The optimal size of a permit market

Journal of Environmental Economics and Management 60 (2010) 133–143

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Journal of Environmental Economics and Management journal homepage: www.elsevier.com/locate/jeem

The optimal size of a permit market Frank C. Krysiak a,, Patrick Schweitzer b a b

Department of Business and Economics, University of Basel, Peter Merian-Weg 6, CH-4002 Basel, Switzerland Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Luxembourg

a r t i c l e in fo

abstract

Article history: Received 7 March 2007 Available online 26 May 2010

Regulating the emissions of non-uniformly mixed pollutants with a permit market carries the risk of hot spot formation, which can be reduced by dividing the regulation area into trading zones. The trading zone approach has been extensively discussed for the full-information case. We consider incomplete information concerning the emitters’ abatement costs, their locations, and pollution dispersion. We derive the optimal number of trading zones and the optimal number of permits per zone and analyze under which conditions a system of independent trading zones is superior to other policy measures. Our results show that appropriately sized permit markets are well-suited to regulating non-uniformly mixed pollutants under informational constraints if firms are not too heterogeneous. Only for substantial heterogeneity and a highly non-linear damage function can it be optimal to use command-and-control strategies. & 2010 Elsevier Inc. All rights reserved.

Keywords: Tradable permits Uncertainty Hot spot Spatial model Regulation Non-uniformly mixed pollutant Asymmetric information

1. Introduction In environmental policy circles, permit markets are increasingly seen as a politically viable alternative to the commandand-control approach that is still ubiquitous in many countries. Although this shift to a potentially cost-effective firm regulation is desirable, it raises problems that need to be addressed. One of these problems is that permit markets regulate only aggregate pollution; they cannot control the spatial distribution of pollution. Whereas this does not matter for uniformly mixed pollutants, it implies the risk of hot spot formation for other pollutants. This problem has been discussed extensively in the literature, see Tietenberg [25]. But the discussion pertains to the case where there is certainty with regard to the locations and production possibilities of all emitters and with regard to pollution dispersion. In this case, we can predict the effects of all permit transactions on ambient pollution and use this information to design the trading system. This full-information setup has been characteristic for the predominantly discussed application of emission trading under the Clean Air Act. There the permit trading involves mostly large, stationary plants that are in use for a long time. Thus, information concerning the emission sources can be gathered and remains valid for a considerable time. Furthermore, extensive research has produced detailed information about the pollution dispersion. But not all permit markets will be designed under such favorable informational conditions. Many pollutants are emitted by sources whose location or technology change frequently, due to the replacement of production equipment or due to market entry and exit. When plants are shut down and replaced by other ones at different locations or with different stack heights, the information about the location or the technical characteristics of emission sources is devalued. If such changes

 Corresponding author. Fax: +41 61 267 04 96.

E-mail address: [email protected] (F.C. Krysiak). 0095-0696/$ - see front matter & 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jeem.2010.05.001

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occur faster than the regulation can be adjusted, for example, due to high rates of technological progress or due to regulatory inertia, the regulation has to be designed under locational or technological uncertainty. Also, the dispersion process induces uncertainty, because pollution dispersion depends on weather conditions, like wind directions or precipitation, that cannot be accurately predicted.1 For the full-information case, there are several well-discussed concepts that provide at least some control over the spatial distribution of pollution. First, ambient permit systems can be used, as in Montgomery [20] and Weber [26]. They achieve a least-cost implementation of a pollution standard with minimal informational requirements for the regulating authority but impose high information and transaction costs on the regulated firms. Second, the permit market can be divided into trading zones with interzone trading either being prohibited or being based on exchange rates, as in Førsund and Nævdal [7] and Klaassen et al. [10]. Such systems typically result in welfare losses compared to an ambient permit system but can achieve welfare gains compared to a conventional permit market. Finally, there is the possibility to use trading rules that impose restrictions on individual permit transactions, as in Krupnick at al. [11] and Oates and McGartland [21]. But in cases of substantial informational constraints, most of these solutions either become inoperable or induce high information or transaction costs. For zonal systems with exchange rates, the exchange rates are usually inferred from the dispersion of the pollutant and the location of the sources. Under locational or dispersion uncertainty, this information is missing.2 For trading rule systems, the information that is required to assess the impact of a given trade is changing rapidly and would thus have to be acquired and verified anew frequently. This would impose costs on permit transactions that would render the system highly inefficient, and perhaps even inoperable. Ambient permits result in high information and transaction costs for the emitters, even in the full-information setup. Under uncertainty, a large number of receptor points would have to be used to provide some safeguard against hot spots. In most applications, this would increase the complexity of the system beyond operability. However, even under informational constraints, it is possible to gain some control over the spatial distribution of emissions by partitioning a permit market into independent trading zones. But the zoning cannot be inferred from known emission locations and dispersion characteristics. This raises a question that has received little attention in the literature: How large should a permit market be? This question is discussed in Tietenberg [24,25], albeit not in a model-based framework. Some studies analyze this problem numerically in the context of specific applications.3 But to our knowledge, there are only two studies that address this question with a general analytic model.4 In Mendelsohn [17], it is shown that under abatement cost and damage uncertainty, a criterion similar to the prices-versus-quantities criterion of Weitzman [27] can be used to discern whether a uniform or a differentiated regulation of heterogeneous pollutants is preferable. In Williams [29], a criterion is derived that describes the optimal number of trading zones in a setup with abatement cost uncertainty. Although these studies provide important insights into the problem of the optimal partitioning of a permit market, they do not account for some potentially important features. In [17], the analysis is focused on the regulation of different types of pollutants. Consequently, pollution dispersion is not explicitly modeled and hot spot induced costs are not fully accounted for.5 In [29], dispersion and hot spots are covered, but only abatement cost uncertainty is considered. So the possibly important problems of hot spots occurring due to unforeseen locational choices or due to imperfectly predicted pollution dispersion are not covered. In this paper, we advance a model that can account for a simple form of abatement cost uncertainty, for locational uncertainty, and for dispersion uncertainty. As a methodological feature, we use a continuous spatial setup that allows us to study hot spot formation without a priori constraints on emitter or receptor locations. In the context of this model, we inquire about the optimal size of a permit market and investigate whether market-based solutions are suitable for regulating the emissions of non-uniformly mixed pollutants under informational constraints. Our results show that the optimal size of a permit market depends strongly on the three types of uncertainty and on the level of technological heterogeneity among emitters.6 More locational or dispersion uncertainty favors smaller permit markets, more abatement cost uncertainty or more heterogeneity render larger markets recommendable. Furthermore, the size of a permit market influences optimal abatement: Smaller zones imply a more lenient regulation. Concerning the suitability of market-based regulation, our analysis shows that market-based policies perform well if either the emitters are not too heterogeneous or if the curvature of the damage function is not too large. Only if hot spots cause large excess

