Emission Permits Trading Across Imperfectly Competitive Product Markets

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1 Emission Permits Trading Across Imperfectly Competitive Product Markets Guy MEUNIER CIRED-Larsen ceco January 20, 2009 Abstract The present paper analyses the efficiency of emission permits trading among several imperfectly competitive product markets. Even if firms are price takers on permit markets the integration of permits markets can decrease welfare because of imperfect competition on product markets. With imperfect competition on product markets the value of emissions should not be equalized between markets and if an integrated permits market is implemented a corrective policy can restore a second best optimum. The issue of permits market integration is then analyzed with uncertainty on markets conditions. JEL Classification: D82, D43, L13, Keywords: emissions permit market, demand uncertainty, market integration. 1 Introduction With the implementation of the European Union emission trading scheme (EUETS) and the Kyoto protocol emissions permits markets are used at an unprecedented scale to regulate an externality. Main sectors concerned by the EUETS are concentrated so they are often considered imperfectly competitive. In particular, several geographically isolated and concentrated electricity markets are concerned by emissions trading. The introduction of an integrated permits market creates relationship between imperfectly competitive markets initially isolated. At the same time as numerous firms are 1

2 active on the emission permits market it can be assumed perfectly competitive. This paper deals with the efficiency of an integrated permits market between two imperfectly competitive outputs markets. The two outputs considered are non-substitutable so markets are initially isolated. The aggregate quantity of emissions is fixed and the issue is the allocation of this constraint between outputs markets, that can be done either with two isolated emissions permits markets or an integrated one. Output markets considered can be isolated in taste, space or time. Hence, the issue addressed concerns the extent of a permits market along several dimensions. The present work is related to the literature on environmental policy and market power 1. Imperfect competition should be considered when designing an environmental policy, as Buchanan (1969) stresses for an environmental tax. An environmental tax should be lower than marginal damage when the regulated market is a monopoly or a symmetric Cournot oligopoly (Barnett; 1980), because it encompasses a subsidy that corrects market power 2. Concerning the influence of market power on the efficiency of a permit market, the literature can be crudely divided in three whether market power is introduced on the permits market, the good market or both simultaneously. This paper belongs to the second strand. Three previous articles have addressed the issue of imperfect competition in the goods market combined with perfect competition in the permit market. Sartzetakis (1997, 2004) analyses the case of a duopoly exercising market power on an output market. He compares the efficiency of a competitive emissions market to a command and control situation. In the benchmark situation emissions of each firm are exogenously given. Emissions trading modifies the allocation of emissions among firms and consequently their production choices. In Sartzetakis (2004), the author shows that welfare can decrease when emissions trading is allowed between asymmetric firms with different abatement and production technologies. The permit price that clears the market is a weighted average of the value of emissions of firms under command and control, therefore, the inefficient firm cost is reduced while the efficient one is increased when permits trade is introduced. The induced reallocation of production on the output markets can offset the efficiency 1 See Requate (2005) for an extensive survey on environmental policy and imperfect competition. 2 With asymmetric firms competing à la Cournot, the optimal tax can be higher than environmental damage(simpson; 1995) because of production allocation among firms effects. 2

3 gains from permits trading. Similarly the trading of permits between sectors can worsen welfare by misallocating emissions. Hung and Sartzetakis (1998) analyse the case of emissions trading between a monopolistic sector and a competitive one. Market imperfection is transmitted from the monopoly to the competitive sector via the emissions market: the monopoly consumes fewer emissions than the optimum quantity and the competitive sector more. In both situations, an integrated permits market can be strictly less efficient than an optimal command and control. I extend this literature by considering emissions permits trading between two imperfectly competitive markets and explicitly analyzing a corrective policy and the effect of asymmetric information on the attractiveness of markets integration. Actually, if an informed benevolent regulator can achieve efficiency the introduction of market mechanisms can never outperform command and control. But, if asymmetric information is considered the regulator may rely on markets to efficiently allocate emissions among agents or at least to perform better than him. And, empirically, in the design of the EUAETS, the allocation of emissions permits to producers of sectors concerned is done on the basis of anticipated demand for outputs. These demands are unknown by the regulator when setting the precise rules of the emissions markets. Furthermore, I address the issue of the effect of the misallocation on the choice of emission cap. I consider two imperfectly competitive markets and analyse how both markets imperfections influence the allocation of emissions. It is shown that the integration of permits markets can decrease welfare. I assume that the regulator cannot directly correct output markets inefficiencies so a second best allocation of emissions is introduced. The first best allocation is defined in the context of perfectly competitive output markets while the second best allocation is defined when output markets are imperfectly competitive. An increase of emissions, besides decreasing costs of production, increases production. Because of market power, this increase of production is valuable. Because of this effect, the second best optimal allocation of emissions cannot be reached in general with an integrated market for emissions even if firms are price takers on this market. The second best allocation can be sustained with an integrated market for emissions if a subsidy or a tax on emissions of one market is introduced. This subsidy is composed of the difference of price cost margins weighted by the sensitivity of production on emissions. These components reflects the value of an emission due to market power. Next, I analyse the attractiveness of an integrated market under asym- 3

