Optimal use of a polluting non renewable resource generating both manageable and catastrophic damages

Transcription

1 Opimal ue of a polluing non renewable reoure generaing boh manageable and aarophi damage Jean-Pierre Amigue, Mihel Moreaux and Kaheline Shuber Thi draf: January, 27h 21 We hank Valérie Nowazyk for ehnial aiane Touloue Shool of Eonomi(INRA, LERNA), Manufaure de Taba, 21 allée de Brienne, 31 Touloue, Frane. [email protected] Touloue Shool of Eonomi (IDEI, IUF and LERNA), Manufaure de Taba, 21 allée de Brienne, 31 Touloue, Frane. [email protected] Pari Shool of Eonomi and Univeriy Pari 1 Panhéon - Sorbonne, Maion de Siene Eonomique, bd de l Hôpial, Pari. [email protected]

2 Conen 1 Inroduion 1 2 The model Aumpion The oial planner problem Shadow o of arbon, ariy of he renewable reoure and ompeiivene of he non-renewable one Cae of an abundan olar energy Cae of a rare olar energy Marginal hadow o of he polluion ok along he opimal pah The renewable reoure i abundan Bu i i more expenive han he non-renewable reoure a he eiling And i i heaper han he non-renewable reoure a he eiling 13 4 The renewable reoure i rare And i i more expenive han he non-renewable reoure a he eiling Bu i i heaper han he non-renewable reoure a he eiling Alhough no heap enough o be exploied before he eiling And heap enough o be exploied before he eiling Time profile of he arbon hadow o and omparaive dynami indued by more evere damage Comparaive dynami indued by more harmful manageable damage Comparaive dynami aben he euriy onrain Comparaive dynami under an effeive eiling onrain Comparaive dynami indued by a more ringen euriy onrain Conluion 26 APPENDIX 28

3 1 Inroduion We are now reaonably ure ha greenhoue gaz aumulaion in he amophere i he aue of global warming, and ha he emperaure inreae reae damage o eoyem and eonomi aiviie. However, he preie relaionhip beween arbon onenraion and damage, he damage funion, i no well known, epeially for imporan inreae of emperaure, when irreveribiliie and non-lineariie are likely o our. For inane, he exenive urvey of he available eimaion of damage in he Sern Review (Sern [26]) highligh he exen of our ignorane a far a he damage funion i onerned. So i make ene o found environmenal poliy no only on he value of unerain marginal damage bu alo on he requiremen o mainain amopheri arbon onenraion under a erain eiling, hoen preiely o preven an exeive rie of he emperaure. Previou work were foued eiher on damage due o arbon aumulaion (Ulph and Ulph [1994], Hoel and Kverndokk [1996], Tahvonen [1997]) or on a onenraion arge (Chakravory, Magné and Moreaux [26a], [26b]). Bu here exi for ure damage before he eiling, o we ombine in hi model a onenraion eiling wih a damage funion under he eiling. A he onenraion eiling i impoed preiely o avoid aarophi diurbane of he limae and hene he eonomy, i make ene o aume ha damage remain relaively mall under he eiling, oherwie he eiling would have obviouly been ill-hoen, and ha hey are relaively well known. We onider a model in whih he energy requiremen of oiey an be aified by wo kind of reoure: a non-renewable one, relaively heap bu polluing, le ay oal, and a renewable one, more expenive bu lean, le ay olar. We hooe a imple framework where he damage funion i linear for he mall damage ouring before he eiling, hi lineariy aumpion ogeher wih he lineariy of arbon aumulaion in he arbon ok 1 enabling u o obain lear reul. We are inereed in he ime pah of exraion of he polluing non-renewable reoure, and of peneraion of he renewable one, bu alo in he iniial value and he ime profile of he arbon hadow o, and in he eniviy of hee variable o he ineniy of he marginal damage and he ringeny of he eiling. In a deenralized e-up, he ounerpar of he arbon hadow o would be a uni arbon ax 1 Thi lineariy aumpion, whih mean ha he naural regeneraion rae of he amophere i onan, i learly reriive and i queioned boh by empirial work and he heoreial lieraure, ee e.g. Farzin and Tahvonen [1996] and Toman and Wihagen [2]. We keep i in hi paper beaue naural regeneraion i no a he hear of he queion we wan o adre. 1

4 applied o he non-renewable reoure onumpion in order o implemen he opimal pah. The iniial value and he ime profile of hi arbon ax are a he hear of limae poliy, a well a i eniiviy o marginal damage and o he amopheri arbon onenraion objeive. Moreover, inuiion ugge ha aking ino aoun damage due o he onumpion of oal and impoing a eiling onrain on arbon onenraion hould aelerae he peneraion of lean renewable ubiue o he non-renewable polluing energy. We wan o know if hi inuiion i orre in our framework where here i no endogenou ehnial progre and where he whole oal ok will be ulimaely exraed and onumed. The main reul are he following. Fir, o maer a lo 2. Whaever abundan or rare, if he renewable reoure i more expenive han he non-renewable one a he eiling, i will never be ued before he end of he phae a he eiling, when he nonrenewable reoure beome o are ha he eiling will never be reahed again. On he onrary, if he renewable reoure i le expenive han he non-renewable one a he eiling, i may even be uffiienly heap o be exploied before he eiling, in whih ae fir he non-renewable reoure i exploied alone, eond boh reoure are ued ogeher before, a and afer he eiling, and finally he renewable reoure i exploied alone, afer he exhauion of he non-renewable one. Seond, in all onfiguraion of o, he arbon hadow value i fir inreaing unil he eiling i reahed, hen i dereae during he phae a he eiling, and finally i abilize a a onan value afer he eiling. The iniial arbon hadow o i an inreaing funion of he value of he marginal damage, and a dereaing funion of he eiling: he higher he eiling, ha i he le ringen he onenraion arge, he lower he iniial arbon hadow o. Laly, onrary o he inuiion, higher marginal damage or a more ringen enenraion arge indue a delay in he peneraion of he lean renewable reoure and alo a delay in he omplee raniion oward lean energy. The paper i organized a follow. Seion 2 preen he model. Seion 3 and 4 are devoed o he udy of he ime pah of exraion of he non-renewable reoure and ue of he renewable one in he wo ae of an abundan and a are renewable reoure. Seion 5 preen he ime profile of he arbon hadow o. Seion 6 onlude. 2 Alhough i i very diffiul o give preie figure for he repeive marginal o of he renewable and he non-renewable reoure. 2

