Regulation via the Polluter-Pays Principle

Transcription

1 Regulaton va the Polluter-Pays Prncple Stefan Ambec Lars Ehlers January 31, 2012 Abstract We consder the problem of regulatng an economy wth envronmental polluton. We examne the dstrbutonal mpact of the polluter-pays prncple whch requres that any agent compensates all other agents for the damages caused by hs or her (polluton) emssons. Wth constant margnal damages we show that regulaton va the polluter-pays prncple leads to the unque welfare dstrbuton that assgns non-negatve ndvdual welfare and renders each agent responsble for hs or her polluton mpact. We extend both the polluter-pays prncple and ths result to ncreasng margnal damages due to polluton. We also compare the polluter-pays prncple wth the Vckrey-Clark-Groves mechansm. JEL classfcaton: C7, D02, D30, D6. Keywords: Regulaton, Polluter-Pays Prncple, Responsblty for Polluton Impact, Externaltes. Ths research started whle the second author was vstng the Toulouse School of Economcs (INRA- LERNA). It receved fnancal support from INRA (France), the ANR (France) through the proect ANR-08- JCJC on Far Envronmental Polces, the SSHRC (Canada) and the FQRSC (Québec). Toulouse School of Economcs (INRA-LERNA-IDEI), France, and Unversty of Gothenburg, Sweden; e- mal: Département de Scences Économques and CIREQ, Unversté de Montréal, Canada; e-mal: (Correspondng author). 1

2 1 Introducton From water management to ar polluton, managng envronmental problems effcently requres well-desgned publc polces or coordnaton among stakeholders (Ostrom, 1990). Envronmental polces are launched to mtgate the falure of market economy due to the presence of negatve externaltes. Yet publc nterventon has an mpact not only on the welfare of the economy as a whole but also on the dstrbuton of welfare. Ths paper addresses the dstrbutonal mpact of envronmental polces n economes wth polluton. The model allows for a varety of negatve externaltes ncludng unlateral or multlateral ones, heterogenous mpacts due to dstance or mtgaton. It formalzes many complex envronmental ssues such as water qualty management n a rver or the reducton of sulfur doxde or greenhouse gas emssons n an nternatonal settng. 1 In ths framework, we defne a regulaton mechansm as ndvdual transfers contngent on polluton emssons. In partcular, we consder the mechansm nspred by a lteral nterpretaton of the polluter-pays (PP) prncple. It states that the costs of polluton should be borne by the entty whch profts from the process that causes polluton. Strctly speakng, t requres that any agent (frm or consumer) compensates all agents who suffer from hs polluton emssons for the damage he causes. The PP mechansm s by constructon budget-balanced. It s also effcent n the sense that t unquely mplements the allocaton of polluton emssons that maxmzes total welfare n Nash equlbrum. Therefore, the PP mechansm shares ths feature wth the mechansms proposed by Duggan and Roberts (2002) and Montero (2008). Note however that here each agent only chooses hs emssons whereas n Duggan and Roberts (2002) each agent chooses hs emsson and reports the emsson of hs neghbor and n Montero (2008) each agent reports hs nverse demand for any level of emssons. The focus of both papers s on the mplementaton of the effcent allocaton under asymmetrc nformaton whereas we are nterested n the dstrbutonal mpacts of a mechansm that mplements the effcent allocaton under perfect nformaton. We examne the propertes of the welfare dstrbuton nduced by regulaton mechansms. We focus on two farness crtera. The frst one s that each ndvdual s welfare s non-negatve. 1 To that respect, t s as rch as the semnal model of Montgomery (1972). 2

3 It s a mnmal acceptablty requrement snce an agent who obtans a negatve welfare does not beneft from the welfare-enhancng economy actvtes exhbtng polluton. The second crtera reles on the concept of responsblty n axomatc theory of ustce (Fleurbaey, 2008). It makes a polluter responsble for ts polluton mpact. More precsely, the welfare dstrbuton should be such that a polluter s assgned the full socal cost due to hs polluton. In partcular, f a polluter modfes the envronmental mpact of hs own emssons n the economy, he should get the full return or loss due to ths change. For nstance, a frm whch flters ts own emssons to reduce ther sulfur content should get the full beneft for the economy of ts cleanng nvestment. A farmer who uses more pestcde and fertlzers leadng to drter waste water should pay the socal cost assocated to ths pestcde and fertlzer ncrease. We show that the welfare dstrbuton nduced by applyng the PP mechansm s the only one that satsfes the two above crtera: non-negatvty and responsblty for polluton mpact. Our frst characterzaton of the polluter-pays welfare dstrbuton reles on the assumpton that margnal damage due to polluton does not depend on polluton concentraton. When margnal damage s ncreasng wth polluton concentraton, the ncremental mpact of each polluter on damage s not straghtforwardly defned because the cost generated by an emtter depends on the other polluters emssons. We extend the polluter-pays prncple to ths framework by makng the polluter pay for the ncremental mpact of hs or her emssons on the total welfare of the other agents when he does pollute and when he does not pollute (at the effcent levels of polluton). As a result, each agent receves n the PP welfare dstrbuton the dfference of socety s welfare when he does pollute and when he does not pollute (but partcpates n the mechansm as a vctm). We then provde a further characterzaton of the PP welfare dstrbuton n the more general framework where margnal damage s ncreasng wth polluton. More precsely, we show that t s the unque welfare dstrbuton that satsfes three crtera: non-negatvty, responsblty for polluton mpact and soldarty upper bounds. The latter bounds requre no agent to obtan more than what he or she would get n absence of polluton of all other agents. We conclude by comparng the polluter-pays regulaton mechansm wth the Vckrey-Clark-Groves (VCG) mechansm appled to the polluton problem. We proceed as follows. Secton 2 ntroduces a smple model of polluton wth constant 3