4 metric information or uncertainty. The regulator does not precisely know the demand on output markets when deciding whether permits markets should be integrated. The integration of markets has two contradictory effects: the first one is the misallocation described above and the second one is the use of information about markets conditions by firms. Without any additive regulation the integration of markets is welfare enhancing if uncertainties about the two markets are sufficiently negatively correlated, in that case the second (positive) effect dominates. It stresses the value of markets as mean to mobilize information, this positive effect of markets should be balanced with imperfections when deciding whether or not to implement an integrated market. Moreover, if accompanied by an ex-ante corrective subsidy an integrated market is always welfare enhancing because the subsidy can correct the first (negative) effect. These results are closed to those obtained on decentralization in a context of asymmetric information and strategic behavior. Concerning the design of the EUAETS, Malueg and Yates (2007) analyze the decentralization process of the allocation of allowances between trading sectors and non trading one within each Member States. In their framework, firms are not strategic but states are. Decentralization allows states to use their private information about abatement costs but also to strategically allocates emissions rights, the trade-off between these two effects determines whether centralization is preferred or not to decentralization. The last issue addressed, that comes naturally to mind, is the effect of the misallocation of the emission cap on the choice of this cap. More generally, the issue is the way internal markets imperfections modify the choice of an optimal macro variables. Intuitively, one may think that to relax the aggregate constraint would correct the misallocation and therefore, the optimal cap should be higher with imperfections than without. The analysis reveals that it is not the case, the relaxation of the emissions cap can either reinforce or soften the misallocation so the sign of the change analyzed is no I began to introduce the model (section) with general cost functions (section 2) and analyse the effect of market power within this framework (section 3). Then, I consider a more simple version with Leontief technologies in order to analyze asymmetric information (section 4) and the choice of the emission cap (section 5). 4

5 2 The model 2.1 Set up I consider two polluting products markets indexed i = 1, 2, with inverse demand function P i (Q i ), i = 1, 2, where Q i is the aggregate quantity of good i produced. Gross surplus from consumption of good i is S i (Q i ) with S i(q i ) = P i. Inverse demand functions satisfy the following assumptions: For each i = 1, 2 there is Q i > 0 such that: P i is strictly decreasing and positive on [ 0, Q i [ P i (Q i ) is null for Q i > Q i P i is twice differentiable and satisfies: P i + Q i P i < 0 for Q i [ 0, Q i [. The last assumption, common in oligopoly literature (?) signifies that the marginal revenue of a firm is decreasing with respect to the production of its rival. It implies that quantities are strategic substitute and ensures existence and uniqueness of Cournot equilibrium when firms have convex cost. On market i = 1, 2, there are n i firms that produce the good. The production of the good requires emissions, individual cost of production are C i (q i, e i ) where q i and e i are quantities of output produced and emissions used by an individual firm. The following assumptions are made on the cost function 3 : q i 0, e i 0: Cost are increasing and convex: C i / q i > 0, 2 C i / q 2 i > 0 It is worth producing: C i (0, e i ) = 0 and P i (0) > C i / q i (0, e i ) Cost and marginal cost are decreasing with respect to emissions: C i / e i < 0 and 2 C i / q i e i 0 The effect of an increase of emissions is decreasing: 2 C i / e 2 i 0 And 2 C i / q 2 i. 2 C i / e 2 i ( 2 C i / q i e i ) I do not introduce upper bound on emissions to ensure that at equilibria considered the emissions constraint is always binding. 5

6 Costs are increasing and convex with respect to output and decreasing and convex with respect to emissions. Marginal costs are also decreasing with respect to emissions. The last assumption ensures that the gross cost of a firm is convex (cf appendix A) by limiting the effect of emissions on marginal cost 4 On each market i = 1, 2 an emission permits market is implemented and the local price of emission is denoted σ i, hence a firm on market i that produces q i with e i emissions has a profit net of initial free allocations: π i (q i, Q i, e i, σ i ) = P i (Q i )q i C i (q i, e i ) σ i e i (1) The aggregate quantity of emissions is e, the quantity of emissions on market i = 1, 2 is E i. Environmental damage is assumed separable and depending only of the aggregate quantity of emissions. As this quantity is fixed I do not explicitly introduce environmental damage. In sections 3 to 5 the issue addressed is the allocation of emissions among sectors and not the choice of the aggregate constraint e that will be assumed fixed. In the last section, I consider how the choice of this quantity is affected by local market imperfections. On each output market, all equilibria considered are symmetric so quantities of output and emissions are equally distributed among firms 5 : individual quantities are q i = Q i /n i and e i = E i /n i. Welfare is the sum of surpluses net of production costs on both polluting sectors: W (Q 1, Q 2, E 1, E 2 ) = i [S(Q i ) n i C i (Q i /n i, E i /n i )] (2) Initial allocation to a market i=1,2 is denoted Êi, it is the quantity of emission available to a market if there is no trade of permits across markets. I begin to consider the first best optimum, i.e. the quantities and emissions on each market chosen by a perfectly informed benevolent regulator able to enforce 4 These assumptions are satisfied for the two common specifications: 1. C(q, e) = c(e/q)q with c < 0, c > 0 2. C(q, e) = q + c(q e) with c > 0, c > 0 for q > e. The first specification represents the choice of a technology: a firm can lower its emission rate (e/q) by increasing its marginal cost. And the second assumes separability between production and abatement (q e), so there is a technology to produce unitary abatement or emissions permits, such as clean development mechanisms or carbon sequestration. 5 I do not consider that the regulator can discriminate among firms in a sector by allocating different quantities of permits. Even if firms are symmetric this could increase welfare as established by Amir and Nannerup (2005). 6