5 2 The model 2.1 Aumpion We onider an eonomy in whih he energy need an be aified by exploiing wo kind of primary reoure, a polluing non-renewable reoure and a lean renewable one. Le oal be he diry reoure and olar be he lean one. Denoe by X() he available ok of oal no ye exploied a ime and by X,X >, he oal iniial endowmen: X X(). Le x() be i inananeou exploiaion rae: Ẋ() = x(). I average exploiaion o i aumed o be onan hene equal o i marginal o, and i denoed by x. Burning oal o produe energy generae a flow of new amopheri polluion proporional o he inananeou exploiaion rae. Le ζ be he uniary polluion onen of oal, o ha he flow of newly generaed polluion amoun o ζx(), adding o he amopheri arbon ok denoed by Z(). The amophere ha ome elf regeneraion apaiy we aume o be proporional o he polluion ok Z(). Le α be hi proporional rae of elf regeneraion. The dynami of he arbon ok i given by he differene beween he flow of newly produed polluion and he elf regeneraion effe: Ż() = ζx() αz(). Le y be he available flow of he renewable reoure and y() i exploiaion rae a ime : y y(). I average exploiaion o i aumed o be onan and i denoed by y. The o x and y inlude all he o neeary o deliver energy direly uable by he uer, ha i he exraion, proeing and delivery o, o ha he wo primary reoure may be een a perfe ubiue, and oal energy onumpion q() an be defined a: q() = x() + y(). The ondiion on x, y and y under whih i i opimal o exploi he diry reoure are deailed in ub-eion 2.3 below. The inananeou onumpion q generae a gro urplu u(q) (meaured in moneary uni). The gro urplu funion i aumed o be wie differeniable, rily inreaing, rily onave and aifying he Inada ondiion lim q u (q) = +. We denoe by p he marginal gro urplu u 3

6 and by q d he demand funion, he invere of u. The amopheri arbon onenraion reae damage. Thee damage are of wo kind. Fir he arbon ok generae a eah ime a manageable welfare lo. To implify we aume ha hi welfare lo i proporional o he urren arbon onenraion Z(). Le hz(),h >, be hi inananeou manageable damage. Seond, would he arbon onenraion be higher han ome eiling Z, oiey would experiene a drai hange of he limae regime leading o aarophi damage. Suh an even an be een a an infinie welfare lo. In order o preven uh an ouome, oiey ik o he objeive of keeping he arbon onenraion Z() permanenly a mo equal o he euriy level Z. Z i of oure greaer han he iniial arbon onenraion Z : Z < Z. All he urplu, o and manageable damage are diouned a ome onan poiive oial rae of dioun ρ. 2.2 The oial planner problem The oial planner problem i o deermine he poliy maximizing he um of he diouned ne urplu while aking are of he euriy onrain, and read: W(X,Z ) = max Ẋ() = x() x() X() = X > given Ż() = ζx() αz() e ρ {u(x() + y()) x x() y y() hz()} d Z() = Z > given, Z > Z, Z Z() y > given, y y(), y(). Denoing by λ() he oae variable of X(), by µ() he oppoie of he oae variable of Z(), by ν() he Lagrange muliplier aoiaed o he eiling onrain and by γ x, γ y and γ y he Lagrange muliplier aoiaed o he poiiviy onrain, he urren value Lagrangian i given by: L = u(x() + y()) x x() y y() hz() λ()x() + γ x ()x() µ() (ζx() αz()) + ν() ( Z Z() ) + γ y() (y y()) + γ y ()y() 4

7 Fir order opimaliy ondiion and omplemenariy lakne ondiion read: u (x() + y()) = x + λ() + ζµ() γ x () (1) u (x() + y()) = y + γ y() γ y () (2) γ x (), x(), γ x ()x() = (3) γ y(), y y(), γ y() (y y()) = (4) γ y (), y(), γ y ()y() = (5) λ() = ρλ() (6) and he ranveraliy ondiion are: µ() = (ρ + α)µ() h ν() (7) ν(), Z Z(), ν() ( Z Z() ) = (8) lim e ρ λ()x() = (9) lim e ρ µ()z() =. (1) A uual wih a onan marginal exploiaion o of he non-renewable reoure, he Hoelling rule redue o a ariy ren of oal λ() having o grow a a proporional rae equal o he rae of dioun. From (6) we ge: λ() = λ e ρ where λ λ(). Thu he ranveraliy ondiion (9) may be rewrien a: λ lim X() =. If oal ha ome value, ha i λ >, hen he oal ok mu be exhaued in he long run. Iniially, he arbon onenraion i below he eiling ine Z < Z. Hene on any iniial ime inerval [,) during whih Z() < Z, ν() = o ha equaion (7) beome µ() = (ρ+α)µ() h and an be inegraed ino: where µ µ(). µ() = [ µ h ρ + α ] e (ρ+α) + h, [,), (11) ρ + α Noe ha if he manageable damage effe i aben, ha i if h =, equaion (11) beome µ() = µ e (ρ+α), whih i nohing bu he expreion 5