4 margnal damages. It also provdes several real-world examples whch ft our framework. Secton 3 descrbes regulaton mechansms and ther nduced dstrbuton rules n equlbrum. It also dscusses several regulatons used n real lfe. Secton 4 ntroduces the polluter-pays regulaton and characterzes ts nduced dstrbuton rule n terms of non-negatvty and responsblty for polluton mpact. Secton 5 generalzes our model to dfferentate polluton and damages and allows for ncreasng margnal damages. We generalze the PP prncple to ths envronment and extend our man results to ths framework. Secton 6 concludes by comparng the PP mechansm wth the VCG mechansm. 2 A model of polluton Consder a set N = {1,..., n} of agents (countres, ctes, farmers, frms, consumers,...). Each agent N s pollutng or s polluted or both. Agent enoys a beneft b (e ) from producton and/or consumpton where e 0 denotes the level of economc actvty hereafter called emssons. The beneft functon b s assumed to be both strctly concave and strctly ncreasng from 0 to a maxmum ê wth b (ê ) = 0 for every N, 2 and twce contnuously dfferentable: for all N and for all 0 e < ê, both b (e ) > 0 and b (e ) < 0. We normalze b (0) = 0 and assume that the margnal beneft at e = 0 s hgh enough (say nfnte) so t s optmal for all agents to produce and/or to consume. Polluton from agent causes a margnal damage a 0 to agent. The parameter a measures the magntude of the polluton mpact of s emsson on. For the moment we consder constant margnal damages. Later we extend our results to envronments wth convex damages and thus, ncreasng margnal damage from emssons. A (negatve) externalty or polluton problem (N, b, a) s defned by a set of agents N, a profle of beneft functons b = (b ) N, and a matrx of externalty/polluton margnal mpacts a = [a ] N N. When there s no confuson, we wrte for short a nstead of (N, b, a). Let R = { N a > 0} denote the receptors of s polluton: the set of agents whch are polluted by. Let R 0 = { N\{} a > 0} denote the receptors of s polluton excludng. We assume that a > 0 for any N wth R 0,.e. f s pollutng other agents, then 2 Ths s wthout loss of generalty snce the maxmum could be ê = + for some N. 4

5 hs polluton also causes some damage at hs locaton. 3 Let S = { N a > 0} denote the set of agents who pollute agent. Let S 0 = { N\{} a > 0} denote the set of agents who pollute excludng. The envronmental damage suffered by n the emsson vector e = (e ) N s therefore d = a e. S The welfare of agent wth emssons e = (e ) N s: b (e ) d = b (e ) S a e. (1) The frst term n (1) s s beneft from hs own emssons whereas the second term s s welfare loss due to polluton. An effcent emssons plan e = (e ) N maxmzes total welfare N [b (e ) d ] = N b (e ) N S a e. It satsfes the followng frst-order condtons for every N: b (e ) = R a. (2) Note that our assumptons on the beneft functon b guarantee that e s unque because b (ê ) = 0 and b s strctly concave and strctly ncreasng between 0 and ê. The margnal beneft of polluton emtted by should be equal to ts margnal damage for socety. Let W (a) = b (e ) a e N N S denote the economy s welfare from the effcent emssons plan e n the problem (N, b, a). A welfare dstrbuton for the problem (N, b, a) s a vector z = (z ) N such that N z W (a). A dstrbuton rule φ assocates wth any problem (N, b, a) a welfare dstrbuton φ(a) for a. Note that a dstrbuton rule dentfes for each problem a welfare dstrbuton whch the socety may wsh to mplement. The externalty problem (N, b, a) exhbts multlateral externaltes f S = R for any N. The problem (N, b, a) exhbts unlateral externaltes f S 0 R 0 = for any N. Let V N denote the set of agents who do not pollute other agents and only suffer from polluton due to other agents actvtes. Formally, for any V, a = 0 for all and 3 For the envronments consdered here, ths assumpton s wthout loss of generalty. 5

6 a > 0 for at least one, or equvalently R 0 = and S 0 ; and wthout loss of generalty, ê = 0. 4 Smlarly, let P N denote the set of agents who do not suffer from other agents polluton: a > 0 for some and a = 0 for all, that s R 0 and S 0 =. Note that any agent n N\V s pollutng the socety from hs economc actvtes. Example 1 (The Rver Polluton Problem) Agents are countres, ctes or factores located along a rver. The set of predecessors of n the rver s S 0 whle the set of followers of s R 0. Each agent emts e unts of polluton whch mpact ts followers downstream: one unt emtted n causes a margnal damage a n. Here the margnal damages a may be decreasng wth respect of the dstance of to,.e. agent s emssons have a hgher polluton mpact on mmedate neghbors than on agents located further downstream the rver. Symmetrcally, agent suffers from polluton emtted upstream by agents n S 0 and by hmself. 5 It s a case of unlateral externaltes: f we take two agents and, ether s upstream or s downstream,.e. R or S. In a sngle canal or one-trbutary rver, agents can be ordered accordng to ther poston from upstream to downstream. In ths case, f N = {1,..., n} and f agents suffer from ther own polluton (e.g. countres), then for any N, R = {1, 2,..., } and S = {, + 1,..., n}. Moreover, for any and, f R then R R. Symmetrcally, f S, then S S. The latter propertes mght not hold n more general rvers. Wth several trbutares that end up on the same man course, for any agent there mght be k, S but both k / S and / Sk. Symmetrcally, for rver deltas or rrgatons dtches orgnated from a source or weed or reservor, we have the reverse: for any agent there mght be k, R but k / R and / Rk. 6 Example 2 (The Internatonal Greenhouse Gas Emssons Game) Players are countres. Each country enoys a beneft b from ts own greenhouse gas emssons e. Greenhouse gases emtted nto the atmosphere cause global warmng that damages countres economes. 4 If ê > 0, then agent s actvty does not have any mpact on socety and hs actvtes can be dsregarded. 5 In the case of a rver, lnearty s a good approxmaton up to the pont at whch the rver becomes so overloaded wth organc materal that oxygen (needed for aerobc bacterologcal decomposton) s depleted. At that pont, [refereed as the rver carryng capacty] the rver s capacty to clean tself s greatly dmnshed. from Kolstad (2000) footnote 2 page See Ambec and Sprumont (2002) and Ambec and Ehlers (2008) for a rgorous analyss of the rver water sharng problem. 6

