The Optimal Pricing of Pollution When Enforcement is Costly

Transcription

1 Unpolshed revson, October 2007 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: ( Fax: ( E-mal: stranlund@resecon.umass.edu. Acknowledgements: We gratefully acknowledge fnancal support from 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.

2 The Optmal Prcng of Polluton When Enforcement s Costly Abstract: We consder the prcng of a unformly mxed pollutant wth a model of optmal, possbly frm-specfc, emssons taxes and ther enforcement. We assume ncomplete nformaton about frms abatement costs, the costs of montorng them for complance, and the costs of mposng sanctons on noncomplant frms. We argue that optmalty requres an enforcement strategy that nduces full complance by every frm, except possbly when a regulator can base the probabltes of detectng and punshng volators on ther actual emssons. Moreover, settng asde several unrealstc specal cases, optmalty requres dscrmnatory emssons taxes except when a regulator s unable to use observable frm-level characterstcs to dstngush ts expectatons of frms abatement costs or the costs of montorng them for complance from other frms. 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 are neffcent. These polces nclude conventonal desgns nvolvng unform taxes and compettve emsson tradng wth freely-allocated or auctoned permts. Keywords: Complance, Enforcement, Emssons Taxes, Montorng, Asymmetrc Informaton, Uncertanty JEL Codes: L51, Q58. 1

3 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 have only ncomplete nformaton about frms costs of controllng ther emssons, the costs of regulatory enforcement, and the damages from polluton. Our model s cast as the jont determnaton of optmal, possbly frm-specfc, emssons taxes and ther enforcement. We demonstrate three new results. The frst two have to do wth the specfcaton of frmspecfc tax/enforcement polces that acheve an uncertan dstrbuton of ndvdual emssons wth mnmum expected enforcement costs. In ths sense we seek tax/enforcement polces that are cost-effectve. We assume throughout that t s more costly to collect emssons taxes from complant frms than to collect penaltes from noncomplant frms. Our frst result s that under a constant expected margnal penalty for tax evason, a cost-effectve tax/enforcement polcy requres suffcent enforcement effort to nduce full complance by all frms. 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 Ths s also true n most theoretcal analyses of the complance and enforcement problem n emssons tradng schemes (e.g., Malk 1990, Keeler 1991, Stranlund and Dhanda 1999, Montero Wthout downplayng the 1 Stll others n ths lterature restrct themselves to only full complance outcomes (Malk 1992, Stranlund and Chavez 2000, Chavez and Stranlund Only Stranlund (2007 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, to reach an exogenous aggregate emssons target the optmal polcy calls for nducng full complance. Under ncomplete nformaton about frm s abatement costs, the full-complance result contnues to hold f a constant margnal penalty for volatons s a constant multple of the prce of emssons permts. 2

4 relevance of examnng ncentve-based polces when enforcement s not suffcent to nduce full complance, our work suggests that these stuatons may nvolve sub-optmal polcy desgns. However, our assumpton of a constant margnal penalty for tax evason s not common n the theoretcal lterature on complance wth ncentve-based polces. Most authors assume expected penaltes that are some combnaton of strctly convex penalty functons, and probabltes of detectng volatons that may depend on a frms emssons reports, the regulator s expectaton of ther emssons, or on ther actual emssons. Consequently, to examne the robustness of our full-complance result we ask whether t s possble to reduce the expected enforcement costs of achevng an uncertan dstrbuton of ndvdual emssons wth a nonconstant expected margnal penalty that results n some level of tax evason. We fnd that a regulator cannot use noncomplance along wth any combnaton of frms emssons reports, ts expectatons of ther emssons, or strctly convex penalty functons to form an effcent tax/enforcement polcy all such strateges result n hgher expected enforcement costs than nducng full complance wth a constant expected margnal penalty. However, montorng effort can be reduced f the probabltes of detectng and punshng frms volatons can be based on ther actual emssons. Whether such a strategy can reduce expected enforcement costs depends on the tradeoff between reduced montorng costs and the expected sanctonng costs that are produced by postve levels of tax evason. Our thrd result comes from determnng optmal emssons taxes under the assumpton that t s not possble to mprove on an enforcement strategy of nducng full complance. Settng asde several unrealstc specal cases, we show that dscrmnatory taxes are optmal except when regulators are unable to use observable frm-specfc characterstcs to dstngush ts expectatons of at least some frms abatement costs or ther montorng costs from at least some other frms. In ths case, a regulator has such poor nformaton about ndvdual frms that t cannot dstngush them from one another n a useful way. Whle such poor nformaton s certanly characterstc of many polluton control settngs, regulators wll not be so ll-nformed n other stuatons. 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. Ths experence s lkely to have provded regulators wth nformaton about the costs of montorng dfferent frms, and may have allowed regulators to determne how observable frm characterstcs lke output, 3

