Régis Breton
email: regis.breton@univ-orleans.fr

Bertrand Gobillard
email: bertrand.gobillard@ens.fr

Robustness of equilibrium price dispersion in finite market games

Keywords: Strategic market games, law of one price, equilibrium selection. JEL Classification: C72, D43, D50

We propose an approach to restricting the set of equilibria in a strategic market game and use it to assess the robustness of the price dispersion results obtained by Koutsougeras [2003, J. Econ. Theory 108, 169-175] in the multiple trading posts setup. More precisely, we perturb the initial game by the introduction of transaction costs and our main results are the following. (i) No equilibrium with price dispersion of the game with costless transactions can be approached by equilibria with positive transaction costs as costs get arbitrarily small. (ii) When this type of perturbation is considered the set of equilibrium outcomes is not affected by the number of trading posts.

Resumé

Nous proposons une approche permettant de réduire l'ensemble des équilibres dans un jeu stratégique de marché, et l'appliquons pour étudier la robustesse du résultat de non uniformité des prix obtenu par Koutsougeras [2003, J. Econ. Theory 108, 169-175] dans le jeu de marché à postes d'échanges multiples. Plus précisément, nous perturbons le modèle initial par l'introduction de coûts de transaction positifs et nos résultats principaux sont les suivants. (i) Aucun équilibre avec dispersion de prix du jeu sans coûts de transaction ne peut être approché par des équilibres avec coûts (strictement) positifs tendant vers zéro. (ii) Lorsque ce type de perturbations est considéré, l'ensemble des allocations d'équilibres est invariant avec le nombre de postes d'échanges.

Introduction

In any economic model, the assumption that trade is costless is to be considered as a simplification. One generally expects that the model's predictions hold approximately for small but positive transaction costs. Formally, the approximation is warranted whenever there is some form of continuity as transaction costs vanish. In contrast, a property that does not hold-even approximately-once arbitrarily small transaction costs are explicitly modeled may be viewed as artificial.

In this paper, we apply this argument to qualify recent results about equilibrium price dispersion in strategic market games. To be precise, we consider the multiple trading posts per commodity variant of two canonical market games (see below). As shown by [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF]Koutsougeras ( , 2003b) ) this framework exhibits-along with equilibria with uniform prices-equilibria where prices are not equalized among posts where the same good is traded. Hence, the 'law of one price' may fail in an exchange economy with costless trade. 1We show that this striking result as well as related properties of the multiple posts setup are not immune to the introduction of arbitrarily small transaction costs. More precisely we perturb the initial game by the introduction of (a simple form of) transaction costs and we obtain two sets of results. Regarding price dispersion, we first show that when transaction costs are positive, any equilibrium must satisfy the law of one price. Our main result then states that an equilibrium with price dispersion of the game with costless transactions cannot be approached by equilibria with positive transaction costs as costs get arbitrarily small-in short, we say that such an equilibrium is not "robust".

We also show that the set of robust outcomes-viz, those outcomes associated with robust equilibria-does not depend on the number of trading posts. This investigation is motivated by the analysis in Koutsougeras (2003a) indicating that the emergence of equilibria with dispersed price is the only difference between the single and the multiple posts variants. In contrast, our irrelevance result suggests that there is no loss of generality in working with the canonical, single trading post market game.

The intuition for why (small) transaction costs restaure the law of one price is as follows. As shown by [START_REF] Gobillard | Wash-sales and the law of one price in multiple trading posts per commodity market games[END_REF], 2 the result of Koutsougeras (2003b) relies on agents placing 'wash-sales' trades, that is bids and offers that cancel each other on a same trading post. Although wash-sales trades matter for the strategic equilibria because they affect the relative thickness of trading posts, any agent is indifferent as to the amount of his own wash-sales (see [START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF][START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF][START_REF] Peck | The market game: existence and structure of equilibrium[END_REF][START_REF] Peck | The market game: existence and structure of equilibrium[END_REF]. In other words, an agent's allocation only depends on his net trades. With strictly positive transaction costs, agents also care about their gross trades, and never choose to be active on both sides of a given post (lemma 1). In the limit where transaction costs vanish, this leads to the selection-among the best response strategies-of the unique strategy minimizing gross trade, that is that without wash-sales. One consequence is that price dispersion is no longer compatible with equilibrium.

We derive our results for two contrasting frameworks: the multiple trading posts extension of the inside money market game of [START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF], and that of the commodity money market game à la [START_REF] Shapley | Trade using one commodity as a means of payment[END_REF] and [START_REF] Dubey | A theory of money and financial institutions. the non-cooperative equilibria of a closed economy with market supply and bidding strategies[END_REF]. Examples of equilibria where the law of one price fails are given by [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF] for the latter setup, and by Koutsougeras (2003a,b) for the former. The key difference between both setups is that in the commodity money case agents' trading strategies are constrained by their money holding. Our result shows that liquidity constraints per se do not induce price dispersion.

Our specification borrows from [START_REF] Rogawski | A strategic market game with transactions costs[END_REF], who introduce transaction costs paid in commodities in the bid-offer market game of [START_REF] Dubey | A theory of money and financial institutions. the non-cooperative equilibria of a closed economy with market supply and bidding strategies[END_REF]. Their main concern is the existence of an equilibrium with active trade when transaction costs are not too prohibitive. Instead, we use transaction costs to define perturbed games and to reduce the set of equilibria. We choose the simplest specification-homogeneous linear costsnecessary to make our point.

From a more general perspective, the paper proposes a natural approach to restricting the set of equilibria in a market game. By eliminating the indeterminacy of best responses associated with wash-sales, our argument cuts down one source for the multiplicity of (Nash) equilibria in market games [START_REF] Peck | The market game: existence and structure of equilibrium[END_REF].3 So far, equilibrium selection in market games has centered around a distinct issue, that is that any market can be active or inactive in equilibrium. In particular, the trivial Nash equilibrium in which all bids and offers are zero always exists [START_REF] Shapley | Non cooperative general exchange[END_REF]. Following [START_REF] Dubey | A theory of money and financial institutions. the non-cooperative equilibria of a closed economy with market supply and bidding strategies[END_REF], one popular approach has been to consider equilibria obtained as the limit of perturbed games in which some outside agency places vanishingly small bids and offers on each trading post. 4 More recently, [START_REF] Matros | Evolution and walrasian behavior in market games[END_REF] use some strong notion of evolutionary stability to rule out equilibria in which (some) markets are inactive.

The paper is organized as follows. In section 2 we introduce the framework and some definitions. We prove the results for the market game with inside money in section 3. In section 4 we show that our results extend to the market game with commodity money. Some general comments are drawn in section 5. One proof is relegated to an appendix.

