The Effect of Emission Permits and Pigouvian Taxes on Market Structure

Transcription

1 The Effect of Emission Permits and Pigouvian Taxes on Market Structure MIGUEL BUÑUEL 1 FUNDACIÓN BIODIVERSIDAD AND UNIVERSIDAD AUTÓNOMA DE MADRID MADRID, SPAIN Abstract Differently from Pigouvian taxes and direct regulation, tradable emission permits can decrease competition in a polluting industry under certain circumstances. Assume a potential entrant who can buy every permit. When permits are given free to current polluters, monopolization occurs if not every polluter foresees it. If foreseen, polluters want to free ride on the entrant s market power, but entry can still occur, although not with certainty. Considering a symmetric, mixedstrategy equilibrium with unconditional bids, the probability of entry decreases as the number of polluters increases. When permits are sold initially, monopolization occurs without more requirements than polluters being financially constrained. JEL classification codes: D42, L12, H89, Q28. Key words: Pigouvian taxes, tradable emission permits, environmental regulation, market structure, monopoly. 1 Miguel Buñuel, Responsable de Estudios y Publicaciones; Fundación Biodiversidad; Pza. Alonso Martínez, 3-4ª planta; Madrid; Spain; mbunuel@fundacion-biodiversidad.es. Miguel Buñuel; Departamento de Economía y Hacienda Pública; Universidad Autónoma de Madrid; Ctra. Colmenar Viejo, Km. 15; Madrid; Spain; mbunuel@hotmail.com. Permanent address: miguel.bunuel.1999@alum.bu.edu. 3

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3 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD 1. Introduction It is well known that a Pigouvian tax and a system of tradable emission permits are equivalent instruments under ideal conditions (e.g. Adar and Griffin (1976)). However, uncertainty and many other practical issues make the choice between those two instruments a very important topic, which has been broadly considered in the literature. Nevertheless, little has been said on the possible effect that the choice between taxes and permits may have on market structure. Market structure has been considered in relation to the internalization of externalities mostly to the extent that a non-competitive market may require to change the usual Pigouvian prescription, as Buchanan (1969) first recognized. More in need of study is the relation between market structure and the internalization of externalities in the opposite direction, that is, the possibility that the choice of instrument for internalizing negative externalities may affect market structure. Among the contributions related to this line of research, von der Fehr (1993) studied how emission permits could be used strategically by oligopolistic firms as an instrument for monopolization and entry deterrence, showing that profits are maximized, under certain circumstances, when a single firm holds all the permits. Also in an oligopolistic setting, Sartzetakis (1997) shows that competition in the product market can be lessened through the use of emission permits as a means to increase rivals costs, in line with results first reached by Misiolek and Elder (1989). The issue explored in this paper may be very important in practice. For instance, von der Fehr (1993) reminds us that the possibility of using permits with exclusionary purposes was one of the main arguments that prevented the implementation of tradable fishing rights on the Norwegian coast line. The same concerns have been raised on the use of emission permits to control sulfur dioxide emissions in the U.K. electricity industry (e.g. von der Fehr (1993)) and water pollution in Scotland s Forth Estuary (e.g. Hanley and Moffat (1992)). The relevance of this problem is also underlined by Joskow and Tirole (2000) in a quite different setting. These authors show that the allocation of transmission rights associated with the use of electric power networks may enhance preexisting market power, and cause production inefficiency and a loss 5

4 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE of welfare. They find that physical rights, which are equivalent to emission permits in our context, may be worse in terms of welfare effects than financial rights, since, as emission permits, they can be withheld from the market. This paper studies whether and how the choice between a Pigouvian tax and a system of emission permits (including direct regulation, which can be regarded as a system of non-tradable permits given free to current polluters), and the choice between different methods for the distribution of permits, may affect the structure of the polluting industry. In particular, the paper examines under what conditions those choices can transform a non-monopolistic output market into a monopolistic one. Similarly to Joskow and Tirole (2000), very important insights are drawn from the extensive literature on takeovers that followed Grossman and Hart s (1980) seminal contribution. After describing the basic model and its assumptions in section 2, the effects of different mechanisms for the internalization of pollution are studied in section 3. The first mechanism considered is a Pigouvian tax on emissions. Secondly, emission permits are introduced into the model, distinguishing between non-tradable (direct regulation) and tradable permits. When permits are tradable, the first case examined is that in which permits are given free to current polluters. It is assumed that current polluters are financially constrained to attempt gaining market power through the purchase of permits, and there exists a single financially-unconstrained potential entrant. Second, the case in which permits are initially distributed by sale is considered, both when there is no potential entrant, and when there is a financially-unconstrained potential entrant and current polluters are financially constrained. Finally, sections 4 presents the main conclusions, and discuss their relevance. 6

5 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD 2. Basic model 2.1. Firms The industry is composed of m identical profit-maximizing firms. In principle, all market structures but monopoly are possible; that is, m {x 2 x "}. The firms problem is max q j 0 Π c j (q j,q j )P d (Q s )q j C(q j ), for j = 1,..., m, where Π j c ((, () is firm j s (Cournot) profit function, q j is firm j s output, j1 m q j q k q l, k1 lj1 Q s q j + q j, P d (() is the market inverse demand function (it is implicit that firms assume market clearance), and C(() is firm j s cost function. It is assumed that P d (() is twice differentiable, P d < 0, and P d 0; C(() is twice differentiable, C > 0, and C 0. The Kuhn-Tucker necessary and sufficient conditions are P d (q * j q j )C'(q * j )q * j P d '(q * j q j )0, q * j >0;P d (q * j q j )C'(q * j )q * j P d '(q * j q j ), (1) where q j * = R j (q j ), where R j is firm j s reaction function, that is, the best action for firm j given that the other firms actions amount to q j. The interpretation of conditions (1) is well known: when optimal output is positive, firms produce until the profit generated by the last unit of output (P C ) equals the loss on profitability of inframarginal output (q j P ), which is caused by the price decrease generated by increasing production by a marginal unit. Notice that the effect of firm j s output on price 7