5 levels and knds of nputs, abatement and producton technologes, etc., are jontly dstrbuted wth ther abatement or montorng costs. We demonstrate that optmalty requres dscrmnatory polluton prces wth ths type of nformaton, even though t s lkely to be ncomplete. An mportant consequence of ths result 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, the polces that drve our conventonal wsdom about the value of ncentve-based polces, lke those nvolvng Pgovan taxes and compettve emssons tradng, are actually suboptmal polces. Whle our result that enforcement costs wll often 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 wll often be optmal to nduce full complance. 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 dscrmnatory taxes reman optmal when regulators have ncomplete nformaton about abatement and enforcement costs. 3 Fnally, Cremer and Gahvar (2002 also nvestgate how recyclng 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 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 4

6 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 neffcent 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 assumes that regulators devote suffcent enforcement resources to nduce full complance n an emsson tradng program, whle we demonstrate that nducng full complance to a tax regulaton s often optmal. In fact, full complance s actually optmal under the assumptons that Malk makes about enforcement strateges. Second, lke Cremer and Gahvar (2002, Malk assumes complete nformaton throughout hs work, whle we consder optmal polluton prcng under 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 and constant expected margnal penaltes. In secton 3, we lay out the components of the costs of enforcng a polcy of emssons taxes, and demonstrate that an optmal tax polcy wth constant expected margnal penaltes must nduce full complance by every frm, even when a regulator has ncomplete nformaton about frms abatement costs and the costs of enforcement. In secton 4, we examne the robustness of our full complance result by askng whether a regulator can combne noncomplance and more 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. 5

7 sophstcated enforcement strateges to reduce expected enforcement costs. In secton 5 we use our full-complance enforcement strategy to determne optmal emssons taxes under ths strategy. The optmal desgn wll typcally nvolve dscrmnatory taxes except when a regulator has very 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 ths result n secton 6, 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 communcates all the elements of the polcy to the regulated frms. The frms then make ther optmal choces of emssons and complance. In ths secton we examne these choces. Throughout consder a fxed set N of n heterogeneous rsk-neutral frms. These frms may or may not belong to the same ndustry, but each emts the same unformly mxed pollutant. A 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. 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 determne the frm s complance status wthout a costly audt. Let π denote the probablty that the regulator s able to make ths determnaton. 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 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. 4 The unt penalty may vary among frms to allow for the possblty that ths may be part of an effcent polcy. However, we assume that t 4 Obvously, under ths specfcaton the expected penalty a frm faces s lnear. Ths assumpton s not common n the lterature on complance wth emssons taxes or tradable emsson permts. We address the value of enforcement strateges that produce alternatve forms of the expected penalty functon n secton 4. 6

8 s restrcted to be no more than a maxmal value φ, whch does not vary across frms. We also assume that φ > t throughout. Ths s a natural assumpton because the penalty can be nterpreted as recoverng evaded taxes plus a puntve element of φ t per unt of underreported emssons. Perhaps more mportantly, ths assumpton ensures that full complance s a possble outcome throughout the paper. 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. Moreover, we assume that each frm has suffcent assets so that the tax or penalty t pays cannot force t nto bankruptcy. Under these assumptons frm chooses t emssons and emssons report to solve: mn ( q, ( ( r C q + tr + πφ q r [1 st.. q r 0, r 0. 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. 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 + λ = 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: q f t π φ r = [5 0 f t > π φ. 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. At ths stage, some may object to our formulaton of the regulator s enforcement strategy on the grounds that t s mplausble that a regulator would not react wth an automatc audt f t 7

