Optimal environmental charges under imperfect compliance

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1 ISSN , England, UK World Journal of Modellng and Smulaon Vol. 4 (28) No. 2, pp Opmal envronmenal charges under mperfec complance Dajn Lu 1, Ya Wang 2 Tazhou Insue of Scence and Technology, Nanjng unversy of Scence and Technology, Tazhou 2253, P. R. Chna (Receved December 18 27, Acceped March 15 28) Absrac. Ths paper proposes a modelng framework for he desgn of opmal envronmenal charges, n an envronmenal managemen problem mplcang frms wh dfferen characerscs decde on boh he level of emssons and her repors. There s an enforcemen agency whose objecve s o conrol polluons under desred levels a recepors. We show ha opmal axes can be acheved even f frms repors are no he same as her own emssons, bu f here are frms hard o monor, axes should be hgher han frms margnal revenues. Moreover, he opmal axes can be fgured ou under complee nformaon. Keywords: envronmenal chargng, mahemacal modellng, monor polcy 1 Inroducon In recen years, envronmenal proecon has been pu more and more mporance n many counres, and become a challenge for her enforcemen agences. Polluon should be conrolled under desred levels a all recepors and revenue from economy of he whole socey should no smaller han srcly necessary. Taxes and sandards are he common polcy nsrumens o regulae he envronmenal qualy. And n radonal approach hese polces are acheved by assumng ha polluers comply wh he envronmenal regulaon [1 3]. However, hs s no guaraneed. In hs paper, we ake hs suaon no consderaon, and exend radonal models by sudyng he opmal axes under he suaon where frms ry o evade he axes by provdng false nformaon of her polluon emsson o maxmze her profs, he oal of whch s assumed as revenue of he whole socey. The hypohess of frms mperfec complance was suded by Sandmo [4] and Inés and Davd [5]. Sandmo explored he condons under whch he effcency propery of axes connues o hold under mperfec complance. Based on, Inés and Davd [5] exend hs model. They suppose ha frms behavors are characerzed by boh he producvy and evason possbles, and suded wo opmal envronmenal polces n and ou of governmen s budge. By learnng frms behavors, we buld he model o desgn axes under he assumpon ha governmen s am s o conrol polluon bu axng, and we do no consder s budge. Followng he model bul by Baumol and Oaes [1], we sudy he opmal emsson axes under one recepor, a whch he polluon should be conrolled under desred level. And on he framework of modellng by Teenberg [2, 3], opmal charges under a polluon conrol sysem of mul-recepor are analyzed. Moreover, we also sudy he model under sochasc condon, n whch he monor probably for each frm s no a consan bu a sochasc varable sasfyng ceran dsrbuon. Wh complee nformaon on polluon benef funcon, monored probably and penaly funcon for every frm, even f frms are ryng o evade her coss of polluon, opmal envronmenal and emsson axes can be acheved. The more hey evade, he hgher axes, whch may be more han her margnal revenues, wll be enforced. However, n mul-recepor sysem, s possble ha envronmenal axes wll be hgher, even f all frms hold he same ably o evade from monorng. Correspondng auhor. wangya@mal.sdu.edu.cn. Publshed by World Academc Press, World Academc Unon