8 obained in Chakravory e al. [26a], [26b], meaning ha he opimal arbon hadow o grow a a proporional rae equal o he um of he oial dioun rae and he naural aborpion rae when he arbon onenraion i below he eiling. In he preen eing, he arbon hadow o remain onan a he level h/(ρ+α) iff µ = h/(ρ+α), while he gap beween µ and h/(ρ + α) inreae a he rae ρ + α iff µ h/(ρ + α). Le u deermine now he ondiion under whih he diry reoure ha o be exploied, and how ha under hee ondiion µ > h/(ρ + α). 2.3 Shadow o of arbon, ariy of he renewable reoure and ompeiivene of he non-renewable one Aume ha from ome ime x onward he non-renewable reoure i no more exploied, o ha onumpion i aified by he renewable one alone. Define ỹ a he oluion of u (y) = y. Then, from ime x on, he quaniy onumed i min {y,ỹ}. If ỹ y, he renewable reoure i abundan, γ y() =, he ariy ren i nil, γ y () =, and he FOC (2) i aified ogeher wih he omplemenary lakne ondiion (4) and (5). If ỹ > y, he renewable reoure i rare, γ y() = u (y ) y >, he ariy ren i poiive, γ y () =, and he FOC (2) and he omplemenariy lakne ondiion (4) and (5) are aified. We denoe by y he value of min {y,ỹ} and by p he orreponding gro marginal urplu : p = u (y). When he renewable reoure i abundan, p = y and no ren ha ever o be harged for i exploiaion, while when i i rare ome ren γ y() ha o be harged one p() i higher han y. In he former ae p() anno be higher han y, in he laer ae p() i a mo equal o p. Le W(X( x ),Z( x )) be he value funion from x onward: W(X( x ),Z( x )) = x {u(y) y y hz()} e ρ( x) d. I inlude wo omponen, he um of he diouned ne moneary urplu (gro urplu minu o) generaed by he onumpion of y, from whih mu be dedued he um of he diouned manageable damage generaed by he ok of polluion inheried from he pa, Z( x ), auming ha 6

9 Z( x ) Z. Beaue he polluing reoure i no more exploied, hen Ż() = αz() o ha Z() = Z( x )e α( x), < x. Hene: and W(X( x ),Z( x )) = 1 ρ (u(y) yy) dw(x( x ),Z( x )) dz( x ) = h ρ + α. h ρ + α Z( x), (12) Wha i noeworhy i ha he burden of he inheried arbon ok i a linear funion of he ok hank o boh he linear form of he manageable damage funion and he linear form of he arbon ok dynami. The hadow marginal o of he inheried ok in value a ime x and equal o h/(ρ + α), i independen of boh Z( x ) and x. Sine boh Z( x ) and x are arbirary, hi implie, by a raighforward reurive argumen, ha he marginal hadow o of he arbon ok Z() in urren value i alo onan and equal o h/(ρ + α). Aume now ha oal i never exploied. The only upply i he olar energy, and i onumpion i equal o y defined above. Le u examine under wha irumane uh a poliy would no be he opimal one Cae of an abundan olar energy Reall ha hi ae orrepond o y ỹ or equivalenly p = y. Conider ome ime inerval [, 1 ], wih 1 >. Over hi inerval, le u redue a eah dae he onumpion of he renewable reoure by ε, < ε y, and inreae he onumpion of he non-renewable reoure by he ame amoun, o ha x() = ε and y() = y ε. 1 Aume ha ε and are uffiienly mall for he arbon onenraion o remain under he eiling. Sine oal onumpion i held onan a he level y, he balane hee of he ubiuion may be redued o wo omponen: a produion o aving reuling from he onumpion of a le oly reoure provided ha x < y ; 1 Exploiing he non-renewable reoure wihou imulaneouly reduing he exploiaion rae of he renewable reoure would reul in a marginal gro urplu u (y +ε) lower han he marginal o of he renewable reoure y. Then i i eaily heked ha he FOC (2) relaive o he ue of hi reoure and he aoiaed omplemenary lakne ondiion (4) and (5) anno be imulaneouly aified. Thu if he non renewable i o be exploied he exploiaion rae of he renewable mu be redued. 7

10 an inreae in he burden of polluion reuling from he exploiaion of a dirier upply. The value funion a ime along he iniial pah i: W(X( ),Z( )) = {u(y) y y hz()} e ρ( ) d. Beaue he emiion of polluion i nil, Z() = Z( )e α( ), and he value funion i given by equaion (12), wih x =. The value funion a ime on he perurbed pah read: W ε (X( ),Z( )) = wih: 1 {u(y) x ε y (y ε) hz()} e ρ( ) d +e ρ( 1 ) ζε αz(), [, 1 ], Ż() = αz(), > 1, 1 {u(y) y y hz()} e ρ( 1) d, implying ha: ( Z( ) ζε ) e α( ), [, 1 ], α Z() = Z( 1 ) e α( 1), 1, where: Z( 1 ) = (13) ( Z( ) ζε ) e α( 1 ) + ζε α α. (14) Hene he following value funion for he perurbed pah: W ε (X( ),Z( )) = 1 ρ (u(y) yy) + ε ( y x h ζ ) (1 ) e ρ( 1 ) ρ α h ρ + α Z( ) + h ζε ( ) 1 e ρ( 1 ). ρ + α α The differene beween he value funion on he perurbed and he iniial pah amoun o: W ε (X( ),Z( )) W(X( ),Z( )) = ε [ ( y x ) ζh ] (1 ) e ρ. ρ ρ + α 8

