The Optimal Pricing of Pollution When Enforcement is Costly

Transcription

1 Unversty of Massachusetts Amherst Department of Resource Economcs Workng Paper No The Optmal Prcng of Polluton When Enforcement s Costly John K. Stranlund 1, Carlos A. Chavez 2 and Maurco G. Vllena 3 Abstract: We consder the prcng of a unformly mxed pollutant when enforcement s costly wth a model of optmal, possbly frm-specfc, emssons taxes and ther enforcement. We argue that optmalty requres an enforcement strategy that nduces full complance by every frm. Ths holds whether or not regulators have complete nformaton about frms abatement costs, the costs of montorng them for complance, or the costs of collectng penaltes from noncomplant frms. Moreover, gnorng several unrealstc specal cases, optmalty requres dscrmnatory emssons taxes except when regulators are unable to observe frms abatement costs, the costs of montorng ndvdual frms, or any frm-specfc characterstc that s known to be jontly dstrbuted wth ether the frms abatement costs or ther montorng costs. In many polluton control settngs, especally those that have been subject to varous forms of envronmental regulaton n the past, regulators are not lkely to be so ll-nformed about ndvdual frms. In these settngs, polces that set or generate a unform polluton prce lke conventonal desgns nvolvng unform taxes and compettve emsson tradng wth freely-allocated or auctoned permts wll not be effcent. Keywords: Complance, Enforcement, Emssons Taxes, Montorng, Asymmetrc Informaton, Uncertanty JEL Classfcaton: L51, Q58 1 John K. Stranlund, Department of Resource Economcs Unversty of Massachusetts, 214 Stockbrdge Hall 80 Campus Center Way, Amherst, MA E: stranlund@resecon.umass.edu P: F: Carlos A. Chavez, Departmento de Economa Unversdad de Concepcon, Concepcon, Chl E: cchavez@udec.cl P: F: Maurco G. Vllena, School of Busness Unversdad Adolfo Ibanez, Santago, Chle E: maurco.vllena@ua.cl P: F:

2 The Optmal Prcng of Polluton When Enforcement s Costly JOHN K. STRANLUND Department of Resource Economcs Unversty of Massachusetts-Amherst CARLOS A. CHÁVEZ Departamento de Economía Unversdad de Concepcón MAURICIO G. VILLENA Escuela de Negocos Unversdad Adolfo Ibáñez Correspondence to: John K. Stranlund Department of Resource Economcs 214 Stockbrdge Hall 80 Campus Center Way Unversty of Massachusetts-Amherst Amherst, MA 01003, USA. Phone: (413) Fax: (413) E-mal: stranlund@resecon.umass.edu. Acknowledgements: We gratefully acknowledge fnancal support provded by Concyt-Chle, under Project Fondecyt No , and Fondecyt Internatonal Cooperaton under Project No Partal support was also provded by the Cooperatve State Research Extenson, Educaton Servce, U. S. Department of Agrculture, Massachusetts Agrcultural Experment Staton, and the Department of Resource Economcs under Project No. MAS Joe Mofftt and John Spraggon provded valuable comments and suggestons. 1

3 The Optmal Prcng of Polluton When Enforcement s Costly Abstract: We consder the prcng of a unformly mxed pollutant when enforcement s costly wth a model of optmal, possbly frm-specfc, emssons taxes and ther enforcement. We argue that optmalty requres an enforcement strategy that nduces full complance by every frm. Ths holds whether or not regulators have complete nformaton about frms abatement costs, the costs of montorng them for complance, or the costs of collectng penaltes from noncomplant frms. Moreover, gnorng several unrealstc specal cases, optmalty requres dscrmnatory emssons taxes except when regulators are unable to observe frms abatement costs, the costs of montorng ndvdual frms, or any frm-specfc characterstc that s known to be jontly dstrbuted wth ether the frms abatement costs or ther montorng costs. In many polluton control settngs, especally those that have been subject to varous forms of envronmental regulaton n the past, regulators are not lkely to be so ll-nformed about ndvdual frms. In these settngs, polces that set or generate a unform polluton prce lke conventonal desgns nvolvng unform taxes and compettve emsson tradng wth freely-allocated or auctoned permts wll not be effcent. Keywords: Complance, Enforcement, Emssons Taxes, Montorng, Asymmetrc Informaton, Uncertanty JEL Codes: L51, Q58. 2

4 The Optmal Prcng of Polluton When Enforcement s Costly 1. Introducton In a frst-best world of envronmental polcy, an optmal tax to control emssons of a unformly mxed pollutant nvolves a unform per unt tax set equal to margnal damage from emssons at the effcent level of aggregate emssons. Alternatvely, a compettve emssons tradng program wth ether freely-allocated or auctoned permts wll generate a unform prce for polluton that s the same as the frst-best tax. In a frst-best world, however, regulatons do not have to be enforced and regulators have complete nformaton about all the benefts and costs of polluton control. These assumptons are always volated n real world applcatons. In ths paper, therefore, we consder the optmal prcng of polluton when complance must be enforced and regulators may have only ncomplete nformaton about frms costs of controllng ther emssons and the costs of regulatory enforcement. The model s cast as the jont determnaton of optmal, possbly frm-specfc, emssons taxes and ther enforcement. We demonstrate two new results wth sgnfcant polcy relevance. Frst, under a constant expected margnal penalty for tax evason and the assumpton that collectng tax revenue from complant frms s cheaper than collectng penaltes from noncomplant frms, the optmal polcy requres suffcent enforcement effort to nduce full complance by all frms. Ths s true even when regulators have only ncomplete nformaton about frms abatement costs and the costs of enforcement. In the theoretcal lterature on complance wth emssons taxes most authors smply assume that full complance s not or can not be acheved (e.g., Harford 1978, 1987; Sandmo 2002, Montero 2002, Cremer and Gahvar 2002, and Macho-Stadler and Perez-Castrllo 2006). Ths s also true n most theoretcal analyses of the complance and enforcement problem 3