2 132 D. Lu & Y. Wang: Opmal envronmenal charges The paper s organzed as follows: n secon 2, we presen he heory of frms ncomplee complance and exend a new corollary for he proposon gven by Inés and Davd [5]. Secon 3 develops radonal models wh one recepor and mul-recepor. Then, we exend he model wh he analyss of sochasc suaon. And numercal examples are presen n secon 4, o llusrae how he adjusmen procedure funcons o acheve he opmal axes, and concluson s n secon 5. 2 Frms behavors In hs secon, we presen frms behavors suded by Inés and Davd [5], and gve a new corollary of her proposon, whch s necessary for he modellng of opmal axes. We suppose ha he frm chooses he level of emsson e, where e [, E]. Hence, E s he emsson level of he frm when polluon s free. If polluon a recepor s under conrol, even when all frms em freely, no axes are needed. So n he followng sudy, we do no ake hs suaon no consderaon. To conrol polluon, Emssons are axed a rae : We suppose ha s exogenously gven; s se by he enforcemen agency. The frm s benef funcon from polluon emsson e s represened as g(e). Here g( ) s ncreasng and concave: g (e) > and g (e) < for all e (, E). Also, as he framework used by Sandmo [4], g () = + and g (E) =. Ths s possble n realy: a small level of emsson has a bg margnal mpac on he frm s profs, whle margnal profs a very hgh emsson levels are very small. If he frm s emsson level s no perfecly monored, and he frm res s bes o evade emsson ax, hen he suaons for he seng of envronmenal axes may change. We denoe by ρ he probably ha he enforcemen agency wll monor and denfy frm s rue emsson. Dfferen from Inés and Davd [5], n hs paper, ρ s he only necessary probably ha an evader s caugh. Snce he man purpose of hs paper s o sudy seng he envronmenal ax, no frm behavor, he probably ha he dffculy n deecng a volaon or fndng srong evdence ha allows he sanconng of frms s no consdered. Jus lke Inés and Davd [5], le z be he repored emsson level for he frm. Snce wll repor no more han s rue emsson level e, hen we have z e. And a penaly Θ( ) wll be mposed o hose whose repors are no denfed wh her emssons. Here Θ( ) s ncreasng and convex n he level of evason: Θ() =, Θ (x) >, and Θ (x) > for x >. Therefor, he expeced prof of he frm wh monor probably ρ, when chooses an emsson level e and repor z, can be wren as: EΠ(e, ) = g(e) z ρ[e z] ρθ(e z) To maxmze he expeced prof, he frm wll choose he opmal emsson level e and repor z under he proposon gven by Inés and Davd [5] : (a) If ρ =, hen e = E and z =. (b) If ρ (, Θ (e )+ ), where e s he opmal emsson level for he frm when do no evade he ax, hen e (e, E) as defned by followng equaon, and z =, wh: g (e) ρ ρθ (e) =. (c) If ρ [ Θ (e )+, Θ ()+ ), hen e = e and z [, e ) as defned by followng equaon: [1 ρ] = ρθ (e z). (d) If ρ Θ ()+, hen e = e and z = e. Defnon 1. ˆρ = +Θ (e ) and ˆρ = +Θ (). Evdenly, when (a): ρ ˆρ, hen he frm wll repor, and emsson more han radonal opmal emsson; when (b): ˆρ ρ ˆρ, hen frm wll repor less han rue emsson level, bu em he same as n radonal model; when (c): ρ ˆρ, wll perfecly comply. Based on he above hypohess and Inés and Davd sudy, we ge he followng concluson: Corollary 1. ˆρ and ˆρ are ncreasng n. Proof. We check he frs order condon: ˆρ = Θ (e ) Θ (e )e ( + Θ (e )) 2 ˆρ = Θ () (Θ () + ) 2 WJMS emal for conrbuon: subm@wjms.org.uk