11 I i rily poiive, whaever he value of he perurbaion ε and he lengh of he ime inerval, provided ha he eiling onrain i no reahed, if and only if: y x > ζh ρ + α, (15) whih imply mean ha he fir omponen of he balane hee of he perurbaion, he o aving omponen, i larger han he eond one, he addiional damage omponen. Thi ondiion i neeary and uffiien for he exploiaion of he non-renewable reoure. In wha follow we aume ha i i aified. We onlude a follow. When he renewable reoure i abundan, ha i y ỹ, a neeary and uffiien ondiion of exploiaion of he diry nonrenewable reoure i ha i exploiaion o x i uffiienly lower han he exploiaion o of he lean renewable ubiue y, he o direpany being a lea equal o ζh/(ρ + α) Cae of a rare olar energy Reall ha hi ae orrepond o y < ỹ or equivalenly p > y. Exploiing he oal ok doe no imply any more o redue he produion of olar energy. Thu he value funion a ime on he perurbed pah read now: W ε (X( ),Z( )) = 1 {u(y + ε) x ε y y hz()} e ρ( ) d + e ρ( 1 ) {u(y) y y hz()} e ρ( ) d. where Z() i given by: e α < Z() = e [ α Z + ζε α (eα e α ) ] < 1 e [ α Z + ζε α (eα 1 e α ) ] 1. Hene ε an be hoen o ha Z() < Z,. For ε uffiienly mall: u(y + ε) u(y) + u (y)ε = u(y) + p ε, 9

12 o ha: W ε (X( ),Z( )) 1 ρ (u(y) yy) + ε ρ ( (p x ) h ζ ) (1 ) e ρ( 1 ) α h ρ + α Z( ) + h ρ + α ζε α ( 1 e ρ( 1 ) ). The differene beween he value funion on he perurbed pah and he iniial pah i now: W ε (X( ),Z( )) W(X( ),Z( )) ε [ (p x ) ζh ] (1 e ρ ). ρ ρ + α I i rily poiive provided ha: p x > ζh ρ + α. (16) We onlude ha i may happen ha i i opimal o exploi he diry non-renewable reoure even if i average o i higher han he o of i lean renewable ubiue when he renewable reoure i rare. I will be he ae when he upply of he renewable y i uffiienly mall, ha i p i uffiienly larger han x, however low y i Marginal hadow o of he polluion ok along he opimal pah Noie ha ζh/(ρ + α) i he marginal damage aoiaed o any addiional ue of he non-renewable reoure, however mall or large. A poined ou above, i i independen of he addiional ue of he non-renewable reoure ε, of he lengh of ime inerval of addiional ue of he diry reoure and of he polluion ok Z(), provided ha i remain below he eiling. Hene wihou any eiling onrain, he arbon hadow o would be preiely equal o h/(ρ + α) a eah dae. Thu he arbon hadow o, when he eiling onrain i aouned for, mu be higher han h/(ρ + α) before he eiling and when a he eiling, and equal o h/(ρ+a) afer he eiling, when he remaining non-renewable reoure ok i o low ha he eiling an never be reahed again. 2 Noe ha for p y ondiion (16) beome (15). 1

13 3 The renewable reoure i abundan We aume in hi eion ha he renewable reoure i abundan: y > ỹ, or equivalenly p = y, and ha ondiion (15) hold. Le u define x a he maximum flow of he polluing reoure onumpion when he arbon ok i a i eiling Z : x = αz/ζ. We denoe by p he orreponding gro marginal urplu: p = u (x). Two ype of opimal pah may arie, aording o he value of y ompared o p. 3.1 Bu i i more expenive han he non-renewable reoure a he eiling Thi ae, orreponding o y > p or equivalenly ỹ < x, i illuraed in Figure 1. y > p mean ha when a he eiling and he non-renewable reoure i he only reoure whih i exploied hen he lean renewable ubiue i no ompeiive. Thu if here mu exi a phae a he eiling during whih he exploiaion of oal i blokaded a x, hi phae mu be a phae during whih only oal ha o be exploied. Thi i implying ha oal i exploied alone from he beginning of he opimal pah when he eiling i no ye hur, nex when a he eiling and laly afer he eiling when he oal exploiaion doe no emi enough new polluion for he eiling onrain o be effeive again. Finally when p() reahe y, he diry oal ok i exhaued and he lean olar energy i exploied alone. < Figure 1 here > The opimal pah ha four phae. During he fir phae [, ) he nonrenewable reoure i he only exploied one. The prie i equal o: p() = x + λ e ρ + ζµ() < p, (17) wih µ() given by (11) beaue he arbon onenraion i under he eiling Z. In (11) µ i greaer ha h/(ρ + α), o ha he ame hold for µ() ielf. During hi phae Z() inreae: p() < p implie ha x() > x and, ogeher wih Z() < Z, hi implie ha ζx() > αz(). Thi phae end a dae when imulaneouly Z( ) = Z and p( ) = p. The eond phae [, ) i he phae a he eiling, wih he non-renewable reoure aifying he whole demand: p() = p and x() = x. A hi prie, 11