General setting and definitions

We consider an exchange economy with a finite set H of agents, indexed by h = 1, ..., H, and L + 1 goods, indexed by i = 1, ..., L + 1. Commodity L + 1 represents money. Each agent h ∈ H has endowment e h ∈ IR L+1 + , and preferences described by a utility function u h : IR L+1 + -→ IR + defined over consumption bundles. We require u h to be continuous, strictly increasing in the consumption of goods 1, . . . , L and non decreasing in money.

Trade is organized as follows. For any commodity i = 1, ..., L, there are K i ≥ 1 trading posts where good i is exchanged for money (good L + 1). Trading posts are indexed by (i, s), where s = 1, ..., K i . We let K = L i=1 K i denote the aggregate number of trading posts in the economy.

An agent's strategy, σ h , specifies for each trading post a non negative offer of commodity, q i,s h , and a non negative bid in term of money, b i,s h . Let S h denote the strategy set of agent h, and S = S 1 × . . . × S H the set of strategy profiles, with generic element σ = (σ h ) h∈H . To single out the strategy of a given agent h, we will sometimes write the strategy profile as (σ h , σ -h ).

Given a strategy profile σ ∈ S, define

B i,s = H b i,s h and Q i,s = H q i,s h ,
and, for a given h ∈ H,

B i,s h = H\{h} b i,s h and Q i,s h = H\{h} q i,s h .
On trading post (i, s), prices are formed according to the standard Shapley-Shubik rule:

p i,s =      B i,s /Q i,s if Q i,s = 0, 0 otherwise. ( 1
)
Subsequently we use the convention 1/p i,s = 0 whenever p i,s = 0. We follow [START_REF] Rogawski | A strategic market game with transactions costs[END_REF] in modeling transaction costs as consuming part of the commodities offered in transaction. We further restrict ourselves to the following linear and homogeneous specification. When an agent offers a quantity q i,s h of commodity i on post (i, s), an additional quantity εq i,s h (ε ≥ 0) is needed in order to place that offer. 5 The assumption that transaction costs do not depend on bids b i,s h is a simplification.

The description of the mapping from strategy profiles to final allocations is deterred to sections 3 and 4 because the details slightly differ between both settings. For ε = 0, our market games reduce to those analyzed in Koutsougeras (2003b) and [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF]. We will refer to the initial market game with inside money (respectively commodity money) as Γ (resp. Γ ) and to the game with a given ε > 0 as Γ ε (resp. Γ ε ).

The rest of this section is devoted to the formal statement of the law of one price, and to our robustness requirement.

Definition 1 A trading post (i, s) is active if p i,s > 0, or equivalently B i,s > 0 and Q i,s > 0.
Definition 2 A strategy profile σ ∈ S satisfies the law of one price if it induces-for any good i = 1, . . . , L-prices that are uniform across active trading-posts:

p i,s p i,r > 0 =⇒ p i,s = p i,r ∀ i, r, s.
We say that a Nash equilibrium of the initial game is robust if it can be approached by equilibria of the perturbed games with strictly positive transaction costs as transaction costs vanish. Formally,

Definition 3 A Nash equilibrium σ of the game Γ is "robust" if there exists a sequence { n ε, n σ} ∞ n=1 where n ε ∈ IR + and n σ (∈ S) is a NE of the perturbed games Γn ε such that lim n→∞ n ε = 0 and lim n→∞ n σ = σ.
3 The market game with inside money

In this section, we derive our results for the market game with inside money introduced by [START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF], and extended to multiple trading posts by Koutsougeras (2003b).

Good L + 1 is an inside money with no direct utility. Agents have no initial money endowment, e L+1 h = 0, and can issue inside money at no cost. To avoid over issuance, it is postulated that an agent that goes bankrupt-that is whose (monetary) gains from sales do not cover his bids-has all his bids and offers confiscated (see [START_REF] Peck | The market game: existence and structure of equilibrium[END_REF] for a discussion).

Formally, agent h chooses his strategy σ h in the set6

S h = b i,s h , q i,s h ∈ IR 2K + | K i s=1 (1 + ε) q i,s h ≤ e i h ,
and, given others' strategies, does not go bankrupt whenever

D h (σ h , σ -h ) := L i=1 K i s=1 b i,s h - L i=1 K i s=1 q i,s h p i,s ≤ 0. (2)
Final allocations are then determined as follows:

x i h (σ) =          e i h -(1 + ε) K i s=1 q i,s h + K i s=1 b i,s h /p i,s if (2) holds, e i h -(1 + ε) K i s=1 q i,s h otherwise.
(3)

It easily follows from (3) that (2) holds with equality at the optimum for h. With a slight abuse in notation, a Nash equilibrium for Γ ε is a strategy profile σ ∈ S such that

∀h ∈ H u h (σ h , σ -h ) = sup σh ∈S h u h (σ h , σ -h ) .

Intermediate results

We start with one intermediate result stating that when ε > 0 it is never a best reply to buy and sell on the same trading post.

Lemma 1 Let ε > 0. Any individual best reply in Γ ε satisfies b i,s h • q i,s h = 0.
PROOF. Assume the contrary, viz b i,s h > 0 and q i,s h > 0 for a candidate best reply σ h . We construct a profitable deviation σh by subtracting a small amount of wash-sales (conveniently defined) on post (i, s). Formally, for η > 0 consider the deviation with bi,s h = b i,s h -η and qi,s h defined by

-q i,s h + bi,s h Q i,s h + qi,s h B i,s h + bi,s h = -q i,s h + b i,s h Q i,s h + q i,s h B i,s h + b i,s h . ( 4
)
For η > 0 small enough, this deviation is admissible. Indeed, b i,s h > 0 and q i,s h > 0 imply that we can choose η > 0 such that bi,s h > 0 and qi,s h > 0. Noting that qi,s h < q i,s h from (4), we have σh ∈ S h . Now σh satisfies condition (2) because substituting σh for σ h affects neither prices nor the (net) commodity bundle obtained through trading. 7 Formally, straightforward manipulations show that D h (σ h , σ -h ) = D h (σ h , σ -h ). Furthermore, by playing σh , agent h gets the same allocation

x j h (σ h , σ -h ) = x j h (σ h , σ -h
) for all goods j = i, and

x i h (σ h , σ -h ) -x j h (σ h , σ -h ) = ε q i,s h -qi,s h > 0. The contradiction follows. 2
When ε = 0, the proof of lemma 1 states the standard result that agent h can obtain his preferred consumption bundle through a continuum of best reply strategies parameterized by the amount of wash-sales (see, e.g., lemma 1 in [START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF].

The next proposition states a necessary condition for best response strategies to be compatible with different prices for a given commodity.