7 it in the next subsection. Then I introduce imperfect competition on output markets and consider a second best optimum: the allocation of emissions that maximizes welfare with market power exercice on output market. 2.2 Optimum and perfect competition The objective of the regulator is to maximize welfare (2) subject to e E 1 + E 2. First best quantities are denoted Q i, E i, i = 1, 2, they satisfy the following first order conditions: S i(q i ) = C i q i (Q i /n i, E i /n i ), i = 1, 2 (3) C 1 e 1 (Q 1/n i, E 1/n 1 ) = C 2 e 2 (Q 2/n 2, E 2/n 2 ) (4) If firms are price takers on both outputs and emissions markets, this optimum can be decentralized by an integrated emissions markets(montgomery; 1972). In that case the permit price is σ = σ 1 = σ 2, and on each markets price taking behavior by firms ensures that (3) and (4) are satisfied. For any initial distribution of emission permits among firms, emissions market integration always improves welfare. If the regulator does not perfectly know each market characteristics such as costs, demand and emission rates an integrated permit market is preferred to a tax or quotas to minimize the cost to reach a given emission cap. Alternatively, if the regulator is informed he can allocate emissions Êi = Ei to each sector and not integrate emissions markets, in that case both local permit prices would be equal. 3 Imperfect competition Imperfect competition on output markets is introduced: firms strategically choose output quantities but are price takers on permits markets. I first describe the equilibrium with two isolated emissions permits market then I consider a second best allocation of emissions among markets with imperfect competition. It is then shown that an integrated market of emissions permits allocates emissions differently and can therefore decrease welfare. The analysis is carried first in the general framework introduced, then some insights are derived from quadratic specifications. 7

8 3.1 General framework On output markets firms compete à la Cournot by strategically choosing output quantities whereas they are price takers on the emissions permits markets. Assumptions on price and cost functions ensure existence of a unique symmetric Cournot equilibrium for any σ i (cf appendix A) on each market. At this equilibrium all firms produce the same quantity of output and consume the same quantity of emissions. The permit price clears the local market for emissions. In order to consider the effect of a change of the quantity of emissions, productions are written as function of emissions. Given individual emissions e i for each firm on market i = 1, 2, each firm maximizes its profit ( 1) when choosing its production. At equilibrium individual production on each market is denoted qi C (e i ) for i = 1, 2. These quantities satisfy the following first order conditions: P i + P i q C i (e i ) = C i q (qc i, e i ), i = 1, 2, (5) and the aggregate production is Q C i (E i ) = n i q C i (E i /n i ). For a local permit price σ i the equilibrium individual demand of emissions of a firm of market i is e C i (σ i ) that satisfies: σ i = C i (qi C (e C i ), e C i ), i = 1, 2, (6) e i the aggregate demand of firms of market i is Ei C = n i e C i. As firms are price takers on the permits market, the initial distribution of permits between firms does not influence the market outcome. So, it is equivalent to consider that the regulator gives an allocation Êi/n i to each firm or an aggregate amount of Ê i to all firms on market i which is allocated between firms by the emission permits market. In the latter case, the price σ i clears the local emissions market so that Êi = Ei C (σ i ). With two isolated permits market, a regulator that can only set the quantity of emissions Êi but not production should consider the effect of the former on the latter. Such an allocation of emissions differs from the first best described in section 2. Definition 1 A second best allocation (E 1, E 2 ) of permits given market power exercise is an allocation that solves: max E 1,E 2 W (Q C 1 (E 1 ), Q C 2 (E 2 ), E 1, E 2 ) subject to E 1 + E 2 e 8

9 I assume that W (Q C 1 (E 1 ), Q C 2 (E 2 ), E 1, E 2 ) is twice differentiable and concave with respect to E 1 and E 2 so that there is a unique second best allocation (E1, E2 ) with E2 = e E1. On each market i = 1, 2 an additive emission increases net surplus of: ( P i C ) i q C i C i q i e i e i The second term is the direct increase of surplus related to the decrease of production costs, the first term is a corrective term related to market power. This indirect effect is composed of two factors: the price-marginal cost difference and the sensitivity of production to emissions. Because of market power, and demand elasticity, this corrective term is strictly positive. At the second best optimum, marginal net surpluses are equalized among markets so the optimal allocation of emissions given the exercise of market power on output markets satisfies the following first order condition: ( P 1 C 1 q 1 ) q C 1 e 1 C 1 e 1 = ( P 2 C 2 q 2 ) q C 2 e 2 C 2 e 2 (7) Quantities produced at the second best allocation are denoted Q 1 and Q 2 they satisfy: Q i = Q C i (E i ) for i = 1, 2 The two corrective terms explain that allocations of emissions with an integrated market for emissions permits can be different than the second best optimal one. Let define the difference of corrective terms: s = P 1 Q 1 q1 C + P Q 2 q2 C 2 (8) n 1 e 1 n 2 e 2 With an integrated market for emissions permits local permits prices are equalized: σ 1 = σ 2 and the equilibrium price σ C clears the permits market: e = E C 1 (σ C ) + E C 2 (σ C ). At this equilibrium: C 1 e 1 (q C 1 (e C 1 ), e C 1 ) = σ C = C 2 e 2 (q C 2 (e C 2 ), e C 2 ), the marginal values of emissions for each firms are equalized across output markets. Therefore, given equilibrium productions the allocation of emissions is optimal but if one considers the effect of emissions on production it is not 9

