is identi…able via the level of litigation costs `C stemming from

v p (1 x)[(1 )h + `C] = 0: (5)
This allows us to consider that …rm e¤ectively chooses `C( h) (instead of p) and x to maximize

2 (`) = G(`C)(v (1 x)(h + y + `C) c(x)): (6)
The level of damages enters the optimization problem only via the constraint. Denote with ^; x the pro…t-maximizing levels of the marginal consumer's litigation costs and safety level, respectively. The former is characterized by 7) is smaller than h and the second-order-conditions are satis…ed. Otherwise, ^= h, or, if h > L, ^= L. The pro…t-maximizing product safety is identi…ed by

@ 2 @`C = g(`C)(v (1 x)(h + y + `C) c(x)) G(`C)(1 x) = 0 (7) if `C solving (
@ 2 @x = G( ^)(h + y + ^ c 0 (x)) = 0; (8)
and, again, is ine¢ ciently high since the …rm incorporates the marginal consumer's litigation costs. In addition, output is ine¢ ciently low if ^< L.

Choice between the scenarios: The …rm chooses between Scenarios 1 and 2 for a given level of . In Scenario 1, the maximal level of pro…ts amounts to and changes with according to

d 1 d = g( h)h(1 x ( ))( h + y) G( h)(1 x ( ))h < 0: (10)
In Scenario 2, the monopolist's pro…ts are increasing in for ^=h and constant for

> ^=h. The product is o¤ered at price p = v (1 x)h + (1 x)( h ^) P max . If ^= h, p = v (1 x)h.
At this price, type N -consumers are willing to buy, too. Hence, for ^=h, the monopolist is clearly better o¤ in Scenario 1.

As 1 ( ) is strictly decreasing in and 2 ( ^) is constant for > ^=h, it may be optimal for the …rm to switch to Scenario 2 at some level of ^=h. Consider = L=h, and note

that 1 (L=h) = 2 (L) 2 ( ^).
There is a threshold 0 L=h such that the …rm strictly prefers Scenario 2 over Scenario 1 for > 0 and vice versa for < 0 .

Consumers' liability preferences Consumers consider their payo¤ as a function of , anticipating how the …rm's instruments adapt to changes in . We assume that consumers know their litigation costs such that they can anticipate their type (i.e., N or S) at speci…c values of . Thereafter, we comment on consumers'liability preferences when consumers are under the veil of ignorance regarding their level of litigation costs.

For Scenario 1 (i.e., for 0 ), the condition (4) yields

dx d = g( h)h( h + y) + G( h)h c 00 (x) > 0: (11) dP max d = h dx d > 0: (12)
With these results at hand, we can consider how consumers'equilibrium payo¤s

U 1 N =0 (13) U 1 S ( ; `) =(1 x ( ))( h `) (14)
change with the level of damages as long as the …rm adopts Scenario 1:

dU 1 N d =0 (15) dU 1 S d = dx d [ h `] + (1 x )h: (16)
Consumers of type N are una¤ected by a variation of as the monopolist always adjusts the price to extract their rent. Type N consumers with `close to h become type S consumers as increases. Consumers of type S obtain a rent. An increase in directly increases their compensation after an accident, but also reduces the probability of being compensated. At low levels of , the former e¤ect unambiguously dominates the latter. In particular, ( 16)

is strictly positive at = 0. Consequently, for low levels of , some consumers support an increase in and no consumer objects. If the direct e¤ect in ( 16) dominates the indirect e¤ect for all levels of , all consumers of (potential) type S prefer the highest implementable in Scenario 1. The direct e¤ect dominating the indirect one can be ensured by having product safety be su¢ ciently unresponsive to changes in , which essentially requires that c(x) is su¢ ciently convex. Assuming that this holds, we obtain that all consumers with `< 0 h have a common bliss point at = 0 .

In Scenario 2 (i.e., for > 0 ), neither …rm's pro…ts nor the level of demand or product safety are a¤ected by a variation of as the share (1 )h directly borne by the consumer will complement the expected harm h directly borne by the …rm in terms of the …rm's price-cost margin. Whereas consumers with litigation costs ` ^always have a zero rent, consumers with litigation costs `< ^earn

U 2 S = (1 x)( ^ `) (17)
where both x and ^are independent of , implying that the rent is independent of the level of damages.

We distinguish two cases with respect to the threshold 0 at which the …rm switches from Scenario 1 to Scenario 2, 0 < L=h, and 0 = L=h. In the latter case, ^= L and x = x (L=h), such that at = L=h, U 2 S = U 1 S for all `. If 0 < L=h, consumers with `2 (`C; 0 h) lose their rent as increases beyond 0 . For consumers with `< `C, there are two e¤ects on their rent as increases beyond 0 , amounting to a di¤erence in their payo¤ of:

(1 x ( 0 ))( 0 h `) (1 x)(`C `). If x x ( 0 )
, all consumers with `< `C lose part of their rent as increases beyond 0 . Otherwise, consumers with `< (>)`C ( 0 h `C)(1 x ( 0 ))=(x ( 0 ) x) are positively (negatively) a¤ected by an increase in above 0 .

To summarize: All consumers weakly prefer some positive level of products liability to no products liability. If c(x) is su¢ ciently convex such that x is not elastic to changes in , all consumers with `< 0 h strictly prefer 0 to lower levels of , and all other consumers are indi¤erent. If x is elastic to changes in , consumers with low litigation costs may prefer < 0 , while consumers with higher litigation costs prefer higher levels. If the …rm does not decrease product safety as it starts serving only consumers S (for > 0 ), all consumers with `< 0 h strictly prefer 0 to higher levels. Again, all other consumers are indi¤erent. Levels higher than 0 are preferred only if the …rm reduces product safety when serving only consumers of type S and only by consumers with low litigation costs.

Under the veil of ignorance -i.e., when litigation costs are revealed only after the preferences regarding have been formed but prior to the buying decision -all consumers agree on the preferred level of . They strictly prefer some positive level of over no products liability.

The possibility to sue for compensation after an accident bene…ts consumers with litigation costs below the marginal consumer's litigation costs. In contrast, consumers receive no rent without products liability. With products liability, consumers are sometimes better o¤ and never worse o¤ than with no products liability. When all consumers weakly prefer 0 to all other values for , it follows that all consumers strictly prefer 0 under the veil of ignorance.

Conclusion

We present a simple framework that describes consumer preferences for products liability, …nding that some consumers support and no consumer objects its introduction despite its ine¢ ciency. We sketch a constellation in which all consumers prefer the same partial compensation of losses. In our framework, products liability reduces e¢ ciency due to litigation costs, distorted product safety levels, and a possible contraction of output.

Our analysis is limited in several ways. We abstract from the preferences of lawyers as this has been the subject of previous research. It can be expected that lawyers will support consumers in their quest for products liability. In addition, we consider a market with a monopolistic supplier, where rents are redistributed from the …rm to consumers. In a simple setting in which competition induces prices equal to expected total costs (including expected liability payments), some consumers gains at the expense of other consumers (and at a loss of e¢ ciency). An interior value of is preferred by the bene…ciaries of this redistribution if exceeding that value induces the …rm to stop delivering the market. More elaborate analyses are left for future research.

Figure 1 :

 1 Figure 1: The monopolist's output as a function of in Scenario 1 (left) and in Scenario 2 (right)

= v (1 x)h c(x) G( h)(1 x)( h + y): (3)

( ) = v (1 x ( ))h c(x ( )) G( h)(1 x ( ))( h + y); (9)