1

Dispersion uncertainty has been analyzed, e.g., in Cabe and Herriges [4] in a non-point source pollution context. Furthermore, as shown in Montero [19], abatement cost uncertainty can render an exchange-rate-based system suboptimal compared to a zonal system without interzone trading. 3 See, e.g., [2,3,5,15]. In Martin et al. [15], not only a spatially but also a temporally differentiated regulation is analyzed and shown to yield substantial welfare gains. 4 There is also literature that addresses the optimal size of a permit market in the context of market frictions, see Liski [13], but it is based on different mechanisms and thus unrelated to our study. 5 In Mendelsohn [17], abatement at different sources can differ with regard to its influence only on the linear part of the damage function. Given that hot spots induce excess costs via non-linearities in the damage function, this approach cannot account for all hot spot related costs. 6 We provide a numerical example that highlights the main results of our analysis for a specific spatial configuration in a supplementary appendix. This appendix is available at JEEM’s online archive, which can be accessed at http://aere.org/journals. 2

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damage due to a highly non-linear damage function and if there is much technological heterogeneity, should market-based solutions be abandoned in favor of a command-and-control regime. 2. The model Our analysis is based on a variant of the linear-quadratic model used in Weitzman [27]. Assume that in a given area A of size A, the emissions of a fixed number of firms N are regulated with a permit market. The emissions of each firm disperse in the area and can contribute to the ambient pollution at every point of the area. But the pollutant is non-uniformly mixed, so that the size of this contribution depends on the distance between emitter and receptor. The ambient pollution causes environmental damage. The objective of the regulation is to minimize the expected total costs resulting from the overall environmental damage and from the firms’ abatement activities. To achieve this, the area is divided into J trading zones. For each zone, a fixed number of emission permits Z are issued that can be used only within this zone.7 The permits are costlessly and evenly distributed to the firms located in this zone. All permit markets are perfectly competitive. The regulation is subject to informational constraints. When designing the regulation, the regulator does not have full information about the number of firms in a given zone, about the location of the firms within a zone, about the firms’ technologies, and about the dispersion of the pollutant. Furthermore, the regulation is fixed for some time after implementation. These assumptions confine the regulation to a second-best outcome. But in many applications, the emitters’ technologies, their locations, and the pollutant dispersion vary over time, and time lags in data availability as well as the duration of administrative decision processes rule out the continuous adjustments that would be necessary to reach the first-best solution. Also, such adjustments may be undesirable, because frequent adjustments of a regulation can distort investment decisions, see Insley [9] and Tarui and Polasky [23], and can provide incentives for strategic under-abatement, see Moledina et al. [18]. The informational constraints have several implications that we want to capture in our model. On the one hand, the locational and technological uncertainty implies that the aggregate demand for permits in a zone is only imperfectly predictable and can differ among zones. The resulting random price differences between zones increase expected total abatement costs, which results in uncertain costs of partitioning the permit market. On the other hand, hot spots result in excess damage if the damage function is strictly convex. But due to the locational and the dispersion uncertainty, hot spots cannot be accurately predicted, so that the benefits of partitioning the permit market are uncertain. An optimal policy has to balance these uncertain costs and benefits by setting the number of trading zones and the number of permits per zone. To analyze this regulatory problem in a tractable model, we use a spatially homogeneous setup: The damage caused by a given level of ambient pollution is the same at all locations and all zones are of equal size. Furthermore, the number of firms in a zone is independent of the regulation.8 Although the actual number of firms may differ across zones, the expected number of firms is the same for all zones. As this has to hold for different numbers of trading zones, we assume, in effect, that the location of a firm is a uniformly distributed random variable. Also, we restrict our analysis to cases where the number of trading zones is small compared to the number of firms (i.e., J 5 N), so that we can neglect situations with no or only a few firms in a trading zone. To gain analytical results, we assume that there are only two types of firms, ‘‘clean’’ and ‘‘dirty’’ firms, with the following abatement costs9: cðaÞ :¼

b 2

a2 ,

cðaÞ :¼ Da þ

b 2

a2 ,

ð1Þ

where a denotes abatement, cðaÞ are the abatement costs of a clean firm and cðaÞ those of a dirty firm. The parameter b 4 0 is the curvature of the abatement cost function and D Z0 describes the cost difference between clean and dirty firms. A ~ firm’s emissions are e ¼ ea, where e~ denotes the emissions in the absence of abatement. As the pollutant is non-uniformly mixed, ambient pollution can differ across locations. The ambient pollution at each point can result from the activities of all N firms. But the contribution of each firm is weighted with a transport coefficient gðj,x; yÞ that depicts the contribution of a unit of pollutant emitted at location x in trading zone j to the ambient pollution at the point y. Denoting the number of firms in zone j by nj and the number of dirty firms in this zone by n j , ambient pollution Q at y can be written as 0 1 nj nj J X X X  @ ð2Þ gðj,xi ; yÞe j þ gðj,xi ; yÞe j A, Q ðyÞ :¼ j¼1

where

e j

and

e j

i¼1

i ¼ nj þ 1

are the emissions of a dirty and a clean firm in zone j, respectively.