10 (in general) because of corrective terms. Even if firms are price takers on the emissions permits market, the integration of emissions permits markets does not increase welfare in general. Proposition 1 If s 0, welfare is strictly lower with an integrated market for emissions than with two isolated markets with initial allocations Êi = Ei, i = 1, 2. The benefit of integration of permits markets depends of initial allocations Êi, if they are (E1, E2 ) the integration of permits market decreases welfare. But if the initial allocation departs from this second best, the welfare effect of markets integration is ambiguous. Welfare might be increased by markets integration if Ê1 is too low or too high. The issue of emissions permits markets integration boils down to the analysis of the choice of initial allocations. These initial allocations can be suboptimal if the regulator lacks some information about market conditions when setting these. Before analyzing this situation, I first analyze how the regulator can correct the integrated permits market in order to reach the second best optimum. Instead of allocating emissions to each sector and keep markets isolated, the regulator can use a price instrument to correct the integrated market and implement the second best optimum. Corollary 1 The second best optimum can be established with an integrated market for emissions permits and a subsidy s of market 1 emissions. The proof is in appendix B. The subsidy s reflects the value of emissions that is not considered by firms while choosing their emissions with an integrated market. It can be either positive or negative, it is positive if market 1 is the underemitting market. This subsidy is an indirect way to correct market power, less efficient than a direct subsidy of production. More generally: strategic firms underproduce given the quantities of inputs they use and, if the regulator cannot directly correct market power by subsidizing production, he can do it indirectly by subsidizing inputs and distorting inputs markets. Here, emissions are the subsidized input and a local subsidy would be P i Q i /n i qi C / e i on market i = 1, 2. The regulator can either set such a subsidy on each market or only a subsidy s on market 1, because the only relevant variable is the difference between the two subsidies 6. 6 So, there is an infinity of corrective policies that could be implemented. The only relevant feature of the corrective policy is that prices of emissions faced by firms on market 1 and on market 2 are different, and the difference should be equal to s. 10

11 The subsidized market is the underemitting one, and it depends of the extend of market power and the relation between production and emissions. The influence of several parameters is investigated in a quadratic setting developed in the next section. 3.2 Quadratic specifications To get a more precise picture of the influence of parameters, a quadratic specification is used. In this section, I content myself with determining which sector underemits with an integrated market for emissions permits and should therefore be subsidized. In the next section the description of the quadratic framework is deepened to understand the influence of uncertainty. It is assumed that on each market i = 1, 2: the marginal production cost (without emissions and abatement) is null and emissions rates are constant and set at 1 and there is a separable abatement technology: On each market i = 1, 2: Consumers surplus on market is: S i (Q i ) = (a i 0.5b i Q i )Q i (9) And cost of production: C i (q i, e i ) = { 0 if qi < e i 0.5c i (q i e i ) 2 otherwise (10) If the marginal cost of abatement is infinite, there is no opportunity to reduce emissions but to reduce production and quantities of outputs and emissions are equal. With two isolated markets, quantities produced can be expressed as functions of local emissions. On market i = 1, 2, if production is strictly positive 7 it satisfies the first order condition : a i b i (n i + 1)q i = max {c i (q i e i ), 0} so aggregate production is: Q C i (E i ) = n i q C i = n ia i + c i E i (n i + 1)b i + c i if n i a i (n i + 1)b i E i n i a i c i (11) 7 It is possible that the quantity of emissions of a sector is negative if this sector abates more emissions and its production. 11

12 Some conditions (cf appendix C should be satisfied by parameters to ensure that the emissions constraint is binding and that both sectors produce a positive quantity with an integrated market. In this case, with an integrated permits market, the allocation of emissions is such that: c 1 (q C 1 e C 1 ) = c 2 (q C 2 e C 2 ), marginal costs of production are equalized. Because of the separability of abatement, the difference of abatement cost does not influence the market outcome 8. Thanks to this separability it is feasible to disentangle the influence of parameters. The marginal effect of individual emissions on individual production is: q C i e i = c i (n i + 1)b i + c i So it is increasing with respect to abatement cost c i and with respect to market size 1/b i and decreasing with respect to the number of firms n i. To determine which sector should be subsidized it is sufficient to compare P i qi C qi C / e i at the uncorrected equilibrium. As mentioned, marginal cost are equalized so P i qi C is determined by the aggregate abatement cost and not by c i. So, if output markets are identical: P 1 = P 2 and n 1 = n 2, the less efficient firms with respect to abatement should be subsidized. At the uncorrected equilibrium firms have similar marginal costs so they produce similar quantities of outputs. To subsidize input of a sector decreases its marginal cost and consequently increases production. So the market that should be subsidized is the sector of which marginal cost is more sensitive to an additive emission and, with quadratic specifications, the effect of emissions on individual marginal cost is c i. So, s > 0 if and only if c 1 > c 2. Similarly, it is worth considering the influence of the number of firms, if price and costs are equal but n 1 < n 2 there are two effects that go in the same directions: the price marginal cost margin is higher on market 1, and the production on market 1 is more sensitive to emissions. Because of these two effects sector 1 should be subsidized 9. And finally, the effect of market sizes 1/b i can be investigated when a 1 = a 2, c 1 = c 2 and n 1 = n 2. At the 8 The aggregate abatement cost is 0.5c(Q 1 + Q 2 e) 2 with c = (n 1 /c 1 + n 2 /c 2 ) 1. The market outcome depends upon c and not the precise values c 1 and c 2. 9 It is not a general result because with general price and cost functions the monotonicity of the factor q C i / e i with respect to n i is unclear. 12