5 n emssons tradng schemes (e.g., Malk 1990, Keeler 1991, Stranlund and Dhanda 1999, Montero 2002). 1 Wthout downplayng the relevance of examnng ncentve-based polces when enforcement s not suffcent to nduce full complance, our work suggests that these stuatons nvolve sub-optmal polcy desgns. Our second man result s that, except for unrealstc specal cases, dscrmnatory taxes are optmal except when regulators are unable to observe frms abatement costs, the costs of montorng ndvdual frms for complance, or any frm-specfc characterstc (e.g., nput and output levels, producton and abatement technologes, etc.) that s known to be jontly dstrbuted wth ether the frms abatement costs or ther montorng costs. In ths case, a regulator s unable to dstngush frms from one another and, therefore, s unable to determne dscrmnatory tax rates. However, n many stuatons, perhaps even most stuatons, regulators wll not be so llnformed. In most real settngs, partcularly n developed countres, frms have been subject to some form of polluton control for many years. Consequently, regulators have pror experence wth and knowledge about the frms that can be used to construct an optmal polcy. For example, past experence s lkely to have provded regulators wth nformaton about the costs of montorng dfferent frms. Moreover, past experence may have allowed regulators to determne how observable frm characterstcs lke output, levels and knds of nputs, abatement and producton technologes, etc., are jontly dstrbuted wth ther abatement or montorng costs. We demonstrate that wth ths type of nformaton, even though t may be ncomplete, optmalty requres dscrmnatory polluton prces. 1 Stll others n ths lterature restrct themselves to only full complance outcomes (Malk 1992, Stranlund and Chavez 2000, Chavez and Stranlund 2003). Only Stranlund (2006) consders the optmal desgn of emsson tradng polces wth costly enforcement. He demonstrates that when regulators are fully nformed about frms costs of controllng ther emssons the optmal polcy calls for nducng full complance. 4

6 An mportant consequence of our work s that when regulators have enough nformaton to dstngush frms from one another n a meanngful way, any emssons control polcy that sets or generates a unform prce cannot be optmal. In partcular, conventonal desgns nvolvng unform tax rates as well as compettve emssons tradng wth freely-allocated or auctoned permts cannot be fully effcent. Whle our result that enforcement costs wll typcally call for dscrmnatory polluton prces s not wdely known, we do not clam that t s entrely new. 2 Cremer and Gahvar (2002), examne the optmal desgn of an emssons tax polcy for homogeneous frms under the assumptons of costly enforcement and that tax revenue s used to offset other taxes n an economy. Whle they obvously obtan an optmal unform tax for a partcular ndustry because of ther assumpton of dentcal frms wthn the ndustry, they also recognze that the tax rate wll be dfferent n other ndustres n part because of dfferences n margnal enforcement costs and abatement costs. One could easly use ther results to argue that dscrmnatory taxes are lkely to be optmal n an ndustry composed of heterogeneous frms. Our work dffers from Cremer and Gahvar s n several mportant ways. Frst, whle they determne optmal taxes and ther enforcement jontly as we do, they lmt ther analyss to polces that generate postve volatons by all frms. In contrast, we argue that the optmal polcy should nduce full complance, whch suggests that Cremer and Gahvar lmted themselves to polces that are, n fact, sub-optmal. The second mportant dfference between our work and Cremer and Gahvar s s that they assume complete nformaton about frms abatement and enforcement costs throughout ther work, whle we nvestgate the extent to whch 2 Beyond the control of unformly mxed pollutants from pont sources, whch s the settng for ths work as well as all of the lterature we dscuss, t s well known that dscrmnatory emssons taxes are optmal when pollutants are spatally dfferentated (see Xepapadeas (1997), chapter 2 secton 8 for references, ncludng zonal taxes) and when emssons are generated by nonpont sources (e.g., Segerson 1988). 5

7 dscrmnatory taxes reman optmal when regulators have ncomplete nformaton about abatement and enforcement costs. 3 Fnally, Cremer and Gahvar (2002) also nvestgate how recyclng emssons tax revenue to reduce other taxes n an economy affects optmal emssons taxes. In contrast, we assume that tax revenue s not used to offset other taxes n order to focus on the roles of costly enforcement and ncomplete nformaton n determnng optmal polluton prces. Malk (1992) provdes an early hnt that polces that generate a unform polluton prce are lkely to be sub-optmal when one accounts for enforcement costs. Malk models a compettve emssons tradng program under complete nformaton that s enforced to acheve full complance and demonstrates that even though the permt market leads to a dstrbuton of emsson control that mnmzes aggregate abatement costs, t does not mnmze the sum of aggregate abatement and enforcement costs. An mportant dstncton between our work and Malk s s that he s concerned wth the optmal dstrbuton of emssons whle we are concerned wth the optmal dstrbuton of emssons prces. The two approaches are obvously complementary, but our prcng approach s nstructve because t llumnates what we beleve s the fundamental reason for the sub-optmalty of emssons tradng that Malk dentfes; that s, a compettve emssons tradng polcy leads to a unform prce, whle enforcement costs typcally call for dscrmnatory prces. Our model dffers from Malk s n other mportant ways as well. Frst, he smply assumes that regulators devote suffcent enforcement resources to nduce full complance n an emsson 3 In a recent contrbuton, Bontems and Bourgeon (2005) consder optmal envronmental taxes under ncomplete nformaton and costly enforcement. They take a standard revelaton approach that reles on elctng truthful reports by frms of ther types. (See Lews (1996) for a revew of ths approach.) In ther work, a polcy conssts of a typespecfc lump sum tax, an emssons standard, a montorng probablty, and a fne for volatng the standard. Ths s very dfferent from the way envronmental economsts and polcymakers usually thnk of emssons taxes. Emssons taxes are usually per unt taxes, no restrctons are placed on frms emssons, and noncomplance occurs f a frm attempts to evade ts tax lablty by under-reportng ts emssons. 6