14 he renewable reoure i no ompeiive and anno relax he eiling onrain. During hi phae µ() i dereaing, and i beome preiely equal o h/(ρ + α) a he end of he phae. Beaue he eiling onrain i igh, he aoiaed Lagrange muliplier ν() i rily poiive and he dynami of µ() i given by (7). Beide, equaion (1) yield: µ() = 1 ζ ( p x λ e ρ). (18) Time differeniaing (18) and ubiuing for µ() given by (7) and for µ() given by (18) we obain: ν() = 1 ζ ( (ρ + α) (p x ) αλ e ρ ζh ). The hird phae [, x ) i again a phae during whih he non-renewable reoure i he only one whih i exploied. The prie i now given by: p() = x + λ e ρ + ζh ρ + α. (19) Thu he prie i higher han p, hene x() dereae and he eiling onrain beome unbinding, o ha µ() = h/(ρ+α) a poined ou above 3. A he end of hi phae p() = y and he renewable reoure beome ompeiive and ake he whole marke. Hene he non-renewable reoure mu be exhaued. The la phae [ x, ) i a phae of a definiively lean eonomy: p() = y and y() = ỹ. There are five endogenou variable o be deermined: λ, µ,, and x. They are given a he oluion of he following yem of five equaion: q d (p 1 ())d + ( ) x + x q d (p 3 ())d = X, (2) Z e α + ζq d (p 1 ())e α( ) d = Z, (21) 3 Noe ha x() < x, >, and Z( ) = Z implie ha Z() < Z, >. However, hi doe no imply ha Z() i monoially dereaing from onward. Bu we are ure ha Z() anno hi again he eiling Z. Hene µ() = h/(ρ+α),, o ha he prie pah, for >, i given by p 3 () defined in equaion (24) below. 12

15 p 1 ( ) = p, p 3 ( ) = p, p 3 ( x ) = y, (22) p 1 () and p 3 () being repeively given by: [ p 1 () = x + λ e ρ + ζ µ h ] e (ρ+α) + ζh ρ + α ρ + α, (23) p 3 () = x + λ e ρ + ζh ρ + α. (24) Equaion (2) i he umulaed demand-upply balane equaion of he nonrenewable reoure. Equaion (21) ae he arbon ok oninuiy a. Equaion (22) ae he prie oninuiy repeively a, and x. I i eaily heked ha for any λ, µ,, and x oluion of hee five equaion here exi value of γ x (), γ y(), γ y (y) and ν() uh ha all he opimaliy ondiion (1) (1) are aified by he above prie and quaniy pah. 3.2 And i i heaper han he non-renewable reoure a he eiling Le u now examine he oher ae: y < p or equivalenly x < ỹ. Here, when he arbon ok i a he eiling and he exploiaion of he non-renewable reoure i bounded by x, he demand anno be aified by hi ole reoure, beaue i would imply a prie p > y and he renewable reoure would be ompeiive. Thu he prie y i he maximum prie whih may prevail. The only poibiliy when a he eiling i ha he prie be equal o y and he demand equal o ỹ. The non-renewable reoure being he le oly one, i mu be ued a muh a poible: x() = x and y() = ỹ x, an immediae impliaion of diouning. By le oly i mu be underood ha x + ζh/(ρ + α) < y, ha i ondiion (15) i aified. Noe ha we mu alo have x + λ e ρ + ζh/(ρ + α) y during hi phae a he eiling. The equaliy hold a he end of he phae, whih mean ha he ok of non-renewable reoure mu hen be exhaued. Afer he end of he phae he prie ay a he ame level y. The opimal pah are illuraed in Figure 2. < Figure 2 here > 13

16 The fir phae [, ) i imilar o he fir phae of he preeding enario, p() = p 1 (), exep ha a he end of he phae, when he arbon ok reahe i eiling, he prie mu now be equal o y. The eond phae [, ) a he eiling i he phae we have previouly deribed: p() = y. Sine he non-renewable reoure ok i exhaued a he end of hi phae, we have now = x. The hird and la phae i he phae of exluive exploiaion of he lean reoure. The prie i he ame han during he eond phae: p() = y, bu now q() = y() = ỹ. The yem of equaion deermining he endogenou variable i given in Appendix A. The main differene wih he preeding enario i ha here he ue of he renewable reoure inreae more progreively: y() = during he fir phae, ỹ x during he eond phae, and ỹ, i maximum and definiive level, during he la phae, inead of wihing direly from o ỹ a in he preeding enario. 4 The renewable reoure i rare We aume in hi eion ha he renewable reoure i rare: : y < ỹ or equivalenly y < p, and ha ondiion (16) hold. A in he abundan ae, y may be eiher larger or maller han p, he prie juifying a demand preiely equal o x, he upper bound of he exraion rae of he diry non renewable reoure when a he eiling. 4.1 And i i more expenive han he non-renewable reoure a he eiling In hi ae where p > y, he analyi i a ligh adapaion of he one developed for an abundan and expenive renewable reoure. The opimal pah are five phae pah illuraed in Figure 3. < Figure 3 here > 14