13 uncorrected equilibrium, individual marginal costs are equal so P i qi C = b i qi C are equal, and the biggest market should be subsidized because its production increases more subsequently to an extra emission. 4 Imperfect information As stated by proposition 1, the integration of emissions permits markets can decrease welfare because of market imperfection on output markets. A perfectly informed regulator is able to perform better than an integrated market by initially allocating the constraint among markets and keeping markets isolated. But, if the regulator lacks information about market conditions he might be unable to set optimal allocations and an integrated market can improve efficiency despite imperfect competition. 4.1 Uncertainty Uncertainty is added to the quadratic framework introduced in the previous section. The approach is similar to the one developed by Weitzman (1974) to compare price and quantity regulatory instruments. Costs are given by (10), and gross consumers surplus on each market is random at the time of allocation of permits: S i (Q i, θ i ), where θ i is a random parameter with Eθ i = 0, uncertainty is assumed additive: S i (Q i, θ i ) = (a i + θ i )Q i b i 2 Q2 i (12) And welfare in a particular state (θ 1, θ 2 ) is : W (Q 1, Q 2, E 1, E 2, θ 1, θ 2 ) = S i (Q i, θ i ) n i C i (Q i /n i, E i /n i ) (13) i=1,2 Parameters θ i can be interpreted in several ways. First, with additive uncertainty and quadratic specifications they can encompass uncertainties about marginal consumers surpluses and marginal costs. Second, the relevant feature of the model is that: when firms decide how much to produce they have more information than the regulator when he designs the policy. So, parameters θ i represent either asymmetric information between the regulator and firms, or, uncertainty about future markets conditions due to the 13

14 time lag between the design of the emissions permits markets and markets interactions. Both are relevant and simultaneously at stake for the EU ETS: it is reasonable to assume that firms have superior knowledge of market conditions than the regulator, and future trends of sectors concerned are uncertain when the regulator designs emissions markets. 4.2 Production In all regulatory options considered the regulator cannot control production but only emissions markets, so it is worth establishing reduce form of welfare as function of emissions. For i = 1, 2 local production depends upon local emissions and market conditions. If a i + θ i is sufficiently high 10, production is similar to (11) so: Q C i (E i, θ i ) = n i(a i + θ i ) + c i E i (n i + 1)b i + c i (14) Given a fixed quantity of emissions, production is adapted to local market condition θ i. Thanks to the linearity of our framework adaptation to θ i and to emissions are not intertwined; the sensitivity of production to emissions is independent of random market condition and denoted α i : α i = QC i E i = c i (n i + 1)b i + c i. (15) Local net surplus can be expressed in a reduced form as a function of emissions: W i (E i, θ i ) = (A i (a i + θ i ) 0.5B i E i ) E i + K i, (16) where coefficients are: A i = α i (n i + 2)b i + c i (n i + 1)b i + c i, B i = b i α i (n i + 1) 2 b i + n i c i n i ((n i + 1)b i + c i ), K i = [ a i 0.5(b i + c i )Q C i (0, θ i ) ] Q C i (0, θ i ). 10 If the quantity of emissions is negative and lower than n i (a i + θ i )/c i ) E i it is not worth producing. And if the quantity of emissions is higher than the unregulated production n i (a i + θ i )/(n i + 1), firms produce this quantity. 14

15 And summing over both markets, a reduce form of aggregate welfare is (with a slight abuse of notation): W (E 1, θ 1, θ 2 ) = W 1 (E 1, θ 1 ) + W 2 (e E 1, θ 2 ) = [A 1 (a 1 + θ 1 ) A 2 (a 2 + θ 2 ) + B 2 e 0.5(B 1 + B 2 )E 1 ] E 1 + [K 1 + K 2 + (A 2 (a 2 + θ 2 ) 0.5B 2 e) e] Within the reduce form of welfare, uncertainties are still additive but weighted by coefficients A i. These coefficients together with B i, encompass the relationship between emissions and productions. Furthermore, on each market i = 1, 2, in any demand states θ i, the demand for emissions permits is Ei C (σ i, θ i ) that satisfies: c i (Q C i Ei C ) = σ i so: Ei C = 1 (α i a i σ i ) (17) β i where: β i = (n i + 1)b i n i α i, i = 1, 2 (18) Some assumptions are required on parameters to ensure that in all configuration considered and in all states, the emission constraint is binding and both goods are produced. Those conditions (listed in appendix C) consist mainly in that a i, i = 1, 2 are sufficiently high and not too different, and the support of θ i, i = 1, 2 should be restricted. 4.3 Regulatory options The regulator does not know the value of random parameters θ i when deciding to allocate emissions among firms. He decides whether or not to implement an integrated market for emissions permits, if markets are integrated, the initial allocation of permits does not influence the market outcome. I analyze both cases whether the regulator set an ex-ante subsidy to partly compensate market power or not. So the three regulatory options are: 1. The regulator allocates emissions Êi to firms of market i = 1, 2 and emissions permit markets are isolated so productions are Q C i (Êi) and expected welfare is: Ŵ = E θ1 θ 2 W (Ê1, θ 1, θ 2 ) 15