7 Alternatively, a firm could be granted a fixed number of permits independently of its location. For the case J 5 N considered below, both approaches yield similar results. 8 A more general model would allow firms to migrate between zones depending on observed permit prices and on migration costs. Under certainty, such migration has been modeled in Markusen et al. [14]. 9 This setup is a special case of the abatement cost model used in Mendelsohn [17], Weitzman [27], and Williams [28,29].

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We assume that the local damage at y is a quadratic function of the ambient pollution at this location DðyÞ :¼ d1 Q ðyÞ þ

d2 2

Q 2 ðyÞ,

ð3Þ

with d1 Z 0, d2 4 0. The overall damage is calculated by integrating this local damage over the regulation area A. With regard to the informational constraints, we assume that the number of dirty firms in a trading zone is random, which depicts technological uncertainty. To model locational uncertainty, we take both the number of firms in a given zone and the location of a firm within a zone as being random variables. The latter implies that the transport coefficients gðj,x; yÞ are random. As the transport coefficient depends on distance and on the dispersion characteristics of the pollutant, it can also reflect dispersion uncertainty. To separate the effects of locational and dispersion uncertainty, we use the following dispersion model:

gðj,x; yÞ ¼ aszðj,x; yÞ:

ð4Þ

Here s 4 0 is a random dispersion parameter, with expected value s^ and variance s2s , that models dispersion uncertainty and that has the same value for all firms. The distance zðj,x; yÞ between source and receptor is firm-specific and random (due to the locational uncertainty). As s tends to zero, the pollutant is perfectly mixed; the distance zðj,x; yÞ becomes irrelevant. We assume that the maximal impact of emissions (a) is sufficiently large, so that gðj,x; yÞ Z 0. To keep the model tractable, we assume that the location and the technology of a firm are independent of each other, of the pollution dispersion, of the locations and the technologies of the other firms, and of the regulation. To simplify our notation, we define

z^ :¼

1 A

Z X J E j ðzðj,x; yÞÞ Aj¼1

s2z ð JÞ :¼ ^

z2 :¼

1 A

1 A

J

dy,

ð5Þ

Z X 2 J ðE j ð z ðj,x; yÞÞE j ðzðj,x; yÞÞ2 Þ dy, J Aj¼1

Z X J X J E j ðzðj,x; yÞÞE r ðzðr,x; yÞÞ A j¼1r¼1

J2

dy,

ð6Þ

ð7Þ

where E j ðzðj,x; yÞÞ is the expected distance of a firm located in trading zone j to the point y. The quantity z^ is the expected distance between a firm located in a given zone and a receptor point, averaged over all zones and all receptor points (i.e., all points in A). It is independent of J, because it is the average expected distance between an emitter and a receptor, and both emitter and receptor locations are not influenced by partitioning the permit market.10 The quantity s2z ð JÞ is the variance of the distance of a firm in a given trading zone to a receptor point, averaged over all zones and all receptor points, and thus a measure of within-zone locational uncertainty. It is decreasing with J, because the smaller a trading zone is, the less uncertainty there is concerning the location of the firms within this trading zone.11 For a finite number of trading zones, we have s2z ð JÞ 4 0, because each zone contains different possible firm ^2 locations. Finally, z is the average of the product of the expected distances of two firms to a given receptor point. It is independent of J for the same reasons as z^ . ^ denote the probability that a given firm uses the dirty technology, where 0 o f ^ o1. In addition to this notation, we let f Furthermore, we define fj :¼ n j =nj as the fraction of dirty firms in zone j. 2.1. The expected abatement costs When designing the regulation the regulator does not know the number of firms in a zone, their type, or their location within the zone. As the expected values of these variables are identical among zones, the regulator will issue the same ~ so that ð1eÞ is the overall proportional reduction goal. As the number of permits per zone. Let this number be Z ¼ eeN=J, total number of firms is known and the firms are uniformly distributed in A, the nj are multinomially distributed. Due to ^ =J and the our homogeneity and independence assumptions, the expected number of dirty firms in zone j is Eðn j Þ ¼ nf ^ ð1f ^ Þ=J. variance of n j is nf The equilibrium permit price in zone j and the individually optimal emissions of a clean (e  ) and a dirty (e  ) firm in this zone are ! N  þ Dfj , pj ¼ be~ 1e ð8Þ Jnj

10 11

with J.

E j ðzðj,x; yÞÞ is the expected distance between a source and a receptor and thus independent of J. 2 E j ð z ðj,x; yÞÞ depends on J, as it measures the variance of the location of a firm within a zone, which is decreasing with the size of a zone and thus

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e  ¼ e~ e

N Dfj  , Jnj b

e  ¼ e~ e

Dð1fj Þ N þ : Jnj b

137

ð9Þ

For calculating the expected abatement costs of a firm in zone j, we need to evaluate Eð1=ðJnj ÞÞ. This expectation exists only if the probability that the zone contains no firm is neglected, that is, if a censored distribution is used. Under the assumption J 5 N, this is a reasonable approximation,12 and we have Eð1=ðJnj ÞÞ  1=N.13 With this approximation, the total expected abatement costs of all firms in all zones, EðCtot Þ, can be written as ! D2 2b 2 ^ ^ ^ ð1eÞ þ e~ fDð1eÞ EðCtot Þ ¼ N e~ ð1f Þf ðNJÞ : ð10Þ 2 2b So the total expected costs increase linearly with J. The more trading zones we have, the fewer firms there are per zone, so differences in the fraction of dirty firms (fj ) between two zones become more likely.14 Different values of fj imply different permit prices and thus different marginal abatement costs, which leads to an increase in total abatement costs. This constitutes the social costs of gaining more control over ambient pollution by using smaller permit markets. It results solely from firm heterogeneity; if D ¼ 0, the social costs of partitioning a permit market are zero. 2.2. The expected damage The expected total damage EðDtot Þ in the region A follows from Eq. (3) by using the expectation operator and by integrating over A: Z  Z  d2 E Q ðyÞ dy þ Q 2 ðyÞ dy : ð11Þ EðDtot Þ ¼ d1 E 2 A A By substituting Eqs. (9) and (4) in Eq. (2) and using the assumption that s and zðj,x; yÞ are independent random variables, we get ! PJ E ðzðj,x; yÞÞ j¼1 j ^ ~ EðQ ðyÞÞ ¼ Ne e as : ð12Þ J To calculate the linear term in Eq. (11), we integrate Eq. (12) with regard to y 2 A and get (with Def. (5)) Z  Q ðyÞ dy ¼ NAe~ eðas^ z^ Þ EðQtot Þ :¼ E