7 The magntude of global warmng depends on total emssons on the earth surface N e. Suppose that total emssons cause a constant margnal damage of δ to country. In ths example, S = R = N and a = a = δ for all, N: all countres exert multlateral externaltes on all other countres of the same magntude. Yet countres dffer on the damage that externaltes cause on ther economy. Semnal papers on nternatonal agreements for greenhouse emsson reducton (Chandler and Tulkens, 1992; Carraro and Snscalco, 1993; Barrett, 1994) rely on these assumptons except that they consder convex damage (or concave beneft of emsson abatements) and, therefore, ncreasng margnal damage. Example 3 (The Internatonal Acd Ran Game) Agents are countres emttng sulfur doxde (SO2) by burnng coal for power producton. Ths causes acd ran whch damages forests and ecosystems n neghborng countres. The parameter a captures the margnal mpact of country s SO2 emssons to acd ran n country. It depends on the fracton of emssons from that s deposted n and ts margnal damage on. Mäler and De Zeeuw (1998) provde estmatons on those parameters for 1990 and 1991 n Europe. For nstance, among the SO2 emssons from Belgum, 19.4% ended up n Belgum, 13.3% n Germany, 9% n France, 4.8% n Netherland and so on. Mäler (1989, 1994) consders an acd ran game wth such heterogeneous transportaton parameters and constant margnal damage. Ths game has been extended by Mäler and De Zeeuw (1998) and Fnus and Tøtta (2003) to envronments wth convex margnal damages. Example 4 (Polluters versus Vctms) Agents n V are ndvduals and those n P are frms and each agent belongs ether to V or to P. Frms emt polluton wthout ncurrng any damage: a = 0 for every. In contrast, any V does not emt polluton but suffers from polluton: ê = 0 for every V and a > 0 for at least one P. In ths case, a can be nterpreted the margnal damage of each unt of frm s polluton causes to person n term of health or envronmental mpact. It depends on technologes, dstance between frms and ndvduals, clmatc condtons, and so on. The vctms of polluton mght be frms nvolved n dfferent sectors than the polluter ones; for nstance hotel and restaurants located close to a lake or sea shore that mght be polluted by local factores. The man dfference wth the prevous examples s that emtters and vctms are dsunct sets of agents. It s a case of 7

8 unlateral externaltes. 3 Regulaton mechansms and dstrbuton rules An mportant polcy tool n polluton problems are regulaton mechansms. A regulaton mechansm t : R N + R N specfes for any emssons a vector of payments (or transfers) t(e) = (t (e)) N. It assgns to agent the transfer t (e) for any emssons plan e = (e ) N. Gven the mechansm t and the emsson plan e, agent s welfare under the vector t(e) s gven by b (e ) d + t (e) = b (e ) S a e + t (e). (3) Of course, each agent chooses hs own emssons and for any problem a, the regulaton mechansm t nduces an emssons game. Let N (t, a) denote the set of (pure) non-cooperatve Nash equlbra n the emssons game under the mechansm t and the problem a. In the non-cooperatve Nash equlbrum of the externalty problem wth the mechansm t, each player maxmzes (3) wth respect to e gven e = (e ) N\{}. Let e t N (t, a) be a Nash equlbrum emsson plan. Agent s equlbrum welfare under e t s: z t = b (e t ) d t + t (e t ), where d t = S a e t. The total welfare s W t (a) = z t = [ b (e t ) d t + t (e t ) ] = b (e t ) d t + t (e t ), N N N N N where n the last expresson the frst term s the total beneft from emsson, the second s the total damage and the thrd s the regulaton mechansm surplus (or defct f negatve). Gven a dstrbuton rule φ and a mechansm t, we say that t mplements φ (n Nash equlbrum) f for all problems a and all e t N (t, a), we have φ (a) = z t = b (e t ) S a e t + t (e t ). A partcular regulaton mechansm s the lassez-fare mechansm t lf defned by t lf (e) = 0 for all N and all e R N +. The lassez-fare mechansm represents stuatons wthout 8

9 regulaton or where socety chooses not to ntervene. It mplements the emssons plan e lf = (e lf ) N satsfyng the followng frst-order condtons, b (e lf ) = a, for every N. Thus, for each problem a, N (t lf, a) s unque and mplctly gven by the above equaltes. In contrast to the effcent emssons plan e, under lassez-fare each agent consders the mpact of hs emssons only on hs own welfare. In partcular, e lf = ê f a = 0. As long as a > 0 for some,.e. s emssons have an mpact on another agent, then e lf > e and therefore dlf > d for every R. Many regulaton mechansms are used n practce. For nstance, consder a norm on polluton emssons mechansm, denoted by t. It defnes upper bounds on emssons ē 0 and penaltes for exceedng these bounds. Formally, let ē = (ē ) N and for all e R N +, 0 f e ē t (e) = F (e ē ) f e > ē for every N where F > 0 s the fne n case of excess polluton (whch can be nfnte or lump-sum). In case of an unform norm, ē = ē and F = F for all, N. If the fne s hgh enough to be persuasve and the norm s bndng n the sense that e lf > ē for all N, then the unque emssons plan mplemented n Nash equlbrum by t are e t = ē for all N. The emsson fee mechansm t f specfes fees f = (f ) N on emssons and, therefore, charges the payment t f (e) = f e from agent. Here f > 0 s polluter s tax rate. The Pgouvan fee s f = R 0 a for every polluter N. It mplements the frst-best emssons e n Nash equlbrum. Alternatvely, the fee can be on ambent polluton rather than on emssons. A polluton fee scheme t F charges F > 0 per unt of emssons at each receptor whch leads agent to pay t F (e) = R a F e. The emsson or ambent polluton fee mechansm can be assocated wth a redstrbuton polcy of the money collected, e.g. through lump-sum transfers or subsdes. A further mportant regulaton nstrument that can be embedded n our model s cap-andtrade or tradable emsson permts. Agents are endowed wth some ntal emssons allowances or permts ē = (ē ) N whch can be traded n a market. They are not allowed to emt more than the amount of permts they own at the end of a pre-polluton tradng phase. Provdng 9