8 tradng program, whle we demonstrate that nducng full complance to a tax regulaton s optmal. Second, lke Cremer and Gahvar (2002), Malk (1992) assumes complete nformaton throughout hs work, whle we consder optmal polluton prcng when regulators have only ncomplete nformaton. The rest of the paper s organzed as follows. In the next secton we develop a model of complance behavor under emssons taxes. In secton 3, we lay out the components of the costs of enforcng a polcy of emssons taxes, whch nclude the costs of montorng frms, and the expected costs of collectng penaltes from noncomplant frms. We then demonstrate that an optmal tax polcy must nduce full complance by every frm, even when a regulator has only ncomplete nformaton about frms abatement costs and the costs of enforcement. In secton 4 we use our full-complance result to determne optmal emssons taxes. The optmal desgn wll typcally nvolve dscrmnatory taxes except when a regulator has rather poor nformaton about frms abatement and montorng costs. In fact, we specfy the exact lmt on the qualty of the regulator s nformaton at whch t becomes optmal to set a unform tax. We dscuss ths nterpretaton as well as other mplcatons of our results n secton 5, and conclude n secton Complance behavor under emssons taxes The regulatory model of ths paper s the standard one n whch a regulator frst commts tself to a tax polcy and ts enforcement and then the regulated frms react to ths polcy wth ther choces of emssons and complance. In ths secton we determne the frms responses to any tax/enforcement polcy. Throughout consder a fxed set N of heterogeneous rsk-neutral frms. These frms may or may not belong to the same ndustry, but each emts the same unformly mxed pollutant. A 7

9 summary of the costs of all the methods frm can use to reduce ts emssons s gven by ts abatement cost functon, C (q ), whch s strctly decreasng and strctly convex n ts emssons q. 4 The frm s emssons are taxed at rate t. The frm s requred to submt a report of ts emssons, r, and t s noncomplant f t attempts to evade some part of ts tax lablty by reportng r <. q The regulator cannot observe the frm s emssons wthout a costly audt. Let π denote the probablty that the regulator s able to determne s complance status. Lke most other authors, we assume that montorng produces a measure of emssons that s accurate enough to judge a frm s complance status wthout error. The detecton probablty s common knowledge and the regulator commts tself to t at the outset. If montorng reveals that has under-reported ts emssons, t faces a unt penalty of φ on q r > 0. 5 We assume that φ > t throughout. Ths allows full complance to be a possble outcome throughout the paper. No upper bound s placed on φ, because none of our results depend on settng arbtrarly hgh penaltes. To smplfy our analyss we restrct t to polces that motvate all frms to reduce ther emssons below what they would chose n the absence of any sort of regulatory control, but that do not cause any frm to choose zero emssons. Under these polces frm chooses t emssons and emssons report to solve: mn ( q, r ) C ( q ) + tr s. t. q r 0, r 0. + π φ( q r ) [1] 4 A frm s abatement cost s the reducton n ts proft that comes from reducng ts emssons below what t would choose n the absence of an nducement to control ts emssons (Montgomery 1972). 5 Our assumpton of a lnear penalty functon s not common n the lterature. Most authors assume that the penalty functon, or at least the expected penalty functon, s strctly convex (e.g., Harford 1978, 1987; Sandmo 2002, Cremer and Gahvar 2002, and Macho-Stadler and Perez-Castrllo 2006). We brefly address the use of lnear penaltes vs. convex penaltes n the next secton. 8

10 Restrctng the frm to q r 0 follows from the fact that a frm wll never have an ncentve to report that ts emssons are hgher than they really are. 6 Let L denote the Lagrange equaton for [1] and let λ denote the multpler attached to the constrant q r 0. Then, the followng frst-order condtons are both necessary and suffcent to determne the frm s optmal choces of emssons and emssons report: L = C ( q ) + π φ λ = 0; [2] q L = t π φ+ λ 0, r 0, r( t πφ+ β ) = 0; [3] r L λ = ( q r) 0, λ 0, λ ( q r) = 0. [4] Makng the common assumpton that a frm wll comply f t s ndfferent between complance and noncomplance, [3] reveals that a frm s optmal emssons report s: r q f = 0 f t t π φ > π φ. [5] Thus, the frm provdes a truthful report of ts emssons when the tax does not exceed the expected margnal penalty. When the tax does exceed the expected margnal penalty, t s cheaper for the frm to report zero emssons and face the expected penalty than to pay the tax. When t φ so that the frm s motvated to report ts emssons truthfully, [3] becomes π t = π φ λ. Combnng ths wth [2] yelds the famlar result that the frm chooses ts emssons to equate ts margnal abatement cost to the tax, that s, C ( q ) + t = 0. However, when t > φ and the frm under-reports ts emssons, [4] ndcates that λ = 0, and [2] becomes C ( q ) + π φ = 0, ndcatng that a noncomplant frms chooses ts emssons to equate ts π 6 If q < r, then the frm could reduce ts abatement costs by allowng ts emssons to ncrease to r wthout ncurrng addtonal costs. 9