6 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE becomes negligible when the number of firms is very large; that is, P d (q j * + q j ) $ 0 as m $ ". In this case, the competitive outcome is achieved (P = C ). The following is also assumed: (i) firms produce a negative externality in the form of pollution; (ii) pollution does not harm firms, only consumers; (iii) there is no feasible abatement technology, and hence reducing output is the only way of cutting emissions. Contrary to what it could seem, assumption (iii) is not very restrictive in our setting, since the conclusions of the model would be the same with abatement technology as long as it cannot abate emissions completely, and, therefore, any polluting firm always needs a certain number of emission permits. The ability of permits to play a role changing market structure depends upon them acting as a barrier to entry into the industry, so that a firm can foreclose competition by buying all the permits. Thus, if the substitutability between abatement technology and permits were perfect, the latter would not be able of altering market structure. Since the possibility of technology abating emissions totally is remote in most instances (if not technologically, at least economically), assuming the nonexistence of abatement technology simplifies the model with little loss of generality Consumers The household sector is formed by n identical utility-maximizing consumers, who suffer damages from pollution generated by the firms. In principle, no restriction is imposed on the size of the household sector, except that there is at least one consumer; that is, n {x 1 x "}. The consumers problem is max q i 0 Υ i (q i,q i )U(q i )P s (Q d )q i D(E(Q d )), for i = 1,..., n, where Υ i ((, () is consumer i s total utility function, q i is consumer i s consumption, i1 n q i q g q h, g1 hi1 8

7 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD U(() is consumer i s function of utility from consumption, Q d q i + q i, P s (() is the market inverse supply function, D(() is consumer i s pollution-damage function, and E(() is the pollutionemission function (it is implicit that consumers assume market clearance). It is assumed that U(() is twice differentiable, U > 0, and U < 0; P s (() is differentiable, P s 0, and P s (0) < P d (0); D(() is twice differentiable, D > 0, and D 0; and E(() is twice differentiable, E > 0, and exhibits constant returns to scale (E = 0), that is, E(Q) = m E(Q m 1 ). The Kuhn-Tucker necessary and sufficient conditions are U'(q * i )P s (q * i q i )q * i P s '(q * i q i )E'(q * i q i )D'(E(q * i q i ))0, q * i >0;P s (q * i q i )U'(q * i )q * i P s '(q * i q i )E'(q * i q i )D'(E(q * i q i )), (2) where q i * = R i (q i ), where R i is consumer i s reaction function, that is, the best action for consumer i given that the other consumers actions amount to q i. Conditions (2) can be interpreted as follows: when optimal consumption is positive, a consumer buys products until the consumer surplus generated by the last unit consumed (U P) equals the loss of consumer surplus derived from inframarginal consumption (q i P ), which is produced by the rise in price that results from increasing consumption by a marginal unit, plus the increase in the pollution damage caused to that consumer by the emissions generated by producing the marginal unit consumed (E D ). Notice that the effect of consumer i s consumption on price and emissions becomes negligible when the number of consumers is * * very large; that is, P s (q i + q i ) $ 0, and E (q i + q i ) $ 0, as n $ ". In this instance, the result is the competitive outcome (P = U ) Market equilibrium The market-clearing equilibrium condition is m! q * n j! q * i Q c P d (Q c )P s (Q c ) P(Q c ), j1 i1 9

8 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE where Q c is the market total output, and P(Q c ) is the market price. Focusing on symmetric equilibrium, Q c * = m q j = n q * * i, for q j = q * * k, k U j, and q i = q * h, h U i. Thus, given conditions (1) and (2), the symmetric equilibrium must satisfy the following whenever market total output is positive: P(Q c )U' Q c n Q c n P s '(Q c )E'(Q c )D'(E(Q c ))C' Q c m Q c m P d '(Q c ). (3) It follows from (3) that firm j s profits are Π c j (q c j )2 P d '(Q c )q c j C'(q c j )C(q c j )>0, c for j = 1,..., m, where q j = Q c m 1 c c. Π j > 0, q j > 0, follows from P d < 0, C > 0, and C 0. Notice how firm j s profits depend on the effect of its output on price, and thus inversely on the number of firms in the market. As m grows larger, firm j s output and its effect on price both decrease. Therefore, profits can be defined as a function of m such that Π c j (m) < 0. In particular, Π c j $ q c j C (q c j ) C(q c j ) as m $ ", that is, in perfect competition. In the case of c constant returns to scale (C = 0), q j C (q c j ) = C(q c j ); the zero-profit result is attained in perfect competition. The equilibrium described above is the long-run market equilibrium because of the following assumptions about costs of entry. It is assumed that there exist fixed costs of entry into the industry, Γ > 0, which are less than the profits that an entrant would make when the number of firms currently in the market is less than m, but larger than an entrant s profits when there are at least m firms in the industry. Therefore, m is the number of firms in the long-run equilibrium; in principle, costs of entry make entry unprofitable for any firm not currently in the market. Since profits decrease as the number of firms increases, the maximum and minimum levels of the assumed fixed costs of entry are lower for larger values of m: 10