9 receves a report of zero emssons. Whle ths s certanly true, we show n the next secton that t wll never be optmal to set a frm s tax and margnal expected penalty so that t reports zero emssons. Thus, even though a report of zero emssons s possble under our specfcaton, t s only possble under a poorly desgned polcy. When t π φ so that the frm s motvated to report ts emssons truthfully, [3 becomes 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 > π φ and the frm under-reports ts emssons, [4 ndcates that λ = 0. In ths case [2 becomes C ( q + π φ = 0, ndcatng that a noncomplant frms chooses ts emssons to equate ts 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 nclude the regulator s expectatons of aggregate montorng costs and the costs of collectng penaltes from 8

10 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 costeffectve enforcement result. More specfcally, we show 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. The regulator s polcy nstruments are ndvdual taxes, detecton probabltes, and unt penaltes for evaded taxes, ( t, π, φ, = 1,, n, whch nduce emssons reports and emssons, (r, q, = 1,, n, accordng to [5 and [6, respectvely. Let the actual cost of montorng frm be the ncreasng and convex functon m ( π, and let aggregate montorng costs be m ( π. The regulator does not have complete nformaton about montorng costs, N because t s not certan of the parameters of m, = 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. 5 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 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. Accused frms may mount costly challenges to any fndng of volaton and the mposton of a penalty, and the government may respond wth ts own costly efforts to fght off these challenges. 6 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 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. 5 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. 6 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 (2007 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. 9

11 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 nc π s. Lke the cost of montorng ndvdual frms, the regulator need not have N complete nformaton about the cost of collectng penaltes from ndvdual frms. 7 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/enforcement polcy, ( t, π, φ, = 1,, n, wth t < φ φ for each. Suppose that frms react to ths polcy wth emssons q, = 1,, n. Ths dstrbuton of emssons s acheved wth mnmum expected aggregate enforcement costs f and only f t = π φ for each = 1,, n. Wth taxes and montorng set n ths way, each frm s complant. Proof: The proof proceeds by frst 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 = π φ ; that s the frm s fully noncomplant. Alternatvely, hold π constant so that q ( aggregate montorng costs do not change, but reduce t so that t = π φ. The frm wll then choose r = π φ so that t s now complant, but t does not change t emssons because q ( q π φ = q ( t (. Moreover, changng t n ths way does not affect the decsons of any of the 7 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. 10

12 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. Therefore, mnmzng expected aggregate enforcement costs requres t = π φ for each = 1,, n. Clearly, gven the tax rates, montorng of all frms can be mnmzed by settng the unt penaltes as hgh as s allowed whle mantanng the equalty between the tax rates and the expected margnal penaltes. Therefore, mnmzng expected aggregate enforcement costs requres t = π φ for each = 1,, n. QED. ( It s mportant to note that the proposton holds even f a regulator s uncertan about the frms abatement costs. Of course, uncertanty about abatement costs mples that the regulator s uncertan about the dstrbuton of ndvdual emssons that wll result from a partcular polcy, 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. That Proposton 1 calls for enforcng frm-specfc emssons taxes so that each frm s complant 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. In contrast, all we requre s that the unt 11

13 penalty each frm faces exceeds ts 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 lmt ther analyss by assumng 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 they 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. 8 Whle lmted enforcement resources are certanly a factor n many real nstances of envronmental polcy enforcement, Proposton 1 suggests that n desgnng a truly optmal tax polcy regulators should allocate suffcent enforcement resources to acheve full complance. We have made three assumptons that dffer from the rest of the lterature on enforcng ncentve-based envronmental polces, but that are crtcal for our full-complance result. Our assumpton that t s costly to collect penaltes from noncomplant frms s crucal, because the fundamental value of nducng full complance s to avod these costs. Moreover, we have gven the regulator the freedom to choose frm-specfc tax rates. All others assume a unform tax rate that s often fxed (e.g., Macho-Stadler and Perez-Castrllo Fnally, no one else to our knowledge specfes enforcement strateges that produce constant expected margnal penaltes. All other authors n ths lterature assume that enforcement produces expected margnal penaltes that depend on the frms choces of emssons, reports, and evaded taxes. In the next secton we examne the robustness of our full-complance polcy recommendaton under non-constant expected margnal penaltes. 8 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. 12