17 The wo fir phae, [, ) and [, ), are he ame han in he abundan ae: a fir phae oward he eiling followed by a phae a he eiling during whih he ole non-renewable reoure i exploied. The hird and fourh phae, [, y ) and [y, x ), are phae during whih he prie pah i he ame: p() = p 3 () defined by equaion (24). Thi i implied by he fa ha he non-renewable reoure i ued during boh phae. During he hird phae [, y ) he non-renewable reoure i ued alone. A ime y he prie reahe y and he renewable reoure beome ompeiive. During he fourh phae [ y, x ) he prie inreae o he level p. Boh reoure are ued now: y() = y and x() = q d (p 3 ()) y. A x, p 3 ( x ) = p and he demand i equal o y. The renewable reoure an aify he whole demand. The ok of non-renewable reoure i exhaued. The fifh and la phae [ x, ) i he phae of exluively lean energy: p() = p, y() = y and x() =. 4.2 Bu i i heaper han he non-renewable reoure a he eiling When y < p, he analyi i lighly more inriae han in he abundan ae. We mu ake are of he fa ha i may happen ha eiher x+y > ỹ or x + y < ỹ. Defining p a he marginal gro urplu generaed by a onumpion rae q = x+y, ha i p = u (x+y ), i may happen equivalenly ha eiher p < y or p > y. Taking for graned ha here mu exi a phae during whih he euriy onrain i aive, when p < y he energy prie mu be equal o he marginal o of he lean renewable ubiue during he phae a he eiling, while when y < p he energy prie mu be equal o p > y during he phae a he eiling. Alhough in boh ae he qualiaive properie of he prie pah are he ame, he mix of reoure are differen. In he fir ae he lean ubiue mu begin o be exploied a he dae a whih he eiling i aained, while in he eond ae i exploiaion mu begin before. Le u examine hee wo ae. 15

18 4.2.1 Alhough no heap enough o be exploied before he eiling In hi ae, p < y 4 Le u how why, when a he eiling, he energy prie mu be preiely equal o y. Aume fir ha, when a he eiling, he energy prie p i lower han y. Then he marginal o of he olar energy y i oo high o be ompeiive ine he marginal gro urplu of energy p = u (y) i lower han y. Thu if uh a prie p were o be he energy prie when a he eiling we hould have y() = and q() = x. Hene p = u (x) = p > y, a onradiion ine we have aumed ha p i lower han y. Nex aume ha when a he eiling he prie p i higher han y bu lower han p, he prie ha prevail when he lean ubiue i he only available upply. For uh a prie p he renewable reoure i ompeiive o ha y() = y. However, ine p < y < p < p, hen q d (p) < ỹ < x + y. Hene a uh prie q d (p) y < x o ha he ue of oal anno be aped by x, meaning ha he eiling onrain i no effeive. Hene again a onradiion. The only remaining poibiliy i ha a he eiling he prie i equal o y. A hi prie, he demand ỹ i aified by a mix of he onrained non-renewable reoure ue x() = x omplemened by a parial exploiaion of i renewable ubiue y() = ỹ x < y. Before he prie y i aained he oal i he only primary reoure having o be exploied. One he prie i higher han y he olar ubiue i fully exploied. We onlude ha he opimal pah are five phae pah, a illuraed in Figure 4. < Figure 4 here > The fir phae [, ) i he phae oward he eiling, wih a prie p() = p 1 () < y, and he non-renewable reoure i he only one o be exploied, ine i i he only one whih i ompeiive a a prie lower han y. A 4 Thu p < y < p. Noe ha p, whih i higher han y, may be eiher lower or higher han p. 16

19 he eiling i aained and imulaneouly he renewable reoure beome ompeiive: p 1 ( ) = y. The eond phae a he eiling i a phae a a prie p() = y during whih boh reoure are exploied. The lea oly reoure, he non-renewable one, mu be exploied a i maximal feaible rae when a he eiling: x() = x. The mo oly, he renewable one, fill he gap beween he demand a prie y, ỹ, and he non-renewable onrained upply x : y() = ỹ x. Sine y() < y, ome par of he renewable reoure poenial y i lef unexploied, explaining why no rariy ren ha o be impued for he exploiaion of he renewable reoure. The nex phae [, x ) i he phae during whih he prie inreae from y o p. Sine he eiling onrain i no more effeive and will never be effeive again, µ() = h/(ρ + α) and p() = p 3 () < p. Sine p() > y hen he renewable reoure ren γ y() = p 3 () y i now poiive and progreively inreae oward i long run level p y. For he ame reaon, ha i p() > y, he whole poenial of he renewable reoure mu be exploied: y() = y. Bu he demand i higher han y ine p() < p, and he omplemen required o balane he demand i upplied by he non-renewable reoure: x() = q d (p 3 ()) y. The exraion of he non-renewable reoure dereae down o zero a he end of hi phae, when p() = p. Then he non-renewable reoure mu be exhaued. The la phae [ x, ) i he lean energy regime. The yem of equaion deermining he endogenou variable i given in Appendix A And heap enough o be exploied before he eiling In hi ae, y < p 5 Now, a he eiling, he prie mu be equal o p. Le u how why. Aume ha a he eiling he prie p i lower han p, hen he onumpion q d (p) would have o be larger han wha a full mobilizaion of he boh reoure, x + y, ould provide ine p < p implie ha q d (p) > q d (p ) = x + y. For prie p higher han p he argumen run a in he preeden paragraph. For p > p hene p > y, he renewable reoure ha o be fully 5 Thu y < p < p. Noe ha p, whih i higher han p, may be eiher lower or higher han p. 17