3 World Journal of Modellng and Smulaon, Vol. 4 (28) No. 2, pp Evdenly, ˆρ >. Snce e = argg () and g( ) s concave, hen e () s decreasng wh, ˆρ >. So wh he ncreasng of, frms complance decrease. 3 Mahemacal model and analyss In hs secon, we exend radonal models by consderng frms mperfec complance decson. Models n hree knds of suaons are suded: when he polluon conrol sysem conans one recepor, mulrecepor and when he monor probably s no a consan for ceran frm bu sasfyng some dsrbuon. 3.1 One recepor In hs secon, when here s only one recepor, we frs analyze an deal model, of whch he characerscs of all frms are he same, such as her benef funcons and he governmen s monor probably for each hem. And hen we generalze hs model, so ha all he characerscs of frms are dfferen. In boh of he models, we do no ake no consderaon of frms dfferen conrbuons o he polluon concenraon of hs recepor, whch s smlar as he suaon of mul-recepor, and we wll sudy n nex secon. We suppose ha he governmen s am s o maxmze he oal economy revenue of he whole socey, whch n hs paper s consdered as he oal revenue from all he frms, and conrol polluon n each recepor. Le = 1,..., n be frms as sources of emsson. Frm chooses he level of emsson e, where e [, E ]. Each frm s benef funcon from polluon emsson e are represened as g (e). So, o make he polluon under desred level a he recepor n an effecve way, we have he followng model: Max s.. g (e) e Q Where, Q s he lmaon of polluon n hs recepor. Snce n an deal model, all frms characerscs are he same, hen g 1 (e 1 ) = g 2 (e 2 ) =... = g n (e n ) = g(e), ρ 1 = ρ 2 =... = ρ n = ρ and E 1 = E 2 =... = E n = E. Make he emsson ax for frm. Followng he four suaons of Inés and Davd [5], s evden ha: In suaon (c) and (d), when ρ > ˆρ, frms may repor less han her rue emssons, envronmenal condon s under conrol even usng he radonal model. So, = = g (e) = τ. Here τ s Lagrangan facor, and hough as he envronmenal ax n mul-recepor sysem. Snce he purpose of envronmenal agency s o proec envronmen bu o ax, I suppose n he process of seng envronmenal ax, complance wll no be consdered and he desred envronmenal qualy s he only am of conrollng polluon for governmen. So, we jus need o dscuss he frs wo suaons. In suaon (b), agency s problem can be wren as: Max s.. g (e) = ng(e) ne Q g (e) ρ ρθ (e) = e E Because g(e) s ncreasng wh e, when Q < nq, by K-T condon, we can ge he opmal ax as: τ = = g (Q /n) ρθ (Q /n) ρ WJMS emal for subscrpon: nfo@wjms.org.uk

4 134 D. Lu & Y. Wang: Opmal envronmenal charges Ths s dfferen from radonal model, of whch = = g (Q /n) = τ. Compared wh radonal model, n hs suaon, he opmal envronmenal ax and emsson ax are no only relave o frms benef funcons bu also governmen s monor probably and penaly funcon for evason. We draw he followng concluson: Proposon 1. When < ρ < ˆρ, emsson ax wll more han frms margnal benef, so ha he polluon a hs recepor s under conrol. Proof. Suppose and τ are he opmal emsson ax and envronmenal ax respecvely n radonal model, hen g (e )τ =. τ Because ρ (, Θ (e )+τ ), hen τ = = [g (e ) ρθ (e )]/ρ, so τ = > [τ (Θ (e ) + τ )]/τ Θ (e ) = τ Tha s when here s frm whose monor probably s no large enough, o conrol he envronmen qualy, he envronmenal ax wll be larger han frm s margnal revenue (MR). In suaon (a),when here s no any monor a all (ρ = ), frms emssons are he same as a consan E, ndependen of emsson ax. Ths means no maer wha he envronmenal ax s, frm wll em as more as hey can and repor o ge he larges prof. In such a suaon, he sraegy of seng envronmenal ax means nohng. I needs more sraegy o conrol he envronmen. So, n he followng chapers, I do no ake hs suaon, where here s such frm, no consderaon. When all frms characerscs are dfferen from each oher, her behavors wll be dfferen because ˆρ for each