16 2. Emissions permits market are integrated and a subsidy s on market 1 emissions is fixed ex ante. Firms choose production and emission once θ 1, θ 2 are revealed, the permit price σ(θ 1, θ 2, s) clears the market for emissions so that: An expected welfare is denoted W I (s): E C 1 (θ 1, σ s) + E C 2 (θ 2, σ) = e (19) W I (s) = E θ1 θ 2 W (E C 1, θ 1, θ 2 ) 3. Emissions permits market are integrated and no corrective policy is implemented. So expected welfare is W I (0). 4.4 Comparison In the benchmark situation the regulator allocates emissions Êi to firms of market i = 1, 2. Firms may trade permits within each markets but not across markets. The regulator maximizes expected welfare: EW (E 1, θ 1, θ 2 ). Hence, he fixes the allocation: Ê i = A ia i A j a j + B j e B i + B j, i, j = 1, 2, j i (20) With this allocation of emissions expected welfare is Ŵ. Emissions are not influenced by random parameters but productions are. With an integrated market for emissions permits, the allocation is conditional on random markets conditions. There are potential gains from market integration that come from the adaptation of emissions to these conditions. If an integrated market is introduced, the initial allocation of permits does not influence the outcome because firms are competitive on the permits market. In order to compare both situations with and without a corrective subsidy on the permit market, I determine a general expression for any subsidy s on market 1 emissions. If a subsidy s is set on emissions on market 1, firms from market 1 face the permit price σ s and firms from market 2 the permit price σ. At equilibrium, the permit price clears the permits market and equation (19) is satisfied. So the equilibrium quantity of emissions is: E1 C 1 (θ 1, θ 2, s) = [α 1 (a 1 + θ 1 ) + s α 2 (a 2 + θ 2 ) + β 2 e] β 1 + β 2 = E 1 (s) + 1 β 1 + β 2 (α 1 θ 1 α 2 θ 2 ) (21) 16

17 This expression consists of a fixed term and a random one. The latter represents the adaptation of emissions allocations to revealed market conditions θ i, i = 1, 2. The nice feature of the quadratic framework is that this effects can be isolated: the subsidy only modifies the expected allocation of emissions and not its random part. Hence, the variation of the allocation can be isolated and injecting expressions into expected welfare allow to isolate effects of both terms on welfare: W I (s) = E [ W ( E C 1, θ 1, θ 2 )] = E [ W (E 1, θ 1, θ 2 ) ] + I (22) = Ŵ 0.5 (B 1 + B 2 ) (Ê1 E 1 ) 2 + I Where I does not depend upon s and contains the effects of adaptation of emissions to market conditions: [( I = E A 1 B ) ( 1 + B 2 α 1 θ 1 A 2 B ) ] 1 + B 2 α1 θ 1 α 2 θ 2 α 2 θ 2 (23) β 1 + β 2 β 1 + β 2 β 1 + β 2 The issue of market integration boils down to the comparison of the uncertainty effect and the misallocation related to market power. With a subsidy the latter can be canceled. Proposition 2 With asymmetric information i With a corrective subsidy expected welfare is greater with market integration if and only if I 0, ii without corrective subsidy, expected welfare is greater with market integration if and only if: (B 1 + B 2 ) (Ê1 E 1 ) 2 2I (24) This proposition stresses the two effects at stake when evaluating market integration. On one hand there is a welfare loss related to market imperfection, but, on the other hand there is an eventual welfare gain related to uncertainty. In a context of asymmetric information, the potential gain from market integration is related to the mobilization of private information of firms when allocating the emission cap. With an integrated the allocation is contingent on market conditions, and this contingency can enhance welfare. So, the main question is the sign of the uncertainty effect I. Without further assumptions on parameters values and distributions, the sign of this term cannot be set. 17

18 Without abatement A special case worth considering is the case without separable abatement: c 1 = c 2 = +. The quantity produced and the quantity of local emissions are identical and at equilibrium: [ Q C 1 = E1 C 1 = (a 1 a 2 ) + n ] b 2 e + s + (θ 1 θ 2 ) (25) β 1 + β 2 n 2 = E C 1 + θ 1 θ 2 β 1 + β 2 where β i = (n i + i) b i /n i can be interpreted as a measure of aggregate market power. The random term represents the adaptation of productions to revealed market conditions θ i, i = 1, 2. Corollary 2 With asymmetric information and no separable abatement, (i) Without any subsidy an integrated market improves expected welfare if and only if: var(θ 1 θ 2 ) > [b 1 (a 1 a 2 + b 2 e) /n 1 b 2 (a 2 a 1 + b 1 e) /n 2 ] 2 [(1 + 2n 1 ) b 1 + (1 + 2n 2 ) b 2 ] (b 1 + b 2 ) (ii) With an optimal ex ante subsidy, markets integration always increases expected welfare. The optimal subsidy is such that Q C 1 (s ) = ˆQ 1 and the welfare gain is : [ var(θ 1 θ 2 ) B 1 ] B 2 2 (b 1 + b 2 ) Calculations are in appendix D. The choice to integrate market is based on the comparison of the two effects mentioned. Without separable abatement, the uncertainty term is always positive and proportional to the variance of the difference of marginal consumers surplus. Once the emission cap is fixed, the determining parameter for the allocation of this cap is not absolute marginal consumers surplus but relative one. The gain from market integration is related to the adaptation of productions to markets conditions in the right direction. Both optimal allocation and the integrated market one are 18