9 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD Q c m1 ΓΓ max (m) m1 ΓΓ min (m) Q c m1 m1 Q c 2 2 P d '(Q c m1 ) Q c m1 m1 C' P d '(Q c m1 ) Q c m1 m1 C' m1 j1 c c m1 Qm1 m1 C Q m1,and c c m1 Qm1 m1 C Q m1,where m1 m1 R j (q j ), and Q c m1 R j (q j ). j1 It is also assumed that there are no costs of exiting the market Efficient equilibrium Assuming symmetric equilibrium, and given the number of firms m, the regulator s problem is the following: maxwnu Q Q0 n nd(e(q))mc Q m, where the welfare function considered, W, is the sum of consumers total utility and firms profits. Given the previous assumptions about U((), D((), C((), and E((), the Kuhn-Tucker necessary and sufficient conditions are U' Q * n ne'(q * )D'(E(Q * ))C' Q * m 0, (4) with strict equality if Q * is positive. The solution to this problem, Q *, is the efficient or sociallyoptimal output, Q o. When Q o > 0, (2) and (4) can be combined as follows: U' Q o n ne'(q o )D'(E(Q o ))C' Q o m P s (Q o ) Q o n P s '(Q o )E'(Q o )D'(E(Q o )). (5) 11

10 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE Therefore, the efficient price is P(Q o )C' Q o m (n1)e'(q o )D'(E(Q o )) Q o n P s '(Q o ). (6) The interpretation of the first equality in (5) is very familiar: efficiency requires that the last unit of output (consumption) generates as much utility as the sum of the damage caused to all the consumers by the pollution created by producing that unit, and the cost of this production. Notice that the right hand side of the second equality in (5) converges to a constant price when the number of consumers is very large (U = n E D + C = P). As Lemma 1 states (proofs of lemmas and propositions are relegated to the appendix), market output in perfect competition is larger than the efficient output. Consequently, perfectcompetition pollution emissions are also larger than the socially optimal emissions, and market price is lower than in the efficient outcome. Lemma 1: m, n $ " ; Q o < Q c ; E o < E c, P(Q o ) > P(Q c ). If the market is not perfectly competitive, output is smaller than in perfect competition. In this case, it is uncertain whether the efficient output is smaller or larger than the market output. However, the only case of interest for the purposes of this paper is that in which Q o < Q c ; otherwise, imperfect competition eliminates the environmental problem, although a problem of loss of welfare remains as long as output is restricted below its optimal level. Therefore, in the remaining of this paper, it is assumed that Q o < Q c, from which it follows that P(Q o ) > P(Q c ), that is, C' Q o m (n1)e'(q o )D'(E(Q o )) Q o n P s '(Q o )>C' Q c m Q c m P d '(Q c ). (7) 12

11 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD 3. Introduction of mechanisms for the internalization of pollution 3.1. Pigouvian tax With a Pigouvian tax on emissions, firms solve the following problem: max q j 0 Π τ j (q j,q j )P d (Q s )q j C(q j )τe(q j ), for j = 1,..., m, where τ is the tax rate per unit of emission, and Π j τ ((, () is firm j s (Cournot) profit given τ. The Kuhn-Tucker necessary and sufficient conditions are P d (q * j q j )q * j P'(q * j q j )C'(q * j )τe j '(q * j )0, q * j >0;P d (q * j q j )C'(q * j )τe j '(q * j )q * j P'(q * j q j ). (8) Assuming firms optimizing behavior, the social planner chooses the optimal rate of the Pigouvian tax by solving the following problem: max τ0 W subject to (8), whose Kuhn-Tucker necessary and sufficient conditions, using (6), can be summarized as τ * τ o >0;τ o (n1)d'(e(q o )) q o j P d '(Q o )q o i P s '(Q o ), E'(q o j ) (9) where the solution to the problem, τ *, is the optimal tax rate, τ o, q j o = Q o m 1, and q i o = Q o n 1. 13

12 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE Notice that in perfect competition (m, n $ ") the optimal tax rate becomes τ o = n D (E(Q o )). This is the usual Pigouvian tax, whose rate equals total marginal damage. Consider now the case in which only n $ ". In this instance, τ o nd'(e(q o )) q o j P d '(Q o ), E'(q o j ) that is, the optimal tax rate is smaller than in perfect competition (since P d < 0, E > 0) because firms restrict their output taking into account the negative effect of increasing output on price. The optimal tax rate is now equal to the total marginal damage caused by the last unit of output minus the decrease in total revenue per unit of increased emissions caused by that last unit of output. Similarly, when only m $ ", τ o (n1)d'(e(q o )) q o i P s '(Q o ). E'(q o j ) In this case, the optimal tax rate is also smaller than in perfect competition (since P s, E > 0) because consumers take into account that increasing their demand augments price and the damage that pollution causes to them, and thus their consumption is smaller than in perfect competition. The optimal tax rate is now equal to the total marginal damage caused to all but one consumer (since each consumer internalizes the damage caused to herself or himself by her or his consumption) by the last unit of output minus the decrease in consumer surplus per unit of increased emissions caused by that last unit of output. Therefore, (9) is the combined result of the two previous cases. The Pigouvian tax attains the efficient level of output, and leads firms to make the following profits: Π τ j (q o j )2 P d '(Q o )q o j C'(q o j )C(q o j )>0, 14