of hem are no he same, and her emssons wll vary. In hs suaon, o desgn he opmal emsson and envronmenal ax, governmen should frs consder frms behavor. From Inés and Davd [5], we know when governmen s monor probably for frm sasfes: ρ < ˆρ, frms emsson wll larger han e, ha s: e = { e ρ ˆρ ρ < ˆρ Here, sasfes he funcon g ( (ρ )) ρ ρ Θ(ρ ) =, ρ (, ˆρ ], and () = E. e subjec o g (e ) =, and ˆρ sasfes ˆρ = +Θ (e ). When, he emsson ax s desgned as, he oal concenrae of polluon a he recepor s: e R = ( + e ) ρ < ˆρ ρ > ˆρ The frs erm llusraes he oal polluon of hose who wll pollue more han n radonal model. And he second represens hose who wll no pollue more bu may also evade ax. So he governmen s problem can be descrbed as: Max g(e ) s.. e R Q Because s ncreasng wh frms emsson e, o solve hs problem, we can also learn from K-T condon ha he opmal ax sasfes he followng funcons: ρ <ˆρ e g ( g (e ) = = τ ˆρ = /[Θ (e ) + ] () = E + ρ >ˆρ e = Q (ρ )) ρ ρ Θ ( (ρ )) = Ths model mproves he frs deal model and we acheve he exenson of proposon 1: WJMS emal for conrbuon: subm@wjms.org.uk

5 World Journal of Modellng and Smulaon, Vol. 4 (28) No. 2, pp Proposon 2. When here exss frms whose monor probables are no large enough, o conrol he envronmen qualy, he opmal ax s larger han ha n radonal model. Proof. Snce n he frs erm of oal emsson, frms rue emssons sasfy he funcon g ( (ρ )) ρ ρ Θ ( (ρ )) =, > e, and er > e under he same ax rae. Because e/ τ <, hen we ge he proposon. 3.2 Mul-recepor In hs secon, we sudy a sysem of mul-recepor. Generally, for beer monor he envronmenal qualy, envronmenal enforcemen always se more han one recepor o monor he envronmenal qualy. So ha governmen s problem s o fnd a seres of envronmenal axes a hese recepors and make he balance of revenue and envronmen. Le m be he number of receporsj = 1,..., m represen each recepor. Then he problem for envronmenal enforcemen s: Max g (e ) s.. α,j e Q j (j = 1,..., m) Here, Q j s he upper lm of polluon level a recepor j, α,j s frm s emsson ransfer coeffcen o recepor j. In radonal model, g (e ) = = m j=1 λ jα,j. Here, λ j s he shadow prce a recepor j. Le τ j = λ j, hen λ j s he envronmenal ax a recepor j, whch s relave o he emsson axes for frms. When all frms are ryng o evade ax, her oal emsson a recepor j wll be: e R j = α,j + α,j e ρ<ˆρ τ, ρ ˆρ τ, Here, ˆρ τ, = [ j τ jα,j ]/[Θ (e ) + j τ jα,j ], and e, e are smlar as dscussed above, subjec o he followng funcons respecvely: g (e ) = = m j=1 α,jτ j and g (e ) m j=1 α,jτ j ρ ρ Θ ( ) =. Then he problem for he governmen s: Max g (e) s.. e R j Q j j = 1,..., m Ths problem can be solved by eraon problem suded by Y. Ermolev, G. Klaassen and A. Nenjes [7, 8]. Le τ = (τ 1,..., τ m) be he nal vecor of envronmenal axes a m recepors, and τ k = (τ k 1,..., τ k m) be he vecor of envronmenal axes a m recepors n sep k eraon. Frm wll change s emsson e k accordng o envronmenal τ k and emsson ax k, o acheve s expeced prof: EΠ (e, ) = g (e ) z ρ [e z ] ρ Θ(e z ) Because he governmen know polluers nformaon compleely, hey can adjus he ax accordng o he followng eraon, by calculang her rue emssons. In whch, τ k+1 j = max{, τ k j + γ k ( e k = e k α,j Q j)} { e k, ρ ˆρ k e k, ρ < ˆρ k WJMS emal for subscrpon: nfo@wjms.org.uk