19 proportional to the difference θ 1 θ 2. It explains that welfare gain is increasing with respect to the variance of relative marginal surplus. This variance is var(θ 1 )+var(θ 2 ) 2cov(θ 1, θ 2 ), it is decreasing with respect to the correlation of random parameters, so most gains from market integration are obtained if markets are negatively correlated. This positive effect emphasizes the role of markets to aggregate information and coordinate decentralized decisions. The negative term due to market power is related to concentration on each market. Some comparative static can be done. First, it should be noticed that with constant production cost an emission rates, first best and second best allocations in any state (θ 1, θ 2 ) coincide. Compared to the optimal allocation, firms of a market produce too much and firms of the other not enough. Market power rather than decreasing production misallocates the constraint. Second, it appears from the expression of production (25) that if firms of market 1 underproduce on average they always do. Therefore, an increase of the number of firms on the underemiting market increases welfare in all demand states and subsequently the appeal of market integration. An increase of the number of firms in the overemitting market has the opposite effect: it worsen the misallocation in all demand states and subsequently the attractiveness of market integration. If an ex ante subsidy is set to correct the effect of market power, an increase of the number of firms in any market increases welfare. With no opportunity of external abatement, the uncertainty effect is positive whatever the distribution of demand states because the optimal production on market 1 and the production with an integrated permits market are both linearly related to the difference θ 1 θ 2. When markets conditions vary, the integrated market quantity e C 1 goes in the right direction: it increases when θ 1 θ 2 does. With abatement With separable abatement, the comparison of regulatory options is described by expressions (22) and (23). First, A i > α i and B i > β i so A i (B 1 + B 2 )α i /(β 1 + β 2 ) > 0. If uncertainties are independently distributed, the uncertainty effect is always positive because if consumers surplus on one market increases, the quantity of emissions permits consume by this market increases. In that case, introduction of abatement does not qualitatively modify the comparison because the information processing rule of market is still positive and can justify the integration of permits market. 19

20 The situation is more interesting when correlation are considered. If market conditions simultaneously varies, i.e. are correlated, and the market moves in the opposite direction direction there is a loss. It can be the case if the optimal contingent allocation and the integrated market one are not proportionally related i.e. A 1 α 1 B 1 + B 2 β 1 + β 2 A 2 α 2 B 1 + B 2 β 1 + β 2, simplications of this expression, it is equivalent to: (n 2 n 1 )b 1 b 2 b 2 c 1 b 1 c 2 Corollary 3 If θ 1 = xθ 2 with x R: If (n 2 n 1 )b 1 b 2 > b 2 c 1 b 1 c 2, there is u > 1 such that: I < 0 α 2 α 1 < x < u α 2 α 1 When abatement is introduced the positive attribute of integrated market vanishes for some distributions of random variables. A particular case is exhibited in corollary (3). Abatement creates a further difference between firms actions and public interest. With abatement firms have two ways to reduce their emissions and the arbitrage they do 5 The choice of the emission cap In previous sections the emission cap was assumed fixed and the issue addressed was how market power on output markets misallocates a fixed emission cap. A corrective policy was determined but it is not likely that such a policy will be put in place in the EUETS in order to limit possible regulatory capture or strategic interventionism by European states. Therefore, a natural question that arises is the effect of such misallocation on the optimal emission cap. Even if the political process that fixes emissions cap can hardly be seen as a benevolent regulator maximizing welfare, such a question is relevant to understand how internal imperfections influence aggregate environmental policies. The issue can be generalized: it is to understand how internal imperfections influence the choice of an aggregate constraint. And 20

21 far from providing any general answer to this question, I establish here with a relatively simple example that it can be in either ways. Internal misallocations of a constraint can imply that the optimal, second best, aggregate quantity is either higher or lower than the first best one (without internal imperfections). In the general case (i.e. with general cost function) three situations should be distinguished: with and without market power and in the former case with and without the corrective policy. As I consider here the linear framework without abatement, the competitive case (first best) coincides with the case of corrected market power (second best). As environmental damage is assumed separable and convex, it is sufficient to analyze the derivative of welfare (2) with respect to e with and without the exercise of market power. In the case of perfect competition, the derivative of welfare (2) is S i(q i ), i=1,2 which is the value of the emission constraint and the price of emissions permits. With Cournot competition on output markets the effect of the cap on quantities produced should be considered and the derivative of welfare is ( S 1 (Q C 1 ) S 2(Q C 2 ) ) Q C 1 / e + S 2. The effect of misallocation on the optimal emissions cap is determined by the sign of the difference of both derivatives: = [ P 1 (Q C 1 ) P 2 (Q C 2 ) ] Q C 1 e + P 2(Q C 2 ) P 2 (Q 2) (26) From this equation it appears that the answer is not obvious for two contradictory effects are at stake. Let s assume that market 1 is the underemitting market, in that case the first term is positive: an increase of the emission cap has a positive effect by increasing the emissions of market 1, but the second term is negative because market 2 is overemitting. Hence, an increase of the emission cap partly corrects the misallocation but has a lower direct effect on welfare. Whether one effect dominates the other is not evident from this equation. It is unclear whether the misallocation implies a more or less stringent environmental policy. With the linear case: P i = a i b i Q i, the difference P 2 (Q C 2 ) P 2 (Q 2) is b 2. ( ( ) Q C 2 Q2) and this is equal to b2 Q 1 Q C 1. Similarly the difference P 1 P 2 can be simply expressed as (b 1 + b 2 ) ( ) Q 1 Q C 1. And finally it appears that the only relevant quantity is the sensitivity of productions to the emissions cap and this sensitivity is determined by the relative extent of market power on both markets. Proposition 3 The optimal emission cap is higher with market power than 21