13 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD τ o for j = 1,..., m. Like before, Π j > 0, q j > 0, follows from P d < 0, C > 0, and C 0; and firms profits depend inversely on the number of firms in the market, since an individual firm s output and its effect on price both decrease as m increases. Again, profits can be defined as a function of m such that Π τ j (m) < 0, and, as m $ " (i.e., in perfect competition), τ o Π j $ q j C (q o j ) C(q o j ), which equals zero in the case of constant returns to scale; that is, the introduction of a Pigouvian tax in the case of perfect competition and constant returns to scale does not change the zero-profit result. The relationship between profits with and without a Pigouvian tax is established in Lemma 2: Lemma 2: If Π j τ, Π j c U 0, Π j τ < Π j c, j. Recall that, once that the market has m firms, the assumed fixed costs of entry into the industry make entry unprofitable for any firm; and notice that the introduction of a Pigouvian tax reduces firms profits, but these are still positive. Only in the case of perfect competition and constant returns to scale profits remain unchanged at zero level. Therefore, there is neither entry into nor exit from the industry; the exogenously given number of m firms will remain constant. However, if m is endogenous, a Pigouvian tax may affect market structure by making some firms exit the market (e.g., Baumol and Oates (1988), ) Non-tradable emission permits given free to current polluters (direct regulation) Consider a system of emission permits that issues exactly E o permits, each of which allows the release of a unit of emission. These permits are allocated to firms free of charge according to some criterion, for instance, in proportion to historical emissions. In our case, regardless of the criterion chosen, each firm receives the same number of permits, E o j, since all o firms are identical: E j = E o m 1. It is assumed that E o o is a multiple of m, so that E j. In this section, the permits considered are non-tradable, which is the same as considering a commandand-control régime, in which the amount of emissions allowed to each firm is directly regulated. 15

14 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE In this setting, firms solve the following problem: max q j 0 Π p j (q j,q j )P d (Q s )q j C(q j ) subject to E(q j ) E o m, for j = 1,..., m, where Π j p ((, () is firm j s (Cournot) profit function given the number of free permits received. The Kuhn-Tucker necessary and sufficient conditions are P d (q * j q j )q * j P d '(q * j q j )C'(q * j )λ* E'(q * j )0, q * j >0;P d (q * j q j )C'(q * j )q * j P d '(q * j q j )λ * E'(q * j ), λ * E(q * j ) E o m 0, (10) where λ is the constraint multiplier. As established in Lemma 3, the constraint of this problem must be binding always, and thus the efficient outcome is implemented. Lemma 3: If Q o < Q c, E(q j * ) = E j o q j * = q j o, j. The need for the constraint to be binding can also be derived from the following: Q o < Q c implies that P(Q o ) > P(Q c ), C (q o j ) C (q c j ), and P d (Q o ) P d (Q c ). Hence, P(Q o ) > C (q o o j ) q j P d (Q o ), which implies that conditions (10) can only be satisfied for λ * > 0. Therefore, the constraint must be binding. 2 From (10) and (6), it follows that the shadow price of emissions is the following: λ * (n1)d'(e(q o )) q o j P d '(Q o )q o i P s '(Q o ), E'(q o j ) 2 If the constraint were not binding, permits would not be required in the first place; there exists a public intervention because firms want to emit more pollution than it is optimal, so it must also be that firms want to use all their permits. 16

15 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD which equals the optimal rate of a Pigouvian tax. Thus, the different cases analyzed with regard to the Pigouvian tax, as well as the interpretation of the tax rates, apply now to the shadow price of emissions. With direct regulation, firms profits are as follows: Π p j q o j [(n1)e'(q o )D'(E(Q o ))q o i P s '(Q o )]q o j C'(q o j )C(q o j )>0, p o for j = 1,..., m. Π j > 0, q j > 0, follows from C > 0, and C 0, which imply that o q j C (q o j ) C(q o j ) 0; and from inequality (7), which imply that (n 1) E (Q o ) D (E(Q o )) > o > q i P s (Q o ). Firms profits still depend inversely on the number of firms in the market, Π p j (m) < 0, but now only because an individual firm s output decreases as m increases. Profits do not depend on the effect of firms output on price any more, but on the output restriction imposed by permits. Thus, the expression for Π p j remains the same even when m $ ". Only when n $ " (or m, n $ "), this expression changes as follows: Π p j q o j ne'(q o )D'(E(Q o ))q o j C'(q o j )C(q o j )>0. Now firms profits are even larger because consumers have no market power. Notice that even in the case of perfect competition and constant returns to scale, when q j o C (q j o ) = C(q j o ), profits are still positive. The relationship between profits with permits, with a Pigouvian tax, and without any public intervention is established in Lemma 4: Lemma 4: (i) Π j c Π j τ 0, j. (ii) Π j p > Π j τ 0, j. (iii) Π j p > Π j c if Q m Q o < Q c ; Π j p > Π j c, Π j p = Π j c, or Π j p < Π j c if Q o < Q m < Q c, j, where Q m is the monopolist s output. Lemma 4 is illustrated in Figure 1 for constant returns to scale and linear demand. In this case, the perfect competition output, Q " c, which satisfies P = C, leads to zero profits ( " c = 0). 17

16 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE Figure 1: Illustration of Lemma 4 for the case of constant returns to scale and linear demand 18