ð13Þ

A

as the expected total pollution. As z^ does not depend on J, EðQtot Þ does not depend on the number of trading zones. Calculating EðQ ðyÞ2 Þ, the non-linear part of Eq. (11), leads to 0 0 12 0 1 nj nj nj nj J J J X X X X X BX @X   2  A  @ Eð Q ðyÞÞ ¼ E @ gðj,xi ; yÞe i þ gðj,xm ; yÞe m þ gðj,xi ; yÞe i þ gðj,xm ; yÞe m A j¼1

0 @

nr X i¼1

i¼1

gðr,xi ; yÞe i þ

j ¼ 1 r ¼ 1,raj

m ¼ nj þ 1

nr X

i¼1

m ¼ nj þ 1

11

gðr,xm ; yÞe m AA:

ð14Þ

m ¼ nr þ 1

With definitions (5)–(7), we can calculate the quadratic part of Eq. (11) ! Z  ^ Þf ^ ðNJÞD2 ð1f ^2 2 2 2 EðQtot Þ :¼ E Q 2 ðyÞ dy ¼ N 2 Ae~ 2 e2 ð a2 2as^ z^ þ z ðs^ þ s2s ÞÞ þ NAðs^ þ s2s Þs2z ð JÞ e~ 2 e2 þ : 2 A Nb

ð15Þ

This part of the expected damage function captures three effects. First, uncertainty concerning the location of the emitters implies a strictly positive spatial variance of ambient pollution, 2 which increases expected damages. This effect is captured by NAe~ 2 e2 s^ s2z ð JÞ and reflects the expected costs of hot spot formation for homogeneous firms in the absence of dispersion uncertainty (D ¼ s2s ¼ 0). It is proportional to the variance s2z ð JÞ of the average distance between an emitter in a given zone and a receptor, and thus decreases with J. To distinguish this effect from the other hot-spot-related effects, we refer to it as a ‘‘locational hot spot,’’ because it results from the spatial proximity of emitters. Second, the dispersion uncertainty increases expected damages, even in the absence of locational uncertainty, as ^2 captured by the term N 2 Ae~ 2 e2 z s2s . Dispersion uncertainty results in uncertain levels of ambient pollution at all points of 12 13

The probability of having no firms in a trading zone is below 0.05% if N Z 10J. For an average of 50 firms per zone, the maximal relative error of this approximation is about 2% and it is below 1% for an average of 100 firms per

zone. 14 Using more zones also renders differences in firm numbers among zones more likely. But for J 5 N, where Eð1=ðJnj ÞÞ  1=N, this effect does not influence the expected total abatement costs.

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A, which increases expected damages, because of the strict convexity of the damage function. Furthermore, as Eq. (15) shows, the dispersion uncertainty augments the effects of the locational uncertainty. Finally, the abatement cost heterogeneity increases the hot spot related costs, as captured by the term ^ Þf ^ ðNJÞD2 =b2 Þs2 ð JÞðs^ 2 þ s2 Þ. If clean and dirty firms trade permits, a dirty firm emits more at one point than a Aðð1f s

z

clean firm would in its place. The resulting higher spatial concentration of pollution increases the expected damage, which can be seen as a ‘‘technological hot spot.’’ In other words, if dirty firms buy permits from clean firms, the firms differ more strongly with regard to their emissions. This difference contributes to a more unequal spatial distribution of emissions and thus to a higher expected damage. Using more trading zones limits such permit transactions15 and thus reduces expected damages. 3. The optimal size of a permit market To characterize the optimal policy,16 we minimize expected social costs with respect to the number of trading zones J and with respect to the emission target e. We restrict the policy to the range ½1,J, where J is a number larger than 1, but much smaller than N. So the decision problem is to minimize  d2 1 ^ 2 ~ ed1 ðas^ z^ Þ EðQtot Þ¼ bðbN e~ 2 ðbð1eÞ2 þ AN d2 e2 ð a2 2as^ z^ þ z2 ðs^ 2 þ s2s ÞÞÞ þ2NbeðA EðCtot Þ þ d1 EðQtot Þ þ 2 2 2b  ^ ÞfD ^ 2 ðNJÞÞ þ Ad ðð1f ^ ÞfD ^ 2 ðNJÞ þ Nb2 e2 e~ 2 Þðs^ 2 þ s2 Þs2 ð JÞ , ^ ð1eÞÞð1f ð16Þ þ fD 2 s z with respect to J 2 ½1,J and e 2 ½0,1. For simplicity, we treat J as a continuous variable.17 To solve the optimization 2 problem, we assume that s2z ð JÞ is twice continuously differentiable on ½1,J, that ds2z ð JÞ=dJ o0, and that d2 s2z ð JÞ=dJ Z 0. These are natural properties, because the size of a trading zone (and thus the within-zone locational uncertainty) is proportional to 1/J, so that increasing J reduces the size of a trading zone at a diminishing rate. Proposition 1. Let J and eð J  Þ be the solution of  @s2z ð JÞ ^ ^ , W1 ð1f Þf ¼ W2  @J  

ð17Þ

J¼J

eð J Þ ¼

^ e~ bd1 Aðas^ z^ Þ þ Df , ^ 2 2 ~ b þAd2 ðNð a2 2as^ z^ þ z ðs^ þ s2s ÞÞ þ ðs^ 2 þ s2s Þs2z ð J  ÞÞÞ eð

ð18Þ

with 2 W1 :¼ D2 ðbAd2 s2z ð J Þðs^ þ s2s ÞÞ, 2

^ Þf ^ ðNJ  ÞÞÞ: W2 :¼ Ad2 ðs^ þ s2s Þðe~ 2 b eð J  Þ2 N þ D2 ðð1f 2

ð19Þ ð20Þ

If J 2 ½1,J and eð J Þ 20,1½, then these values represent the unique optimal policy. 