14 4. On the Robustness of the Optmalty of Full Complance The focus on postve volaton choces by Cremer and Gahvar (2002, Macho-Stadler and Perez-Castrllo (2006, and many others, s accomplshed, n part, wth the assumpton that expected margnal penaltes are functons of the frms choces of emssons, emssons reports, and evaded taxes. For example, Harford (1978 and 1987 and Sandmo (2002 assume strctly convex expected penalty functons that nvolve montorng probabltes that depend on frms emssons and ther emssons report, and penalty functons that are strctly convex n the frms volatons. That s, they assume expected penaltes of the form π ( qr, f( q r, where f s a penalty functon. Malk (1990 and vanegteren and Weber (1996 do the same n the context of emssons tradng wth r replaced wth a frm s permt holdngs. Stranlund and Dhanda (1999 and Stranlund (2007 n the case of emsson tradng, and Macho-Stadler and Perez-Castrllo (2006 n the case of emssons taxes, assume that the probablty of detecton s ndependent of frm s choces but that the penalty functon s strctly convex. Of course the regulator s montorng actvty cannot depend drectly on a frm s actual volatons, because ts emssons are unknown untl t s actually audted. Sandmo (2002 recognzes ths, but notes that unusually hgh emssons levels mght gve rse to observatons (dead fsh floatng ashore, hgh ncdences of dsease near a producton ste, etc. that would n turn gve cause for frequent nspectons (pp Thus, Sandmo justfes condtonng a regulator s montorng strategy on a frm s actual emssons by assumng that emssons produce observable correlates that a regulator can use to allocate ts montorng effort. Another alternatve s that the regulator can form an expectaton of the frm s emssons gven a tax/enforcement polcy and then use ths expectaton to refne ts montorng strategy. (Malk (1990 suggests ths approach. Moreover, t makes sense that a regulator wll look for ways to use a frm s emssons report to conserve montorng effort. In ths secton we examne the robustness of our polcy prescrpton to nduce full complance wth a constant expected margnal penalty by askng the followng queston: Can a regulator devse an enforcement strategy under whch a frm wll be noncomplant that results n lower expected enforcement costs than nducng complance wth a constant expected margnal penalty? Snce aggregate expected montorng and sanctonng costs are lnear n the costs of montorng and sanctonng ndvdual frms, we can answer ths queston for a sngle frm and apply the results n the aggregate. 13

15 To answer our queston consder two polces of the form [ t, ( qˆ, r, f( q r π, where ˆq s ether the frm s actual emssons, q, or the regulator s expectaton of the frm s emssons, E( q. (E denotes the expectaton operator throughout the paper. Snce we are focusng on a sngle frm, we have dropped the frm-dentfyng subscrpt. Under both of these polces the regulator expects the frm to be noncomplant. To smplfy our analyss we make two assumptons about the enforcement strateges under [ t, ( qˆ, r, f( q r π. Frst, we assume that the detecton probablty π ( qr ˆ, s a lnear functon of ˆq and r. Ths mples that the regulator s pror expectaton of the detecton probablty t wll have to mantan s E[ π ( q, r ( Eq (, Er ( = π. Smlarly, we assume that the margnal penalty, f ( q r, s lnear so that [ ( ( ( E f q r = f E q r. We compare the montorng effort of each of these polces to the montorng effort requred under a polcy that nduces the same expected emssons from the frm, but that features a constant expected margnal penalty that motvates the frm to truthfully report ts emssons. c c Denote ths thrd polcy type as t, π, φ. The superscrpt c ndcates that the emssons tax and the detecton probablty are chosen to nduce full complance, gven the unt penalty φ. From Proposton 1, motvatng the frm to be complant wth mnmal montorng requres t c = π c φ. So that our comparsons do not depend on dfferences n margnal penaltes, we suppose that the c c constant margnal penalty under t, π, φ s equal to the expected equlbrum margnal penalty under [ t, ( qˆ, r, f( q r π ; that s, we wll mpose φ = f ( Eq ( r. To be clear, we should note here that we are not nterested n a three-way comparson of the polces. Instead we compare expected montorng effort under [ t, ( q, r, f( q r c c c π under,, π to the t π φ that produces the same expected emssons from the frm, and do the same c c π and another t, π, φ. The results of these comparsons are summarzed n the followng proposton. for [ t, ( E( q, r, f( q r Proposton 2: Holdng the regulator s expectaton of a frm s emssons fxed, a tax/enforcement polcy that the regulator expects wll result n the frm s noncomplance and that 14

