A comparison between emission intensity and emission cap regulations

Energy Policy xxx (xxxx) xxx

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Energy Policy journal homepage: http://www.elsevier.com/locate/enpol

A comparison between emission intensity and emission cap regulations☆ Kosuke Hirose a, *, Toshihiro Matsumura b a b

Osaka University of Economics, 2-2-8, Ohsumi, Higashiyodogawa-Ku, Osaka, 533-8533, Japan Institute of Social Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

A R T I C L E I N F O

A B S T R A C T

JEL classification: Q52 L13 L51

We revisit command-and-control regulations and compare their efficiency, particularly an emission cap regu­ lation that restricts total emissions and an emission intensity regulation that restricts emissions per unit of output under emission equivalence. We find that in both the most stringent target case, in which the target emissions level is close to zero, and the weakest target case, in which the target emissions level is close to business as usual, the emission intensity regulation yields greater welfare than does the emission cap regulation. However, in the moderate target cases, an emission cap regulation may be better in terms of welfare than an emission cap regulation because the emission intensity regulation causes over-production. Our results suggest that the effi­ ciency of a policy measure depends on the target level of emissions.

Keywords: Near-zero emission industry Emission cap Emission intensity Emission equivalence

1. Introduction Global warming is one of the most serious risks that society faces, and many countries have recently voluntarily committed to reducing CO2 emissions under the Paris Agreement on climate change. A standard policy for reducing CO2 emissions is the introduction of so-called “market-based instruments,” such as a carbon tax or an emission tax. A number of theoretical studies in environmental economics encourage the use of indirect regulations rather than command-and-control ap­ proaches from the perspective of cost-effectiveness. However, in prac­ tice, command-and-control instruments are now at the center of the regulatory debate for several reasons: political difficulty of introducing the optimal tax; provision of a simple, certain way to achieve a desirable goal; and incomplete enforcement and high monitoring costs for marketbased instruments.1 In such a case, direct regulations can play an important role. In this study, we compare two representative direct regulations, an emission cap regulation that restricts total emissions and an emission intensity regulation that restricts emissions per unit of output, and we

examine their efficiency.2 Many studies have shown the differences between these two regulations and evaluated their performance; how­ ever, the comparison is theoretically ambiguous because different in­ struments provide different incentives for firms (Besanko, 1987; Helfand, 1991; Lahiri and Ono, 2007; Kiyono and Ishikawa, 2013; Amir et al., 2018). For example, Montero (2002) focused only on firms’ incentive under four regulations, including an emission cap and an emission intensity regulation, and evaluated environmental R&D rank­ ings in a symmetric duopoly model with general demand. As for social welfare, Lahiri and Ono (2007) compared an emission intensity regu­ lation and an emission tax under a Cournot oligopoly and showed the advantage of the emission intensity regulation over the emission tax in the weakest target case (the target emission level is close to business as usual). Amir et al. (2018) extended Montero (2002) analysis to rank regulatory instruments with respect to social welfare but provided nu­ merical examples by using simple specification. None of these studies advocate either form of emission regulation, especially when imple­ menting a drastic reduction of emissions. Under the oligopolistic market, we show that the emission intensity

We are grateful to Hong Hwang, Hiroaki Ino, Keisaku Higashida, Atsushi Kajii, Kuo-Feng Kuo, Noriaki Matsushima, Takao Ohkawa, Cheng-Hau Peng, and the participants of seminars at Kwansei Gakuin University, National Taiwan University, The University of Tokyo, and Kochi University for their helpful comments and suggestions. We are indebted to two anonymous reviewers for their valuable and constructive suggestions. This work was supported by JSPS KAKENHI (18K01500) and the Murata Science Foundation. * Corresponding author. E-mail addresses: [email protected] (K. Hirose), [email protected] (T. Matsumura). 1 For a more detailed argument, see B} ohringer et al. (2017), Cohen and Keiser (2017), and Demirel et al. (2018). 2 Ellerman and Sue Wing (2003) provided a non-technical introduction to the differences between an emission cap regulation and an emission intensity regulation, giving real examples of each regulation. ☆

https://doi.org/10.1016/j.enpol.2019.111115 Received 27 April 2019; Received in revised form 3 October 2019; Accepted 12 November 2019 0301-4215/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Kosuke Hirose, Toshihiro Matsumura, Energy Policy, https://doi.org/10.1016/j.enpol.2019.111115

K. Hirose and T. Matsumura

Energy Policy xxx (xxxx) xxx

regulation yields greater welfare than the emission cap regulation does in the most stringent target case, in which the target emission level is close to zero (Proposition 1). Although the target level is extreme, the hypothetical target plays a relevant role. For example, under the Paris climate agreement, many countries, such as the UK, France, Germany, and Japan, plan to reduce CO2 emissions drastically by 2050 (about 80% reduction at least against a business-as-usual scenario). To achieve this goal, in several industries, such as electric power and transportation, authorities may impose an emission constraint that is close to zero emissions. Thus, the most stringent case we discuss is important. Japa­ nese electric power companies have committed to CO2 emissions per kilowatt-hour, not total emissions, and our result suggests that this could be an efficient commitment. In addition, we examine the opposite case, in which the emission target is significantly weak. Again, we show that the emission intensity regulation improves social welfare more than the emission cap regulation does (Proposition 2). However, we find that the welfare ranking may be reversed, meaning that the emission cap regu­ lation may yield greater welfare than the emission cap regulation under a moderate emission target (Proposition 3). We also show that under an emission intensity regulation, the stricter regulation may reduce abatement investment, whereas the stricter regulation always increases abatement investment under an emission cap regulation. In other words, the relationship between the degree of regulation and emission abatement activity depends on the regulation measure. This result suggests that a larger abatement investment might not imply stricter regulation in the industry.3 The rest of this paper is organized as follows. Section 2 formulates the basic model, and Section 3 compares emission intensity and emission cap regulations. Section 4 checks the robustness of the main results. Section 5 concludes the paper.