11 margnal abatement cost to the expected margnal penalty t faces. Thus, a frm s optmal choce of emssons s: q ( t ) C ( q ) + t = 0, f t π φ q = [6] q ( π φ) C ( q ) + π φ = 0, f t > π φ. Snce we assume that an optmal tax/enforcement polcy s desgned under ncomplete nformaton about frms abatement costs, t s worthwhle at ths pont to specfy what a regulator can and cannot know about frm behavor. Although a regulator knows how frms wll choose ther emssons n response to a tax and enforcement strategy, t cannot know exactly what a frm s emssons wll be because t lacks complete nformaton about ts abatement costs. Moreover, f the regulator chooses not to nduce complance by a frm, t s uncertan about the extent of the frm s volaton because ths depends on the frm s choce of emssons. However, snce the regulator controls the varables that fully determne a frm s complance status;.e., the tax and the expected margnal penalty, the regulator does know whether a frm wll be complant or not. 3. The costs of enforcng emssons taxes and the optmalty of full complance The regulatory objectve of ths paper s to choose a system of ndvdualzed emssons taxes and enforcement strateges to mnmze the expected socal costs of the regulaton. These costs nclude the regulator s expectatons of aggregate abatement costs, polluton damage, and enforcement costs. In ths secton we focus on expected enforcement costs, whch consst of the regulator s expectatons of aggregate montorng costs and the costs of collectng penaltes from noncomplant frms. Under the assumpton that collectng tax revenue from complant frms s cheaper than collectng penaltes from noncomplant frms, we state and prove a sort of cost- 10

12 effectve enforcement result. More specfcally, we ask how a tax/enforcement polcy should be desgned to mnmze the expected enforcement costs of nducng an arbtrary, fxed, and lkely unknown set of ndvdual emssons, and show that a polcy that mnmzes expected enforcement costs requres that each ndvdual frm s tax be equal to the expected margnal penalty t faces. Consequently, all frms are fully complant. In the next secton we ncorporate ths cost-effectve enforcement result nto the determnaton of taxes that result n the optmal dstrbuton of ndvdual emssons. The unt penalty for evaded taxes s fxed throughout ths paper, so the regulator s polcy nstruments are ndvdual taxes and detecton probabltes, ( t, π ), = 1,, n, whch nduce emssons reports and emssons, (r, q ), = 1,, n, accordng to [5] and [6], respectvely. Assume that the actual cost of montorng frm s the ncreasng and convex functon m ( π ), and aggregate montorng costs are m ( π ). The regulator does not have complete N nformaton about montorng costs because t s not certan of the parameters of, = 1,, n. We allow the costs of montorng the frms to vary to reflect the possblty that the regulator wll fnd t harder to determne the complance status of some frms than others. 7 The lnearty of aggregate montorng costs n ndvdual montorng costs s not essental. To obtan the results of ths paper t s suffcent to assume that aggregate montorng costs are ncreasng and convex n the ndvdual detecton probabltes. We assume throughout that tax revenue and revenue from penalzng noncomplant frms are smple transfers wth no real effects. Despte ths, socety s not ndfferent about collectng them; n partcular, penalzng noncomplant frms may nvolve sgnfcant costs. These nclude m 7 Ths assumpton s closely related to the assumpton that ndvduals vary n ther probabltes of apprehenson, whch was frst analyzed by Bebchuk and Kaplow (1993). Macho-Stadler and Perez-Castrllo (2006) assume heterogeneous probabltes of apprehenson n ther study of enforcng emssons taxes. 11

13 the government s costs of generatng suffcent evdence to get a court to agree wth ther fndng of volaton and the mposton of a penalty. Moreover, accused frms may mount costly challenges to any fndng of volaton and the mposton of a penalty, to whch the government may then respond wth ts own costly efforts to fght off these challenges. 8 On the other hand, a complant frm reports the full extent of ts emssons, and n dong so, essentally admts lablty for these emssons. Wth ths admsson of lablty, the government does not need to generate the evdence that would be necessary to mpose a penalty for noncomplance. Moreover, a frm that admts ts lablty s not lkely to challenge the mposton of the tax. Therefore, we thnk that t s reasonable to assume that mposng and collectng penaltes from noncomplant frms s more costly that collectng taxes from complant frms, and ncorporate ths assumpton nto our model. To do so n a smply way, suppose that collectng emssons taxes s costless, but that s > 0 s the cost of collectng the penalty from frm f t s caught evadng ts tax lablty. Although s may be a functon of the sze of the frm s penalty, our results do not depend on specfyng such a relatonshp. Let the aggregate expected costs of collectng penaltes be lnear n the costs of collectng penaltes from ndvdual frms. If nc N denotes the subset of frms that are noncomplant, the aggregate expected costs of collectng penaltes are π N nc s. As wth aggregate montorng costs, the lnearty of aggregate expected sanctonng costs s not essental. For our results t s suffcent to assume that aggregate expected sanctonng costs are montoncally ncreasng n the costs of sanctonng ndvdual frms. 8 Although t s perfectly reasonable to assume that penalzng frms s costly, none of the work n the lterature on enforcng emssons taxes that we are aware of deals explctly wth the costs of collectng penaltes. In the lterature on enforcng emssons tradng polces, only Stranlund (2006) assumes that mposng and collectng penaltes s costly. Costly sanctons are also not very common n the much larger lterature on optmal law enforcement; however, see Polnsky and Shavell (1992) for an analyss of how costly sanctons affect the determnaton of optmal law enforcement. 12