16 features any combnaton of a strctly convex penalty functon, the frm s emssons report, or the regulator s expectaton of the frm s emssons wll produce hgher expected enforcement costs than a polcy that motvates the frm to be complant wth a constant expected margnal penalty. However, f the regulator s probablty of detectng and punshng the frm when t s noncomplant s a strctly ncreasng functon of ts actual emssons, then expected montorng costs are lower when the regulator expects the frm to be noncomplant. Expected enforcement costs are also lower f and only f the reducton n expected montorng costs s greater than the ncrease n expected sanctonng costs. Proof: The frm s expected costs under [ t, ( qˆ, r, f( q r π are Cq ( + tr+ π ( qr ˆ, f( q r. [7 Assumng that t chooses postve emssons, emssons report, and a non-negatve volaton, ts optmal choces of these values are determned by: C ( q + π f( q r + π( qˆ, r f ( q r λ = 0; [8 q t+ π f( q r π( qˆ, r f ( q r + λ = 0; [9 r ( q r 0, λ 0, λ( q r = 0, [10 where, λ s the Lagrange multpler attached to the constrant q r 0, and π q and π r are the constant margnal effects of the frm s emssons and report on π ( qr ˆ,. It s mportant to note that when qˆ = E( q, the regulator commts to π ( Eq (, r and communcates ths to the frm before the frm makes ts choces. Consequently, the frm treats E(q as a constant. Ths mples that when qˆ = E( q, the frm makes ts choces accordng to [8 [10, but wth π = 0. The regulator does not know exactly how the frm wll respond to [ t, ( qˆ, r, f( q r q π, because t has ncomplete nformaton about the frm s abatement cost functon. It can, however, use [8 [10 to form expectatons of the frm s emssons, Eq, ( ts emssons report, Er ( and ts volaton, Eq ( r. Snce the regulator expects the frm to be noncomplant under [ t, π ( qˆ, r, f( q r, set λ = 0 n [8 and [9 to obtan: EC ( q πq f( Eq r π ( Eq Er f ( Eq r t π f ( E( q r π ( E( q, E( r f ( E( q r 0, r ( + ( + (, ( ( = 0; [11 + = [12 15

17 where, wth some abuse of notaton, E( C ( q s the regulator s expectaton of the frm s margnal abatement cost functon evaluated at ts expectaton of the frm s choce of emssons. c c Polces of the form t, π, φ [ t, ( qˆ, r, f( q r nduce the same expected emssons E( q as under π, but motvate the frm to report ts true level of emssons. The frm s complance s ensured by t c = π c φ, and the regulator s expectaton of the frm s emssons s + = To determne the tax that nduces the same expected emssons c determned by E( C ( q t 0. as under [ t, ( qˆ, r, f( q r E( C q t πq πr f ( E q r c and then combne [13 and E( C ( q t 0 π, frst combne [11 and [12 to obtan ( + + ( + ( = 0, [13 + = to obtan ( c t = t+ ( π + π f E( q r. [14 q r Settng t c n ths way and enforcng t so that t c [ t, ( qˆ, r, f( q r π. = π c φ yelds the same ( E q as under c c Let us now compare the detecton probabltes under t, π, φ to the expected detecton probabltes under [ t, π ( q, r, f( q r and [ t, ( E( q, r, f( q r and t c π c c = φ to obtan t πφ ( πq πr f ( E( q r π. To do so use [14 = +. Substtute ths nto [12 to obtan ( Eq Er f ( Eq r f( Eq r c π (, ( ( + π ( = π φ. [15 q Equaton [15 allows us to compare the regulator s expectaton of the detecton probablty under [ t, ( qˆ, r, f( q r c c c π to π under t, π, φ. So that the dfferences between ( Eq (, Er ( π and c π under the two comparsons do not depend on dfferences n margnal penaltes, substtute φ = f ( Eq ( r nto [15 and rearrange terms to obtan ( ( c π π( E( q, E( r = π f E( q r f E( q r. [16 Recall that 0 q π = under [ t, ( E( q, r, f( q r q π. Thus, [16 ndcates that the detecton c c probablty under t, π, φ s the same as the expected detecton probablty under [ t, π ( E( q, r, f( q r. Consequently, snce [ t, ( E( q, r, f( q r π nvolves an expected 16