differentiable for xi � 0, increasing, and strictly convex for xi > 0. We further assume that Kð0Þ ¼ K0 ð0Þ ¼ 0.6 This assumption guarantees that the social optimal level of abatement is never zero and that the profit function is smooth. Total social surplus (firms’ profits plus consumer surplus minus the loss caused by the externality) is given by Z Q n n X X Wðq1 ; …; qn Þ ¼ π i þ CS ηðEÞ ¼ PðzÞdz ðCðqi Þ þ Kðxi ÞÞ 0

i¼1

i¼1

ηðEÞ; Pn where η : Rþ 7!Rþ is the welfare loss of emissions, and E ¼ i¼1 ei : We assume that the environmental target E for total emissions is exogenously given (regulated by the government). E may depend on the administrative cost, political pressure, or international commitment, and thus, for simplicity, we treat the target as an exogenous variable. We assume that E 2 ð0; EB Þ where EB is the profit-maximizing emission level without a binding emission target (business-as-usual level). Let qB be the profit-maximizing output without a binding emission target. E is attained through an emission intensity or emission cap. Because there is no heterogeneity among firms, we focus on the symmetric equilibrium in which all firms choose the same actions in equilibrium. 3. Analysis 3.1. Emission intensity regulation Let α be the upper bound of the emission per unit of output. Firm i chooses its output, qi , and abatement level, xi , to maximize its profit subject to

2. The model

α�

We consider an industry with n symmetric polluting firms. The firm produces a single commodity for which the inverse demand function is given by P : Rþ 7!Rþ . We assume that PðQÞ is twice continuously differentiable and P0 ðQÞ < 0 for all Q as long as P > 0. Let Cðqi Þ : Rþ 7! Rþ be the cost function of each firm, where qi ði ¼ 1; 2:; …; nÞ is the output of firm i ði ¼ 1; 2; …; nÞ. We suppose C is twice continuously differentiable, increasing, and convex for all qi .4 We assume that the marginal revenue is decreasing in rivals’ outputs (i.e., P0 ðQÞ þ P00 ðQÞqi < 0). These assumptions are standard and guarantee that the second-order condition is satisfied. Emissions are associated with production, which yields a negative externality. After emissions have been generated, they can be reduced through investment in abatement technologies.5 Thus, firm i’s net emissions are ei : ¼ gðqi Þ xi , where g : Rþ 7!Rþ represents emissions associated with production and xi ð2 Rþ Þ is firm i’s abatement level. We assume that g is twice continuously differentiable, increasing, and convex for all qi . The firm’s profit is PðQÞqi Cðqi Þ Kðxi Þ, where the third term represents the abatement cost. We suppose that K is twice continuously

ei gðqi Þ xi ¼ : qi qi When the constraint is binding,7 firm i’s optimization problem is

maxPðQÞqi qi

Cðqi Þ

Kðgðqi Þ

(1)

αqi Þ:

Let superscript EI denote the equilibrium outcomes under the emis­ sion intensity regulation. Define πEI Cðqi Þ Kðgðqi Þ i ðqi ;Q i Þ : ¼ PðQÞqi αqi Þ. Focusing on the symmetric equilibrium, the equilibrium output, qEI ðαÞ, is characterized by the following first-order condition: � � ∂π EI i ¼ P0 nqEI qEI þ P nqEI ∂qi

C0 qEI

�

� � K 0 xEI g0 qEI

�

α ¼ 0:

(2)

The second-order condition is satisfied. We obtain xEI ðαÞ ¼ gðqEI ðαÞÞ xEI ðαÞ ¼ αqEI ðαÞ. Differentiating (2) yields � EI ∂2 π EI ∂qi ∂α dq i . ¼ > 0; (3) P 2 EI . dα 2 ∂2 πEI ∂ q þ ∂qi ∂qj i i j6¼i ∂ π i

αqEI ðαÞ and eEI ðαÞ ¼ gðqEI ðαÞÞ

2 EI 2 where we use ∂2 πEI i =∂qi ∂α ¼ K’ þ ðg’ αÞK’’qi > 0 and ∂ π i =∂qi þ P 2 EI 2 00 C’’ g’’K’ ðg’ αÞ K þ nðP’ þ P’’qi Þ < 0. j6¼i ∂ π i =∂qi ∂qj ¼ P’

An increase in α relaxes the emission restriction and reduces the mar­ ginal cost of production, which increases qEI . The government sets the emission intensity α ¼ α such that neEI ðαÞ ¼ E. Let ðqEI ðEÞ; xEI ðEÞÞ be the pair of equilibrium output and abatement and WEI ðEÞ be the equilibrium welfare under the emission intensity

3 For empirical work on the relationship between abatement investment and environmental performance, see Guti�errez and Teshima (2018). 4 We can relax this assumption. Our results hold if C’’ P’ > 0 for all Q as long as.P > 0: 5 These are called end-of-pipe technologies. An alternative approach to reduce emissions is to change the production process. For a recent discussion of the relationship between mandatory regulation and this type of innovation, see Matsumura and Yamagishi (2017). We discuss a more general technology in Section 4.3.

6 The form of the abatement (R&D) cost function is a standard assumption in industrial organization and environmental economics (D’Aspremont and Jac­ quemin, 1998; Amir et al., 2018). 7 The constraint is always binding because E < EB .

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regulation when. α ¼ α:

producing output (i.e., E ¼ α ¼ 0), all emissions are reduced by the abatement activities and there are no emissions in the industry. Regardless of output level, the total emissions are zero if and only if the emissions per unit of output are zero. Therefore, when E ¼ 0, the emission cap regulation and emission intensity regulation yield the same outcome. Let qZ and xZ be common q and x under the zero-emission constraint (i.e., when E ¼ 0). We now present the result when E is close to zero.

3.2. Emission cap regulation Next, we consider the case under an emission cap regulation. Firms are symmetric and the number of permits is uniformly allocated, E= n. Let superscript EC denote the equilibrium outcomes under the emission cap regulation. Then, the profit function of firm i under the emission cap regulation is defined by πEC Cðqi Þ Kðgðqi Þ E =nÞ. i ðqi ; Q i Þ : ¼ PðQÞqi The equilibrium output, qEC ðEÞ, is characterized by the following firstorder condition: � � ∂π EC i ¼ P’ nqEC qEC þ P nqEC ∂qi

C’ qEC

�

� � K’ xEC g’ qEC ¼ 0:

Proposition 1. If E is sufficiently close to zero, the emission intensity regulation yields greater welfare than the emission cap regulation does. Proof. See the Appendix. The intuition behind the result is as follows. As explained after Lemma 1, given α > 0, each firm has a stronger incentive to expand its output under the emission intensity regulation than under the emission cap regulation, because under the former regulation, each firm can in­ crease the upper limit of emissions. However, this problem does not exist when E ¼ α ¼ 0. Therefore, qEC ¼ qEI and xEC ¼ xEI when E ¼ α ¼ 0. An increase in α relaxes the restriction on emissions. This leads to an increase in emissions, resulting in larger disutility from the emissions (emission effect). However, by the assumption of emission equivalence between two regimes, the emission effect is the same for the regimes. An increase in E and α affects the per-firm output, q, and the per-firm abatement, x, (allocation effect). As stated above, the emission in­ tensity regulation yields larger q and x than the emission cap regulation does. Given the emission level, under the emission cap regulation, the marginal social cost of the reduction of emissions by the reduction of q is P=g0 and that by the increase of x is K’. The marginal private cost for meeting the constraint by the reduction of q is ðP þP0 qÞ=g0 and that by the increase of x is K’. Thus, both x and q chosen by the firms are too small from the welfare viewpoint. Given the emission level, under the emission intensity regulation, the marginal private cost for meeting the constraint by the reduction of q for each firm is ðP þP0 qÞ=ðg’ αÞ and that by the increase of x is K’. When α is small, both x and q chosen by the firms are still too small from the welfare viewpoint, but both are larger than those under the emission cap regulation. Therefore, the emission intensity regulation is better for welfare than the emission cap regulation. We believe that the most stringent case discussed in Proposition 1 is important. As mentioned in the introduction, under the Paris climate agreement, many countries, including the UK, China, France, Germany, and Japan, aim to reduce CO2 emissions drastically by 2050 (about an 80% reduction against a business-as-usual scenario). To achieve this goal, in several industries, such as electric power and transportation, a severe constraint that is close to zero emissions might be imposed. Thus, our result in the most stringent case has important implications for the debate on regulation. Next, we examine the opposite (loosest constraint) case in which E is close to EB .

(4)

The second-order condition is satisfied. We obtain xEC ðEÞ ¼ gðqEC ðEÞÞ E=n. Differentiating (4) yields � ∂2 πEC ∂qi ∂E dqEC i . > 0; (5) ¼ P 2 EC . 2 dE ∂2 πEC ∂ q þ ∂qi ∂qj i i j6¼i ∂ π i where P 2 j6¼i ∂

we

use

∂2 π EC i =∂qi ∂E ¼ ðK’’g’Þ=n > 0

πEC i =∂qi ∂qj ¼ P’

C’’

g’’K’

2

and

2 ∂2 πEC i =∂qi þ

g’ K’’ þ nðP’ þ P’’qi Þ < 0.

Similar to the emission intensity case, an increase in E increases qEC . 3.3. Comparison

In this subsection, we compare the two instruments. As expected, the equilibrium output is larger under the emission intensity regulation than under the emission cap regulation. Result 1.

qEI ðEÞ > qEC ðEÞ.

Proof. See the Appendix. Result 1 suggests that the equilibrium output under the emission intensity regulation is larger than that under the emission cap regula­ tion. This result implies that the emission intensity regulation yields greater consumer surplus than the emission cap regulation does. We now present our result on each firm’s profit. Let πl ðEÞ :¼ π li ðql ðEÞ; ðn 1Þql ðEÞÞ denote each firm’s equilibrium profit, l ¼ EI; EC. Regarding firm profits, Helfand (1991) showed that the standard as a set level of emissions yields a higher profit for monopoly than the standard as emissions per unit of output. We obtain the same result in the basic oligopoly setting. Lemma 1. The emission cap regulation yields a higher profit than the emission intensity regulation does; that is, πEC ðEÞ > πEI ðEÞ. Proof. See the Appendix. We explain the intuition behind Result 1 and Lemma 1. Under the emission intensity regulation, given α, an increase in qi increases the upper limit of emissions. Therefore, each firm has a stronger incentive to increase its output under the emission intensity regulation than under the emission cap regulation, resulting in the larger equilibrium output (Result 1). However, anticipating this behavior of each firm, the gov­ ernment sets a stricter regulation to meet the emission target E, which reduces the firms’ profits. We now discuss the welfare comparison. The emission intensity regulation is superior for consumer welfare compared to the emission cap regulation, but is less profitable for the firms. Thus, it is generally ambiguous which regulation is socially preferable. Let WEI ðEÞ and WEC ðEÞ be the equilibrium welfare under the emission intensity regu­ lation and the emission cap regulation, respectively. We present two cases in which the emission intensity regulation yields greater welfare than the emission cap regulation does; that is, WEI ðEÞ > WEC ðEÞ. First, we consider the case with the most stringent target case (E is close to zero). When the firms are not allowed to pollute in the process of

Proposition 2. Suppose that E is sufficiently close to EB . The emission intensity regulation yields greater welfare than the emission cap regulation does. Proof. See the Appendix. We explain the intuition. Because of emission equivalence, the emission effect is the same between the two regimes. When E ¼ EB , qEC ¼ qEI ¼ qB and xEC ¼ xEI ¼ 0. Because K0 ð0Þ ¼ 0, this abatement level is too low for welfare, and a marginal reduction of emissions by an increase in x is much more efficient than that by a reduction in q for welfare. In other words, given the emission, q is too large and x is too small for welfare. A marginal decrease in α increases x and reduces q under both emission cap and emission intensity regulations, which im­ proves welfare. The magnitude of this effect is stronger under the emission intensity regulation. Note that qEI > qEC , and thus, xEI > xEC for 3

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K. Hirose and T. Matsumura

E 2 ð0; EB Þ. In Propositions 1 and 2, we show that when the target level is close to the strictest and loosest cases, the emission intensity regulation is better for welfare than the emission cap regulation. The emission intensity regulation stimulates production and mitigates the problem of subop­ timal production and abatement investment, which improves welfare under emission equivalence. Because the emission intensity regulation is better for welfare in the two polar cases, it might be natural to consider that the emission intensity regulation is better for any E 2 ð0; EB Þ. However, this is not true. Let ðx� ðEÞ; q� ðEÞÞ be the pair of the second-best abatement and output level (social optimum x and q given E ¼ E). The derivation is as follows. Let q ¼ ½q1 ; q2 ; …; qn � and x ¼ ½x1 ; x2 ; …; xn � denote an output profile and abatement profile, respectively. Given E, the problem is Z Q n X maxW ¼ PðzÞdz ðCðqi Þ Kðxi ÞÞ ηðEÞ ðq;xÞ

s:t:

0

Fig. 1. Welfare difference (n ¼ 3; a ¼ 6; b ¼ 1; e ¼ 2 and k ¼ k

E = n ¼ gðqi Þ

Proof See the Appendix. ~ ¼ lim Because lim k

xi :

E→0

The second-best output level, q ðEÞ, is characterized by the following first-order condition: C’ðq� ðEÞÞ

K’ðx� ðEÞÞg’ðq� ðEÞÞ ¼ 0 ði ¼ 1; 2; …; nÞ:

To see this fact, we now describe the relationship between E and the welfare difference between emission intensity and emission cap regu­ ~ always holds. Therefore lations in more detail. If k < k, then k < k