20 exploied, y = y, while q d (p) < x + y. Thu we hould have x < x and he exploiaion rae of he non-renewable reoure would no be onrained. We onlude ha, when a he eiling, he only poibiliy i ha p = p. The opimal pah are five phae pah, illuraed in Figure 5. < Figure 5 here > The wo fir phae, [, y ) and [ y, ), are he phae oward he eiling. The prie pah i p 1 () beaue he non-renewable reoure i ued during he boh phae. Bu, a y, p 1 ( y ) = y and he renewable reoure beome ompeiive, alhough he eiling i no ye reahed. Thu, while he nonrenewable reoure i he only one o be exploied before y, afer y he upply beome a mix: x() = q d (p 1 ()) y and y() = y. A ime, p 1 ( ) = p and he eiling i reahed. Then begin he hird phae [, ) a he eiling. During hi phae p() = p, x() = x and y() = y. The fourh phae [, x ) i a phae during whih he prie i p3 (), beaue he non-renewable reoure i ill exploied and he eiling onrain will never beome effeive again. Hene he arbon hadow o i h/(ρ+α). A x, p 3 ( x ) = p and he whole demand an now be upplied by he renewable reoure. The non-renewable reoure i exhaued. A hi dae begin he la phae [ x, ) of lean energy. The yem of equaion deermining he endogenou variable i given in Appendix A. 5 Time profile of he arbon hadow o and omparaive dynami indued by more evere damage The qualiaive properie of he ime profile of he arbon hadow o µ() are he ame in all he enarii, he renewable reoure being eiher abundan or rare and eiher expenive or heap. A illuraed in Figure 6, he arbon hadow o iniially inreae up o a maximum level aained when 18

21 he eiling onrain begin o be effeive, nex dereae during he phae a he eiling, down o h/(ρ + α) and la i fla forever a hi level. Thi hadow o of he arbon ok may be pli ino wo omponen, a manageable damage omponen h/(ρ+α) and a euriy eiling omponen µ() h/(ρ + α). < Figure 6 here > The hadow o of he amopheri arbon ok µ() an be een a he urren ax whih would be harged per uni of arbon rejeed in he amophere a ime in a deenralized eonomy. Noe ha hi ax rae ha o hold even afer x, he dae a whih oal i exhaued. In a deenralized eonomy, oal would be exhaued a dae x a a reul of he whole ime profile of he ax rae, and peifially beaue polluing emiion are axed before x and would be axed aferward. Anoher poin worh emphaizing i ha one he eiling i reahed he ax revenue µ()x() begin o dereae. During he phae a he eiling, he ax rae dereae while exraion remain onan a he level x. Afer he eiling, he ax rae i fla bu exraion dereae oward zero. However, hing are ambiguou before he eiling: he ax rae inreae bu oal onumpion dereae. A far a polluion i onerned he problem i o deermine how more evere polluion damage affe he opimal pah. Eonomi inuiion ugge ha hampering he ue of he diry oal reoure hould reul ino a lower ariy ren of hi reoure ogeher wih a higher hadow o of he arbon ok. Under he aumpion of he preen model hi i aually he ae. Bu a we hall how he ne effe of he dereae of λ and he inreae of µ may be ha he dae a whih he lean olar ubiue mu begin o be exploied i delayed. Alo in he ae of a rare olar ubiue, more evere arbon damage may depre he ariy ren of he olar energy during he raniory phae of imulaneou exploiaion of he wo reoure. 5.1 Comparaive dynami indued by more harmful manageable damage In order o gain a good underanding of he omeime ambiguou effe of more harmful manageable damage, he be i o examine fir wha would happen aben any euriy onrain. 19

22 5.1.1 Comparaive dynami aben he euriy onrain Abundan renewable reoure The opimal pah are wo phae pah. During a fir phae [, x ), only oal i exploied, wih p() = p 3 () < y,x() = q d (p 3 ()) and y() =. A ime x,p 3 ( x ) = y and he oal ok i exhaued. Nex begin he eond phae [ x, ), he olar energy phae: p() = y,x() = and y() = ỹ. A een in eion 2 and illuraed in Figure 6, here µ() = µ = h/(ρ + α), >. Only λ and x have o be deermined. They are he oluion of: x q d (p 3 ())d = X and p 3 ( x ) = y. (25) The inuiion i he following. More inenive manageable damage (dh > ) have wo effe on he prie p 3 (), an indire effe on he mining ren λ e ρ and a dire effe on he ax having o be paid per uni of oal onumpion ζh/(ρ + α). Higher damage inreae he ax on oal onumpion o ζ(h+dh)/(ρ+α). Hene, would λ remain unaffeed, he new prie pah p 3 ()+dp 3 () would be permanenly above he iniial one o ha he dae a whih p 3 () + dp 3 () = y would be advaned (d x < ). Bu x + d x i alo he dae a whih oal mu be exhaued, and hen oal i now exploied a a lower rae (dp 3 () < ) over a horer period, whih i a onradiion given he umulaed reoure onrain. We onlude ha λ mu be affeed, and more preiely ha dλ <. Now aume ha dλ ζdh/(ρ + α). Then dp 3 () = e ρ dλ + ζdh/(ρ + α) <, >, reuling ino ome d x >. Now he oal onumpion rae would be higher han along he iniial pah over a larger ime inerval, implying a umulaive oal onumpion larger han he iniial endowmen X, a onradiion again. Thu we mu have dλ < ζdh/(ρ + α), whih mean ha dp 3 ()/dh >, ha i he new prie pah i iniially above he referene one. Bu learly hi anno hold for any ime before x + d x for he ame reaon han he one pu forward in he preeding paragraph. Hene here mu exi ome dae uh ha < < x + d x, before whih dp 3 () > and afer whih dp 3 () <, a illuraed in Figure 7. Toally differeniaing (25) wih repe o λ, x and h, we how in Appendix B ha he above inuiion i righ: dλ dh <, dλ dh < ζ ρ + α dp 3() dh > and d x dh >. (26) 2