18 c c volaton and expected sanctonng costs whle t, π, φ does not, the expected enforcement cost of nducng a gven ( c c t, π, φ Eq s strctly greater under [ t, ( E( q, r, f( q r. However, snce f ( Eq ( r and f ( Eq ( r π than under are both strctly postve, [16 c ndcates that π > π( Eq (, Er ( f and only f π > 0. Therefore, f π ( qr, s a strctly ncreasng functon of the frm s actual emssons, the expected detecton probablty under [ t, ( q, r, f( q r c c π s lower than the detecton probablty under t, π, φ. Thus, t s possble that the polcy [ t, ( q, r, f( q r q c c π nvolves lower expected enforcement costs than t, π, φ. Ths wll be so f and only f expected montorng costs are reduced by a greater amount than the ncrease n expected sanctonng costs. Ths completes the proof of Proposton 2. QED. Proposton 2 provdes strong results that clarfy the value of enforcement strateges that are commonly assumed n the lterature. Despte the sophstcaton of these strateges, t appears that the only way a regulator can mprove on the smple strategy of nducng full complance wth a constant expected margnal penaltes s f the probabltes of detectng and punshng noncomplant frms are strctly ncreasng functons of the frms actual emssons. However, one should be clear about the crcumstances under whch ths s possble. We have already noted that a regulator cannot base ts montorng strategy on frms actual emssons, because these are hdden untl they are actually audted. We also noted the possblty that hgher emssons from a frm may produce an observable outcome that the regulator can use to ntensfy ts montorng of the frm. Perhaps hgher emssons are assocated wth more smoke leavng a statonary polluton source, or elevated ambent concentratons of a pollutant can be lnked to hgher emssons from a partcular source. Another possblty s that the probabltes that frms volatons are detected and punshed may be ncreasng n ther volatons f the lkelhood of convctng noncomplant frms s hgher when ther volatons are hgher. Snce ther volatons are functons of ther actual emssons, a regulator may be able to explot an ncreasng lkelhood of punshng noncomplant frms to reduce ts montorng effort. (We thank an anonymous revewer for pontng ths possblty out to us. Whle there may be crcumstances n whch the lkelhood of detecton and punshng noncomplance can depend on frms actual emssons, one cannot say that ths s always 17

19 possble. Not all knds of emssons wll have observable correlates that a regulator can use to refne ts montorng strategy, and the lkelhood of punshng a noncomplant frm wll not always be a functon of ts volaton. Moreover, t bears repeatng that even f a regulator can use the frms actual emssons and ther noncomplance to reduce montorng effort, ths wll be optmal only f the value of reducng montorng effort outweghs the addtonal costs of penalzng noncomplance Optmal emsson taxes under ncomplete nformaton and costly enforcement In ths secton we determne optmal emssons taxes when a regulator has ncomplete nformaton about frms abatement costs, the costs of enforcement, and the damage from polluton, under the assumpton that t s not possble to mprove on the enforcement strategy of nducng full complance wth a constant expected margnal penalty. Our prmary focus now s on whether an optmal tax polcy nvolves dscrmnatory taxes or whether a regulator should set a sngle tax that apples 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 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 9 We have not fully explored ths tradeoff here, so t should be the topc of future work. And we need to recognze that there may be other features of the regulatory envronment that we (and most others have assumed away that mght gve regulators the opportunty to conserve enforcement costs by allowng noncomplance. We have taken a standard approach to modelng regulatons by assumng that a regulator commts to a polcy and communcates ths to the frms before they make ther decsons. However, regulators mght be unable or unwllng to commt to polces that produce full complance. Moreover, regulators may be motvated to keep elements of a polcy hdden, n partcular ts montorng strategy. If frms are averse to the ambguty about the lkelhood that they wll be nspected, a regulator may be able to use ther noncomplance to conserve enforcement costs. In a smlar fashon, regulators may be able explot the rsk averson of frm managers. Fnally, we ve assumed that montorng produces a perfectly accurate judgment of a frm s complance status. Inducng full complance when montorng produces errors may not be desrable, partcularly gven that some complant frms may be punshed for volatons they dd not commt. All of these features deserve rgorous nvestgaton. 18