14 Fnally, lke the cost of montorng ndvdual frms, the regulator need not have complete nformaton about the cost of collectng penaltes from ndvdual frms. 9 Despte our rather weak assumptons about the expected costs of enforcng emssons taxes, we are able to prove the followng proposton concernng the optmal enforcement of these polces. Proposton 1: Consder a tax/montorng polcy, ( t, π ), = 1,, n, and an exogenous constant margnal penalty that satsfes φ > t, = 1,, n. Suppose that frms react to ths polcy wth emssons q, = 1,, n. Ths dstrbuton of emssons s acheved wth mnmum expected aggregate enforcement costs only f t = π φ for each = 1,, n. Wth taxes and montorng set n ths way, each frm s fully complant. Proof: The proof proceeds by showng that any polcy nvolvng t π φ for some can be modfed to reduce enforcement costs wthout changng the dstrbuton of emssons. Frst suppose that t > π φ for some. Then, [5] ndcates that r = 0, and [6] ndcates that q = q ( π φ) ; that s the frm s fully noncomplant. Alternatvely, hold π constant so that aggregate montorng costs do not change, but reduce t so that t = π φ. The frm wll then choose r = ( φ) so that t s now complant, but t does not change t emssons because q π q π φ) = q ( t ). Moreover, changng n ths way does not affect the decsons of any of the ( t 9 Any enforcement strategy s lkely to nvolve fxed montorng costs and sanctonng costs, whch we do not model. Addng these fxed costs does not change any of the results of our work as long as they are not so hgh that t s optmal to forego regulaton altogether. 13

15 other frms. However, reducng t to t = π φ elmnates the expected costs of penalzng the frm; hence, aggregate expected enforcement costs are reduced. Now suppose that t < π φ for some. In ths case, [5] and [6] reveal that r = q t ) ; that ( s, the frm s complant. However, f π s reduced so that t = π φ, the frm does not change ts choce of emssons and t remans complant. Ths change n π does not affect the decsons of the other frms, but aggregate expected enforcement costs are reduced because expected montorng costs decrease. QED. It s mportant to note that the proposton holds even f a regulator s uncertan about frms abatement costs. Of course, uncertanty about abatement costs mples that the regulator cannot be certan about the dstrbuton of ndvdual emssons that wll result from a partcular polcy, ( t, π ), = 1,, n, but the proposton does not depend on knowng how ndvdual emssons are dstrbuted. Whatever q, = 1,, n, results from a partcular polcy, the expected enforcement costs of holdng the frms to ths dstrbuton of emssons are mnmzed by choosng t = π φ, = 1,, n. Moreover, the proposton holds even f the regulator s uncertan about the parameters of enforcement costs. All the regulator has to know s that expected aggregate montorng costs are ncreasng n ndvdual montorng levels, and expected aggregate sanctonng costs are ncreasng n the costs of penalzng ndvdual frms. Because the strategy t = π φ, = 1,, n, mnmzes the expected enforcement costs of nducng a fxed set of ndvdual emssons, we shall ncorporate ths strategy nto our determnaton of optmal taxes and ther enforcement n the next secton. 14

16 That ths strategy nduces full complance by each frm s a new result wth sgnfcant relevance for the determnaton of optmal emssons taxes. As noted n the ntroducton, the related lterature has not dealt squarely wth the possblty that nducng full complance may be a component of an optmal tax polcy. For example, Montero (2002), n hs study of prce vs. quantty regulaton wth costly enforcement, explctly assumes that montorng costs are large enough and penaltes are restrcted enough so that full complance s not socally optmal (page 439). Although the unt penalty s not restrcted n our work, all that s requred s that t exceed each frm s tax rate. Gven ths assumpton, optmalty requres full complance regardless of montorng costs. To be sure, montorng costs wll affect the optmal tax rates, but they do not affect the decson to nduce full complance. Smlarly, but not as explctly, Cremer and Gahvar s (2002) work s clearly focused on optmal taxes and ther enforcement n stuatons nvolvng less than full complance. They smply sdestep the possblty that optmalty may call for full complance by gnorng the natural zero-boundary constrant on frms unreported emssons. Instead they assume that frms choces of unreported emssons are always nteror choces (see ther frst-order condton for an emssons report, (8a) on page 391). Our proposton suggests that Cremer and Gahvar may have lmted themselves unnecessarly to suboptmal enforcement strateges. Macho-Stadler and Perez-Castrllo (2006) also focus solely on mperfect complance by assumng a constant emssons tax and the regulatory objectve of mnmzng aggregate emssons wth a fxed enforcement budget that s not suffcent to nduce full complance. 10 Whle lmted enforcement resources are certanly a factor n many real nstances of 10 Garve and Keeler (1994) examne the choces of a budget-constraned enforcer under emssons standards, whle Stranlund and Dhanda (1999) do the same n the context of compettve emssons tradng. 15

17 envronmental polcy enforcement, our proposton suggests that n desgnng a truly optmal tax polcy regulators should allocate suffcent enforcement resources to acheve full complance. The focus on postve volaton choces by Cremer and Gahvar (2002), Macho-Stadler and Perez-Castrllo (2006), and several others (e.g., Harford (1978 and 1987) and Sandmo (2002)), s accomplshed, n part, wth the assumpton that expected penalty functons are strctly convex. We know of no emprcal justfcaton for ths assumpton, nor are we aware of any effcency reason to prefer strctly convex expected penalty functons for tax evason to lnear expected penalty functons. 11 Moreover, expected penaltes are strctly convex ether because montorng probabltes or penaltes are strctly convex n unreported emssons. The frst case nvolves a questonable assumpton because a regulator s montorng actvty would then depend on actual volatons: but, of course, a frm s volaton cannot be known untl t s montored. If montorng probabltes are ndependent of unreported emssons, then strct convexty of the expected penalty functon comes from margnal penaltes that are ncreasng n volaton levels. We are not aware of any tax polcy that specfes an ncreasng margnal penalty. On the other hand, a constant margnal penalty s a very natural choce of penalty because the margnal ncentve to evade the tax s the constant tax rate. Our readng of the lterature suggests that the prmary justfcaton for strctly convex expected penaltes s to allow analysts to focus on stuatons nvolvng postve volatons. In contrast, we assume that the expected margnal penalty functon s lnear, whch forces us to take the boundary condton of zero volatons serously, and ultmately leads us to the optmalty of nducng full complance. Whle our choce to model constant expected margnal penaltes s one reason we fnd that nducng full complance s optmal, t s not the only reason. Our 11 Although, for a compettve emssons tradng program that seeks to acheve an aggregate emssons target costeffectvely, Stranlund (2006) demonstrates that a lnear penalty and montorng to nduce full complance s preferred to a strctly convex penalty and montorng that allows noncomplance. 16