Proposition 1 characterizes the optimal policy. It shows that the optimal number of trading zones results from balancing the abatement cost effect, the technological hot spot effect, and the locational hot spot effect. The abatement cost and the technological hot spot effect are captured on the left-hand side of Eq. (17) by the two terms with opposing signs in the welfare weight W1. The locational hot spot effect is captured by the welfare weight W2 on the right-hand side of Eq. (17). The abatement cost effect measures the abatement cost increase incurred by the trading restrictions that are implied by a larger number of trading zones. Using more zones implies larger price differences between zones and thus larger differences between the marginal abatement costs of firms located in different zones (see Section 2.1). These differences increase the total expected abatement costs. The technological hot spot effect captures the consequences of partitioning on the difference between the emissions of clean and dirty firms. Having fewer firms per zone implies that there is less trade between clean and dirty firms, because with fewer firms per zone the variance of the fraction of dirty firms among zones is higher; we can expect to have more zones with few or many dirty firms. This results in fewer transactions (on average) between the firm types, so that the emissions of these firm types become more equal. As discussed in Section 2.2, this reduces expected damages. The locational hot spot effect describes the effect that partitioning has on the spatial distribution of emissions. Increasing the number of trading zones implies that emission caps are set for smaller regions. This enforces a more even distribution of emissions and thereby reduces the danger of locational hot spots and thus expected damages. 15 For J = 1, all firms can trade with each other; the expected difference in emissions is maximal. For J = N, there would be no trading; all firms would have the same emissions. 16 The proofs of the propositions are given in an appendix that is available at JEEM’s online archive (which can be assessed at http://aere.org/ journals). 17 In the online appendix, we show that this is indeed an approximation of the case where J is an integer.

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Note that dispersion uncertainty alone is not a reason to partition a permit market, as it cannot be reduced by using smaller trading zones. But dispersion uncertainty augments the effects of locational uncertainty. By Eqs. (19) and (20) an increase in s2s results in a higher W2 and a lower W1 and thus, for a given e, in a higher optimal number of trading zones. The same holds for more locational uncertainty, for example, due to a larger regulation area. In contrast, for a given ^ Þf ^ ) renders emission target, more heterogeneity (i.e., a higher D) or more technological uncertainty (i.e., a higher ð1f larger permit markets desirable. These dependencies are intuitive. The optimal number of trading zones balances three effects and each of the above variations influences only one or two of them, so that the optimal number of zones changes. For instance, higher dispersion uncertainty increases the danger of hot spots. Predictions as to how emissions of different firms add up are less accurate with higher dispersion uncertainty, so that spatial concentrations of emissions cause higher expected damages. As a finer partitioning provides more control over such spatial concentrations (by setting emission caps for smaller regions), the benefits of partitioning are larger. But dispersion uncertainty does not influence the impact of partitioning on abatement costs, so that partitioning yields higher gains at the same costs. Consequently, the optimal number of trading zones increases. As noted in Section 2, the function s2z ð JÞ is decreasing in J and as we have e 40, the optimal target eð J Þ depends positively on J. It follows that the optimal abatement level depends on the partitioning of the permit market. Corollary 1. The more trading zones are used, the more permits should be issued per firm. So the interdependency of J and e has a distinctive and intuitive direction. The more a uniform distribution of emissions is enforced, the smaller is the expected marginal damage, because hot spots are less likely. More trading zones facilitate such an enforcement. Thereby they allow for higher emissions. Locational uncertainty influences expected marginal damages (and thus the optimal emission target), because it influences the impact of emissions on pollution. More locational uncertainty implies that we know less about the impact of an additional unit of emissions on ambient pollution. We do not know where this unit is emitted and thus to what extent it increases pollution in already highly stressed areas. As the damage function depends on ambient pollution and is strictly convex, this increases the damage of an additional unit of emissions and therefore the marginal damage. More abstractly, locational uncertainty is a multiplicative uncertainty. In contrast to additive uncertainty (as in Weitzman [27]), multiplicative uncertainty can alter optimal emission targets, see Adar and Griffin [1]. Inspection of Eq. (16) reveals that the expected social costs are strictly decreasing in J for sufficiently small b, which leads to the following result: 2

Corollary 2. If b oAd2 ðs^ þ s2s Þs2z ð2Þ, then it is always optimal to use more than a single trading zone. Corollary 2 shows that there are cases in which it is always optimal to partition a permit market, regardless of the occurrence of locational hot spots. The reason for this is that the price variations among trading zones, which are a consequence of partitioning the permit market, are not always bad from a social perspective. On the one hand, these price differences result in differences in marginal abatement costs between firms located in different trading zones and thus increase expected total abatement costs. On the other hand, these price differences reduce technological hot spots. Prices are higher in zones with many dirty firms. The high permit price induces firms in these zones (which are mostly dirty firms) to abate more, while firms in zones with many clean firms (and thus mostly clean firms) face a low permit price and abate less. Thus the price differences among zones reduce the gap between the emissions of clean and dirty firms, leading to less strong technological hot spots and thus to less expected damage. Which of these opposing effects (abatement costs or technological hot spots) dominates, depends on the curvatures of the abatement cost and damage functions. The greater the curvature of the cost function, the more costly it is for firms in different zones to have differing marginal abatement costs. In contrast, the greater the curvature of the damage function, the greater the damage caused by technological hot spots. So, if b is small compared to d2 ,18 the technological hot spot effect dominates, and it is optimal to partition the permit market. This is similar to Weitzman’s [27] prices-versus-quantities criterion. There, the criterion stems from a trade-off between reducing expected abatement costs by allowing total emissions to vary and reducing expected damage by stabilizing total emissions. In our setup, total emissions are constant, so that this trade-off does not exist. Rather, there is a trade-off between equalizing marginal abatement costs (which reduces total abatement costs) and equalizing firm emissions (which reduces technological hot spots). As in [27], the relative importance of these objectives depends on the curvatures of the abatement cost and damage functions. Finally, we compare a partitioned permit market with taxes and a command-and-control approach. For this, we need some additional assumptions. First, in our setup all firms and locations are equal from the perspective of the regulator. Accordingly, we assume that a command-and-control approach implements a uniform emission standard and that an emission tax does not discriminate between locations. Second, to rule out the well-discussed prices-versus-quantities effect of Weitzman [27], we take the 18 As Corollary 2 shows, the threshold for b=d2 below which it is always optimal to partition a permit market depends on the likelihood that hot spots occur, which is influenced by the dispersion of the pollutant and the locational uncertainty.