6 136 D. Lu & Y. Wang: Opmal envronmenal charges Here, ˆρ k = [ j τ j kα,j]/[θ (e k ) + j τ j kα,j], and e k and e k sasfy respecvely he funcon: g (ek ) = = m ) m ) =. γ k s he sep sze of eraon facor, k j=1 α,jτj k and g (ek j=1 α,jτj kρ ρ Θ (e k sasfyng γ k >. Evdenly, when τj k + γ k( n ek α,j Q k+1 j ) <, here s τj =. The he convergence of vecor τ k = (τ1 k,..., τ m)k k s relave o γ k. Learnng from [7], when seres {γ k } sasfyng γ k >, k= γ k = and k= γ2 k 3.3 Improvemen for monor probables <, hs mehod s effecve. The above model s bul on he assumpon ha he monor probably for each frm s dfferen from each oher bu consans. In hs secon, we exend he above model by consderng ha he probables are no consan, bu sasfyng he same sochasc dsrbuon. Suppose he governmen s monor probables for each frm s sochasc and ndependen. They subjec o he same dsrbuon. Because hese frms are under he same governmen, he assumpon of he dsrbuon s reasonable. Oher condons are he same as suded before. Le x be he sochasc varable of monor probably of frm, and f(x ) be he densy funcon. I s evden ha f(x ) = ρ, and le F (x ) be he dsrbuon funcon. Le x [x, x ] = σ be he doman of varable x, when ρ ˆρ. Therefore, when frm, = 1,..., n ryng o evade ax, her expecaon emsson s: Tha s: Ee = Ee = x x ˆρ In hs suaon, he problem for governmen s: (x)f(x)dx + [1 F (σ )]e (ρ )ρ dρ + [1 F (σ )]e Max s.. g (E(e )) α,j E(e ) Q j j = 1,..., m To solve hs problem, we need o know he relaonshp beween frms expeced emsson Ee, and governmen s ax. We have he followng concluson: Proposon 3. Under model s presupposon, he expecaon of frm s rue emsson s decreasng wh emsson ax, alhough he s ryng o evade ax. Proof. Assume here are wo sandard of emsson ax: 1 2, and 1 < 2. Then Ee 1 = ˆρ 1 1 (ρ)ρdρ + [1 F (σ 1)]e 1, and Ee 2 = ˆρ 2 2 (ρ)ρdρ + [1 F (σ 2)]e 2. Because ˆρ = Θ (e )+ s ncreasng wh, ˆρ 1 < ˆρ 2. So σ 1 < σ 2, hen Ee 1 and Ee 2 can be wren n he followng forms respecvely. Ee 1 = ˆρ 1 1 (ρ)ρdρ + ˆρ 2 ˆρ 1 e 1 (ρ)ρdρ + [1 F (σ 2)]e 1. Ee 2 = ˆρ 1 2 (ρ)ρdρ + ˆρ 2 ˆρ 1 2 (ρ)ρdρ + [1 F (σ 2)]e 2. So Ee 1 Ee 2 = ˆρ 1 (e 1 e 2 )ρdρ + ˆρ 2 ˆρ 1 (e 1 e 2 )ρdρ + [1 F (σ 2)](e 1 e 2 ). Le he hree erms of he above funcon be A1, A2, A3. Evdenly A1 > and A3 >. Because frms emsson s decreasng wh ax, when he monor probably s ceran, we know when ρ s fxed, (e 1 e 2 ) >, hen A2 >, hence Ee 1 > Ee 2. Accordng o hs proposon, o desgn he opmal ax, eraon mehod can also be used. The nal vecor of envronmenal axes a recepors s τ = (τ 1,..., τ m), and he adjusmen procedure follows: WJMS emal for conrbuon: subm@wjms.org.uk

7 World Journal of Modellng and Smulaon, Vol. 4 (28) No. 2, pp τ k+1 j = max{, τ k j + γ k ( E(e k )α,j Q j)} Here, {γ k } sll sasfes he condons: γ k convergency of eraon. >, k= γ k = and k= γ2 k <, o guaranee he 4 Numercal analyss In hs secon, we gve wo numercal examples o llusrae models gven above, and o compare our models and radonal model. There are wo pars of hs secon. The frs par gves opmal ax under radonal model, of whch here s no evason by frms, and we ge he crcal value of governmen monor probably ˆρ for each frm. In second par, we make a numercal analyss of opmal ax, when frms are ryng o evade ax. 4.1 A numercal example of radonal model We consder such suaon ha here are 5 frms who em n a dsrc. Ther upper lms of emsson are as follows: E 1 E 2 E 3 E 4 E The benef funcons of hese frm are g(e ) = a [ 1 e 1 Q ] = 1,..., 5 Here a sand for a 1 a 2 a 3 a 4 a Four recepors are se n hs dsrc o conrol polluon. Here we jus dscuss pon polluon. The upper lms of polluon a each recepor are: And he ransfer coeffcen α,j sasfy: Q 1 Q 2 Q 3 Q α,j = 1 = 2 = 3 = 4 = 5 j = j = j = j = Lack of real daa, all he daa above are chosen randomly by compuer n a reasonable range. We acheve emsson axes for frm ( = 1,..., 5): Then he opmal emsson for each frm are: e 1 e 2 e 3 e 4 e Suppose he dfferenal funcon of governmen s penaly funcon for frms evason s Θ (e z) = 2[e z]. WJMS emal for subscrpon: nfo@wjms.org.uk