17 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD The monopolist determines its output, Q m, by equalizing its marginal cost and marginal revenue (C = R 1 ). Since Q m arg max (Q), m = max (Q). When 1 < m < ", firms also equalize their marginal cost and marginal revenue (C = R m ), but since their output effect on price is smaller than the monopolist s ( slope(r 1 ) > slope(r m ) ), their aggregate output, Q c m, is larger than Q m c. As a result, firms profit is m such that m c c > m > " = 0. Consider three cases of negative externalities which produce marginal damages D A, D B, and D C. The efficient levels of output are such that price equals marginal cost plus marginal damage, Q o A, Q o B, and Q o C, which are assumed to be smaller than Q c m. For Q o A > Q m, o A > c m. For Q o B, Q o C < Q m, o B > mc o, and C < mc. The most dramatic change produced by direct regulation can be observed in the case of perfect competition and constant returns to scale. In this instance, permits lead firms to make positive profits, when otherwise their profits are zero. In general, the output restriction imposed by the permits increases firms profits beyond what is possible for their market power. Without permits, firms produce the Cournot output. A smaller aggregate production would increase firms profits toward the monopolist s profit, but it is not attainable, because, if all firms but j decrease their output, j would increase its own. By forcing firms to restrict their output, permits implement an aggregate production that, although provides larger profits, would not be a (non-cooperative) equilibrium otherwise. Only when the restriction created by permits is so acute that it brings total production enough below the monopolist s output, profits can decrease with the introduction of direct regulation. Notice that although profits could now be larger than the cost of entry into the market, the structure of the industry will not change, since permits also act as a barrier to entry Tradable emission permits given free to current polluters (grandfathering) Consider the same system of emission permits from last section, with the only difference that permits are tradable after their initial free allocation. Assume that all the firms in the market are financially constrained so that none of them can attempt to gain market power through the purchase of permits, and there is a potential entrant who is financially unconstrained to become a monopolist in the output market by acquiring all the permits. Notice that the amount of money 19

18 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE required to buy all the permits in a real-world system of tradable emission permits may be very large. As the number of firms in the market increases, the size of each of them is smaller, and hence it is less likely that any of these firms possesses the financial resources needed to ever acquire all the permits. The assumption of only one financially-unconstrained firm is a key assumption; the effects on the model results of assuming that more than one firm is financially unconstrained are briefly discussed at the end of this section. It is also assumed that each permit is permanent, the total number of permits (E o ) will not change in the future, the firms and the entrant discount future profits at the same factor, δ, and the potential entrant s cost function, Φ((), is such that Φ(Q e ) =! j C(q j ), for Q e =! j q j, where Q e is the entrant s output. The latter assumption can be justified by the replication argument, and implies that Φ(() is twice differentiable, Φ > 0, and Φ 0. It is useful to define formally the game that arises when a potential entrant can bid for the firms permits in the setting just described. The structure of the game is as follows: it is a two-stage game with m + 1 risk-neutral players; m current polluters who own E o j permits, and a potential entrant who desires to become the monopolist in the output market through purchasing all the permits. The entrant s strategy is a bid per permit, b = (ρ N), where ρ is the price per permit offered to current polluters, and N is the minimum number of permits that must be tendered for the bidder to buy the tendered permits at price ρ, and thus enter the output market. Denote the entrant s strategy set B; the entrant chooses b B, where B = {(x y) x, 0 y E o }. The strategy of current polluters is a number of permits they tender to the entrant, which will be denoted t j, for j = 1,..., m. Thus, each firm s strategy set is T j = {x 0 x E o j }, and t j T j. The price per marginal permit offered to current polluters by the potential entrant can be restricted by an upper and a lower bounds: ρ min ρ ρ max. The lower bound is the value of each permit when market output is not restricted beyond the efficient output. When the production function exhibits constant returns to scale, this value is the discounted stream of profits per permit obtained by each firm given the output restriction imposed by the permits: 20

19 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD ρ min 1 1δ Π p j E o j. This value, which is the same for each permit given constant returns to scale, is the permit price required for a firm which does not foresee that the entrant is trying to become a monopolist to be indifferent between selling and not selling its permits. In the case of decreasing returns to scale, ρ min is not constant, because each marginal unit of output generates a smaller increase in profits than the one before, and thus the price that equals the discounted stream of profits generated as a result of holding a permit at the margin increases as more permits are sold: ρ min (x) 1 1δ E 1 (x) E 1 (x1) [P d (y)yc(y)]dy, x [1, E j o ], where ρ min (x) is the value of the (E j o x + 1) th permit sold, that is, the discounted stream of profits generated by the output allowed (indirectly by allowing emissions) by that permit, and E 1 (() is the inverse pollution-emission function. Hence, ρ min (x) > ρ min ( y), x < y. The upper bound is the discounted stream of the monopolist s profit attributable to each permit. Again, this is a constant if the production function exhibits constant returns to scale: ρ max 1 1δ m E o. Analogously to the case of ρ min, when returns to scale are decreasing, ρ max is a variable: ρ max (X) 1 1δ E 1 (X) E 1 (X1) [P d (Y)YC(Y)]dY, X [1, E o ], where ρ max (X ) is the value of the X th permit hold by the monopolist. Thus, ρ max (x) > ρ max (y), x < y. If the entrant acquires all the permits at ρ max, the monopoly profits would be entirely transferred from the entrant to the original polluting firms. 21