18 assumpton that t s costly to collect penaltes from noncomplant frms s crucal. In fact, the fundamental value of nducng full complance s to avod these costs. 4. Optmal emsson taxes under ncomplete nformaton Havng characterzed an optmal enforcement strategy, we ncorporate ths strategy nto the determnaton of optmal taxes when a regulator has only ncomplete nformaton about frms abatement costs and the costs of montorng them for complance. The focus of our analyss s on whether an optmal tax polcy nvolves dscrmnatory taxes or whether a regulator should set a sngle tax that apples unformly to all frms. Clearly, f a regulator s nformaton about ndvdual frms s so poor that t s unable to dstngush them from one another n a meanngful way, t has no bass for choosng dscrmnatory taxes. Whle such poor nformaton may be characterstc of some polluton control settngs, t certanly s not a unversal feature. In fact, we suspect that n many stuatons, partcularly those n whch frms have been subject to control polces n the past, regulators can probably observe ndvdual frm characterstcs, lke output, nputs, abatement equpment, and producton technologes, that provde some nformaton about ther unknown abatement and montorng cost parameters. For example, past experence may have provded regulators wth a great deal of nformaton about the dffculty of montorng the emssons from dfferent frms. Perhaps they know that larger facltes wth more ponts of dscharge are harder to montor than smaller facltes. Regulators may have also learned that certan knds of abatement and producton technologes make montorng more or less dffcult than other technologes. Past experence may have also allowed regulators to derve estmates of the parameters of frms abatement costs as functons of observable frm characterstcs. 17

19 Consequently, we let x denote a vector of observable characterstcs of frm, and assume that a regulator knows that these characterstcs are jontly dstrbuted wth the parameters of the frms abatement and montorng costs n the populaton of regulated frms. Knowledge of the jont dstrbuton of the frms cost parameters and at least some of ther observable characterstcs allows the regulator to form a jont probablty densty functon of the unknown cost parameters for each frm that s condtoned on the frm s observable characterstcs. The regulator can then use these condtonal dstrbutons to determne optmal ndvdual tax rates and ther enforcement. Ths polcy s determned to maxmze the regulator s condtonal expectaton of the sum of aggregate abatement costs, aggregate montorng costs, and the damage from the frms emssons. Frst consder expected abatement costs. Let the condtonal expectaton of frm s abatement cost functon be E( C( q) x ), where E denotes the expectaton operator throughout. Recall from Proposton 1 that t s optmal to set a frm s tax equal to ts expected margnal penalty ( t = π φ) so that t s complant, and from [6], a frm wll then choose ts emssons so that ts margnal abatement cost s equal to the tax rate. Therefore, the regulator s condtonal expectaton of the frm s choce of emssons under the tax t s ( ( ) ) ( ( ) ) E q t x = q E C q x = t. [7] Usng [7], the regulator can specfy ts condtonal expectaton of the frm s abatement costs n terms of the tax; ( ( ) ) ( ( ( )) ) E C t x = E C q t x. [8] Then, gven the regulator s observatons of all x, = 1,, n, ts condtonal expectaton of the aggregate abatement costs generated by ts tax polcy t, = 1,, n, s 18

20 E( C( t) x). [9] From here on, summatons are over all regulated frms. The margnal effect of changng a sngle tax, t, on the regulator s condtonal expectaton of aggregate abatement costs s smply the margnal effect on the regulator s expectaton of the abatement costs of that frm. From [8] ths effect s E( C ( q) q ( t )) x ) knows the frm wll choose ts emssons so that C ( q ) = t, ( ( ) x ) de C t dt ( x). Snce the regulator = te q( t)) > 0. [10] The sgn of [10] follows because the regulator s condtonal expectaton of the frm s margnal emssons response to a change n ts tax s negatve. That s, from [7], ( ) ( ) E q ( t ) x = E 1/ C ( q ) x < 0. [11] Lke the frms abatement costs the regulator s uncertan about the costs of montorng ndvdual frms for complance, but t can evaluate the condtonal expectatons of the frms montorng costs. For frm ths s E( m ( ) x ) π. Snce optmal enforcement requres t = π φ, the regulator s expectaton of the cost of montorng n terms of the tax s ( ( / φ) x ) E m t, and ts expectaton of how the cost of montorng changes wth t s ( ( / φ) ) ( ( π ) ) de m t x E m x = > 0. [12] dt φ The regulator s expectaton of the montorng cost of mantanng complance by the frm ncreases wth ts tax to offset the frm s ncreased ncentve to evade a hgher tax. Gven the regulator s observatons of x, = 1,, n, ts condtonal expectaton of aggregate montorng costs n terms of ts tax polcy t, = 1,, n, s 19