8 138 D. Lu & Y. Wang: Opmal envronmenal charges Then we have he crcal value of ˆρ (e ): ˆρ 1 ˆρ 2 ˆρ 3 ˆρ 4 ˆρ When ρ ˆρ, frms may be sll no complance wh axng polcy, bu her emssons are under conrol. In he followng, we llusrae our model by he resul above. 4.2 Numercal analyss of our model In hs secon, we use daa o llusrae he opmal ax when frms are ryng o evade axes. Frs, we suppose he monor probables for each frm are: ρ 1 ρ 2 ρ 3 ρ 4 ρ Tha s ρ < ˆρ, = 1, 4, and ρ > ˆρ, = 2, 3, 5. We acheve emsson axes for frm ( = 1,..., 5): Then he margnal revenue of each frm are: e 1 e2 e 3 e 4 e And frms rue emssons e are as follows: e 1 e 2 e 3 e 4 e Compared wh radonal model, when monor probables for some frms are no large enough, o conrol he polluon, emsson ax need o be much larger han ha n radonal model, and n hs suaon, more frms choose o evade ax. In hs example, all frms rue emssons sasfy e < e, ha s z = = 1,..., 5, and her repors are all. Ths proves he heores suded above. Compared wh he frms emsson, we fnd ha he emssons of frm 1 and 4 ncrease, and ohers decrease. Tha s, when monor probables are no large enough, governmen s polcy end o reduce he frms polluon, who s easy o monor. Ths s smlar as he resul n [5]. 5 Summary and conclusons Ths paper s suded on he bass of frms behavors o ax and monor polcy, and exend radonal model wh one recepor and mul-recepor. Our models are bul on he assumpon ha frms are ryng o evade axes and make false repors f s possble. Ths s more reasonable han radonal models. Mahemacal opmal models are bul, and adjusmen procedure mehod s used o solve hese opmal program. New conclusons on he relaonshp of frms emsson and emsson ax are drawn. In secon 4, numercal examples are gven o llusrae our heores. The assumpon of hs paper s on he condon of complee nformaon, however, some me, governmen can no know perfecly he benef funcons of frms. In such suaon, how o desgn he opmal ax sll needs o be furher suded. WJMS emal for conrbuon: subm@wjms.org.uk

9 World Journal of Modellng and Smulaon, Vol. 4 (28) No. 2, pp References [1] W. Baumol, W. Oaes, The Theory of Envronmenal Polcy: Exernales, Publc Oulays and The Qualy of Lfe, Englewood Clffs, NJ: Prence-Hall, [2] Tom Teenberg, Envromenal and Naural Resource Economcs, Addson Wesley Longman. 2. [3] T. Teenberg, Conrollng polluon by prce and sandard sysem: A general equlbrum analyss, Swedsh Journal of Economcs 75, , [4] A. Sandmo, Effcen envronmenal polcy wh mperfec complance, Envronmenal Resource Economcs, 23 (22) [5] Inés Macho-Sadler, Davd Pérez-Casrllo, Opmal enforcemen polcy and frms emsson and complance wh envronmenal axes, Journal of Envronmenal Economcs and Managemen 51, 26, [6] S. Clemhou, H. Y. Wan, Dynamc common propery resources and envronmenal problems, Journal of Opmzaon Theory and Applcaons 46(4), 1985, [7] Y. Ermolev, G. Klaassen, A. Nenjes, Incomplee nformaon and he cos-effcency of amben charges, WP-93-72, IIASA, Laxenburg, Ausra, [8] Y. Ermolev, G. Klaassen, A. Nenjes, Adapve cos-effecve amben charges under ncomplee nformaon, Journal of Envronmenal Economcs and Managemen 31, 37-48, WJMS emal for subscrpon: nfo@wjms.org.uk