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aggregate abatement costs of an unpartitioned market as being certain.19 This implies that there is no difference with regard to expected social costs between an emission tax and a conventional permit market. Whereas these assumptions assure that the above analysis encompasses a conventional permit market and an emission ~ that is, all tax (for J=1), the command-and-control regulation needs to be analyzed separately. For this, we set e i ¼ e i ¼ ee; firms have to meet the same target. We get the expected social costs EðSC cc Þ by substituting this abatement target into Eqs. (1) and (2), setting J =1 (because the command-and-control approach limits individual emissions but has no control 2 over the spatial distribution of emissions), and calculating EðCtot Þ, EðQtot Þ, and EðQtot Þ as in the preceding sections. This leads to ~ eN ^2 2 2 ^ ÞÞ: ðAe~ d2 e2 ðNð a2 2as^ z^ þ z ðs^ þ s2s ÞÞ þ ðs^ þ s2s Þs2z ð JÞÞ þ2Ad2 eðas^ z^ Þ þ ð1eÞðe~ bð1eÞ þ 2fD 2

EðSC cc Þ ¼

ð21Þ

Finally, we restrict our analysis to a setup where all instruments implement the same emission target e.20 For a given value of e, we define 2

2

Dtax ð JÞ :¼

Dcc ð JÞ :¼

Ad2 Ne~ 2 b e2 ðs^ þ s2s Þðs2z ð1Þs2z ð JÞÞ ^ Þf ^ ðð J1ÞbAd ðs^ 2 þ s2 ÞððN1Þs2 ð1ÞðNJÞs2 ð JÞÞÞ ð1f 2 s z z

2 2 Ad2 N e~ 2 b e2 ðs^ þ s2s Þðs2z ð1Þs2z ð JÞÞ : ^ ðNJÞðAd ðs^ 2 þ s2 Þs2 ð JÞbÞ ^ Þf ð1f 2

s

,

ð22Þ

ð23Þ

z

Proposition 2. For a given 0 o e o1, (a) a permit market with J trading zones induces at most the same expected social costs as a command-and-control strategy, if and only if either D2 r Dcc ð JÞ or Dcc ð JÞ o0; (b) a permit market with J trading zones results in at most the same expected social costs as a conventional permit market or an emission tax, if and only if either D2 r Dtax ð JÞ or Dtax ð JÞ o 0; 2 (c) if b oAd2 ðs^ þ s2s Þs2z ð1Þ, then Dtax ð JÞ o0, for all 1 o J rJ; a partitioned permit market is always preferable to an emission tax and a conventional permit market; 2 (d) if b 4Ad2 ðs^ þ s2s Þs2z ð1Þ, then Dcc ð JÞ o 0, for all 1o J r J; a partitioned permit market is always preferable to a commandand-control strategy.

Proposition 2 shows that a partitioned permit market is desirable if firms are not too heterogeneous. This is due to the differing influences that the instruments have on the three effects discerned above. A partitioned permit market can reduce locational hot spots, because it sets regional emission caps. Both taxes and a command-and-control approach cannot influence the formation of locational hot spots, as they only regulate emissions per firm. But each of these instruments can address one of the other effects better than a partitioned permit market. Taxes minimize expected total abatement costs by assuring that all firms have the same marginal abatement costs, which cannot be achieved with a partitioned permit market. A command-and-control approach eliminates technological hot spots, because it assures that all firms have the same emissions, regardless of their technology. Again, this cannot be achieved with a partitioned permit market. Without heterogeneity (D ¼ 0), only locational hot spots matter. All firms use the same technology, so that there is no technological hot spot effect. Also, the fraction of dirty firms in a zone is irrelevant for permit demand (firms do not differ), so that the permit price is the same in all zones21; partitioning does not increase expected total abatement costs. As a partitioned permit market is the only one of the above instruments that can reduce locational hot spots, it is the preferred instrument in this case. With firm heterogeneity, the other two effects gain in importance. The more heterogeneous are firms, the higher is the increase in expected total abatement costs if marginal abatement costs are not equalized among firms. Also, the more heterogeneous are firms, the more their emissions vary, leading to stronger technological hot spots if emissions are not equalized among firms. In contrast, locational hot spots result from the spatial proximity of emitters and are independent of technological heterogeneity. So, from a certain level of firm heterogeneity onwards, either the abatement cost effect or the technological hot spot effect will become more important than the locational hot spot effect. As discussed above, which of these effects is stronger, and thus dominates the locational hot spot effect, depends on the relationship between the curvatures of the abatement cost function (b) and the damage function (d2 ). P 19 ^ , that is, whenever the fraction of clean and dirty firms in a given zone is uncertain but not their total This is the case whenever ð1=JÞ Jj ¼ 1 fj ¼ f numbers in the regulation area. 20 According to Corollary 1, using the same e for all instruments is not optimal. But the conditions of Proposition 2 are sufficient for the optimality of a partitioned permit market if we use the target e that is optimal for a conventional permit market, an emission tax, and a command-and-control strategy (as can be easily shown, this target is the same for all these instruments). If these conditions are met for this e, a partitioned permit market that implements this suboptimal target leads to lower expected social costs than the other instruments. A partitioned permit market with an optimally set e will reduce social costs even further. 21 The influence of variations of the total number of firms per zone is negligible for J 5N.