20 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE The number of permits tendered by the firms currently in the market is a function of the bid per permit offered by the entrant: t j = t j (b), j. It is assumed that t j (x) = 0, x = (y z) such that y ρ min (E j o ). The payoff function of the potential entrant, ψ e ((), is the following: ψ e (b) 1 1δ Π e (Q e ( j t j (b),e 1 (E o j t j (b))),e 1 (E o j t j (b))) j ρt j (b)γ, where E 1 (E o! j t j (b)) is the output produced by current polluters based on the permits that they do not sell to the entrant, Q e ((, () is the entrant s output, which is a function of the number of permits bought and the output produced by current polluters, Π e ((, () is the entrant s profit, and Γ is the cost of entry. The current polluters payoff function is as follows: ψ f j (b) 1 1δ Π f j E 1 (E o j t j (b)), Q e ( j t j (b),e 1 (E o j t j (b)))e 1 ( kuj [E o k t k (b)]) ρt j (b), for j = 1,..., m, where ψ f j (() is firm j s payoff function, Π f j ((, () is firm j s market profit given the number of permits sold to the entrant by firm j and the other m 1 firms, and E 1 (! k U j (E o k t k (b))) is the output produced by the other m 1 current polluters based on the permits that they did not sell to the entrant. In the first stage of the game, the entrant chooses b B. The second stage of the game is a subgame in which the m current polluters choose simultaneously t j (b) T j, given the entrant s choice of b. If firm j plays a pure strategy, it is denoted σ j (b). In this case, the vector of pure strategies is σ(b) = (σ 1 (b), σ 2 (b),..., σ m (b)). If firm j plays a mixed strategy, it is a choice of a cumulative probability distribution over the set T j, which is denoted F j (( b). The vector of mixed strategies is F(( b) = (F 1 (( b), F 2 (( b),..., F m (( b)). 22

21 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD Consider the subgame-perfect equilibria of this game. An equilibrium is a vector (b, F(( b)) such that: (i) F j (( b) = arg max {Ε(ψ f j (() F k (( b)), k U j}, b, and (ii) b = arg max {Ε(ψ e (() F(( b))}, where Ε(( () is the conditional-expectation operator. Initially, consider the case in which current polluters do not foresee that the bidder is trying to become the monopolist in the output market. 3 In this instance, and if the production function exhibits constant returns to scale, the entrant only needs to bid slightly more than ρ min to make all firms sell their permits, thus exiting the industry and leaving the entrant as the monopolist in the output market (of course, also in the permit market). In the case of decreasing returns to scale, the price bidden must be slightly larger than ρ min (1) to attain the same outcome. Since the firms can be led to selling their permits with probability one by choosing the minimum price per permit, the entrant does not need to condition its bid on any minimum number of tendered permits. In fact, the choice of such a minimum number would be revealing of the entrant s intended monopolistic behavior, which would contradict that firms do not foresee its profits. Hence, the bid can be considered unconditional, that is, N = 0. For economy of notation, let b = ρ when N = 0. Define the market-aggregate profit function (Q) = P d (Q) Q Φ(Q), where Q is total output, and Φ(Q) =! j C(q j ). is continuous, twice differentiable, and < 0. Notice that a potential monopolist cannot profit from entering the market if the cost of entry is larger than or equal to the difference between the discounted stream of the monopolist s profits and the discounted stream of market-aggregate profits with permits: Γ (1 δ) 1 [ m (Q o )]. When Γ < (1 δ) 1 [ m (Q o )], and if the production function exhibits constant returns to scale, the only requirement for the industry to become a monopoly is that the monopolist s output is lesser than the efficient one, as Proposition 1 states. The same result can be obtained for decreasing returns to scale if the bidding mechanism allows the entrant to pay different prices for the permits. If the price is unique, this result may still hold in some cases, but not necessarily; in 3 This may occur because the market for permits is such that sellers do not know that a single buyer is bidding for all the permits. 23

22 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE some instances, the equilibrium will have the entrant buying only part of the permits (hence, not becoming the monopolist), and in other cases entry will not occur. Proposition 1: If firms do not foresee the potential monopolist s profits, the production function exhibits constant returns to scale, and Γ < (1 δ) 1 [ m (Q o )], a pure-strategy equilibrium of the game is a vector (ρ min + ε, (E o j, E o j,..., E o j )), for ε > 0 and ε $ 0, if and only if Q m < Q o. A vector (ρ, (0, 0,..., 0)), ρ [0, ρ min ], is a pure-strategy equilibrium if and only if Q m Q o. The intuition behind Proposition 1 is as follows: if the efficient output, which is the maximum output allowed by the permits, is smaller than or equal to the monopolistic output, the aggregate-market profits are already at their maximum possible level. Thus, entry is impossible, since an entrant cannot make profits larger than those of the current polluters combined. Entry is only possible when the monopolist s output is lesser than the efficient one. In this case, total market profits can be increased by restricting output beyond the level allowed by the permits. If current polluters do not foresee that the buyer of its permits is going to restrict output, and entry costs are covered by the difference between the monopolist s profit and the amount paid for all the permits, then entry will occur, and the entrant will become the output-market monopolist. Assuming that firms do not foresee the entrant s profit, and thus sell their permits by slightly more than the benefit they were obtaining, is very restrictive. If firms are completely and perfectly informed, they must foresee that the potential entrant is trying to acquire market power (and therefore force market price to increase). In this case, the current polluters situation is analogous to that of the shareholders of a firm subject to a takeover bid who foresee that the bidder is going to implement improvements in the company (which will increase the price of shares). Grossman and Hart (1980) analyze this situation when the number of shares that each shareholder owns is small enough for one shareholder s decision of not selling not to compromise the buyer s decision of taking the firm. In this instance, shareholders try to free ride on the bidder s effort to improve the firm, willing to sell their shares only at a price that completely extracts the buyer s gains of introducing improvements. As a result, the takeover bid 24