Proposition 1 Let ε > 0. Consider a candidate equilibrium of Γ ε with p i,s > p i,r > 0. Consider the set B (i, s) := h ∈ H | b i,s h > 0 of (equilibrium) bidders on post (i, s). Then B i,s h B i,s h + b i,s h > B i,r h B i,r h + b i,r h ∀h ∈ B (i, s) . ( 5
)
PROOF. The proof proceeds by contradiction. Consider h ∈ B (i, s) such that

B i,s h B i,s h + b i,s h ≤ B i,r h B i,r h + b i,r h . ( 6
)
First note that q i,s h = 0 by lemma 1. We consider a deviation shifting a small amount of money from (i, s) to (i, r). For η > 0, consider the deviation σh defined by the substitution of bi,s

h = b i,s h -η and bi,r h = b i,r h + η for b i,s h and b i,r
h in σ h . This is well defined for η > 0 small enough, because b i,s h > 0. We first check that σh satisfies the no bankruptcy condition D h (σ h , σ -h ) ≤ 0. By the definition of equilibrium, we have that D h (σ h , σ -h ) = 0. Using qi,s h = q i,s h = 0 and qi,r h = q i,r h , straightforward manipulation then yields

D h (σ h , σ -h ) = D h (σ h , σ -h ) + q i,r h p i,r -pi,r . (7)
Now,

p i,r -pi,r = B i,r h + b i,r h Q i,r - B i,r h + bi,r h Q i,r = b i,r h -bi,r h Q i,r = - η Q i,r < 0 (8) so that D h (σ h , σ -h ) < 0.
It remains to show that this admissible deviation (for η small enough) is indeed preferred by h. First note that final consumption for good

j = i remains unchanged. Define xi h (η) := x j h (σ h (η) , σ -h ). Clearly xi h (0) = x i h (σ h , σ -h
), the (putative) equilibrium consumption. Furthermore,

dx i h (η) dη | η=0 + = - ∂x i h (.) ∂b i,s h + ∂x i h (.) ∂b i,r h = -Q i,s b i,s h +B i,s h B i,s h B i,s h +b i,s h + Q i,r b i,r h +B i,r h B i,r h B i,r h +b i,r h = - B i,s h b i,s h +B i,s h 1 p i,s + B i,r h b i,r h +B i,r h 1 p i,r .
It easily follows from p i,s > p i,r and (6) that this is strictly positive, so that for η small enough u h (σ h , σ -h ) > u h (σ h , σ -h ) , contradicting the assumption that σ h is a best reply. 2

Condition ( 5) is stated in [START_REF] Gobillard | Wash-sales and the law of one price in multiple trading posts per commodity market games[END_REF] for the case ε = 0 under the additional restriction that agents are precluded by assumption to act simultaneously on both sides of a trading post. This restriction is not needed when ε > 0. Furthermore the proof in that paper uses the full apparatus of constrained optimization, while ours uses simple deviations and does not require differentiability.

A direct consequence of proposition 1 is that the law of one price cannot be violated when transaction costs are positive.

Proposition 2 Let ε > 0. Any equilibrium of the market game Γ ε satisfies the law of one price.

PROOF. Assume the contrary, that is there exist i, s and r such that p i,s > p i,r > 0. First note that by ( 5

) b i,s h > 0 implies b i,r h > 0, so that B (i, s) ⊆ B (i, r). Now, H -1 = H B i,s h B i,s = B(i,s) B i,s h B i,s + H\B(i,s) 1 > B(i,s) B i,r h B i,r + H\B(i,s) 1 ≥ B(i,r) B i,r h B i,r + H\B(i,r) 1 = H -1.
where the first inequality comes from (5). A contradiction. 2

Main results

We are now ready to state our (non-)robustness results. Our first main result shows that the law of one price holds for any robust equilibrium.

Theorem 1 Price dispersion is not a robust property. More precisely if an equilibrium of Γ features price dispersion, then it is not robust.

PROOF. The proof is by contradiction. Assume there exists a robust equilibrium * σ of Γ with dispersed prices. Then, there exist i, s and r such that By robustness, there exists a sequence { n σ} ∞ n=1 of equilibria of perturbed games with vanishing costs such that n σ → * σ, implying in particular

lim n→∞ n B i,s n Q i,s = * B i,s * Q i,s and lim n→∞ n B i,r n Q i,r = * B i,r * Q i,r . ( 10
)
Now, theorem 2 implies that

n B i,s n Q i,s = n B i,r n Q i,r
∀n. By unicity of the limit, we have that lim n→∞

n B i,s n Q i,s = lim n→∞ n B i,r n Q i,r
, which is in contradiction with (9). 2

The example in Koutsougeras (2003b) shows that price uniformity is not a (necessary) property of equilibria for the economy in the limit, Γ. In contrast, theorem 1 states that price uniformity does hold for the limit economy obtained as transaction costs vanish. 8We now turn to our second main result. Proposition 5 and 6 in Koutsougeras (2003a) assert the equivalence between (equilibrium) allocations of the one trading post market game and uniform prices allocations of the multiple trading post variant. Theorem 1 in turn implies that robust equilibrium allocations-viz, allocations associated with robust equilibria-are a subset of uniform price allocations. This suggests that the set of robust equilibrium allocations does not depend on the number of trading posts. We show that this intuition is valid.

Theorem 2 Provided that K i ≥ 1, the number of trading posts is irrelevant for robust allocations.

PROOF. See the appendix. 2

Finally, note that not all "uniform price" equilibria are robust. Intuitively, only those equilibria where agents do not place wash sales trades may be immune to our robustness requirement. The general implication of our robustness test on the structure of the set of equilibria of the single trading post setup is beyond the scope of this paper, though. We simply state the following:

Proposition 3 If an equilibrium is robust, then b i,s h • q i,s h = 0 ∀i, s, h.
PROOF. First note that only relative bids matter in equilibrium. Accordingly, we consider without loss of generality equilibria with normalized bids, in the set

S =    b i,s h , q i,s h ∈ IR 2KH + | K i s=1 q i,s h ≤ e i h ∀i ∀h, h,i,s b i,s h ≤ 1    .
The set of strategy profiles without wash sales

Ŝ = b i,s h , q i,s h ∈ S| b i,s h • q i,s h = 0 , is a closed subset of S. Now consider a converging sequence n σ → σ. Lemma 1 implies that n σ ∈ Ŝ ∀n. As Ŝ is closed, we have σ ∈ Ŝ. 2
4 The market game with commodity money

In this section, we show that our results extend to the multiple post extension of the market game with commodity money of [START_REF] Dubey | A theory of money and financial institutions. the non-cooperative equilibria of a closed economy with market supply and bidding strategies[END_REF]. There are two reasons to this analysis. First, [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF] demonstrates that the law of one price may fail in this setup too. Secondly, given that the distinctive feature of this framework lies in the presence of (money) liquidity constraints, one might expect price dispersion to obtain under weaker conditions. We show that this is not the case.