10 that the permt market s compettve (mplyng that agents are prce takers), the tradable emsson permt regulaton s as f each agent faces a transfer scheme t tp (e) = p(ē e ) where p s the equlbrum prce of permts. Ths prce s unquely determned by the frstorder condtons b (et ) a = p for every N\V and the market clearng condton N ē = N et. The ntal allocaton of permts mpacts the level and dstrbuton of welfare. emsson ē Under grandfatherng, each agent s assgned a share of hs or her lassez-fare = αe lf wth 0 < α 1. A lower α means lower emssons n the economy. When permts are auctoned by the government, t s as f those who get the revenue from ths aucton are endowed wth the permts. For nstance, f the money s used exclusvely to reduce or compensate the damage at agent h s locaton, then t s as f agent h obtans all permts and trades them wth polluters n a compettve market,.e. ē h = N\V et. Emsson allowances can also be defned on receptors emssons, each agent potentally ownng ē emsson allowances at receptor that can be exchange aganst other emsson allowances for the same receptor. Gven the abundance of dfferent regulaton mechansms n realty, a socety would lke to dstngush between them accordng to desrable crtera. The followng wll be two very basc requrements any socety would lke any regulaton to comply wth. Effcency requres that the frst-best outcome s mplemented n Nash equlbrum. Effcency: For all problems a and all e t N (t, a), we have e t = e. The second property requres that the payments of the mechansm are budget-balancng at Nash equlbrum. Budget Balance: For all problems a and all e t N (t, a), we have N t (e t ) 0. A budget balanced regulaton mechansm t where N (t, a) s a sngleton for any a, say N (t, a) = {e t }, nduces a dstrbuton rule φ t of the total welfare. For any problem a, the 10

11 dstrbuton rule mplemented by the budget balanced mechansm t sets: φ t (a) = b (e t ) S a e t + t (e t ). Any of the above regulaton mechansms s budget balanced and has a unque Nash equlbrum, and hence, nduces a correspondng dstrbuton rule. We now focus on a partcular regulaton mechansm, the one nspred by the polluter-pays prncple. 4 A Characterzaton of the Polluter-Pays prncple In ths secton, we frst descrbe the polluter-pays mechansm and show two of ts propertes, namely budget balancedness and effcency. Second, we examne the propertes of the welfare dstrbuton rules mplemented by regulaton mechansms n Nash equlbrum, and, n partcular by the polluter-pays welfare dstrbuton rule. 4.1 The Polluter-Pays Mechansm Many countres have adopted the polluter-pays (PP) prncple as a regulaton mechansm. It bascally renders the polluter responsble for the damage t causes to the envronment. It requres that the costs of polluton should be borne by the entty whch profts from the process that causes polluton. In order to satsfy the polluter-pays prncple, the entty who pollutes should compensate those who suffer from ths polluton for the damages t causes. If a vctm s not fully compensated then he or she pays part of the cost of someone else s polluton. Hence, strctly speakng, the PP prncple mposes not only that polluters pay for the damage caused to socety, but also that vctms are fully compensated for those damages. In our model, an arbtrary agent who pollutes should compensate every agent R 0 for the caused damage a e. Agent pays a e to every R 0. Therefore, as a vctm of polluton, agent receves the compensaton a e from each agent S 0 who pollutes hm. Summng up all these sde-payments, the polluter-pays prncple leads to the regulaton mechansm t P P (e) defned as follows for any agent N: t P P (e) = a e a e = d a e a e = d a e. (4) S 0 R 0 R 0 R 11

12 Agent receves the net transfer from the cost of polluton he suffers mnus the cost of polluton he causes to socety. Snce the polluter-pays prncple nvolves sde-payments among agents, the payments n the PP-mechansm sum up to zero. It s therefore budget-balanced. Agent s welfare under the payments t P P (e) wth emsson plan e s: b (e ) R a e (5) Snce agent pays for the margnal damage caused to others and s compensated from the margnal damage caused by others, hs welfare under the PP-mechansm n (5) s the socal beneft from hs economc actvty. Therefore, agent has ncentve to emt the effcent level e for any gven emssons emtted by other agents. Formally, maxmzng (5) wth respect to e leads to the frst-order condton (2) whch mples e t = e for every N. Ths mples that the PP-mechansm mplements the effcent emsson plan e n Nash equlbrum,.e. N (t P P, a) = {e }. A partcular feature of regulaton through the PP-mechansm wth constant margnal damages s that, snce any ndvdual s payoffs depend only on the agent s own choce (no externalty), the effcent emsson plan s a domnant strategy equlbrum, whch s an equlbrum concept whch s less demandng n terms of cogntve sklls than Nash equlbrum. Therefore, the effcent emssons plan remans the unque Nash equlbrum when the parameters a are publcly known but the beneft functons are prvate nformaton. One can even check that the effcent emssons plan s robust to colluson,.e. t remans the unque equlbrum n the PP mechansm even f we allow coaltons to ontly change ther emssons. 7 We wll denote by φ P P the polluter-pays (PP) dstrbuton rule assocatng wth each problem (N, b, a). Its polluter-pays welfare dstrbuton φ P P (a) s gven by agent s equlbrum welfare for every N: φ P P (a) = b (e ) R a e. (6) The result below follows straghtforwardly from our dscusson. Proposton 1 The polluter-pays mechansm s an effcent and budget-balanced regulaton mplementng the polluter-pays dstrbuton rule. 7 Ths s easly seen by the followng argument: for any non-empty coalton S N we have that (e ) S P solves max (e ) S 0 S [b(e) + tp ((e ) S, (e ) N\S )]. 12

13 4.2 The Polluter-Pays Dstrbuton Rule The followng are two desrable crtera a socety would lke to be satsfed by the welfare dstrbutons mplemented va a regulaton mechansm. The frst crterum requres that any agent should receve a non-negatve payoff. Non-Negatvty: For all problems a and all N, φ (a) 0. In the absence of polluton or emsson actvtes, any agent s welfare s zero and the state of no polluton may be nterpreted as status quo. Non-negatvty smply requres that nobody should be worse off under polluton than wthout polluton. The second crterum renders the polluter responsble to any change of hs polluton mpact on the economy. Responsblty for Polluton Impact (RPI): Consder any arbtrary agent N. Suppose that agent s polluton mpact s reduced from (a ) N to (a ) N wth a a for all N, and all other polluton mpacts beng unchanged (a l = a l for all l N\{} and all N). The dstrbuton rule φ renders agents responsble for ther polluton mpact f for any N, any reducton a of s polluton mpact from a, φ (a ) φ (a) = W (a ) W (a). Responsblty for polluton mpact (RPI) requres to assgn to any agent the full return or loss of any change of hs own polluton mpact. In addton to beng a farness prncple, RPI has attractve ncentve propertes. Suppose that an agent s able to reduce hs polluton mpact at some cost by swtchng to a greener technology, reducng or cleanng ts wastes, mprovng energy effcency or usng less toxc nputs. By assgnng the full return of ths polluton reducton, RPI provdes effcent ncentves to nvest n polluton mpact reducton. Symmetrcally, f an agent benefts from ncreasng hs polluton mpact per unt of emssons (e.g. usng hgher sulfur content coal), RPI assgns to ths agent the economc cost of ths extra polluton. Among the above regulatons, the Pgouvan fee regulaton mechansm s effcent. It s 13