(6)

x� ðEÞ is derived from E ¼ nðgðq� Þ x� Þ: Differentiating (6) yields � ∂2 Wðq� ; …; q� Þ ∂qi ∂E dq� . . > 0; ¼ P dE ∂2 Wðq� ; …; q� Þ ∂q2i þ j6¼i ∂2 Wðq� ; …; q� Þ ∂qi ∂qj where we use ∂2 W=∂qi ∂E ¼ ðg’K’’Þ=n > 0, ∂2 W=∂q2i ¼ P’

C’’

WEI ðEÞ > WEC ðEÞ for all E 2 ð0; EB Þ. Fig. 1 describes how E affects the welfare difference between the two regimes. Fig. 1 suggests a non-monotonic relationship between the welfare difference and the target emission level. ~ may hold, depending on E. Therefore, WEI ðEÞ < If k > k, then k > k

(7) g’’K’

g’2 K’’ < 0, and ∂2 W=∂qi ∂qj ¼ P’ < 0. As discussed above, qEC ðEÞ < q� ðEÞ and thus, xEC ðEÞ < x� ðEÞ. In the two polar cases (strictest and loosest cases), ðxEC ðEÞ; qEC ðEÞÞ ¼ ðxEI ðEÞ; qEI ðEÞÞ: Except for the two polar cases, ðxEC ðEÞ; qEC ðEÞÞ < ðxEI ðEÞ; qEI ðEÞÞ holds (Result 1). As long as ðxEC ðEÞ; qEC ðEÞÞ < ðxEI ðEÞ; qEI ðEÞÞ < ðx� ðEÞ; q� ðEÞÞ, the outcome under the emission intensity regulation is closer to the second-best outcome than that under the emission cap regulation, and thus, the emission intensity regulation naturally yields greater welfare than the emission intensity regulation does. However, it is possible that ðxEI ðEÞ; qEI ðEÞÞ > ðx� ðEÞ; q� ðEÞÞ. In that case, the emission intensity yields excessive production and excessive abatement invest­ ment, and thus the emission cap regulation might be better than the emission intensity regulation for welfare. We present the case wherein the emission cap regulation could be better than the emission intensity regulation for welfare.

WEC ðEÞ may hold (see Fig. 2). Solving equation WEI ðEÞ WEC ðEÞ ¼ 0 for E 2 ð0; EB Þ, we obtain two solutions, E1 and E2 , with E1 < E2 : rffiffiffiffi ! � � � H bn þ e2 k an e b2 n2 2n 8 þ 2be2 kðn 1Þ þ e4 k2 k � � E1 ¼ ; 2b b2 n2 ðn þ 1Þ þ 2be2 k n2 þ n þ 2 þ e4 k2 ðn þ 1Þ ! rffiffiffiffi � � � H an e b2 n2 2n 8 þ 2be2 kðn 1Þ þ e4 k2 þ bn þ e2 k k � � E2 ¼ 2b b2 n2 ðn þ 1Þ þ 2be2 k n2 þ n þ 2 þ e4 k2 ðn þ 1Þ

where H ¼ ðbn þ 2b þ e2 kÞð8b2 n þ 8b2 be2 kn þ 6be2 k e4 k2 Þ. 8 EI EC Then, W ðEÞ W ðEÞ < 0 if and only if E 2 ðE1 ; E2 Þ. Again Fig. 2 suggests the non-monotonic relationship between the welfare difference and targeted emission level. Fig. 3 suggests that the range for welfare advantage of the emission cap regulation compared to the emission intensity regulation is broader when k is larger. We now discuss the properties of the equilibrium abatement. From

Proposition 3. Suppose that P ¼ a bQ, C ¼ 0, g ¼ eqi , and K ¼ kx2i = 2. ~ where k ~ :¼ Then, WEI > WEC if and only if k < k,

� b 2e2

αeðn þ 2Þ þ α2 ðn þ 2Þ

~ ¼ ∞, k < ~ k holds (and thus WEI > WEC

E→EB k

holds) for any k if E is sufficiently close to 0 or EB . This result is a special case of Propositions 1–2 under the specific setting. Therefore, Proposi­ tions 1–2 and Proposition 3 are consistent. However, when the target ~ emerges (and thus WEI < WEC holds). emission level is moderate, k > k

�

∂W ¼ Pðnq� ðEÞÞ ∂qi

0:1).

i¼1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 4e4 þ 4αe3 n þ α2 e2 ðn2 þ 4Þ 2α3 eðn þ 2Þ2 þ α4 ðn þ 2Þ2 : 2αe2 ðe αÞ

~ ¼ ∞. Moreover, ~ The critical value ~ k satisfies limE→0 ~ k ¼ limE→EB k k is U-

shaped with respect to E and has the lower bound k : ¼ ðbð6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 68 þ 20n þ n2 Þ=2e2 > 0.

nÞ þ

8 Equivalently, solving equation k ~ k ¼ 0 for α 2 ð0; αB Þ, we obtain two so­ lutions, α1 and α2 , with αðE1 Þ ¼ α1 and αðE2 Þ ¼ α2 . Then, W EI W EC < 0 if and only if α 2 ðα1 ; α2 Þ.

4

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Lemma 2. (i) x� and xEC are decreasing in E. (ii) xEI is decreasing in E if E is sufficiently close to EB . (iii) xEI is non-monotone with respect to E if g0 K0 > qZ ðP0 C00 g00 K0 þ nðP0 þ P00 qZ ÞÞ. Proof See the Appendix. Lemma 2 suggests that the effect of emission regulation on the abatement activity depends on the regulatory regimes. With regard to Lemma 2(iii), it is difficult to judge under which conditions this inequality is satisfied. Here, we again specify the model as we did in Proposition 3 and derive the conditions. Lemma 3. Suppose that P ¼ a bQ, C ¼ 0, g ¼ eqi , and K ¼ kx2i =2. Then, xEI is non-monotone with respect to E if ke2 > ðn þ1Þb and decreasing in E otherwise. Proof See the Appendix. When e is larger, an increase in q increases the emission level more. An increase in E substantially increases the output under the emission intensity regulation, which induces larger abatement to offset the in­ crease in emissions. The output expansion effect is stronger when b or n is smaller. Specifically, if the demand is elastic or the number of firms is small, a firm’s output is more sensitive to a change in emission regula­ tion. Therefore, an increase in E more likely increases x when b or n is smaller. When k is larger, an increase in E reduces the marginal cost of production, including the emission abatement cost, more significantly, and thus, an increase in E more significantly increases the output. Therefore, an increase in E more likely increases x when k is larger.

Fig. 2. Welfare difference (n ¼ 3; a ¼ 6; b ¼ 1; e ¼ 2 and k ¼ k þ 0:1).