Good L + 1 is an outside commodity money that may enter utility. Any agent has an initial money endowment, e L+1 h ≥ 0, and cannot bid more than this initial money holdings. His strategy space is:

S h =    b i,s h , q i,s h ∈ IR 2K + | K i s=1 (1 + ε) q i,s h ≤ e i h , L i=1 K i s=1 b i,s h ≤ e L+1 h    .
Final allocations are determined for any commodity i ∈ {1, .., L} by

x i h (σ) = e i h -(1 + ε) K i s=1 q i,s h + K i s=1 b i,s h /p i,s , (11)
and, for the L + 1 th commodity (money), by:

x L+1 h (σ) = e L+1 h - L i=1 K i s=1 b i,s h + L i=1 K i s=1 q i,s h p i,s . (12)
We now argue that all results of the previous section extend in a straightforward way to Γ . For this purpose, we rewrite final money holdings (12) as

x L+1 h (σ h , σ -h ) = e L+1 h -D h (σ h , σ -h ) , ( 13
) with D h (σ h , σ -h ) introduced in (2). Lemma 2 Let ε > 0. Any individual best reply in Γ ε satisfies b i,s h • q i,s h = 0
PROOF. Consider the deviation in the proof of lemma 1. It satisfies the liquidity constraint because it reduces the agent's aggregate bids. Further, the consumption of money is unaffected,

xL+1 h = x L+1 h (σ h , σ -h ) because D h (σ h , σ -h ) = D h (σ h , σ -h
) by the definition of wash-sales. The proof follows. 2 Proposition 4 Let ε > 0. Consider a candidate equilibrium of Γ ε in which p i,s > p i,r > 0.Then the relative weight condition (5) holds ∀h ∈ B (i, s) .

PROOF. The deviation σh in the proof of proposition 1 amounts to shifting a small quantity of money from one post to another. Thus, σh satisfies the liquidity constraint because the amount of bids is unchanged. Further, 7) and (8). The proof follows. 2 Other proofs are (almost) unaffected. In particular, we have:

x L+1 h (σ h , σ -h ) -x L+1 h (σ h , σ -h ) = D h (σ h , σ -h ) -D h (σ h , σ -h ) ≥ 0 by (
Theorem 3 If an equilibrium of Γ features price dispersion, then it is not robust.

Theorem 4 The set of robust allocations for Γ is independent of the number of trading posts (provided that K i ≥ 1).

Proposition 5 If an equilibrium is robust, then b i,s h • q i,s h = 0 ∀i, s, h
The fact that we obtain similar results for the market game with money liquidity constraints of [START_REF] Dubey | A theory of money and financial institutions. the non-cooperative equilibria of a closed economy with market supply and bidding strategies[END_REF] and for that with perfect costless inside money of [START_REF] Postlewaite | Approximate efficiency of non-walrasian equilibria[END_REF] suggests the following observations. First, the price dispersion results illustrated in Koutsougeras (2003b) for the latter framework and in [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF] for the former have the very same source (namely, wash sales). Secondly, the existence of money liquidity constraints per se does not induce price dispersion in this framework. The intuition for this hinges on the assumption that there is a unique means of transaction (money). Hence, although the trading structure allows for one good to be purchased or sold on different locations-and potentially at different prices-there is only one way to transact. This amounts to assuming an upper bound on the degree of price inconsistency.

Concluding remarks

The present paper introduces a natural approach to restricting the set of equilibria in a market game. To be precise, we require equilibria to be robust to the introduction of arbitrarily small transactions costs. In the context of (two versions of) the multiple trading posts variant of the canonical market game, we show that the uniformity of prices holds in any equilibrium satisfying the requirement. In short, the failure of the law of one price-emphasized by [START_REF] Koutsougeras | A remark on the number of trading posts in strategic market games[END_REF]Koutsougeras ( , 2003b))-is not a robust property. At this point it is worth mentioning that the equilibrium price dispersion result of [START_REF] Amir | A strategic market game with complete markets[END_REF], which is of a different nature, should not be affected by our perturbations.

Our results may be used in assessing the usefulness of the multiple trading posts variant, as opposed to the canonical, single trading post market game. In this respect, we suggest the following interpretation. In some sense, the multiple trading posts variant might be seen as a generalization of the canonical market game. However this generalization is misleading, in that its unique impact is to give rise to unreasonable equilibria (in the precise sense that all "new" equilibria are killed by the introduction of arbitrarily small transaction costs). Thus, in our view, there is no loss of generality in working with the single trading post version in which the law of one price is posited.

More generally, the approach also has implications for the canonical market game. In essence, only equilibria in which agents do not place wash sales trades should be robust (proposition 3). The analysis of the structure of the set of robust equilibria is the subject of future research. An interesting result is that the indeterminacy result of [START_REF] Peck | The market game: existence and structure of equilibrium[END_REF], obtained for interior Nash equilibria, does not extend to robust equilibria.

A Proof of proposition 2

We first introduce additional notations to make explicit the dependence on the number of trading To this end, denote

K = K 1 , . . . , K L ∈ IN L + . Further define, in strategy space, NE ε (K) = set of equilibria of Γ K ε , NE (K) = set of equilibria of Γ K , RE (K) = subset of NE (K) that are robust,
and, in allocation space,

NA ε (K) = x ∈ IR LH + | ∃σ ∈ NE ε (K) x = x (σ) , NA (K) = x ∈ IR LH + | ∃σ ∈ NE (K) x = x (σ) , RA (K) = x ∈ IR LH + | ∃σ ∈ RE (K) x = x (σ) ,
where x (σ) := (x i h (σ)). Proposition 2 then rewrites:

Proposition 6 Let K , K ∈ IN L + . Then RA (K ) = RA (K) .
We first show the following intermediate result.

Lemma 3 Let ε > 0 and K , K ∈ IN L + . Then NA ε (K ) = NA ε (K).
PROOF. (By induction and permutations), it is sufficient to prove the result for the case

K = K 1 -1, K 2 , . . . , K L and K = K 1 , . . . , K L .
First note that NA ε (K ) ⊇ NA ε (K) because one can always add an inactive post. Hence, we need to show that NA ε (K ) ⊆ NA ε (K). Fix an allocation x ∈ NA ε (K ), and let σ ∈ NE ε (K ) be one equilibrium such that x = x (σ ). Without loss of generality we assume that all (1, s) posts are active at σ (otherwise, a mere permutation in the label of posts is sufficient). Consider the strategy profile σ∈S (K) constructed from σ by transferring all trades posted on post (1, K 1 ) to post (1, 1) :

b 1,1 h , q 1,1 h = b 1,1 h + b 1,K 1 h , q 1,1 h + q 1,K 1 h , b i,s h , q i,s h = b i,s h , q i,s h ∀h ∀ (i, s) = (1, 1) .
We claim that σ is an equilibrium and that it implements x. We proceed in two steps.