20 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 producton and abatement technologes, or levels and knds of nputs and outputs. Consequently, let x denote a vector of observable characterstcs of frm. Let the abatement cost functon of frm be Cq (, x, ε, where ε s known to the frm but s a random varable from the regulator s perspectve. Note that we have dropped the frm-dentfyng subscrpt on C, ndcatng that the functonal form of abatement costs s constant across frms. Ths s not necessary for our results, but t does reduce our notatonal burden. Though the form of C does not vary, ndvdual abatement cost functons dffer accordng to dfferences n ther observable characterstcs and the realzatons of the random parameter. Smlarly, let the cost of montorng frm be m( π, x, μ, where μ s a random parameter. We can gnore expected sanctonng costs from here on because the regulator wll nduce full complance. Fnally,, damage s an mperfectly known, ncreasng functon of aggregate emssons, D( q, δ where δ s a random varable. [From here on, summatons are over all regulated frms. The regulator knows the jont dstrbuton of ( ε1, K, εn, μ1, K, μn, δ so t can form an expectaton of the socal costs of polluton and ts control, condtonal on ts observatons of x, = 1, K, n: { (,, ε + ( π,, μ + (, δ }. [17 E C q x m x D q Snce the regulator wll enforce the optmal polcy so that all frms are complant, usng Proposton 1 t constrans the mnmzaton of [17 by choosng ( t, π, = 1, K, n, so that t = πφ, = 1, K, n. [18 Moreover, even though the regulator does not know exactly how frms wll respond to the polcy, t does know that they choose ther emssons to equate ther margnal abatement costs to the taxes they face. That s, C ( q, x, ε + t = 0, = 1, K, n, [19 q whch mplctly defne the frms emssons as q = q( t, x, ε = 1, K, n. [20 Substtutng [18 and [20 nto [17 gves us the regulator s condtonal expectaton of the socal cost functon n terms of well-enforced frm-specfc tax rates: 19

21 { ( (,, ε,, ε + ( / φ,, μ + ( (,, ε, δ }. [21 E Cqt x x mt x D qt x We assume that [21 s strctly convex n ( t1, K, t n and that optmalty calls for a postve tax for each frm. Then, the followng frst-order condtons unquely dentfy the optmal tax rates: ( q( (,, ε,, ε t(,, ε E C q t x x q t x ( π ( / φ,, μ E m t x + φ ( ( ε δ t ε + E D q( t, x,, q ( t, x, = 0, = 1, K, n. Substtute [19 nto these and rearrange the results to obtan ( ( ( ( π E D q( t, x, ε, δ qt( t, x, ε E m ( t / φ, x, μ t = +, = 1, K, n. [22 E q ( t, x, ε φe q ( t, x, ε ( t t Usng the defnton of the covarance between random varables, the frst term on the rght hand sde of equatons [22 s ( ( (,, ε, δ, t(,, ε Cov D q t x q t x E( D ( q( t, x, ε, δ +, = 1, K, n, [23 E q t x ( (,, ε t where Cov denotes the covarance operator. Moreover, use [20 to obtan q ( t, x, ε = 1 C ( q, x, ε, = 1, K, n. [24 t qq These ndcate that the frms margnal responses to ther taxes are equal to the recprocal of the slopes of ther margnal abatement cost functons. Substtute [23 and [24 nto [22 to obtan ( ( (,,, t = E D q t x ε δ ( π ( / φ,, μ ( 1 qq (,, ε ( ( (,, ε, δ, 1 qq(,, ε E( 1 Cqq ( q, x, ε Cov D q t x C q x E m t x, 1,, n. φe C q x = K [25 Note that the frst term on the rght hand sde s the expectaton of margnal damage. Snce ths term appears n all of the equatons n [25, our fnal proposton follows mmedately. Proposton 3: An optmal polcy of well-enforced emssons taxes under ncomplete nformaton about frms abatement and montorng costs nvolves a unform tax f and only f 20