14 budget balanced f the revenue collected s redstrbuted to agents. The welfare dstrbuton t mplements does not satsfy non-negatvty snce vctm-only agents (.e. agents V ) are not compensated for the envronmental damage they ncur. The welfare dstrbuton wth the Pgouvan fee regulaton mechansm also satsfes RPI. An emsson norm ē = e wth a persuasve fne (e.g. nfnte) s effcent and budget balanced but ts welfare dstrbuton does not satsfy RPI and non-negatvty. A cap-and-trade system (tradable polluton allowances) for polluton at each receptor wth grandfatherng s effcent and budget balanced but the welfare dstrbuton t leads to does not satsfy non-negatvty snce vctms are not compensated entrely. It mght or mght not satsfy RPI dependng on the ntal allocaton of permts. A smlar cap-and-trade system where permts are auctoned satsfes effcency and RPI but t s not budget balanced unless the money collected s redstrbuted. Theorem 1 The polluter-pays dstrbuton rule s the unque dstrbuton rule that satsfes non-negatvty and responsblty for polluton mpact. Proof. Frst, we show that f a dstrbuton rule satsfes non-negatvty and responsblty for polluton mpact, then t must be the polluter-pays dstrbuton rule φ P P. Consder another dstrbuton rule φ and let a be a problem. Let φ(a) = z and φ P P (a) = z P P. Let N z = W. Suppose that z z P P. Snce N zp P W W (a). Thus, N z N zp P N. Note that for all V, z P P = W (a) and z s a welfare dstrbuton, we have whch, combned z z P P forces z < z P P for some = 0 and by non-negatvty of φ, z 0. Thus, we must have N\V and both a > 0 and ê > 0. Let a be such that a s a polluton mpact reducton for agent from a such that a < a and everythng else remans dentcal,.e. a l = a l for all l, N such that l. Pck a suffcently large such that b (e lf ) < z P P z (7) where N (t lf, a ) = {e lf }. Let φ(a ) = z and φ P P (a ) = z P P denote the dstrbutons chosen by φ and φ P P for the problem (N, b, a ). By responsblty for polluton mpact, z z = z P P z P P Rearrangng terms and usng the defnton of z P P ths leads to z P P z = b (e ) a e z, (8) R 14

15 where e denotes the effcent emsson plan for (N, b, a ). By non-negatvty of φ, z 0. Now snce b (e lf ) b (e ), a 0 for all R, and z 0, we obtan from (7), z P P z > b (e lf ) b (e ) a e z, R whch contradcts (8). Second, we show that φ P P satsfes non-negatvty and responsblty for polluton mpact. For non-negatvty, = b (e ) a e = max b (e ) a e b (0) a 0 = 0, R R R z P P e 0 where the nequalty follows from the fact that agent can always choose e = 0 (no emsson or producton). For responsblty for polluton mpact, for any agent, consder any reducton of s polluton mpact from a to a : a a for all N and (a k) N = (a k ) N for any k. Let φ P P (a) = z P P and φ P P (a ) = z P P. Let W (a) and W (a ) denote the correspondng total welfare n (N, b, a) and (N, b, a ), respectvely. Note that by effcency of t P P, we have both W P P (a) = W (a) and W P P (a ) = W (a ). Smlarly, denote by e and e the effcent emsson plan of (N, b, a) and (N, b, a ), respectvely. By defnton, z P P z P P = b(e ) a e R b(e ) R a e. (9) Snce a k = a k for every k, the effcent emsson levels are not affected by the change of matrx of polluton mpacts from a to a whch mples e k = e k we have: W (a ) W (a) = b(e ) a e R b(e ) R a e for every k N\. Therefore, whch, combned wth (9), leads to z P P z P P = W (a ) W (a). Because for any problem a, φ P P (a) s an effcent welfare dstrbuton, Theorem 1 shows that non-negatvty and responsblty for polluton mpact mply effcency,.e. for any problem the total welfare s dstrbuted among the agents. 15

16 Secton 5 of Ambec and Ehlers (2010) s concerned wth the acceptablty of the PP prncple. 8 Ths s an mportant ssue snce envronmental polces emerge as a collectve choce n democratc socetes. They show that the PP mechansm fals to be acceptable n general snce, for nstance, the most upstream polluter of a rver prefers lassez-fare to the applcaton of the PP prncple. They nevertheless show that the polluter-pays welfare dstrbuton satsfes all non-cooperatve core lower bounds n a symmetrc polluton envronment,.e. where all beneft functons are dentcal and polluton mpacts are the same across all agents. 9 Ths s for nstance the case n most theoretcal models examnng nternatonal agreements for greenhouse gas emsson reducton ncludng Chandler and Tulkens (1992), Carraro and Snscalco (1993) and Barrett (1994). 5 Generalzaton to ncreasng margnal damage We now consder the polluter-pays prncple wth convex damage functons whch requres a slght modfcaton of the model. We dfferentate emssons from polluton and damage. The emsson plan e generates a polluton level p at s locaton (to receptor ) defned by: p = S a e. (10) The matrx a defnes now the transfer coeffcents that translates emssons of nto polluton of (e.g. waste water released by nto water polluton concentraton on ). Polluton at level p causes damages d (p ) to wth d beng ncreasng and convex: d (0) = 0, d (p ) > 0 and d (p ) 0 for every p R + and N\P. 10 The welfare of agent wth emssons e = (e ) N s: b (e ) d (p ), (11) where p s defned by (10). A polluton problem s now descrbed by (N, b, a, d). 8 Demange (2004) studes the acceptablty of dstrbutons n herarches and networks when blockng s only allowed by consecutve or connected coaltons. 9 Formally, for all, N, both b = b and a = a. 10 Recall that P s the set of only polluter agents. 16