Step 1. We first check that σ leads to the final allocation x. First note that by theorem 2, σ satisfies the of one price, so that

p 1,1 = B 1,1 Q 1,1 = B 1,K 1 Q 1,K 1 = p 1,K 1 . (A.1)
Prices induced by σ are thus given by

p i,s = B i,s Q i,s = p i,s ∀ (i, s) = (1, 1) , (A.2) p 1,1 = B 1,1 + B 1,K 1 Q 1,1 + Q 1,K 1 = p 1,1 = p 1,K 1 , (A.3)
where the last equality follows from (A.1). One can easily check using (A.2) and (A.3) that the strategy profile σ satisfies all the relevant constraints, and that x (σ) = x (σ ) .

Step 2. We now show that σ ∈ NE ε (K). Assume the contrary. Then there exists one agent, say h, and one deviation σh ∈S h (K) such that u h (σ h , σ -h ) > u h (σ h , σ -h ). We shall use σh to construct a profitable deviation σ h to the equilibrium σ . Define σ h ∈S h (K ) by b 1,1 h , q 1,1 h = τ b b1,1 h , τ q q1,1 h , (A.4) b 1,K 1 h , q 1,K 1 h = (1 -τ b ) b1,1 h , (1 -τ q ) q1,1 h , (A.5) b i,s h , q i,s h = bi,s h , qi,s h ∀h ∀ (i, s) = (1, 1) , (1, K 1 ) , (A.6) where the weights τ b and τ q are given by

τ b = B 1,1 h B 1,1 h + B 1,K 1 h , τ q = Q 1,1 h Q 1,1 h + Q 1,K 1 h . (A.7)
To show that σ h is feasible and yields the same allocation as σh , we first compare prices. In view of (A.6), we have p i,s = pi,s ∀ (i, s) = (1, 1) , (1, K 1 ). Using successively Eq. (A.4)-(A.5), Eq. (A.7) and the definition of σ, the price on post (1, 1) can be computed as

p 1,1 = b 1,1 h +B 1,1 h q 1,1 h +Q 1,1 h = τ b b1,1 h +B 1,1 h τ q q1,1 h +Q 1,1 h = τ b τq b1,1 h +B 1,1 h +B 1,K 1 h q1,1 h +Q 1,1 h +Q 1,K 1 h = τ b τ q b1,1 h +B 1,1 h q1,1 h +Q 1,1 h = τ b τ q p1,1 . (A.8)
Similarly, on post (1, K 1 ) :

p 1,K 1 = 1 -τ b 1 -τ q p1,1 . (A.9)
Now, to see that σ h is feasible, compute

D h σ h , σ -h -D h (σ h , σ -h ) = q1,1 h p1,1 -q 1,1 h p 1,1 -q 1,K 1 h p 1,K 1 = 0,
where the last equality follows from (A.4)-(A.5) and (A.8)-(A.9). To see that σ h yields the same allocation as σh , we simply need to compute

x 1 h σ h σ -h -x 1 h (σ h , σ -h ) = b 1,1 h p 1,1 + b 1,K 1 h p 1,K 1 - b1,1 h
p1,1 = 0 by (A.4)-(A.5) and (A.8)-(A.9). Hence, we have that u h σ h , σ -h > u h σ h , σ -h , contradicting the fact that σ ∈ NE ε (K ). Thus, σ ∈ NE ε (K). 2

We now prove the result. Let x ∈ RA (K). There exists σ ∈ RE (K) with x (σ) = x and a sequence { n ε, n σ} converging to (0, σ) with n σ ∈ NEn ε (K). Lemma 3 implies that for any n, there exist n σ ∈ NEn ε (K ) such that x ( n σ ) = x ( n σ). Now, any n σ might be viewed as an element of the compact set S (K ). Compactness implies that the sequence { n σ } ∞ n=1 contains a subsequence { z σ } ∞ z=1 which converges to an element σ of S. By continuity, (i) u h ., σ -h is maximized for σ h for any h, so that σ ∈ NE (K ), and, (ii) x (σ ) = lim z→∞ x ( z σ ) = lim z→∞ x ( z σ) = x. Further, by construction σ ∈ RE (K ), whence x ∈ RA (K ). This terminates the proof. 2

And, importantly, in an environment in which agents face no liquidity constraint in their arbitrage strategy(Koutsougeras,

2003b).2 See also proposition 9 in[START_REF] Bloch | Trade fragmentation and coordination in strategic market games[END_REF].

In a very different context[START_REF] Peck | Large market games with demand uncertainty[END_REF] shows that the multiplicity of best responses is not robust to the introduction of arbitrarily small demand uncertainty.

 4 It is worth noticing that the selection of a non trivial Nash equilibrium is not guaranteed as shown by[START_REF] Cordella | Nice" trivial equilibria in strategic market games[END_REF] and[START_REF] Busetto | Very Nice" trivial equilibria in strategic market games[END_REF].

That is, some proportional fraction of goods disappears in transaction. This corresponds to an "iceberg cost" specification.

To ease the exposition, we do not write S ε h for the strategy set although it does depend on ε. Note that strategy sets for any ε > 0 are included into strategy sets for ε = 0, so that any equilibria belongs to this larger set, S 0 1 × • • • × S 0 H . No confusion should result.

In that sense, (b i,s h -bi,s h , q i,s h -qi,s h ) are 'wash-sales' trades.

See Gale (1986) for the discussion of 'the economy in the limit' and 'the limit economy'.

Acknowledgements

We thank Vincent Bignon, Yannick Kalantzis and Mariana Rojas Breu for comments on a previous draft. We also wish to thank audiences at the Journée concurrence at University of Paris X (june 2006, Paris), and at the 17th International Conference on Game Theory (july 2006, SUNY, Stony Brook). Remaining errors are our sole responsibility.


Jacques Drèze
email: brechet@core.ucl.ac.be.

Keywords: No-regret options, pollution, regulation JEL codes: D20; Q50

Empirical evidence support the existence of pollution abatement possibilities at negative costs, the so-called 'no-regret options'. We provide a microeconomic rationale for the existence of such potential at the firm's level under environmental regulation. An econometric application confirms that marginal pollution abatement cost curves with no-regret options are compatible with a standard production function, as stated in our theoretical model.

Introduction

In this paper we provide a microeconomic rationale for the existence of 'noregret pollution abatement options' at the firm's level, that is, the opportunity to reduce pollution while increasing profits.