4. Robustness check 4.1. Differentiated Bertrand competition In this study, we assume that firms face Cournot competition in a homogeneous product market. We can show that our results do not depend on this competition mode and that Lemma 1 and Propositions 1–2 hold in the following Bertrand competition model.9 Assume that there are n symmetric firms that produce differentiated products. The direct demand function for product i ði ¼ 1; 2; …; nÞ is given by Di ðp1 ; p2 ; …; pn Þ : Rþ 7!Rþ . We assume that D is twice continu­ ously differentiable for all pi > 0. The demand is downward sloping, ∂ Di =∂pi < 0; and ∂Di =∂pj > 0; j 6¼ i as long as D > 0. The latter condition means that goods are substitutes. In addition, we assume that the direct effect of a price change dominates the indirect effect, � Pn P �� 2 � 2 2 m¼1 ð∂Di =∂pm Þ < 0 and ∂ Di =ð∂pi Þ þ j6¼i �∂ Di =∂pi ∂pj � < 0. We

Fig. 3. (n ¼ 3; a ¼ 6; b ¼ 1; e ¼ 2; and k 2 ½k; k þ 1Þ).

further assume that demand has increasing differences, ∂2 Di =∂pi ∂pj � 0; which implies that the price-setting game is supermodular. These are standard assumptions in a price-setting oligopoly with differentiated products.10 Except for the demand system, we follow the same structure in the quantity competition analysis. The firm i0 s profit is πi ¼ pi Di ðpi ; p i Þ CðDi ðpi ; p i ÞÞ Kðxi Þ. 4.2. Emission cap regulation and emission tax Fig. 4. Abatement level comparison.

In this study, we compare emission intensity and emission cap reg­ ulations. However, our analysis can be applied to the comparison be­ tween the emission intensity regulation and the emission tax policy. Suppose that the government imposes emission tax t � 0 and then firms compete, given the tax rate. We can show that this emission tax policy

Fig. 4, we observe that x� and xEC are decreasing in E. This is intuitive. A stricter regulation (smaller E) stimulates emission abatement. By contrast, Fig. 4 suggests that xEI can be increasing in E. This is because an increase in E relaxes the emission constraint, thereby naturally reducing abatement. However, an increase in E enlarges the output, which leads to the large abatement. Because the output expansion effect is strong under the emission intensity constraint, the latter effect might dominate the former effect, and thus, xEI can be increasing in E. Under the general conditions, this discussion can be reduced to the following result.

9

The proofs are available upon request from the authors. Examples of the demand system include linear and constant elasticity of substitution demand and the logit demand model (see Vives, 1999; Anderson et al., 2001). 10

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Energy Policy xxx (xxxx) xxx

yields the same equilibrium outcome as the emission cap regulation.11

4.3.2. Emission cap regulation Firm i’s optimization problem is

Lemma 4. Given E, there exists tðEÞ such that the emission tax policy yields the same equilibrium outcome as the emission cap regulation.

maxPðQÞqi qi ;xi

Proof See the Appendix. Therefore, our Propositions 1–2 suggest an advantage of the emission intensity regulation over the emission tax policy in both the most stringent target and the weakest target cases. Also, our Proposition 3 implies that the emission tax policy can be better for welfare when the target emission level is moderate.

The emission constraint is binding for the same reason discussed in the emission intensity regulation case. From the emission constraint and b by the implicit function theorem, we obtain x ¼ b x ðq ; EÞ. Focusing on i

∂πEC i ¼ P’ðQÞqi þ PðQÞ ∂qi

¼ P’ðQÞqi þ PðQÞ

(11)

eðqi ; xi Þ ¼ 0 (and thus,

4.3.3. Comparison We obtain the following results. (i) qEI ðEÞ > qEC ðEÞ. (ii) The emission cap regulation yields a higher profit than the emission intensity regulation does. (iii) If E is sufficiently close to zero or EB , then the emission intensity regulation yields greater welfare than the emission cap regulation does. The proofs are presented in the Appendix. 5. Conclusion and policy implications

(8)

eðqi ; xi Þ : qi

In this study, we compare two direct-regulation tools, an emission cap regulation and an emission intensity regulation. We find that profitmaximizing firms always prefer the emission cap regulation. However, the emission intensity regulation always yields greater consumer wel­ fare. Moreover, we present two cases in which the emission intensity regulation yields greater welfare than the emission cap regulation does: the case of the strictest target, which is close to a zero-emission target, and the case of the loosest target, which is close to business as usual. Our result suggests that the government should adopt the emission intensity regulation, especially to achieve a zero-emission society efficiently. However, if the targeted emission level is moderate, then the emission cap regulation can yield greater welfare than the emission intensity regulation because the latter may lead to excessive production. Our study neglects any uncertainty of demand or cost. If the gov­ ernment chooses the direct regulation before knowing the demand parameter, an increase of the degree of demand uncertainty increases the advantage of the emission intensity regulation over the emission cap regulation for both the welfare and profits of firms. This is because the firms can expand (shrink) their output more flexibly under the emission intensity regulation than under the emission cap regulation when de­ mand is high (low). We consider this as the reason why some companies, such as Japanese electric power companies, choose the emission in­ tensity regulation as their favored form of self-regulation. Comparing the two tools after introducing demand uncertainty is left for future

Because cðqi ; xi Þ is strictly increasing in xi and eðqi ; xi Þ is strictly decreasing in xi , this constraint is binding. From the emission constraint and by the implicit function theorem, we obtain xi ¼ b x i ðqi ; αÞ. Focusing on the symmetric equilibrium, then, the equilibrium output, qEI ðαÞ, is characterized by the following first-order condition:

∂c ∂qi 2

¼ P’ðQÞqi þ PðQÞ

3 ∂e 6 ∂c ∂c 6∂qi 7 7 ¼ 0; þ ∂qi ∂xi 4 ∂e 5 ∂xi

sume that the second-order condition is satisfied. We also obtain b x i ðqEI ; EÞ. xEC ðEÞ ¼ b

cðqi ; xi Þ

∂c ∂qi

∂c ∂xi ∂xi ∂qi

where we use the emission cap constraint E=n

Firm i’s optimization problem is

∂πEI i ¼ P0 ðQÞqi þ PðQÞ ∂qi

i

∂xi =∂qi ¼ ð∂e =∂qi Þ=ð∂e =∂xi Þ by the implicit function theorem). We as­

eðqi ; xi Þ : qi

s:t: ​ α �

∂c ∂qi 2

4.3.1. Emission intensity regulation Firm i chooses its output, qi , and abatement level, xi , to maximize its profit subject to

qi ;xi

i

the symmetric equilibrium, then, the equilibrium output, qEC ðEÞ, is characterized by the following first-order condition:

In this study, we assume that production and abatement costs are additively separable. However, this assumption is for simplicity. We can show that Result 1 and Propositions 1–2 hold under a more general model formulation. Consider an oligopoly market as discussed in the previous sections, except for the assumptions on cost and emission functions. cðqi ; xi Þ is firm i’s cost, including production and abatement costs. Then, firm i’s profit is PðQÞqi cðqi ; xi Þ: We assume cðqi ; xi Þ is twice continuously differentiable. We also assume ∂c=∂qi > 0 and ∂c=∂xi > 0 for xi > 0 and that the function is strictly convex. Moreover, we assume ∂cðqB ; 0Þ= ∂xi ¼ 0 to ensure that the equilibrium abatement level is positive. eðqi ; xi Þ is firm i’s emission function. We assume eðqi ; xi Þ is twice continuously differentiable, ∂e=∂qi > 0 and ∂e=∂xi < 0, and that the function is convex.

maxPðQÞqi

(10)

s:t: ​ E=n � eðqi ; xi Þ:

4.3. Alternative model formulation of abatement activities

α�

cðqi ; xi Þ

∂c ∂xi ∂xi ∂qi 3

∂e α ∂c 6 ∂qi 7 6 7 ¼ 0; ∂xi 4 ∂e 5 ∂xi

(9)

where we use the emission constraint αqi eðqi ; xi Þ ¼ 0 (and thus, ∂xi = ∂ qi ¼ ðα ∂e =∂qi Þ=ð∂e =∂xi Þ by the implicit function theorem). We assume that the second-order condition is satisfied. We also obtain xEI ðαÞ ¼ b x i ðqEI ; αÞ and. eEI ðαÞ ¼ αqEI ðαÞ: The government sets the emission intensity α ¼ α such that neEI ðαÞ ¼ E. Let ðqEI ðEÞ; xEI ðEÞÞ be the pair of equilibrium output and abatement and WEI ðEÞ be the equilibrium welfare under the emission intensity regulation when. α ¼ α: 11 However, we emphasize that Lemma 4 depends on the assumption that no uncertainty exists in our model. See Weitzman (1974).

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Energy Policy xxx (xxxx) xxx

research. Appendices. Proof of Result 1 �

� � � ∂π EC i � ¼ K’ xEI g’ qEI ∂qi �qi ¼qEI

α

�

� � K’ xEI g’ qEI < 0:

This inequality implies that qEI is excessive from firm i’s profit-maximizing viewpoint under the emission cap regulation.■ Proof of Lemma 1 We obtain EC

�

� � � � K g� qEC � �E n C �qEC EI EI q K g� q C� q �E n K g qEI C qEI αqEI ¼ πEI ðEÞ;

π ðEÞ ¼ P nq�EC qEC EI

> P�nq ¼ P nqEI qEI

EI

where the inequality follows from the fact that qEC ðEÞ ¼ argmaxfqi g PðQÞqi

Cðqi Þ

Kðgðqi Þ E =nÞ and qEI 6¼ qEC :■

Proof of Proposition 1 For l ¼ EC; EI, we obtain � � n � X dW l �� ∂W l dql ∂W l dxl ∂W l ¼ þ þ � ∂qi dE ∂xi dE dE E¼0 ∂E i¼1 ¼ n P nqZ ¼ n P nqZ

�

¼ n P nqZ

�

�

C’ qZ

C’ qZ

�� dql dE

C’ qZ

�

�� dql

nK’ xZ

dE nK’ xZ

�

� dxl

η’ð0Þ

dE

� � dql g’ qZ dE

1 n

�

η’ð0Þ

� �� dql � K’ xZ g’ qZ þ K’ xZ dE

η’ð0Þ;

where we use gðqi Þ xi ¼ E=n (and thus, dxl =dE ¼ g’ðdql =dEÞ E 2 ð0; EB Þ, we obtain � � dqEI �� dqEC �� > > 0: dE �E¼0 dE �E¼0 From (4), we obtain P � � dW EI �� dW EC �� > : � dE E¼0 dE �E¼0

C’

K’g’ > P þ P’qi

C’

1=nÞ, and ðqEC ; xEC Þ ¼ ðqEI ; xEI Þ ¼ ðqZ ; xZ Þ when E ¼ 0. Because qZ < qEC < qEI for all

K’g’ ¼ 0. Under these conditions, we obtain

Because WEI ¼ WEC when E ¼ 0, we obtain Proposition 1.■ Proof of Proposition 2 For l ¼ EC; EI, we obtain � � n � X ∂W l �� ∂W l dql ∂W l dxl ¼ þ ∂qi dE ∂xi dE ∂E �E¼EB i¼1 ¼ n P nqB

�

C’ qB

¼ n P nqB

�� dql

�

dE

nK’ xB

C’ qB

�� dql dE

� dxl dE

η’ E B

�

η’;

where we use qEC ¼ qEI ¼ qB and xEC ¼ xEI ¼ 0 when E ¼ EB and K’ð0Þ ¼ 0: Because qEC < qEI < qB for all E 2 ð0; EB Þ, we obtain � � dqEI �� dqEC �� 0< < : � dE E¼EB dE �E¼EB From (4), we obtain P

C’ > 0. Under these conditions,

7

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Energy Policy xxx (xxxx) xxx

� � ∂W EI �� ∂W EC �� < : ∂E �E¼EB ∂E �E¼EB Because WEI ¼ WEC when E ¼ EB , we obtain Proposition 2.■ Proof of Proposition 3 First, we consider the equilibrium outputs for each regime. From (2) and (4), we obtain qEI ¼

a ð1 þ nÞb þ kðe

αÞ2

; ​ qEC ¼

a þ keE : ð1 þ nÞb þ e2 k

Substituting the equilibrium outputs into total surplus, we obtain W EI ðαÞ ¼

anðaðbðn þ 2Þ þ kðe

αÞ2 Þ

2ðbðn þ 1Þ þ kðe

W EC ðEÞ ¼

αÞ2 ÞÞ

2ηαðbðn þ 1Þ þ kðe

;

2 2

αÞ Þ

a2 n2 ðbðn þ 2Þ þ e2 kÞ þ 2aeknEðbðn þ 2Þ þ e2 kÞ

2

bkE bðn þ 1Þ2 þ e2 kn

2nðbn þ b þ e2 kÞ2

�

ηE:

Using EEI ðαÞ ¼ αqEI ¼ E, WEC ðEÞ can be rewritten as a function of α. Thus, we obtain W EI ðαÞ