22 without if and only if: (n 1 n 2 ) ( Q 1 Q C 1 ) 0 (27) Calculations are straightforward to establish that (b 1 + b 2 ) QC 1 b e 2 is proportional to n 1 n 2. The difference of markets concentrations is sufficient to compare both effects mentioned above. If market 1 is under emitting and n 1 n 2 the corrective influence of the emission cap dominates and the optimal emission cap is higher with market power than without, i.e. the environmental policy is less stringent. But if the concentration on market 1 is higher than on market 2, the misallocation due to market power implies a more severe environmental policy. Hence, with two similar markets (P 1 (Q) = P 2 (Q)) the optimal emission cap is always increased by market imperfection. 6 Conclusion In this paper I analyzed how imperfect competition on two output markets influences the efficiency of an integrated market of emissions permits. If output markets are imperfectly competitive the integration of permits markets can decrease welfare. Market power on output markets explains that the allocation of the emissions constraint is suboptimal with an integrated permits market. This misallocation can be corrected by a subsidy on emissions of one sector that encompasses the effect of emissions on production and market power distorsion. A perfectly informed regulator can do strictly better with two isolated permits markets than an uncorrected integrated market. However, it is more realistic to consider that there is asymmetric information between firms and the regulator about market conditions. By introducing, in a quadratic specification of the model, additive uncertainty on demand conditions the issue of permits market integration under asymmetric information has been analyzed. Additionaly to the misallocation previously stressed, an uncertainty effect is introduced. An integrated markets for emissions has the additional feature of adapting the allocation of the emissions cap to revealed market conditions. The decision whether to integrate markets for emissions should be based on the comparisons of the two effects. If there is no possibility of abatement, the uncertainty effect is always positive and decreasing with markets corelations. An integrated markets have 22

23 Many markets concerned by the regulation of emissions are concentrated such as electricity markets, and more firms are present on the emissions market than on output markets, hence, it is justified to analysis how local imperfections interacted via emissions markets. The scope of an emission market can therefore be limited in order to avoid imperfection contagion. References Amir, R. and Nannerup, N. (2005). Asymmetric Regulation of Identical Polluters in Oligopoly Models, Environmental and Resource Economics 30(1): Barnett, A. (1980). The Pigouvian Tax Rule under Monopoly, American Economic Review 70(5): Buchanan, J. (1969). External Diseconomies, Corrective Taxes, and Market Structure, American Economic Review 59(1): Hung, N. and Sartzetakis, E. (1998). Cross-Industry Emission Permits Trading, Journal of Regulatory Economics 13(1): Malueg, D. A. and Yates, A. J. (2007). Strategic Behavior, Private Information, and Decentralization in the European Union Emissions Trading Scheme, SSRN elibrary. Montgomery, W. (1972). Markets in Licenses and Efficient Pollution Control Programs, Journal of Economic Theory 5(3): Novshek, W. (1985). On the Existence of Cournot Equilibrium, The Review of Economic Studies 52(1): Requate, T. (2005). Environmental policy under imperfect competition a survey, Christian-Albrechts-Universität Kiel, Department of Economics: Economics Working Paper ( ). Sartzetakis, E. (1997). Tradeable Emission Permits Regulations in the Presence of Imperfectly Competitive Product Markets: Welfare Implications, Environmental and Resource Economics 9(1):

24 Sartzetakis, E. (2004). On the Efficiency of Competitive Markets for Emission Permits, Environmental and Resource Economics 27(1): Simpson, R. (1995). Optimal pollution taxation in a Cournot duopoly, Environmental and Resource Economics 6(4): Weitzman, M. (1974). 41(4): Prices vs. Quantities, Review of Economic Studies A Existence of uniqueness of Cournot equilibrium Let i {1, 2}, I establish that for any permit price σ i there exists a unique equilibrium which is symmetric on market i. To do so I consider the gross cost function Γ (q i, σ i ) = min e i {C i (q i, e i ) σ i e i }. This function is convex with respect to q i by derivation: Γ q i = C i q i (q i, e i ) and 2 Γ q 2 i = 2 C i q 2 i ( 2 C i q i e i ) 2 ( ) 2 1 C i 0 e 2 Thanks to the assumption on the price function P i + P i Q i, Q i there is a unique Cournot equilibrium for all σ i (Novshek; 1985). Equilibrium quantities satisfy the two equations: P i + P i q i = C i q i σ i = C i e i Hence, for any permit price the strategy ( q C i (e C i ), e C i (n i, σ i ) ) is the unique (symmetric) equilibrium of the game considered. B Proof of corollary 1 With a subsidy s on emissions of market 1 and an integrated market for emissions with a price σ, the price of emissions faced by firms on market 24

25 1 is σ s and by firm on market 2 σ, therefore, at equilibrium respective emissions per firm are e C 1 (n 1, σ s) and e C 2 (n 2, σ)and therefore: C 1 e 1 = C 2 e 2 + s (28) For a subsidy s, emissions satisfy the first order condition of the second best equation (7), so, thanks to the assumption of welfare concavity, it is the second best allocation. C Conditions with quadratic specifications For the emissions constraint to be binding the following condition should be satisfied: n 1 a 1 + n 2a 2 e (C1) (n 1 + 1)b 1 (n 2 + 1)b 2 And, to ensure that both sectors produce a positive quantity with an integrated market, the equilibrium permit price should be lower than a i ( n i n1 (a i a j ) e + a j + n ) 2 (C2) (n i + 1)b i c 1 c 2 D Proof of proposition 2 With quadratic expressions welfare could be written (22) as in the main text. When an integrated market is implemented equilibrium production on market 1 is given by expression (refeq:prodcournot): Q C 1 = Q C 1 +(θ 1 θ 2 ) /B and the expected production is: 1 B(n 1, n 2 ) [ (a 1 a 2 ) + n ] b 2 e + s n 2 (29) Quadratic expressions and additive uncertainty on marginal surpluses allow to separate effects of expected production and random term as in expression (refeq:welfarecournotquad). The effect of expected production is: ) EW (Q C1, Q C2, θ 1, θ 2 = Ŵ b 1 + b 2 2 ( ) 2 Q C 1 Q 1 25

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