Many studies provide empirical evidence for such options, but most of them at countries or sectoral levels, rarely at the firm's level. The potential of no-regret opportunities to reduce greenhouse gases emissions, for example, has been highlighted by the Intergovernmental Panel on Climate Change: "Half of the potential reductions may be achieved by 2020 with direct benefits (energy saving) exceeding direct costs (net capital, operating and maintenance costs)." (IPCC, 2001, p. 40). More recently, a set of in-depth analyses of greenhouse gases abatement costs have been performed on behalf of the World Bank in several developing and transition countries. Carried out on a common bottom-up methodological framework, these studies identify noregret options in most countries 1 . Another example is a study undergone by the German Fraunhoer Institute for the Belgian federal government: it establishes that Belgium could meet its the Kyoto commitment (i.e. a 7.5% abatement in 2010 with respect to the 1990's level) by negative or zero cost measures. A recent a very complete empirical analysis at the firm's level is the one by [START_REF] Isaksson | Abatement costs in response to the Swedish charge on nitrogen oxide emissions[END_REF] with a plant-level data set comprising 162 abatement measures for nitrogen oxide emissions. It shows that "extensive emission reductions have been possible at very low or even zero or negative costs" (Isaksson, 2005, p. 118), though only one of the 162 measures reported costs-savings larger enough to outweigh the costs.

One cannot look down the fact that most economists feel uncomfortable with this evidence because it seems to conflict with the assumption of rational behavior. The standard microeconomic theory fails to explain what a so-called free lunch. The famous economists' metaphor cited by Porter and van der Linde (1995) is that $10 bills will never be found on the ground because someone would have already picked them up. As stressed out by [START_REF] Palmer | Tightening the environmental standards: The Benefit-Cost or the No-Cost Paradigm?[END_REF] in the same journal, the question is neither to claim that firms are "ever-vigilantly perched on their efficiency frontier" (p. 120), nor to generalize the idea of free lunch on the simple basis on few empirical examples.

The objective of our paper is not to provide a rational for the Porter hypothesis, that is, to the hypothesis that tightening environmental standards may increase firm's profit2 . Instead, we show that pollution abatement is always costly but also that implementing internal environmental management may lead to increases in productivity. Thus, when comparing situations without, and with environmental regulation, it may happen that a firm gain from being green, what is called a no-regret option. It is clear that the possibility for such options to occur may have major implications for the debate on environmental regulation, competitiveness and growth (see e.g. [START_REF] Barbera | The impact of environmental regulations on industry productivity: direct and indirect effects[END_REF], [START_REF] Jaffe | Environmental regulation and the competitiveness of U.S. manufacturing: what does the evidence tell us?[END_REF] and the survey of [START_REF] Ambec | Can environmental regulation be good for business? An assessment of the Porter hypothesis[END_REF]).

When looking for empirical evidence for no-regret options the debate rapidly faces methodological issues. How do engineers calculate abatement costs? Is it coherent with standard economic concepts? Answering the former issue goes beyond the objective of this paper (note that the World Bank project presented above constitutes a good example of transparent and publicly available methodology). [START_REF] Stoft | The economics of Conserved-Energy 'Supply' Curves[END_REF] is probably the only author who tried to tackle the latter issue by developing a framework to reconciliate the economists' concept of production function with marginal abatement cost curves as built up by engineers. Without justification, however, Stoft assumed that the firms are below their production frontier, thus introducing ad hoc no-regret abatement options at the firm's level. Actually, two main strands of economic literature may be related to this issue. The first one quantifies the so-called X-inefficiencies and follows Farrell's (1957) definition of technical efficiency. These studies extract information from large bodies of data at the firm's level to determine the best production frontier and calculate the distance between each firm and this frontier. These approaches are characterized by parametric or non-parametric methods. They calculate but they do not explain. The second strand of literature focuses on the barriers to energy efficiency. The authors exhibit a panel of reasons for which the firms seem inefficient, while being fully rational but facing some hidden costs, that is, costs which are neglected in standard static and deterministic analyses. These costs are related, e.g., to the uncertainty on future energy prices, to technological lock-in effects, to uncertainty on the characteristics on equipment goods... A survey of this literature can be found in Sorrel (2004). How useful these analyses are, they fail to propose a general rational for the possible existence of no-regret options at the firm's level under environmental regulation.

Within a standard static microeconomic framework our paper provides a rational for no-regret options to exist, even though these options are susceptible to appear only in very specific contexts or for some firms only. The issue is to understand in which context these options may appear. Our starting point is that pollution operates as a hidden -and neglected-factor in the production process, along with the usual production factors. While neglecting this factor, the firm may bear an opportunity cost. Put differently, the firm may gain by taking into account the productive contribution of pollution within the whole optimization process, thus enlarging its production set. We show how environmental regulation and management help internalizing this productive contribution and why they may lead to a profit increase when pollution is reduced. Using our theoretical framework we propose a graphical illustration (in the spirit of [START_REF] Stoft | The economics of Conserved-Energy 'Supply' Curves[END_REF]. We also provide empirical evidence with a simple econometric application in the glass industry in Wallonia (Belgium).

The article is organized as follows. Section 2 presents the firm's behavior in terms of environmental management under regulation. The opportunity cost of neglecting pollution and the existence of no-regret options at the firm's level are proved. Section 3 provides a graphical illustration and the econometric application is presented in Section 4. Section 5 concludes.

Environmental regulation and management

We shall consider two scenarios. The first one (called business-as-usual ) corresponds to a situation in which pollution is neglected by the firm. Such situations prevailed some years ago for many pollutants for which even a mere reporting was not organized. It still prevails today for some pollutants in some sectors or firms (e.g. carbon dioxide emissions at firms' level are largely neglected in some sectors like transportation, buildings or market services). Under this scenario we consider a firm which maximizes its profit while neglecting pollution. In the second scenario (called the green scenario), firms are committed to a pollution quota, thus discovering (or recognizing) that they do have an impact on the environment. As a consequence, the firm would like to have a clear idea about its emission level and about its own ability to reduce it in order to comply the pollution constraint. In this purpose we assume that the firm sets up an environmental management system and hires an environmental manager. Actually, this scenario is the one many firms face today while environmental regulation and awareness spread. What we are interested in is the productive implications of the fact that the firm now takes pollution into account.

Production technology and pollution

The firm produces an homogeneous good taken as numeraire with a wellbehaved (increasing, concave and homogeneous of degree one) production function F (X) where X is the vector of N production factors, X = {X 1 , ..., X N }.