17 The frst-order condtons that characterze the effcent emsson plan e (whch maxmzes the total welfare N [b (e ) d (p )]) are for every N: 11 b (e ) = R a d (p ) = R a d ( a l e l ). (12) l S The margnal beneft of agent s emsson should be equal to ts margnal cost for socety whch depends on ts margnal mpact on polluton a and the margnal damage of polluton at each receptor R. Each unt of emsson from agent leads to a unts of polluton at receptor whch causes margnal damages evaluated to a d (p ). The total welfare wth the effcent emsson plan e s: W (a) = N [b (e ) d (p )] = N [b (e ) d ( S a e )]. In contrast wth constant margnal damages (.e. the frst-order condton n (2)), wth ncreasng margnal damage the effcent level of s emsson (the frst-order condton n (12)) depends on what s emtted by the other polluters of wth beng a receptor of s polluton ( R). Margnal damage beng ncreasng wth polluton concentraton, agent s emsson has more mpact on damages at when pollutant emtted by other polluters n R 0 ncreases. Because a polluter s margnal mpact depends on polluton concentraton due to other polluters, applyng the polluter-pays prncple n ths framework s not straghtforward. One needs to defne each polluter s responsblty on the damage caused to socety when computng the cost of polluton of one entty on others. Wth only one sngle polluter, t s easy: agent should pay the damage d (a e ) to vctm. However, wth more than one polluter at a receptor, say and k, the PP prncple does not tell us how to share d (a e + a k e k ) (the overall cost at ) among and k. If polluter s held responsble for the frst a e unts of polluton, he has to pay d (a e ). If polluter s responsble for the last ones, he has to pay d(a e + a k e k ) d (a k e k ) whch s larger than d (a e ) by convexty of d. It s also 11 The exstence of the effcent emsson plan e s guaranteed by Brouwer s fxed pont theorem: defne g : N [0, ê ] N [0, ê ] by g(e) = ((b ) 1 ( R ad ( l S a le l ))) N. Snce b s strctly concave, b tends to nfnty at zero, and b tends to zero at ê, g s a well defned functon. Our assumptons on damages ensure that g s contnuous. Now snce N [0, ê ] s compact and convex, Brouwer s fxed pont theorem mples that the functon g must have a fxed pont whch s a soluton to (12). Unqueness of e follows from strct concavty of the benefts and the convexty of the damages. 17

18 ncreasng wth the other polluter k s emssons. One can thnk about several ways to share the damage d (p ). For nstance, t could be assgned proportonally to a polluter s share on total polluton, each polluter payng a e p d (p ) to for every R. Such a dvson of the damage s defned for gven emssons by and k. Yet, snce emssons are substtutes for receptor, the presence of s emssons at leads to a reducton of k s emssons e k at the frst-best. The nter-connecton of polluters effcent emssons wth convex damage creates a further cost of polluton on socety: s emsson do not only cause damage at, t also encroaches on k s emsson at the frst-best. In ths framework, we nterpret the PP prncple of makng payng the cost of polluton of one entty on others by chargng a polluter the ncremental mpact of hs emssons on other agents. Due to ncreasng margnal damage, we can dstngush between two mpacts. A frst one s an ncrease of damage at each receptor R. The second one s due to the substtuton between polluters emssons for each receptor : f emts more polluton, then each polluter k S should emt less at the frst-best. We also nterpret the PP prncple by compensatng each agent exactly for the damage caused by others emssons n absence of hs emsson. Let us denote by e 0 the effcent emsson plan wthout s emsson for every N (wth fxng e 0 = 0). Notce that e 0 s the effcent plan of an economy wthout s emsson but wth s damage functon d (.e. agent s then a vctm only ). It maxmzes the total welfare of the problem (N, b, a, d) where by b mplctly means that agent becomes a vctm. The polluter-pays regulaton mechansm t P P s defned for every N by: t P P (e) = d (p 0 ) [ b (e 0 ) d (p 0 ) (b (e ) d (p )) ]. (13) The transfer s decomposed n two terms. The left-hand term s agent s damage at the frst-best wthout s emssons. The summaton s the economc loss due to s emsson for all other agents. A polluter who s vctm of s polluton, the transfer s smply the loss of beneft b (e 0 ) b (e ). For a polluter who s a vctm of s polluton t s the change of welfare ncludng damage b (e 0 ) d (p 0 ) (b (e ) d (p )). For a vctm only agent V, the PP transfer reduces to the frst term whch s the damage at the frst-best d (p ). The PP mechansm yelds to agent a total welfare of (notng b (e 0 ) = b (0) = 0): b (e ) d (p ) + t P P (e) = [ b (e ) d (p ) (b (e 0 ) d (p 0 )) ]. (14) N 18

19 Agent s welfare under the PP regulaton mechansm s hs emsson s contrbuton to total welfare for any emsson plan. Snce each agent nternalzes the mpact of hs own emssons on total welfare gven the other agent s emssons, the PP prncple mplements the effcent emsson plan e. Indeed, gven other agent s emssons e t, the maxmzaton of agent s welfare b (e ) R d (a e + l S\ a l e t ) + N\ b (e t ) N\R d (p t ) N (b (e 0 ) d (p 0 )) wth respect to e leads to the frst-order condtons n (12) of the effcent emsson plan e. Therefore, e t = e for any e t N (t, a). Agent s equlbrum welfare s thus by (14): φ P P (a) = b (e ) d (p ) + t P P (e ) = W (a) (b (e 0 ) d (p 0 )). (15) N where W (a) = N (b (e ) d (p )). Smlarly as before, we call φp P the polluter-pays dstrbuton rule (nduced by t P P for convex damages). Agent s welfare s the ncremental contrbuton of hs emssons at the frst-best. For a vctm only agent V, t smplfes to zero snce he s fully compensated for the damage d (p ). A polluter only agent P obtans hs frst-best beneft b (e ) net of the negatve mpact of hs emssons on socety [ ] b (e ) d (p ) (b (e 0 ) d (p 0 )). Note that snce e 0 = e wth constant margnal damages for every the PP mechansm defned n (13) s a generalzaton of the one defned n (4) to convex damage functons. The next proposton shows that t P P s budget-balanced. Proposton 2 The polluter-pays mechansm s an effcent and budget-balanced regulaton mplementng the polluter-pays dstrbuton rule. Proof. It remans to be shown that t P P s budget-balanced. Snce t P P s effcent, t suffces to show N tp P (e ) 0. Note that snce e 0 s an effcent emsson plan of the problem (N, b, a, d) whle the emsson plan (e, 0) can be mplemented n (N, b, a, d), we have N [ b (e 0 ) d (p 0 ) ] d (p a e ) + [ b (e ) d (p a e ) ]. Multplyng both sdes wth -1, we combne the above nequalty wth the defnton of t P P (e) n (13) at the frst-best and use b (e 0 ) = b (0) = 0 and a = 0 for / R, and obtan: t P P (e ) d (p a e ) [ d (p ) d (p a e ) ] R 0 19