21 E( m( t / φ) x). [13] The margnal affect on the regulator s condtonal expectaton of aggregate montorng costs of a change n a sngle tax, t, s gven by [12]. Lastly, the regulator must evaluate how ts polcy affects ts condtonal expectaton of polluton damage. We assume that the frms emssons are unformly mxed so damage s an ncreasng and convex functon, D, of aggregate emssons. For smplcty, suppose that the regulator knows the damage functon; however, t remans uncertan about how much damage ts polcy wll produce, because t s uncertan about how each frm wll respond to ts tax. It does, however, know the condtonal expectaton of how each frm wll respond, E q ( t ) x, = 1,, n, so polluton damage at the regulator s condtonal expectaton of aggregate emssons s, D( E( q( t) x) ) ( ). [14] The regulator s condtonal expectaton of the change n damage from a change n the tax on frm s ( ) ( ) D E( q ( t ) x ) E q ( t ) x < 0. [15] We now have all the components of socal costs. Let Wt ( 1,, t x1,, x n ) denote the regulator s expectaton of socal costs n terms of a polcy of well-enforced ndvdual emssons taxes, t, = 1,, n, condtonal on ts observatons of x, = 1,, n. Note that n Wt (,, t x,, x n ) s the sum of [9], [13], and [14]; that s, 1 n 1 D( E( q( t) x) ). ( ) ( φ ) Wt ( 1,, tn x1,, xn) = ECt ( ) x + Emt ( / ) x + [16] Suppose that Wt ( 1,, t x1,, x n ) s strctly convex n t, = 1,, n, for all possble realzatons n of x, = 1,, n, and that an optmal polcy nvolves strctly postve ndvdual taxes. Then, the 20

22 optmal tax rates are determned by W / t = 0, = 1,, n. Snce W / t s equal to the sum of [10], [12], and [15], the frst-order condtons for the optmal taxes can be wrtten as: ( ( π ) x ) E m te( q ( t)) x) + = D ( E( q( t) x) ) E( q ( t) x), = 1,, n. [17] φ The left-hand sde of [17] for a partcular frm s the regulator s expectaton of the margnal costs of ncreasng the frm s tax, whch conssts of the mpact of the ncreased tax on the regulator s expectaton of the frm s abatement costs and on ts expectaton of the costs of montorng the frm. The rght hand-sde s the regulator s expectaton of the margnal reducton n aggregate damage from ncreasng s tax. Now, take any two frms, and j, and use [17] to calculate ( ( π ) x) ( ) ( j( π j) xj) ( ) 1 E m E m t tj = φ E q ( t) x E q j( tj) xj. [18] Our last proposton follows drectly from [18], so t s offered wthout formal proof. Proposton 2: An optmal polcy of emssons taxes under ncomplete nformaton about frms abatement and montorng costs nvolves dscrmnatory taxes f and only f ( ( π ) ) ( ( ) ) E( m j( π j) xj) E( q j( tj) xj) E m x E q t x for some and j. 5. Dscusson of Proposton 2 Proposton 2 reveals the condton under whch enforcement costs nduce optmal dscrmnatory taxes and, n dong so, helps us specfy condtons that would lead to a unform tax. Several of the condtons that nduce a sngle tax rate are qute unrealstc. For example a unform tax wll be optmal f the frms margnal montorng costs and ther margnal emssons responses to taxaton are the same. Ths s true as well f the frms observable characterstcs are all the same. 21

23 Another unrealstc possblty s that the frms margnal montorng costs are all zero. In ths case, [17] reduces to t D ( E( q( t) x) = ), whch s the famlar result that optmalty requres a unform tax that s set equal to margnal damage at the effcent level of the regulator s expectaton of aggregate emssons. 12 However, t seems to us that the most relevant condton under whch a unform tax s optmal s when a regulator s unable to observe any frm characterstc that t knows to be jontly dstrbuted wth the parameters of the frms abatement and montorng costs. In ths case the regulator s expectatons of the frms margnal montorng costs and emssons responses are the same, because t has no way to dstngush the frms from one another. Thus, we would argue that the fundamental justfcaton for settng a unform tax to control a unformly mxed pollutant s that a regulator has very poor nformaton about ndvdual frms. 13 As we ve already noted, we do not beleve regulators wll be so ll-nformed n many real settngs. When a regulator has somewhat better, but stll ncomplete, nformaton about ndvdual frms, [18] ndcates that ther tax rates wll vary except n unrealstc specal cases. Admttedly, the number of dstnct optmal tax rates may be small f a regulator has only coarse nformaton about ndvdual frms. For example, suppose that a control stuaton nvolves the frms from a number of ndustres, and that the regulator knows somethng about how montorng or abatement costs dffer across the ndustres, but s unable to dstngush frms wthn partcular ndustres. In ths case, the number of dstnct tax rates may smply be equal to the number of 12 Wth [17] t s straghtforward to demonstrate that costly enforcement generally mples that ndvdual tax rates are below margnal damage. Consequently, ndvdual and aggregate emssons are hgher than f enforcement was costless. 13 We recognze, however, that even when regulators have suffcent nformaton to mpose dscrmnatory taxes, other consderatons may lmt ther ablty to do so. Legal prohbtons aganst dscrmnatory taxaton may prevent regulators from mplementng a fully optmal tax polcy. Even f dscrmnatory taxes are lawful, they may not be poltcally feasble. In all lkelhood some frms wll perceve dscrmnatory taxes as unfar and lobby aganst ther use. Successful lobbyng efforts could lead to a unform tax when dscrmnatory taxes are approprate. 22