W EC ðEÞ ¼

a2 knαðe

αÞH

2ðbn þ b þ e2 kÞ2 ðbðn þ 1Þ þ kðe

where H : ¼ 2b2 ðn þ 1Þ þ bkð2e2

αÞ2 Þ2

;

eðn þ 2Þα þ ðn þ 2Þα2 Þ þ e2 k2 αðα

eÞ. WEI ðαÞ

WEC ðEÞ is positive if and only if H > 0 and

H > ð<Þ0 ​ if k < ð>Þ~k; ~ :¼ where k � b 2e2 αeðn þ 2Þ þ α2 ðn þ 2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 4e4 þ 4αe3 n þ α2 e2 ðn2 þ 4Þ 2α3 eðn þ 2Þ2 þ α4 ðn þ 2Þ2 : 2αe2 ðe αÞ

k also depends on the emission target via α. This implies that WEI ðαÞ > ð<ÞWEC ðEÞ if k < Remember that α is determined by EEI ðαÞ ¼ E, and thus, ~ ~ ¼ ∞, we obtain lim k ~ ¼ lim B k ~ ¼ ∞. k ¼ limα→e k ð>Þ~ k. Because limα→0 ~ E→0 E→E ~ Differentiating k with α, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � � 2 2 4e4 þ 4barαe3 n þ α2 e2 ðn2 þ 4Þ 2α3 eðn þ 2Þ2 þ α4 ðn þ 2Þ2 ∂~k bðe 2αÞ 2e þ αen α n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ ∂α α2 ðe αÞ2 4e4 þ 4αe3 n þ α2 e2 ðn2 þ 4Þ 2α3 eðn þ 2Þ2 þ α4 ðn þ 2Þ2 ~ ∂α is negative (positive) when α < ð>Þ e=2, kðEÞ ~ ~ is minimized when α ¼ e= 2, we Because ∂k= is U-shaped and minimized at α ¼ e=2. Because k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 B 2 obtain k ¼ bð6 n þ n þ 20n þ 68 Þ=2e . Note that αðEÞ is increasing, αð0Þ ¼ 0; and αðE Þ ¼ e.■ Proof of Lemma 2 (i) For l ¼ EC; EI, and *, we have l

dx dql ¼ g’ dE dE

1 ; n

where we use g x ¼ E. Combining the equation with (7), we obtain � � ðg’K’’Þ=n dx� 1 ¼ g’ 2 n dE nP’ þ P’’q C’’ g’’K’ g’ K’’ � � 1 g’2 K’’ ¼ 1 < 0: n nP’ C’’ g’’K’ g’2 K’’ Likewise, from (5), we obtain � � dx ðK’’g’Þ=n 1 ¼ g’ n dE P’ C’’ g’’K’ g’2 K’’ þ nðP’ þ P’’qi Þ � � 1 g’2 K’’ ¼ 1 < 0: n P’ C’’ g’’K’ g’2 K’’ þ nðP’ þ P’’qi Þ EC

Therefore, x� and xEC are decreasing in E in general demand and cost functions. (ii) From (3), we obtain 8

K. Hirose and T. Matsumura

Energy Policy xxx (xxxx) xxx

dxEI dqEI dα 1 ¼ g’ dα dE n dE � 1 � dqEI 1 EI dqEI q þα ¼ g’ dα n dα � 1 EI � 1 dq dqEI ¼ g’ qEI þ α n dα dα

1 ; n ! 1 ;

where we use E=n ¼ αqEI ðαÞ when E > 0. Thus, dxEI > ð<Þ0 ​ if ​ and ​ only ​ if ​ Θ � ðg’ dE

αÞ

dqEI dα

qEI > ð<Þ0:

If E ¼ EB , we obtain ΘjE¼EB ¼ ðg’

αÞ

� ¼ ðg’

αÞ

dqEI dα

qB

P’

C’’

g’’K’

� ¼ qB

�

αÞK’’qB ðg’ αÞ2 K’’ þ nðP’ þ P’’qi Þ

ðg’

αÞ2 K’’ ðg’ αÞ2 K’’ þ nðP’ þ P’’qi Þ

ðg’ P’

C’’

g’’K’

qB

� 1

< 0;

where we use qEI ¼ qB ; xEI ¼ 0, and K’ð0Þ ¼ 0 when E ¼ EB . Thus, xEI is decreasing in E if E is sufficiently close to EB . (iii) If E ¼ 0, we obtain

� ¼ g’

dqEI ΘjE¼0 ¼ g’ dα P’

C’’

qZ

K’ þ g’K’’qZ � g’’K’ g’2 K’’ þ n P’ þ P’’qZ

� qZ ;

where we use qEI ¼ qZ , α ¼ 0 when E ¼ 0. Thus, we obtain ΘjE¼0 > ð<Þ0 ​ if ​ and ​ only ​ if ​ g’K’ > ð<Þ

qZ P’

C’’

�� g’’K’ þ n P’ þ P’’qZ :

(12)

This implies Lemma 2(iii).■ Proof of Lemma 3 From (12), we obtain EI

dx > ð<Þ0 ​ if ​ and ​ only ​ if ​ ke2 > ð<Þðn þ 1Þb: j dE E¼0 This and Lemma 2(ii) imply that xEI is non-monotone with respect to E if ke2 > ðn þ 1Þb. Suppose that ke2 � ðn þ1Þb: We show that xEI is decreasing in E, which is the latter part of Lemma 3. We obtain Θ ¼ ðg’

αÞ

dqEI dα

qEI ¼ ðe

¼

αÞ

2akðe

αÞ

ðn þ 1Þb þ kðe

2 �2

αÞ

a ðn þ 1Þb þ kðe

αÞ2

� ðn þ 1Þb < 0: � 2 ðn þ 1Þb þ kðe αÞ2

aðkðe

αÞ2

Proof of Lemma 4 The profit of firm i is given by π i ðqi ; xi ; Q i Þ :¼ PðQÞqi Cðqi Þ Kðxi Þ tei : Let superscript T denote the equilibrium outcomes under the emission tax policy. qT ðtÞ, and the equilibrium abatement, xT ðtÞ, are characterized by the following first-order conditions: � � ∂πi ¼ P’ nqT qT þ P nqT ∂qi

C’ qT

�

� tg’ qT ¼ 0:

� ∂πi ¼ K’ xT þ t ¼ 0: ∂xi

9

K. Hirose and T. Matsumura

Energy Policy xxx (xxxx) xxx

The second-order conditions are satisfied. From the second equation, we obtain that xT ¼ xEC when t ¼ K’ðxEC Þ. We now show that t ¼ K’ðxEC Þ yields qT ¼ qEC . When t ¼ K’ðxEC Þ, the first-order conditions are summarized as follows: � � ∂π i ¼ P’ nqT qT þ P nqT ∂qi