Under the business-as-usual scenario, the output level is noted Ŷ . This activity results in the emission of a pollutant in quantity P = ψ Ŷ (ψ > 0), pollutant which is disregarded by the firm. Let assume that this pollution output ratio can be considered from a technological viewpoint as an increas-ing continuous function ψ(z) of a technological index z,

ψ(z) = P F (X) (1)
Without loss of generality we assume that the function ψ(.) is defined and inversible on R + . The cost of pollution abatement is expressed in terms of output losses by choosing the index z applied to the output level [START_REF] Stokey | Are there limits to growth?[END_REF],

Y = zF (X) (2)
The modeling à la Stokey (1998) allows to easily compare the two situations, with and without environmental regulation. We assume that the abatement

cost (1 -z)F (X) is defined in [0, F (X)], i.e z ∈ [0, 1].
Eliminating z between ( 1) and ( 2) allows us to define a (N + 1)-factor production function Φ, homogeneous of degree one, in which pollution is considered as an input for production,

Φ(X, P ) = ψ -1 ( P F (X) )F (X) (3)
Thus, the production function changes depending on whether environmental management is operative (z < 1) or not (z = 1). The overall production function is given by Y = G(X, P ) = min{F (X), Φ(X, P )}. As a min-function, G(X, P ) is not differentiable at the point where the two terms F (.) and Φ(., .)

are equal, i.e. for z = 1.

The unexpected effects of environmental management

Under the business-as-usual scenario the output level is given by the N -factor production function Ŷ = F (X). Assuming perfect competition the vector of input prices p = {p 1 , ...p N } is given and the firm's programme writes

max {X} π = F (X) -pX (4)
leading to N first-order conditions of the form

F X i ( X) = p i , ∀i ∈ N .
The pollution level is given by P = ψF ( X).

Under the green scenario the firm is subject to an emission quota P such that 0 < P ≤ P . To tackle this new constraint the firm hires an environmental manager whose mission is twofold 3 . Firstly, she has to report the actual emissions level, i.e. the one under the business-as-usual scenario.

Secondly, she has to identify the technological opportunities to comply with the pollution constraint. We represented these technological opportunities with the function ψ(z) of the technological index z. Using (1) the constraint P ≤ P can be re-written as ψ(z)F (X) ≤ P . Under environmental regulation the problem of the firm now reads max

{X,z} π = zF (X) -pX ( 5
)
3 In our analysis we neglect the hiring and wage costs of the environmental manager.

This boils down to assume that these cost are small compared to the total production cost of the firm. Still, considering a positive cost would not change the outcome of the analysis.

subject to ψ(z)F (X) ≤ P . We denote µ the Lagrangian multiplier of this constraint and this Lagrangian writes

L = zF (X) -pX + µ( P -ψ(z)F (X)) (6)
The value of X and z solutions of this problem are solutions of the following FOCs,

(z -µψ(z))F X i (X) = p i , ∀i ∈ N (7) 1 -µψ (z) ≤ 0, (= 0 if z < 1) (8)
µ( P -ψ(z)F (X)) = 0 (9)
By doing her job the environmental manager reveals that pollution operates as a hidden factor within the production process and interacts with the other production factors. In the FOCs, this is reflected by the fact that the factors' marginal productivity differs from their price this productive interaction being the multiplicative term z -µψ(z). The function z -µψ(z) is concave and its derivative is positive at z = 0 (Jouvet, Michel and Rotillon, 2005). If the constraint is not binding, i.e. P = P , there exists a range [0, µ] of values of µ compatible with this pollution target. As long as 0 ≤ µ ≤ µ, z = 1 and all factors' level remain unchanged in comparison with the business-as-usual scenario (i.e. X).

Considering the relation ( 8), the highest value of µ compatible with no abatement is defined by

µ = 1 ψ (1) (10)
Knowing the value of µ and using equation ( 7) the maximal impact of the environmental factor on the marginal productivity of the input X i (∀i ∈ N )

writes (1 -µψ( 1))F X i ( X), which is lower than F X i ( X) = p i . So, under the business-as-usual scenario, the contribution of pollution to production is embodied in the other production factors' contribution, and it is unpaid. Following Worcester's terminology [START_REF] Worcester | Pecuniary and technological externality, factor rents, and social costs[END_REF] we can disentangle this contribution as the combination of the two parameters identified above, namely a technological rent, ψ(1), and a pecuniary rent, µ. Thus, discovering the productive contribution of pollution opens the door to a revaluation of the productive contribution of each factor X i , the latter being valued (at most) by the multiplicative term 1 -µψ [START_REF] Ambec | Can environmental regulation be good for business? An assessment of the Porter hypothesis[END_REF]. Since the actual marginal productivity of the production factors is lower than their cost, there exists an opportunity cost associated to the fact that the environmental factor was neglected. This is summarised in the following proposition.

Proposition. A firm may experience a profit increase while committed

with a pollution constraint. The maximal profit increase is given by Ω = µψ(1)F ( X).

Example 1 The Cobb-Douglas case. Consider a Cobb-Douglas produc-

tion function Y = AK α L 1-α (with 0 < α < 1). The firm is price-taker and output is taken as numeraire . Under the business-as-usual scenario the pollution level is P = ψ Y (with ψ > 0). Let the pollution function be

P/Y = ψz β (with 0 < z ≤ 1 and β > 0). The cost of pollution abatement is (1 -z)AK α L 1-α .
Under the green scenario the pollution function writes

P AK α L 1-α = Ψ(z) = ψz β+1 (11)
By substitution, the previous equations allow us to define the technological index z as a function of pollution, capital and labor,

z = P ψAK α L 1-α 1 1+β (12)
and we get a three-factor production function, homogeneous of degree one of capital, labor and pollution,

Φ(K, L, P ) = P ψ 1 1+β A β 1+β K αβ 1+β L (1-α)β 1+β ( 13
)
As shown in the previous section, the highest value of µ compatible with z = 1

reads µ = 1/(ψ(1 + β)).
As long as 0 ≤ µ ≤ µ, z = 1 and both capital and labor levels remain unchanged with respect to the business-as-usual situation.

While considering an emission constraint P such that 0 < P ≤ P , the firm's profit at the optimum is given by π * ( P ) = Φ(K * , L * , P ) -wL * -RK * with

L * = (1-α)β 1+β Φ(K * ,L * , P ) w and K * = αβ 1+β Φ(K * ,L * , P ) R
. Hence, the profit becomes positive and is given by

π * ( P ) = 1 1 + β Φ(K * , L * , P ) (14)
This profit function is increasing with P . It can easily checked that lim P → e P π( P ) = Ω.