20 Summng up all transfers t P P N leads to: t P P (e ) (d (p a e ) [ d (p ) d (p a e ) ] ) N R 0 Rearrangng terms yelds: N t P P (e ) (d (p a e ) [d (p ) d (p a e )]). (16) N S 0 Consder any N. Wthout loss of generalty, let S = {1,..., s}. Snce p = S a e, we can rewrte d (p ) by: d (p ) = s k 1 [d (p a e ) d (p k=1 =1 k a e )] (17) =1 Note that for any k = 1,..., s, p k 1 =1 a e (p k =1 a e ) = a ke k = p (p a ke k ) Thus, by convexty of d, for any k = 1,..., s, k 1 d (p a e ) d (p =1 k a e ) d (p ) d (p a k e k ) (18) =1 Combnng (17) wth (18) for any k = 1,..., s leads to: s d (p ) (p k=1[d ) d (p a k e k )] = [d (p ) d (p a e )]. S By S = S 0 {}, ths s equvalent to: d (p a e ) S 0 [ d (p ) d (p a e ) ]. The last nequalty combned wth (16) leads to the desred concluson. Notce that as along as two polluters mpact the same receptors, the PP dstrbuton rule does not dstrbute total welfare n the sense that N φp P (a) < W (a). To see that, suppose that N = {1, 2, 3} wth polluter 1 and 3 pollutng only a vctm 2,.e. a 2 > 0 for = 1, 3. Then polluter 1 has to pay the ncremental damage at 2, formally d 2 (a 12 e 1 +a 32e 3 ) d 2(a 32 e 01 3 ) as well as the loss of beneft for 3, that s b 3 (e 01 3 ) b 3(e 3 ). Smlarly polluter 3 has to pay for ncrement damages at 2 and beneft loss for 1 due to hs emssons. The vctm 2 receves a compensaton equals to the damage d 2 (p 2 ). Yet, the total payment by 1 and 3 more than 20

21 offsets the compensaton to 2: t P 1 P (e ) + t P 3 P (e ) + t P 2 P (e 2 ) < 0 because tp P 1 (e ) t P 3 P (e ) > t P 2 P (e 2 ) = d 2(a 12 e 1 + a 32e 3 ). The PP regulaton mechansm exhbts a fnancal surplus and, therefore, the PP regulaton rule dstrbutes strctly less than total welfare. To characterze the PP dstrbuton rule φ P P wth margnal ncreasng damages, we ntroduce a further farness prncple, called soldarty upper bounds. Its motvaton reles on polluters mnmal clams when applyng the PP prncple. To mnmze hs payment, a polluter would clam responsblty only on the damage mpact due to hs own emsson n absence of any other polluton at each receptor R (ncludng hmself). Under ths nterpretaton of the PP prncple each agent would enoy an ndvdual welfare of max e 0[b (e ) R d (a e )] for a gven emsson plan e. On the other hand, applyng the PP prncple requres to fully compensate any agent for the damage d (p ). Wth (strctly) convex damage functons we have S d (a e ) < d (p ) = d ( S a e ) whenever S > 1, and such an nterpretaton of the polluter pays prncple would lead to unbalanced transfers. One way to reconcle a dstrbuton rule (or budget-balanced transfers) wth the above clams s to mpose that, by soldarty, no agent should get more than the clamed stand-alone welfare. Ths soldarty prncple s gven n the followng requrement. Soldarty Upper Bounds: For all problems a and all N, φ (a) max e 0[b (e ) R d (a e )]. A second, and more fundamental, ustfcaton of soldarty upper bounds fnds ts roots n Mouln s noton of group externalty (Mouln, 1990). Under ncreasng margnal damage, the presence of polluton from other sources mght reduce the ablty of a polluter to emt. Formally, let us denote by e 0 polluter s effcent emsson when s the only polluter to emt (e = 0 for every ). It s the effcent emsson plan to the problem (N, b, a, d) (where b means that all agents n N\ become vctms). It s also the soluton to the maxmzaton problem n the soldarty upper bounds property. Note that f there exst R and k S wth k, then e 0 > e : agent could pollute more n the absence of k. Dong so, under the PP regulaton mechansm t P P, he could enoy a welfare of b (e 0 ) d (p 0 ) + t P P (e 0 ) = max e 0[b (e ) R d (a e )], whch s hgher than hs welfare under the emsson plan 21

22 e. In Mouln s terms, the presence of other polluters exhbts a negatve group externalty on polluter. Soldarty upper bounds requres that every polluter who creates ths negatve group externalty should take up a share of t. For vctm only polluters V, the soldarty upper bound s equal to zero whch s ther welfare under the PP mechansm. We now provde our characterzaton of the PP prncple generalzed to ncreasng margnal damages. Theorem 2 The polluter-pays dstrbuton rule s the unque dstrbuton rule that satsfes non-negatvty, responsblty for polluton mpact and the soldarty upper bounds. Proof. Let φ be a dstrbuton rule satsfyng non-negatvty, responsblty for polluton mpact and the soldarty upper bounds. Let a be a problem, φ(a) = z and φ P P (a) = y. Suppose that z y. Note that for all V, by non-negatvty and soldarty upper bounds, z = 0 = y. Thus, there exsts N\V such that z y. Let a > a. Consder the problem where a changes to a and everythng else remans dentcal,.e. a = (a, a ). Let φ(a ) = z, φ P P (a ) = y, and e denote the effcent emsson plan for a. By RPI, z z = W (a) W (a ) = y y. Now we may take lmts,.e. lm z a + z = lm W (a) W a + (a ) and we obtan = lm a + y y, z lm a + z = W (a) lm W a + (a ) = y lm y a. + Note that lm a + max e 0[b (e ) R d (a e )] = 0. Therefore, by non-negatvty and soldarty upper bounds, both lm a + z = 0 = lm a + y. But now we obtan z = W (a) lm W a + (a ) = y, 22