C’ qT

�

� � K 0 xT g’ qT ¼ 0:

(13)

From (4) and (13), we obtain that the condition is satisfied when xT ¼ xEC ðEÞ ¼ gðqEC Þ ernment sets t ¼ K’ðxEC Þ.■

E=n. This implies ðqEC ; xEC Þ ¼ ðqT ; xT Þ holds if the gov­

Proofs of the Results in Section 4.3 (i) qEI ðEÞ > qEC ðEÞ: From (9) and (11), we obtain

� 2 EI EI 3 � α ∂eðq ; x 7 � � ∂cðq ; x ∂cðq ; x 6 ∂π ∂qi 6 7 ¼ P’ nqEI qEI þ P nqEI 4 ∂eðqEI ; xEI � 5 ¼ 0; ∂qi ∂qi ∂xi ∂xi �3 2 ∂eðqEC ; xEC � � EC EC EC EC 7 � � ∂cðq ; x ∂cðq ; x 6 ∂πEC ∂qi i 6 7 ¼ P’ nqEC qEC þ P nqEC þ 4 ∂eðqEC ; xEC � 5 ¼ 0: ∂qi ∂qi ∂xi ∂xi EI

EI i

EI

�

EI

EI

Suppose qi ¼ qEI . Then, xi ¼ xEI holds because of the emission equivalence. We obtain � � ∂π EC ∂cðqEI ; xEI Þ α i � ¼ < 0: ∂eðqEI ;xEI Þ ∂qi �qi ¼qEI ∂xi ∂xi

This inequality implies that qEI is excessive from firm i’s profit-maximizing viewpoint under the emission cap regulation.■ (ii) The emission cap regulation yields a higher profit than the emission intensity regulation does: Using the resulting profit and emission equivalence, we obtain � � EC π ðEÞ ¼ P nqEC qEC c qEC ; xEC � EC �� EC EC b EC x q ;E ¼ P nq q c q ;b � �� bx qEI ; E�� > P nqEI �qEI c qEI ; b EI EI EI EI ¼ P nq� q c q ; bx� q ; E ¼ P nqEI qEI c qEI ; xEI ¼ πEI ðEÞ; b x i ðqi ; EÞÞ and qEI 6¼ qEC ; and the fourth line follows from the emission where the third line follows from the fact that qEC ðEÞ ¼ argmaxfqi g PðQÞqi cðqi ; b b equivalence. Note that eðq; b x ðq; EÞÞ ¼ E=n for any q and eðq; b x ðq; EÞÞ ¼ E=n when q ¼ qEI .■ (iii) If E is sufficiently close to zero or EB , then the emission intensity regulation yields greater welfare than the emission cap regulation does: First, we discuss the case where E is sufficiently close to zero. For l ¼ EC; EI, we obtain � � n � X dW l �� ∂W l dql ∂W l dxl ∂W l ¼ þ þ � ∂ q ∂ x dE E¼0 ∂E i dE i dE i¼1 � � l � Z Z � Z Z l � ∂cðq ; x ∂cðq ; x dx dq ¼ n P nqZ η’ð0Þ n ∂qi ∂xi dE dE �� �� � � � ∂cðqZ ; xZ ∂cðqZ ; xZ ∂xi dql ∂xi dql ¼ n P nqZ η’ð0Þ n þ ∂qi ∂xi ∂qi dE ∂E dE 02 3 1 ∂e � � � Z Z � Z Z B6 l l 7 � ∂cðq ; x ∂cðq ; x B6∂qi 7 dq dq 1 C C η’ð0Þ ¼ n P nqZ n þ @ 4 5 A ∂ e ∂ e ∂qi ∂xi dE dE n ∂xi ∂xi 0 2 31 ∂c � � ∂e Z Z Z Z C dql ∂xi B � ∂cðq ; x ∂cðq ; x 6 ∂qi 7 Z 6 7 C B ¼ n@P nq 4 ∂e 5A dE ∂e η’ð0Þ; ∂qi ∂xi ∂xi ∂xi where we use eðqi ; xi Þ ¼ E=n, the implicit function theorem, and ðqEC ; xEC Þ ¼ ðqEI ; xEI Þ ¼ ðqZ ; xZ Þ when E ¼ 0. Thus, the difference between the two regulations is dql =dE only. Because qEC < qEI for all E 2 ð0; EB Þ, we obtain 10

K. Hirose and T. Matsumura

Energy Policy xxx (xxxx) xxx

� � dqEI �� dqEC �� > : dE �E¼0 dE �E¼0 From (11), we obtain P nqZ

�

Z

Z

∂cðq ; x ∂qi

� � ​ > P’ nqZ qZ þ P nqZ

2 3 � ∂e 7 ∂cðq ; x 6 6∂qi 7 4 ∂e 5 ∂xi ∂xi

�

Z

Z

∂cðq ; x ∂qi

Z

�

Z

2 3 � ∂e 7 ∂cðq ; x 6 6∂qi 7 ¼ 0: 4 ∂e 5 ∂xi ∂xi Z

Z

Under these conditions, we obtain � � dW � dW EC �� > : dE �E¼0 dE �E¼0 EI �

Because WEI ¼ WEC when E ¼ 0, we obtain the first result. Next, we discuss the case where E is sufficiently close to EB . For l ¼ EC; EI, we obtain � � n � X dW l �� ∂W l dql ∂W l dxl ∂W l ¼ þ þ � ∂ q ∂ x dE E¼EB dE dE ∂E i i i¼1 � � � ∂cðqB ; 0Þ dql ∂cðqB ; 0Þ dxl ¼ n P nqB η’ð0Þ n ∂qi ∂xi dE dE � � � ∂cðqB ; 0Þ dql ¼ n P nqB η0 ð0Þ; ∂qi dE where we use ðqEC ; xEC Þ ¼ ðqEI ; xEI Þ ¼ ðqB ; xB Þ when E ¼ 0 and ∂cðqB ; 0Þ=∂xi ¼ 0. Because qEC < qEI for all E < EB , we obtain � � dqEI �� dqEC �� < : dE �E¼EB dE �E¼EB From (11), we obtain P nqB

�

� � ∂cðqB ; xB Þ > P’ nqB qB þ P nqB ∂qi

∂cðqB ; 0Þ ¼ 0: ∂qi

Under these conditions, we obtain � � dW EI �� dW EC �� < : � dE E¼EB dE �E¼EB Because WEI ¼ WEC when E ¼ EB , we obtain the second result. ■

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