22 ( ( ( ( π ( Cov D q( t, x, ε, δ, 1 Cqq( q, x, ε E m ( t / φ, x, μ E 1 C ( q, x, ε φe 1 C ( q, x, ε s the same for all = 1, K, n. qq qq [26 6. Dscusson of the optmalty of dscrmnatory taxes Proposton 3 ndcates that there are three potental sources of varaton n optmal ndvdual emssons taxes. These are the regulator s condtonal expectatons of the margnal costs of montorng the frms, ( E mπ ( t / φ, x, μ, = 1, K, n; ts condtonal expectatons of the recprocal of the slopes of the frms margnal abatement cost functons, E( Cqq q x ε 1 (,,, = 1, K, n, and the covarances between margnal damage and 1 C ( q, x, ε, = 1, K, n. qq Before we dscuss these sources of varaton n detal let us hghlght what appears to us to be fundamental effcency justfcaton for choosng a unform tax. Ths s when the frms observable characterstcs do not provde the regulator wth any nformaton about the varaton of the slopes of the frms margnal abatement costs, or the margnal costs of montorng them for complance. 10 In ths case the terms n equaton [26 do not vary across frms, and the regulator chooses a unform tax, because t cannot dstngush the frms from one another. Thus, Proposton 3 suggests 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. 11 As we ve already noted, regulators may not be so ll-nformed n many real settngs. When a regulator has somewhat better, but stll ncomplete, nformaton about ndvdual frms, ther optmal tax rates wll vary. Admttedly, the number of dstnct 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 10 There are several unrealstc specal cases under whch [26 would not vary across frms. Suppose that the covarance term n [26 s zero. Then, optmalty calls for a unform tax f margnal montorng costs are equal to zero, the frms margnal montorng costs and the slopes of ther margnal abatement costs are the same, or f ther observable characterstcs are the same. 11 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 effcent. 21

23 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 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 can use to dstngush the frms abatement or montorng costs, the number of 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. Now let us look nto the sources of varaton of ndvdual tax rates when a regulator can dstngush at least some frms from at least some others. The frst term of [26 s nterestng because t does not depend at all on the costs of enforcng emssons taxes. That s, even f one assumes zero enforcement costs, ncomplete nformaton about frms abatement costs can produce varaton n optmal tax rates when the regulator has some nformaton that allows t to dstngush the slopes of the frms margnal abatement cost functons from one another. Note that the frst term of [26 s zero for every frm f the covarances between margnal damage and the recprocal of the slopes of the frms margnal abatement cost functons are all zero. Ths would occur f margnal damage s a known constant, or f the slopes of the frms margnal abatement cost functons are known constants. Ths latter assumpton s mportant because t s common to model uncertanty about abatement costs as a random shft of only the ntercept of margnal abatement costs. Under ths assumpton and the equally common assumpton that enforcement s not costly, optmalty requres a unform tax set equal to the regulator s expectaton of margnal damage at ts expectaton of the frms aggregate emssons under the tax However, we are manly nterested n how enforcement costs nduce dscrmnatory emsson taxes, whch s captured by the varaton n Em ( π ( t/ φ, x, μ φe( 1 Cqq( qx,, ε, = 1,, n. These values ndcate that ndvdual tax rates wll vary f there s varaton n the regulator s condtonal expectatons of the ndvdual margnal montorng costs assocated wth enforcng the frms taxes, f there s varaton n ts condtonal 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 22