24 ndustres nvolved. Or, magne a control settng nvolvng the emssons of the frms n a sngle ndustry that use only a small number of dstnct abatement technologes to control ther emssons. If ths pece of nformaton s the only characterstc that a regulator knows s jontly dstrbuted wth the frms abatement or montorng costs, the number of dstnct tax rates may be equal to the number of avalable control technologes. Dependng on the degree of heterogenety n the populaton of regulated frms, more detaled nformaton about each of them may lead to a greater number of dstnct tax rates. When a regulator can dstngush frms from others, ther tax rates wll vary wth the varaton n E( m ( π ) x ) E( q ( t ) x ), = 1,, n. Recall from [11] that E( q ( t ) x ) = E(1/ C ( q ) x )< 0. Therefore, ndvdual tax rates vary f there s varaton n the regulator s condtonal expectatons of the frms margnal montorng costs, or f there s varaton n ts expectatons of the recprocal of the slopes of ther margnal abatement costs, or both. It s partcularly noteworthy that optmal emssons taxes wll lkely vary across frms even f the margnal cost of montorng each of them for complance s the same. Therefore, smply recognzng that montorng s costly wll often be suffcent justfcaton for mposng dscrmnatory taxes. Snce E( q ( t ) x ) < 0 and E( m ( π ) x ) > 0, ther rato s negatve. From [18], then, t < E( mj( π j) xj) E( qj( tj) xj) < t j f and only f E( m ( π ) x ) E( q ( t ) x ) ( π ) ( ). For frm, E m ( ) x E q ( t ) x reflects the regulator s expectaton of the margnal montorng costs assocated wth nducng lower emssons from the frm wth a well-enforced tax. Inducng a margnal decrease n the emssons of a frm wth a more steeply sloped margnal abatement cost curve (.e., hgher C ( q ) ) requres a relatvely greater ncrease n ts tax and, consequently, a 23

25 relatvely greater ncrease n montorng to mantan the frm s complance. Therefore, to conserve montorng costs, optmal taxes wll tend to be lower for frms that the regulator expects have relatvely steep margnal abatement cost functons. For the same reason, optmal tax rates wll tend to be lower for frms that the regulator expects are more dffcult to montor, and hence, have hgher margnal montorng cost functons. One of the most mportant mplcatons of Proposton 2 s that, when regulators have suffcent nformaton to set dscrmnatory taxes, any polcy that sets or generates a sngle prce for polluton cannot be fully effcent. These polces nclude conventonal desgns nvolvng unform emssons taxes, or transferable permt programs wth freely-gven or auctoned permts. But t s the sngle polluton prce that drves much of our understandng of prce-based control and that leads to the most mportant reason for desgnng and mplementng prce-based polces. That reason s the ablty of these polces to nduce a dstrbuton of ndvdual emssons control that mnmzes the aggregate abatement costs of achevng some aggregate emssons target, even when a target cannot be guaranteed because of ncomplete nformaton. A sngle polluton prce motvates frms to choose emssons so that ther margnal abatement costs are equal to the prce. Snce they all face the same prce, all of ther margnal abatement costs are equal n equlbrum, whch forms the set of necessary condtons for mnmzng aggregate abatement costs. Clearly ths result requres a unform polluton prce, and just as clearly, when t s optmal to set dscrmnatory prces, aggregate abatement costs wll not be mnmzed. Usng our approach t s easy to demonstrate that achevng an aggregate emssons target (at least n expectaton) wth lowest possble expected aggregate abatement costs plus expected enforcement costs cannot be acheved wth a sngle prce mnmzng ths sum requres dscrmnatory prces. 24

26 6. Concluson We have examned the optmal prcng of a unformly mxed pollutant when enforcement s costly and regulators have ncomplete nformaton about frms abatement costs and the costs of enforcement, and have demonstrated two new results. Frst, an optmal polcy calls on regulators to devote suffcent enforcement resources to nduce full complance by all frms. Ths s true regardless of the qualty of nformaton regulators have about ndvdual frms abatement costs and the costs of enforcement. Second, enforcement costs wll typcally nduce dscrmnatory polluton prces, except when regulators have very poor nformaton about ndvdual frms. More specfcally, unrealstc specal cases asde, a polcy that sets or generates a unform prce wll be optmal only when regulators are unable to use observable frm characterstcs to form condtonal expectatons of ndvdual frm s abatement costs and the costs of montorng them for complance. Although there are certanly polluton control settngs n whch regulators have such poor nformaton, there are many others, partcularly those that have been subject to envronmental controls n the past, n whch regulators are lkely to have some nformaton that allows them to dstngush frms from one another. In these cases, unform prcng of polluton cannot be effcent. There are, of course, several ways to extend the results of our work. For example, gven the poltcal appeal of emssons tradng polces, t s probably worth explorng whether tradng schemes can be modfed n straghtforward ways to generate the correct frm-specfc prces. Modfcatons to these polces to nduce the correct prces may nvolve permt tradng ratos among frms that dffer from the one-for-one tradng schemes that characterze conventonal emssons tradng. Another opton mght be to combne one-for-one permt tradng wth frmspecfc taxes to account for dfferences n the optmal polluton prces. 25

27 Reexamnng our results n more dffcult control stuatons may also be worthwhle. In ths work we have focused on the control of unformly mxed pollutants, largely because ths s the settng that drves much of what we understand about prce-based polluton control. Moreover, t s the one settng n whch conventonal wsdom calls for settng a unform emssons tax, or mplementng transferable or auctoned permt schemes that generate a unform permt prce. Of course, n contrast to what we perceve to be conventonal wsdom, we ve argued that the presence of enforcement costs wll typcally call for dscrmnatory prcng of a unformly mxed pollutant. There are other control stuatons that call for dscrmnatory prcng, even under the assumpton of costless enforcement. These nclude nonpont polluton problems and problems nvolvng spatally-dfferentated pollutants. In these cases t s the nature of the pollutant that leads to dscrmnatory prcng. Addng the enforcement dmenson to these control problems wll lkely generate two motvatons for dscrmnatory prcng. How enforcement costs and the nature of the pollutant nteract to determne optmal polluton prces s lkely to be a frutful area for future research. 26

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