23 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD fails. Analogously, current polluters would try to free ride on the entrant s market power, anticipating that output would be restricted beyond the limitation imposed by permits. By not selling their permits (and staying in the product market) current polluters can benefit from the increase in price that the entrant s output restriction would cause. Therefore, current polluters would be willing to sell their permits only at a price that extracts all of the entrant s profit; as the takeover bidder fails to buy the firm, the potential monopolist cannot enter the market. In the previous setting, suppose that s firms in the market have perfect and complete information, that is, foresee the entrant s profit, and therefore try to free ride on the entrant s market power, for 0 s < m. If s = 1, the single free rider will not be willing to sell its permits for anything less than or equal to the discounted stream of its profits when the only producers in the market are the entrant and itself. This discounted stream of profits approaches the share of the discounted stream of the monopolist s profit equal to the free rider s current share of the market as the number of current polluters approaches infinity. Thus, the total payment that the entrant needs to disburse to acquire the free riders permits decreases as the number of firms decreases, but it will be assumed to remain close enough to the share of the discounted stream of the monopolist s profit equal to the free riders current share of the market. 4 Like in section 3.2, the free riders output, q s j, is all of what they can produce given the permits they receive: s q j = q o j, and, thus, their aggregate production is Q s o = s q j = s m 1 Q o. Then, entry occurs (with the m s firms without perfect and complete information selling their permits) if and only if the combined output of the s free riders plus the optimal output of the entrant is less than the efficient output, as Proposition 2 asserts. Proposition 2: If s firms have perfect and complete information, for 0 s < m, Γ < (1 δ) 1 [Π e (m s) m 1 (Q o )] when Π e > (m s) m 1 (Q o ), Γ > (1 δ) 1 m α, 4 Notice that it cannot exist an equilibrium where the free riders sell their permits even if the entrant pays slightly more than what they would obtain if staying in the market as free riders. Selling by this amount is not a Nash equilibrium because if s 1 free riders do it, the s th free rider would deviate from the strategy of selling, since staying in the market means obtaining something close enough to the share of the discounted stream of the monopolist s profit equal to this free rider s current share of the market. Therefore, no free rider would sell. 25

24 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE where α is close enough to (1 δ) 1 m, and the production function exhibits constant returns to scale, a pure-strategy equilibrium of the game is a vector (ρ min + ε, (0, 0,..., 0, E o j, E o j,..., E o j )), for ε > 0 and ε $ 0, where σ * k (ρ min + ε) = 0, k = 1,..., s, and σ * l (ρ min + ε) = E o j, l = s + 1,..., m, if and only if Q * + Q s < Q o, for Q * > 0, where Q * is the entrant s optimal output. A vector (ρ, (0, 0,..., 0)), ρ [0, ρ min ], is a pure-strategy equilibrium if and only if Q * + Q s < Q o implies that Q * = 0, or Q * + Q s = Q o, for Q * > 0. Proposition 2 reflects, similarly to Proposition 1, that entry can only occur if the efficient output is not so small that restricting output even further does not enable the entrant to cover all the costs of entry. Notice that Proposition 2 presents two extreme cases. The first one is the case in which there are no free riders; Proposition 1 is the special case of Proposition 2 in which s = 0, and thus Q s = 0 and Q * = Q m, which makes the condition for entry in Proposition 2 equal to that condition in Proposition 1. In the other extreme case of Proposition 2, the number of free riders approaches the total number of firms in the industry: s $ m implies that Q s $ Q o, which tends to make the condition for entry (Q * + Q s < Q o, for Q * > 0) impossible to satisfy. 5 The result of Proposition 2 maximizes the entrant and the free riders payoffs given the static nature of the two-stage game considered. If the entrant could bid a second time for the permits not tendered, it would buy the free riders permits too, becoming a monopolist, as Proposition 3 states. Again, for a potential entrant to become the monopolist in the output market in a multiple bidding process it is required that the combined output of the s free riders plus the 5 Notice that Proposition 2 applies only to unconditional bids, that is, b = (ρ 0) = ρ. If the entrant can condition the payment of ρ per permit tendered to a minimum number of permits sold, it is no longer optimal for the entrant to set N = 0. In this case, even if all the firms have complete and perfect information, the entrant can force them all to sell their permits by choosing b = (ρ min + ε E o ), for ε > 0 and ε $ 0, since firms obtain (1 δ) 1 Π p j if not all the permits are tendered, and (ρ min + ε) E o j = (1 δ) 1 Π p j + ε E o j > (1 δ) 1 Π p j if all the permits are tendered. However, the conditional strategy loses relevance because it is hardly credible; if all the permits are not tendered but the number tendered is enough to make entry profitable, the entrant would like to deviate from its conditional strategy and enter the market. 26