23 whch contradcts z y. Second, we show that φ P P satsfes RPI, non-negatvty and soldarty upper bounds. From (15) t s straghtforward that φ P P satsfes RPI because e 0 s optmal for both (N, b, a, d) and (N, b, a, d) whenever s polluton mpact s reduced (wth a l = a l for all l N\{} and all N). Snce e 0 can be mplemented as an emssons plan n the problem (N, b, a, d), W (a) N (b (e 0 ) d (p 0 )) and, therefore, by (15), φp P satsfes non-negatvty. For soldarty upper bounds, frst note that by convexty of d, 12 we have for any e 0, d (a e ) d (a e + p ) d (p ), where p = k R\ a ke k. Therefore, for any N and e 0, b (e ) ( ) d (a e + p ) d (p ) b (e ) d (a e ). R R Maxmzng both sdes of the nequalty wth respect to e leads to: b (e ) ( d (p ) d (p a e ) ) max [b (e ) d (a e )]. (19) e 0 R R Second, snce e 0 maxmzes d (p ) + N\ (b (e ) d (p )) whle (0, e ) s a possble emsson plan for (N, b, a, d), t yelds a hgher total welfare: d (p 0 ) + (b (e 0 ) d (p 0 )) d (p a e ) + (b (e ) d (p a e )). N\ N\ Multplyng both sdes by 1, addng W (a) to both sdes, and usng the defnton of φ P P n (15) yelds: φ P P (a) b (e ) R(d (p ) d (p a e )). The last nequalty combned wth (19) shows that soldarty upper bounds holds for all N. For lnear damages, by Theorem 1, non-negatvty and responsblty for polluton mpact mply that for any problem the total welfare s dstrbuted among the agents. For ncreasng margnal damages, non-negatvty, responsblty for polluton mpact (RPI) and soldarty upper bounds mply that not for any problem the total welfare s dstrbuted among the agents. Here RPI does not mply effcency. 12 Note that d (0) = 0 and therefore, d s superaddtve: d (u) + d (v) d (u + v) for any u, v R +. 23

24 6 Concluson: PP versus VCG We conclude by comparng the PP mechansm wth the Vckrey-Clark-Groves (VCG) mechansm appled to economes wth externaltes. A VCG mechansm would make each agent pay or receve hs mpact on total welfare. Let e denote the effcent emsson plan that maxmzes the total welfare wthout defned by (b (e ) d (p )) wth p = l S\ a le l. Note that wthout here means both wthout s emsson and wthout s damage. The VCG mechansm t V CG assgns to every N: t V CG (e) = (b (e ) d (p )) (b (e ) d (p )). Under the VCG mechansm, each agent obtans the total welfare net of the welfare wthout at the frst-best. 13 Therefore, agent nternalzes the mpact of hs emsson on socety whch means that the VCG mechansm s effcent,.e. N (t V CG, a) = {e }. It leads to the VCG-dstrbuton rule φ V CG defned for every N by: φ V CG (a) = W (a) (b (e ) d (p )). Each agent N obtans the dfference between the welfare wth and wthout hm at the frstbest. In case of unlateral externaltes, for a vctm-only agent V, φ V CG (a) < 0 because s presence n the economy reduces total welfare. Therefore, non-negatvty of the dstrbuton rule nduced by t V CG s volated. Indeed, agent does not only brng hs damage d to the economy whch dmnshes welfare, t also forces polluters S to reduce ther emssons. Hence, n addton to not beng compensated for the damage d (p ), a vctm has to pay for the loss of welfare whch hs presence causes to the polluters, namely S [(b (e ) d (p )) (b (e ) d (p ))]. For a polluter-only agent P, the PP and VCG welfare concde because e 0 = e for any P. for every N\ whle d = 0 for any P. Therefore φ P P (a) = φ V CG (a) Wth multlateral externaltes polluton s a publc bad and the polluton problem s closer to the publc good provson framework n whch the Vckrey-Clark-Groves 13 In a settng wth a fnte set of alternatves and where agents reveal ther utltes, Mouln (1986) characterzed VCG mechansms by strategy-proofness (agents reveal ther true utltes) and other propertes (see also Thomson (1976) for the case of two alternatves). Note that here agents smply choose ther emsson (lke n real lfe) and do not report utlty functons and the set of emssons s nfnte. 24

25 mechansm has been put forward. Although the VCG mechansm satsfes responsblty for polluton mpact (RPI) and the soldarty upper bounds, t fals to satsfy non-negatvty. An agent adds both new emsson e and new damage d to the welfare. Agent pollutes other agents and forces n addton them to reduce ther own emssons from e to e for every S and Sk\ where k R for convex damage functon d. Therefore, under multlateral externaltes, we may have t V CG (e ) < 0. It s easy to found examples n whch the negatve mpact of hs presence t V CG (e ) to socety s not compensated by the s net beneft b (e ) d (p ) at the frst-best so that φv CG (a) < 0 for every N. Under the PP prncple, agents pay for the negatve mpact of ther emssons on socety not ther damage. They are ndeed compensated for that. There are mportant dfferences between the VCG and the PP mechansm. Although both mplement the effcent allocaton of polluton emssons as a unque Nash equlbrum, only the PP prncple dstrbutes the welfare to satsfy responsblty for polluton mpact and non-negatvty (for constant margnal damages) and soldarty upper bounds (for convex damages). Smlarly, the mechansms proposed by Duggan and Roberts (2002) and Montero (2008) allocate polluton emssons effcently. The advantage of the PP mechansm s that, n addton to achevng effcency, t dstrbutes the welfare farly n the sense of the above three requrements. Our results strongly support the use of our nterpretaton of the PP prncple n polluton envronment. 25

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