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If b is large compared to d2 , total expected abatement costs increase more with firm heterogeneity than do the damages attributable to technological hot spots. As taxes (and a conventional permit market) minimize abatement costs, they are a better choice in this case than a command-and-control approach. Thus we have to compare the partitioned permit market with them; the relevant boundary for D2 is Dtax ð JÞ. If firm heterogeneity exceeds this boundary, the abatement cost increase due to partitioning outweighs the benefits of partitioning, that is, the reduction in expected damage due to less strong locational and technological hot spots.22 In contrast, for low values of b (relative to d2 ), the damage attributable to technological hot spots increases more with firm heterogeneity than do total expected abatement costs. As a command-and-control approach minimizes technological hot spots, we have to compare a partitioned permit market with this instrument. The appropriate boundary for D2 is Dcc ð JÞ. This is the level of firm heterogeneity above which the higher expected damage attributable to technological hot spots outweighs the lower expected total abatement costs achieved by allowing for some trading and the lower expected damage due to the reduction of locational hot spots. So a partitioned permit market is the instrument of choice for low levels of firm heterogeneity, where locational hot spots are most important, but not for high levels of heterogeneity, where either technological hot spots or higher abatement costs dominate instrument choice. We have stated Proposition 2 for an arbitrary number of trading zones. But it encompasses the comparison between an optimally partitioned permit market and the other instruments; we only need to replace Dtax ð JÞ and Dcc ð JÞ by D tax

maxJ2½2,J  D

ð JÞ and D

cc

cc

tax

:¼

23

:¼ maxJ2½2,J  D ð JÞ, respectively.

Altogether, our analysis shows that market-based approaches to regulating non-uniformly mixed pollutants can perform well, even under considerable informational constraints. The optimal policy balances the effect that partitioning has on abatement costs with its effects on locational and technological hot spots. The number of trading zones and the optimal emission target are interrelated; using more trading zones facilitates higher emissions per firm. Finally, a partitioned permit market is the instrument of choice if firms are not too heterogeneous. Of course, these results hold only for the specific model used in our analysis. However, the main effects identified by our analysis are intuitive and it seems likely that they are relevant in more general settings. Furthermore, as shown in Weitzman [27], the cost functions that we have used can be seen as approximations valid for small emission variations,24 which, in our case, translates to a limit on firm heterogeneity.25 Thus, particularly for those cases in which a partitioned permit market is the instrument of choice, our results should be robust.26 4. Conclusions In this paper, we have analyzed the question of how large a permit market should be if the regulated pollutant is not uniformly mixed and if there is uncertainty about abatement costs and about the danger of hot spot formation. We have derived a condition for the optimal size of a permit market, which balances the expected benefits of emissions trading with the expected costs of having less control over the spatial distribution of emissions. Furthermore, we have characterized cases in which a partitioned permit market is superior in terms of expected social costs to other commonly discussed instruments of environmental policy. In comparison to Mendelsohn [17] and Williams [29], we have covered additional types of uncertainty, which, as our results show, have a considerable influence on the optimal design and on the comparative merits of the trading zone approach. Compared to [17], we have also explicitly accounted for pollution dispersion and the damages resulting from hot spots. In comparison to [29], we have used a different concept of zoning. In [29], an optimal assignment of firms to the zones is analyzed, whereas we have presumed that the allocation of firms to the zones is based solely on firms’ locations. Thus, our results are not a generalization of those of [29] but should rather be seen as being complementary. In our analysis, we have assumed that all trading zones are of equal size. This is reasonable for the case of spatial homogeneity considered in this paper (indeed, it is optimal). However, there will be some spatial heterogeneity in applications. This implies that a uniform zoning is usually suboptimal, raising the question of how a spatially heterogeneous area should be partitioned; not only the number of zones but also their relative size and their locations are decision variables. The situation is somewhat similar to the analysis of horizontal mergers among heterogeneous firms, see, for example, Levin [12], McAfee and Williams [16], or Polasky and Mason [22]. Such mergers result in increased concentration but may reduce production costs due to scale effects. Similarly, increasing one trading zone at the expense of others will tend to reduce expected abatement costs but might increase expected damages. The literature on mergers shows that, with heterogeneous firms, not only the average size of firms needs to be considered but also the question 22 The stronger is the locational hot spot effect (numerator of Eq. (22)), and the smaller is the abatement cost effect minus the technological hot spot effect (denominator), the more heterogeneity is needed before a partitioned permit market ceases to be preferable to a tax. tax cc 23 As ½2,J  is a compact set, and Dtax ð JÞ and Dcc ð JÞ are bounded, these maxima exist and thus D and D are well-defined. 24 The additional terms used in [27] can be added without significant changes to our results. 25 If D is not too large, emissions do not vary to strongly between clean and dirty firms and between zones. 26 In the case where the partitioned permit market is the instrument of choice (small levels of firm heterogeneity), our cost specifications are not restrictive, because they are good approximations to general (twice differentiable) cost functions [27].

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which firms will and should merge. With spatial heterogeneity, an optimal partitioning of permit markets needs to investigate which trading zones should be extended and which ones downsized.27 An important difference between the regulation problem considered here and the merger literature is that we have no incentive constraints; increasing one zone at the expense of others is a regulatory decision not a decentralized outcome. Thus, if there is spatial heterogeneity, either with regard to pollution diffusion, with regard to local sensitivity to pollution, or with regard to emitters (e.g., some locations might attract more firms or more emission-intensive firms), it is optimal to adjust the relative size and location of trading zones to this heterogeneity. It is obvious that an optimization of zone sizes and locations renders zoning more attractive compared to the case of uniform zone sizes. However, it is not possible to say whether such optimization leads to more or less zones.28 Given that even our simple, spatially homogeneous setup has proven to be mathematically intricate, it seems likely that the problem of optimal zoning in a spatially heterogeneous setup can only be solved numerically. Although our results are more applicable than applied, they have implications for environmental policy. They show that market-based instruments can be used to control ambient pollution even for pollutants that are far from being uniformly mixed and even under considerable informational constraints. In many cases, dividing a regulation area into independent trading zones mitigates the risk of hot spot formation while retaining some of the cost advantages of emission trading. Our analysis also indicates that there are limits to market-based solutions. If there is a high level of firm heterogeneity and if hot spots cause large damage due to a highly non-linear damage function, then it can be optimal to forego the advantages of emission trading and to use a command-and-control approach.

