Introduction. 1. The Model

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2 H23, Q5

3 Introducton In the feld of polluton regulaton the problems stemmng from the asymmetry of nformaton between the regulator and the pollutng frms have been thoroughly studed. The semnal works by Wetzman (1974) and (1978) focused on the relatve performance of prce and quantty nstruments when beneft or abatement cost functons are mperfectly known and on ths lne of research another mportant contrbuton was Roberts and Spence (1976) proposal of a med ta and lcences scheme. A dfferent approach was poneered by Kwerel (1977) who proposed a mechansm to nduce truthful revelaton of abatement cost functon by pollutng frms: t encompasses ssung the optmal number of transferable lcences and payng a subsdy for lcences hold n ecess of emssons. Some shortcomngs of ths scheme were evdenced by Dasgupta Hammond and Maskn (1980) who proposed alternatve solutons drawng heavly on the lterature on ncentve compatble mechansms for the provson of publc goods. In the followng decades several works, ngenous and theoretcally elegant, addressed the problem n dfferent contets: among them, to cte only few, Varan (1994), Duggan and Roberts (2002) and Montero (2008). Notwthstandng the relevance of such theoretcal results one cannot refran from notng, wth Chavez and Stranlund (2009, p.138) or Montero (2008, p.497), that n the real world none of the most sophstcated mechansms has never been mplemented. Startng from ths premse ths paper adopts a pragmatc perspectve nvestgatng the robustness of the ncentve for pollutng frms to report a false margnal abatement cost functon and presentng a very smple mechansm wth nterestng propertes: the basc dea behnd ths mechansm s that the regulator s free to choose ether an effluent fee or a standard but the pollutng frms don t know n advance whch nstrument wll be chosen when they are requested to communcate ther margnal beneft functon (or margnal abatement cost functon). Its theoretcal underpnnngs are found n the well known Wetzman (1974) result that prce and quantty nstruments for polluton control acheve opposte outcomes when there s uncertanty about margnal abatement cost functons; another mportant ngredent s drawn from a paper by ulckaen (1997) where t s demonstrated that, under an effluent fee, the gan for a pollutng frm to hde ts true abatement cost functon s not unbounded f the frm s commtted to behave n accordance to the reported abatement cost functon. 1. The Model Ths secton presents some results whch can also be found n Kwerel (1977) or ulckaen (1997); they are repeated here to ease the readng of the paper. There are N pollutng frms wth cost functons ndcated by: (,, ˆ C y ) =1,..,N where y s output, polluton, supposed to be verfable by the regulator, and ˆ a parameter known only to the frm; they are obvously ncreasng n y and t s assumed they are decreasng and conve n. The correspondng margnal abatement cost functons, suppressng the output varable, are ndcated by: C C (, ˆ ) =1,..,N They are negatve and ncreasng: 2 C 2 C ˆ (, ) 0 =1,..,N In contrast wth most of the lterature t s chosen to deal wth the correspondng margnal beneft functons whch are the opposte of the margnal abatement cost functons and wll be denoted by: C (, ˆ ) =1,..,N

4 They are postve (so t s more ntutve to defne over or under reportng) and decreasng, that s: 2 (, ˆ C ) 0 =1,..,N 2 The damage functon, whch s assumed to be common knowledge, s: D ( ) wth N 1 It s ncreasng and conve for >0, that s: D D D ( ) ( ) 0 D ( ) ( ) The regulator has the objectve to mamze socal net beneft and asks each frm to communcate ts beneft functon (whch can be dfferent from the true functon) ndcated by: (, ) =1,..,N Aggregate mamum beneft functon s the soluton to the followng program: MA (, ) s.t. and the resultng functon wll be ndcated by: (, θ) θ,.., ) Its dervatves are: 0 where 0 ( 1 N 0 =1,..,N s the effect on margnal beneft functon of a reported parameter dfferent fromˆ. Mamum socal net beneft s: MA (, θ ) D( ) Optmal polluton level () s gven by the frst order condton: D ( ) (, θ) (1) To acheve ths and the correspondng optmal polluton levels by each frm, the regulator can set an effluent fee or a standard accordng to the followng condtons:

5 t( ( θ)) D ( ( θ)) ( ( θ), θ) (2) D ( ( θ)) (, ) ( ( θ), θ) (3) y the mplct functon theorem we get from (1): d d D 0 =1,..,N (4) From (2), usng (1) we get: dt D d d D d D =1,..,N (5) If the regulator sets an effluent fee, by dfferentatng wth respect to the equlbrum condton at ndvdual frm level t ( θ ) ( ( θ ), ), t s obtaned: dt d =1,..,N d d hence, by substtuton of (5): d 1 d dt d D 0 =1,..,N (6) Reported parameter s determned by optmzng behavour of pollutng frms. If the regulator sets a standard t requres: MA ( ( ), ˆ θ ) =1,..,N wth frst order condton: ( ˆ d ) 0 =1,..,N (7) d Gven (6), condton (7) requres ( ˆ) 0 mplyng an over reportng of the true beneft functon n order to let the regulator choose current (unregulated) polluton level, ndcated by 0. If the regulator sets an effluent fee the objectve s: MA ˆ ( ( θ), ) t( θ) ( θ) =1,..,N and the resultng frst order condton for s: ( ˆ d dt d ) t 0 =1,..,N (8) d d d Substtutng (5) and (6) nto (8) we get:

6 D ( ˆ ) ( ) D 0 =1,..,N hence: ˆ ( ) ( ) D 0 =1,..,N (9) These results confrm, as s well known snce Kwerel (1977), that pollutng frms have, n ether case, an ncentve to msrepresent ther true beneft functon. In partcular, condton (9) mples that for ˆ the margnal beneft from under reportng s strctly postve; nonetheless, as ponted out n ulckaen (1997), the calculaton of the optmal level of reported parameter requres the knowledge of the slope (at ) of the margnal damage functon and the slope (at each ) of margnal beneft functons of all other frms thus showng a crucal role played by the nformaton avalable to each frm. 2. Proposed Mechansm Gven the conclusons of the prevous secton t s nterestng to evaluate the performance of a very smple mechansm whch has a partcular feature whch makes t dfferent from any other mechansm proposed n the lterature: the nformaton provded by the frm wll be used by the regulator, accordng to ts objectve, to determne ether an effluent fee per unt of emsson or a standard (whch can also be nterpreted as a number of unmarketable permts) but the frms don t know n advance whch nstrument wll be chosen by the regulator. Assumng as reasonable that each frm assgns an equal probablty to each nstrument, ts epected net beneft, as a functon of the reported parameter, s gven by: ˆ E ( θ) 1/ 2 2 ( ( θ), ) t( θ) ( θ) and the frst order condton for a mamum, after some straghtforward substtutons, s: de d ˆ 2( ) ( ) D 0 =1,..,N (10) Calculatng (10) for ˆ, the condton for under or over reportng s obtaned: ˆ f (ˆ ) D 0 (11) The reader wll notce a resemblance wth the well known results of Wetzman (1974): here, for a gven margnal beneft functon, the ncentve to under (over) report s stronger the steeper (flatter) the margnal damage functon and the flatter (steeper) the margnal beneft functon wth respect to the reported aggregate functon. A partcular feature of the proposed mechansm emerges: the same nformaton that n ulckaen (1997) are requred to compute the optmal level of under reportng are now necessary to determne whether t s optmal to under or over report: the uncertanty about the nstrument chosen by the regulator balances the ncentve to under report the true beneft functon when an effluent fee s selected and the opposte ncentve f a standard s chosen. The above condton (11) can be reformulated n a slghtly dfferent manner as follows: de (θ) d ˆ D

7 The term D, computed at the optmal level of, s the relatve slope of margnal socal damage and aggregate beneft functons and ts absolute value s greater equal or less than 1: let s ndcate ts absolute value by α. Smlarly let s ndcate by the absolute value of whch s greater equal or less than 1: The above epresson can thus be reformulated as: de ( θ) d ˆ 1 (12) Thus the choce to under or over report the true beneft functon s determned by the nteracton of and : the former s greater than 1, for any gven level of margnal beneft, the greater the correspondng level of and the steeper the margnal beneft functon; alternatvely, for any gven level of, the steeper the margnal beneft functon and the lower the level of correspondng margnal beneft. The latter s greater than 1 the steeper the margnal socal damage functon wth respect to the aggregate margnal beneft functon. Further nsghts emerge under the addtonal assumpton, often found n the lterature, of lnear margnal beneft functons of the form: b 0 b =1,..,N n whch case the only parameter to communcate to the regulator s the (constant) slope of such functons gven that emssons are observable. The resultng aggregate margnal beneft functon, necessary to determne the optmal effluent fee or standard, wll be a pecewse lnear functon: f the optmal level of polluton s strctly postve for each frm the equaton of such functon s: b0 b 1 ( ) 1 1 b b (13) Condton (11) now yelds: de ( θ) b b 0 1 d b 0 =1,..,N (14) where: b0 b (15) 0 1 b Gven that s the same for each frm, epresson (14) makes more eplct than (11) that for some frms t mght be optmal to over report ther true margnal beneft functons whle for others the opposte s true. Specfcally, f mamum margnal beneft for frm s below (above) the weghted average ( 0 ) parameter s less (greater) than one; f, n addton the margnal damage functon s flatter (steeper) than the aggregate margnal beneft functon, for frm s unambguously optmal to over (under) report ts true margnal beneft functon. Conversely f the condtons for the sgn of are the same as before but those for the sgn of are reversed t mght be optmal for each frm to over or under report ts true margnal beneft functon. Interestng results emerge under the alternatve assumpton of lnear and proportonal margnal socal damage functon. Computng (10) for ˆ yelds:

8 de d 1 (16) whose sgn depends on the relatve weght of frm n total polluton and n overall curvature of the aggregate beneft functon. If a relatvely small optmal polluton level and curvature dentfes a relatvely effcent pollutng frm the above epresson requres that for a relatvely effcent (neffcent) pollutng frm t s optmal to over (under) report ts margnal beneft functon. As a result of such optmal behavour the resultng overall polluton level mght even be not so far from optmal but not the dstrbuton among pollutng frms, wth a resultng gan for relatvely effcent frms at the epense of neffcent ones n terms of abatement costs. A general concluson whch can be drawn form the results of ths secton s that, under the proposed mechansm, the decson to under or over report the true margnal beneft functon crucally depends on detaled nformaton about the margnal beneft functon of all other frms: f such nformaton s not avalable a prudent choce mght be truthful revelaton. Moreover n many crcumstances t mght be true that for some frms t s optmal to over report whle for others the opposte s true: n such cases the resultng aggregate margnal beneft functon mght also be not so far from the true one and consequently also the aggregate polluton level attaned by ths mechansm. Summary and Conclusons The paper addresses the problem of asymmetry of nformaton between a regulator and some pollutng frms startng from the recognton that none of the mechansms proposed n the lterature for a truthful revelaton of the relevant functons by the pollutng frms have been mplemented n the real world. It proposes a very smple mechansm wth some appealng propertes: ts man ngredent s the possblty for the regulator to choose, wthout communcatng n advance to the frms, among two nstruments: an effluent fee and a standard. Ths added uncertanty mples that each frm requres detaled nformaton on the margnal beneft (abatement cost) functons of all other frms n order to choose whether t s optmal to under or over report ts true functon and ths nformaton mght not be avalable or mght be costly to acqure: as a result n a real world settng ths nformatonal gap mght nduce frms to a truthful revelaton. Under the addtonal assumptons of lnearty of margnal beneft or margnal socal damage functons t s demonstrated that, n many cases, the resultng optmal behavour mght be under reportng for some frms and over reportng for others so that the resultng margnal aggregate beneft functon mght be not so far from the true one: consequently aggregate polluton level attaned by the mechansm mght be not so far from the optmal one but not the dstrbuton of ths aggregate among pollutng frms: n other words an almost optmal aggregate polluton level mght be reached wth a not cost mnmzng dstrbuton among pollutng frms; fnally, f there s a suffcently sharp dstncton among relatvely effcent and neffcent frms the resultng dstrbuton of polluton favours the formers n terms of abatement costs. References ulckaen, F., (1997), Emssons Charges and Asymmetrc Informaton: Consstently a Problem?, Journal of Envronmental Economcs and Management, Vol.34, pp Chavez, C. and Stranlund, J.K,. (2009) A Note on Emssons Taes and Incomplete Informaton, Envronmental and Resource Economcs, Vol. 44, pp Dasgupta, P., Hammond, P. and Maskn, E., (1980 On Imperfect Informaton and Optmal Polluton Control, Revew of Economc Studes, Vol. 47, pp Duggan, J. and Roberts, J., (2002) Implementng the Effcent Allocaton of Polluton, Amercan Economc Revew, Vol. 92 pp Kwerel, E., (1977) To tell the Truth: Imperfect Informaton and Optmal Polluton Control, Revew of Economc Studes, Vol. 44, pp Montero, J.P., (2008) A Smple Aucton Mechansm for the Optmal Allocaton of the Commons, Amercan Economc Revew, Vol. 98 pp

9 Roberts, M.J. and Spence, M., (1976) Effluent charges and lcences under uncertanty, Journal of Publc Economcs, Vol. 5, pp Varan, H.L., (1994) A Soluton to the Problem of Eternaltes when Agents are Well-Informed, Amercan Economc Revew, Vol. 84 pp Wetzman, M.L., (1974) Prces vs Quanttes, Revew of Economc Studes, Vol. 41, pp Wetzman, M.L., (1978) Optmal Rewards for Economc Regulaton, Amercan Economc Revew, Vol. 68 pp

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