25 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD optimal output of the entrant is less than the efficient output. If that is the case, the entrant will pay the free riders an amount close enough to the share of the discounted stream of the monopolist s profit equal to the free riders current share of the market. The entrant buys even at this price because the remaining share of the monopolist s profit is larger than the entrant s profit in the case that the free riders stay in the market. This result is a Nash equilibrium, since no free rider can gain more by deviating from this strategy, and the entrant minimizes its payment for the permits. The intuition for this result is that the entrant finds worthwhile buying the free riders permits in the second bid, even if it cannot make any profit on them, because becoming the monopolist allows the entrant to increase its profit on the permits bought from the non-free-riding firms in the first bid. Proposition 3: If s firms have perfect and complete information, for 0 s < m, Γ < (1 δ) 1 [Π e (m s) m 1 (Q o )] when Π e > (m s) m 1 (Q o ), and the production function exhibits constant returns to scale, a pure-strategy equilibrium of the game played twice is: (i) in the first game, a vector (ρ min + ε, (0, 0,..., 0, E o j, E o j,..., E o j )), for ε > 0 and ε $ 0, where σ k* (ρ min + ε) = 0, k = 1,..., s, and σ l* (ρ min + ε) = E o j, l = s + 1,..., m, and (ii) in the second game, a vector (ρ, (E o j, E o j,..., E o j )), where ρ is close enough to ρ max, j = 1,..., s; if and only if Q * + Q s < Q o, for Q * > 0, where Q * is the entrant s optimal output. A vector (ρ, (0, 0,..., 0)), ρ [0, ρ min ], in both games, is a pure-strategy equilibrium if and only if Q * + Q s < Q o implies that Q * = 0, or Q * + Q s = Q o, for Q * > 0. In the previous analysis, it has been assumed that not all the firms currently in the market foresee the entrant s profits. If all of them had perfect and complete information, the conclusions of Grossman and Hart (1980) should imply that it is impossible that entry occurs. However, as it has been pointed out in some of the abundant body of literature on takeovers that emerged after Grossman and Hart s seminal contribution (e.g., Bagnoli and Lipman (1988); Bebchuk (1988); Holmström and Nalebuff (1992); and Noe (1995)), there is a paradox behind Grossman and Hart s argument. Placing this argument in the context of this paper, the paradox is as follows. If entry is expected to occur, firms will free ride: σ j* (y) = E o j, for ρ min + z y ρ max, where z is close enough to ρ max ρ min, j; and σ * j (ρ) = 0, ρ < ρ min + z, j. But if all the firms foresee 27

26 MIGUEL BUÑUEL THE EFFECT OF EMISSION PERMITS AND PIGOUVIAN TAXES ON MARKET STRUCTURE the entrant s profits, their free-riding behavior will make entry impossible. However, if entry is not possible firms will obtain a profit equal to ρ min per permit, and the expectation of this event will make them sell their permits for anything more than ρ min : σ * j (ρ min + x) = E o j, x > 0, j; and σ * j (ρ) = 0, ρ ρ min, j. Therefore, Grossman and Hart s insight tells us that when all the firms have perfect and complete information there cannot be an equilibrium where entry occurs with probability one, since none would sell in this case, but it does not say what is an equilibrium. In particular, no equilibrium can exist in which entry fails with probability one, because if that were the case, all the firms would want to sell. The conclusion is that there cannot be a pure-strategy equilibrium with unconditional bids when all firms have perfect and complete information. Thus, our attention should be redirected towards finding mixed-strategy equilibria. Similarly to Bagnoli and Lipman (1988), Bebchuk (1988), Holmström and Nalebuff (1992), and Noe (1995), consider a symmetric, mixed-strategy equilibrium with an unconditional bid. Assume that the bid can be placed twice, that is, if the first bid results in the entry of the bidder, a second bid is placed that makes the entrant become the output-market monopolist, as shown by Proposition 3. For simplicity, assume that the minimum price per permit that would lead all firms to sell their permits in the second bid is ρ max rather than ρ min + z, where z is close enough to ρ max ρ min. Assume that each permit allows the emission of E o j units of emission; that is, each firm holds only one permit. Therefore, firms strategy set is reduced to T = {0, 1}, and, assuming that the production function exhibits constant returns to scale, ρ min = (1 δ) 1 Π p j, and ρ max = (1 δ) 1 m 1 m. Notice that the entrant buys the permits tendered in an unconditional bid even if entry is not profitable. In this case, assume that the entrant resells the permits bought to the firms that tendered them by exactly their market value with no entry, that is, ρ min. 6 Assume that if the entrant is indifferent between entering and not it will enter, and denote the minimum number of permits that make it indifferent M(m), 7 which satisfies (1 δ) 1 M(m) m 1 m = = M(m) ρ + Γ, that is, 6 This assumption leaves current polluters payoffs unchanged, and decreases the loss of the potential entrant when entry is unprofitable by avoiding entry costs. 7 It is implicitly assumed that Γ < (1 δ) 1 [ m (Q o )]. 28

27 WORKING PAPERS ON ENVIRONMENT AND ECONOMICS 1/2001 FUNDACIÓN BIODIVERSIDAD M(m) Γ ρ max ρ. (11) M (() > 0, since the number of permits increases, but the emissions per permit decreases, as the number of firms in the market increases. Assume that the minimum fraction of total permits required for entry converges to a constant γ as the number of firms in the industry grows larger, that is, M(m) m 1 $ γ as m $ ". Assume that the probability of a firm other than firm j not selling its permit is θ ; that is, F k (0 ρ) = θ, and F k (1 ρ) = 1, k U j. Firm j will use the same randomizing strategy if and only if the payoff of tendering, which is ρ with certainty given that the bid is unconditional, equals the expected payoff of not tendering; that is, ρ 1 1δ Ε[Π f j t j (ρ)0] 1 1δ {Pr[entry t j (ρ)0]ε[π f j t j (ρ)0, entry] Pr[no entry t j (ρ)0]ε[π f j t j (ρ)0, no entry]}, which is equal to the following: ρ 1 1δ Pr kuj t k (ρ)m(m) m m Pr kuj t k (ρ)<m(m) Pr kuj t k (ρ)m(m) ρ max Pr kuj t k (ρ)<m(m) ρ min. p j (12) Substituting for the probabilities of entry and no entry conditional on j not tendering, condition (12) can be written as follows: ρ m1 km(m) m1 k M(m)1 (1θ) k θ m1k ρ max k0 m1 k (1θ) k θ m1k ρ min. (13) 29