Acknowledgments We are indebted to two anonymous reviewers, Arun S. Malik, and Charles F. Mason whose comments helped considerably to improve the paper. References [1] Z. Adar, J.M. Griffin, Uncertainty and the choice of pollution control instruments, J. Environ. Econ. Manage. 3 (1976) 178–188. [2] S.E. Atkinson, T.H. Tietenberg, The empirical properties of two classes of designs for transferable discharge permit markets, J. Environ. Econ. Manage. 9 (1982) 101–121. [3] S.E. Atkinson, T.H. Tietenberg, Economic implications of emission trading rules for local and regional pollutants, Can. J. Econ. 20 (1987) 370–386. [4] R. Cabe, J.A. Herriges, The regulation of non-point-source pollution under imperfect and asymmetric information, J. Environ. Econ. Manage. 22 (1992) 134–146. [5] F. Destandau, P. Point, Cheminement d’impact et partition efficace de l’espace, Revue e´conomique 51 (2000) 609–620. [6] M. Finus, Game theoretic research on the design of international environmental agreements: insights, critical remarks and future challenges, Int. Rev. Environ. Resource Econ. 39 (2008) 357–377. [7] F.R. Førsund, E. Nævdal, Efficiency gains under exchange-rate emission trading, Environ. Resource Econ. 12 (1998) 403–423. [8] F. Hackl, G.J. Pruckner, How global is the solution to global warming?, Econ Modelling 20 (2002) 93–117. [9] M.C. Insley, On the option to invest in pollution control under a regime of tradable emissions allowances, Can. J. Econ. 36 (2003) 860–883. [10] G.A.J. Klaassen, F.R. Førsund, M. Amann, Emission trading in Europe with an exchange rate, Environ. Resource Econ. 4 (1994) 305–330. [11] A.J. Krupnick, W.E. Oates, E. van de Verg, On marketable air-pollution permits: the case for a system of pollution offsets, J. Environ. Econ. Manage. 10 (1983) 233–247. [12] D. Levin, Horizontal mergers: the 50-percent benchmark, Amer. Econ. Rev. 80 (1990) 1238–1245. [13] M. Liski, Thin versus thick CO2 market, J. Environ. Econ. Manage. 41 (2001) 295–311. [14] J.R. Markusen, E.R. Morey, N.D. Olewiler, Environmental policy when market structure and plant locations are endogenous, J. Environ. Econ. Manage. 24 (1993) 69–86. [15] K.C. Martin, P.L. Joskow, A.D. Ellermann, Time and location differentiated NOx control in competitive electricity markets using cap-and-trade mechanisms, Working Paper 2007-4, MIT, 2007. [16] R.P. McAfee, M.A. Williams, Horizontal mergers and antitrust policy, J. Ind. Econ. 40 (1992) 181–187. [17] R. Mendelsohn, Regulating heterogeneous emissions, J. Environ. Econ. Manage. 13 (1986) 301–312. [18] A.A. Moledina, J.S. Coggins, S. Polasky, C. Costello, Dynamic environmental policy with strategic firms: prices versus quantities, J. Environ. Econ. Manage. 45 (2003) 356–376. [19] J.-P. Montero, Multipollutant markets, RAND J. Econ. 32 (2001) 762–774. [20] W.D. Montgomery, Markets in licenses and efficient pollution control programs, J. Econ. Theory 5 (1972) 395–418. [21] W.E. Oates, A.M. McGartland, Marketable pollution permits and acid rain externalities: a comment and some further evidence, Can. J. Econ. 18 (1985) 668–675. [22] S. Polasky, C.F. Mason, On the welfare effects of mergers: short run vs. long run, Quart. Rev. Econ. Finance 38 (1998) 1–24. [23] N. Tarui, S. Polasky, Environmental regulation with technology adoption, learning and strategic behavior, J. Environ. Econ. Manage. 50 (2005) 447–467.

27 Another line of literature where such questions arise is the analysis of international environmental agreements, where feasible coalitions have to be identified among heterogeneous countries, see, for example, [6,8]. 28 For example, if locations differ only with regard to pollution sensitivity and if there are few zones, an optimization of relative zone sizes usually increases the expected marginal benefits of adding an additional zone compared to a uniform zoning. The change from one to two zones reduces expected damages more strongly, if relative zone sizes are optimized. However, for many zones, such optimization can have the opposite effect. If the already existing zones have optimal relative sizes an additional zone may achieve less reduction in expected damages compared to a case where the already existing zones are not optimized in this way. Consequently, optimizing the relative size of zones increases the optimal number of zones, if this number is small, but might reduce it, if this number is large.

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[24] T.H. Tietenberg, Transferable discharge permits and the control of stationary source air pollution: a survey and synthesis, Land Econ. 56 (1980) 391–416. [25] T.H. Tietenberg, Tradeable permits for pollution control when emission location matters: What have we learned?, Environ Resource Econ. 5 (1995) 95–113. [26] M.L. Weber, Markets for water rights under environmental constraints, J. Environ. Econ. Manage. 42 (2001) 53–64. [27] M.L. Weitzman, Prices vs. quantities, Rev. Econ. Stud. 41 (1974) 477–491. [28] R.C. Williams III, Prices vs. quantities vs. tradable quantities, Working Paper 9283, NBER, 2002. [29] R.C. Williams III, Cost effectiveness vs. hot spots: determining the optimal size of emission permit trading zones, University of Texas at Austin, Working Paper, 2003.