No-regret options for pollution abatement

We can illustrate the fact that emission reductions can be associated with an increase of the firm's profit by using the marginal abatement cost curves (MAC, hereafter). For the sake of simplicity, linear functions will be considered in the figures. In Fig. 1, the horizontal axis represents the emissions level and the vertical axis represents the marginal profit for a given firm. In the absence of environmental regulation the pollution level ( P ) corresponds to the zero marginal profit condition. The profit of this firm is the area OA P . When subject to the environmental regulation, the firm re-evaluates the marginal productivity of its factors and the monetary value of the maximal potential benefit for this firm without changing emissions level is AB P .

The vertical jump at P = P reflects that a discontinuity occurs and the size of this jump is given by µψ [START_REF] Ambec | Can environmental regulation be good for business? An assessment of the Porter hypothesis[END_REF].

Insert figures 1 and 2 about here

The building of the MAC curve stems from the previous Figure . In Figure 2 pollution abatement is indicated on the horizontal axis as ∆P = P -P and the marginal abatement cost is shown on the vertical axis. In the business-as-usual scenario the MAC curve starts from the origin and its slope is the slope of the marginal profit curve as given in Figure 1. Under environmental regulation, and if no-regret options exist for pollution abatement, then two phenomena occur. Firstly, the MAC curve moves downward and a discontinuity appears at the origin, the maximal value of the latter being given by the value of µψ [START_REF] Ambec | Can environmental regulation be good for business? An assessment of the Porter hypothesis[END_REF]. Secondly, the slope of the MAC curve changes: it is all the flatter as the opportunity cost associated to pollution is important.

The area below the horizontal axis and above the MAC curve represents the potential of no-regret pollution abatement measures and the point C where the MAC curve crosses the axis indicates the optimal abatement level for this firm, i.e. the one at which the marginal abatement cost is zero and the total benefit is maximal.

An econometric application

In this section we apply our theoretical model to the glass industry in Wallonia (Belgium) by carrying out an econometric estimation. The dataset was collected during auditing procedures at the firm's level on behalf of the regional administration in charge of the environmental policy4 . They provide a set of technical abatement options for carbon dioxide and the fixed and operational costs associated. These costs are annualized and discounted to be expressed in 2010, the reference year. Then these measures are ranked in increasing order of marginal cost. All in all 32 abatement options are available, of which four at a negative cost. Using the Cobb-Douglas specification presented in Section 2, the cost of pollution abatement is given by

(1 -z)F (K, L) = AK α L 1-α - P ψ 1 1+β A β 1+β K αβ 1+β L (1-α)β 1+β (15)
and the marginal abatement cost writes

M AC = - 1 1 + β ψ -1 1+β P -β 1+β A β 1+β K αβ 1+β L (1-α)β 1+β (16)
The marginal abatement cost curve to be estimated rises from equation ( 16),

ln(-M AC) = lnθ-θlnψ+(1-θ)lnA-(1-θ)lnP +(1-θ)[αlnK +(1-α)lnL] (17)
with θ = 1/(1 + β). Considering that the marginal abatement cost may eventually be negative for first pollution abatement efforts, we add a positive constant T in the logarithm of the left hand side of this expression such that (-M AC +T ) > 0. Considering that θ, ψ and A are parameters we can define the two following constants ξ 1 = lnθ -θlnψ + (1 -θ)lnA and ξ 2 = (1 -θ).

Finally, the equation to be estimated writes

ln(-M AC + T ) = ξ 1 -ξ 2 lnP + ξ 2 [αlnK + (1 -α)lnL] + ε ( 18
)
where ε is the error term. Equation ( 18

Conclusion

In this paper we provide a microeconomic rational why no-regret pollution abatement options (i.e. pollution reductions at negative costs) may exist at the firm's level. By recognizing that pollution is a production factor, we show how neglecting its interactions with the other production factors in the production process may represent an opportunity cost for the firm. Put differently, enlarging the production set with this polluting factor may lead to a profit increase at the firm's level. The basic argument may be illustrated with a revisited version of the Porter and van der Linde's metaphor quoted at the beginning of this article: $10 bills may perfectly stay on the ground of the cellar if there is nobody to switch on the light. Switching on the light is the environmental manager's mission. Our econometric application confirms that marginal abatement cost curves with no-regret options, as built up by engineers, are fully compatible with a standard production function, as used in economics. So, in contrast with previous works (essentially Porter and van der Linde (1995) and Stoft (1995)), our paper provides a framework for analysing no-regret options which is both formal and general. Importantly, in our setting pollution abatement always has a cost, but this cost may be outweighed, to a certain extend, by productive improvements revealed when implementing internal environmental management.

One of the avenues for further research would be to analyze the firm's capacity to benefit from this opportunity cost. It may be possible for a firm to identify no-regret measures but not to be able to benefit from them.

The reason is that the firm's ability to increase its profit will depend on the markets' structure and on the firm's capacity to exert its market power (if there exists). Hence, the very existence and the implementation of no-regret options are not only a matter of technological choice but also a matter of adequate internal management considering the markets' structure. Another natural extension would concern policy implications. These are threefold.

First, there should be a serious re-examination of macroeconomic costs of pollution abatement in the case where net benefits can be expected in some sectors or firms. Second, the existence of no-regret measures would certainly question the relative efficiency of policy instruments. In particular this may provide a rational to the firms' participation in voluntary pollution abatement programmes 5 . These ones should be chosen so as to extract this potential as much as possible, whenever it exists. Third, and finally, the capture of this rent raises redistributive issues among firms.

  ) is estimated by OLS. All the coefficients are statistically significant and have the expected sign: ξ 1 = 0.703 (t-stat: 64.3), ξ 2 = 3.956 (t-stat: 155.0) and α = 0.301 (t-stat: 13.3). This result provides empirical evidence for two results. First, no-regret abatement options as represented by MAC curves built by the engineers are compatible with a standard well-behaved three-factor production function (a Cobb-Douglas function in this example). Second, this estimation evaluates the current potential for no-regret carbon dioxide abatement options in the glass industry in Wallonia. Knowing that ξ 2 = (1 -θ) we get that θ = 0.297 and β = 2.367. So the value of µ can also be computed. Considering the current emission and output levels in the glass industry in Wallonia in 2003, we calculate that the maximal profit increase Ω associated to pollution abatement at negative costs amounts to 29% of the output value.

Figure 2 :

 2 Figure 1: The maximal potential benefit

Part of the National CDM/JI Strategy Studies (NSS), all these analyses are publicly available at http://www.worldbank.org.

A vast literature has been devoted to the Porter hypothesis. For a survey, see Ambec and Barla (2005).

The data were collected by ECONOTEC, a consultancy agency specialized in energy audits and technico-economic evaluation in the field of energy and the environment (www.econotec.be).

An assessment of these programmes has been made by the OECD: see OECD (2003).

& & & & & & & & & & & & & & & & & & & & & µψ(1)∆P