23 < Figure 7 here > To onlude, aben he euriy onrain and for he ae of an abundan olar energy, a more agreive manageable damage, dh >, indue: - an iniial prie inreae over ome ime inerval [, ), < x, generaing an iniial dereae of he exploiaion of oal hene of energy onumpion and an iniial dereae of polluion; - followed by a prie dereae over he ime inerval (, x + d x ),d x >, generaing an inreae of he exploiaion of oal hene of energy onumpion. The ime a whih he lean energy beome ompeiive i delayed. More agreive manageable damage are poponing he arrival of he lean ubiue! The rare olar energy ae When olar energy i rare, i exploiaion mu begin before he exhauion of he oal ok: y < x. The opimal pah ha hree phae. During he wo fir phae [, y ) and [ y, x ), he prie pah i he ame: p() = p 3 (). The fir phae end when he prie reahe he average o of he renewable reoure, p 3 ( y ) = y, and he eond phae when he prie aain p, p 3 ( x ) = p. In beween, he ren of he renewable reoure p 3 () y eadily inreae. During he fir phae only oal i exploied, x() = q d (p 3 ()), and during he eond phae boh reoure are ued, x() = q d (p 3 ()) ỹ and y() = ỹ. The la phae i he uual lean phae. We how in Appendix B ha he effe of dh on λ, x and p 3 () are he ame a in he ae of an abundan reoure, ha i (26) hold again. A he ime around whih he prie pah i pivoing afer ome dh >, on he iniial prie pah he prie level p 3 ( ) may be eiher higher han y in whih ae y <, or lower han y in whih ae < y. In Figure 8, he former ae i illuraed by y = y1 < p 3 ( ) and y = y1 <, and he eond ae by y = y2 > p 3 ( ) and y = y2 >. < Figure 8 here > 21

24 In he ae y < p 3 ( ) ( y1 < p 3 ( ) in Figure 8), an inreae of he manageable damage dh >, indue an earlier dae of arrival of he lean ubiue, d y /dh <, while i i he onrary in he ae y > p 3 ( ) ( y2 > p 3 ( ) in Figure 8). In boh ae he dae a whih he eonomy wihe o a purely lean eonomy i delayed Comparaive dynami under an effeive eiling onrain Le u onider he imple enario ariing under an effeive eiling onrain, he enario of ub-eion 3.1. Solar energy i abundan bu oly: y > p. The haraeri of he opimal prie pah are illuraed in Figure 1. λ, µ,, and x are oluion of he yem (2)-(22), whih ha o be oally differeniaed. Conerning he variaion of p 1 (), he fir par of he prie pah before he eiling, we how in Appendix C (laim () of Lemma 1) ha dp 1 ()/dh >, whih mean ha when manageable damage are ronger he prie pah i iniially hifed upward. Aume ha p 1 () i hifed upward over he whole inerval [, ). Thi would reul ino a horer iniial phae [, +d ),d <, beaue p i lef unaffeed by dh. The diry reoure would be exploied a a lower rae and he emiion flow would be le inene a eah dae over a horer ime inerval. Hene if, on he referene pah, Z() reahe he riial level Z a, i anno be he ae a + d on he hifed pah, a onradiion. We mu onlude ha here exi ome ime 1, < 1 <, uh ha dp 1 ()/dh > for < 1 and dp 1 ()/dh < for > 1 reuling ino a poiive d, a illuraed in Figure 9 and 1. A for he variaion of p 3 (), laim (e) of Lemma 1 in Appendix C ae ha (26) hold a in he ae wihou eiling. Thu here exi ome pivoal dae we denoe by 3, < 3 < x, before whih dp 3 ()/dh > and afer whih dp 3 ()/dh <. However i may happen ha eiher 3 > or 3 <. In he former ae illuraed in Figure 9, an inreae of he manageable damage reul in an earlier dae for he end of he phae a he eiling, while in he eond ae illuraed in Figure 1, he ame inreae reul ino a laer dae. In boh ae he dae of wihing o a purely lean energy world x i delayed. Thu in he former ae d( x )/dh >. We how in Appendix C (laim (h) of Lemma 1) ha he ame hold in he eond ae. Hene in boh ae he duraion of he po-eiling phae inreae. Noe alo ha in he ae 3 > he duraion of he phae a he eiling i horened. < Figure 9 here > 22

25 < Figure 1 here > Muai muandi he ame analyi applie o he oher kind of opimal pah. In he ae of an abundan bu heap enough lean olar ubiue examined in ub-eion 3.2 and illuraed in Figure 2, hank o he fa ha y, he prie prevailing a he eiling, doe no depend upon h, a poiive dh reul ino: a delayed dae of arrival a he eiling, ha i he dae a whih p 1 () = y, whih i alo he dae a whih he lean ubiue begin o be exploied; a delayed dae of he end of he phae a he eiling, ha i he dae a whih p 3 () = y, whih i alo he dae of exhauion of he diry reoure and he dae a whih he eonomy wihe o a purely lean energy regime. For he ae of a rare and oly lean olar energy of ub-eion 4.1 illuraed in Figure 4, a in he ae examined a he beginning of he paragraph: eiher 3 < and boh y, he dae a whih olar energy beome ompeiive, and x, he dae a whih oal i exhaued and olar energy upplie he whole demand, are delayed; or < 3 and i may happen ha: eiher y < 3 o ha olar energy begin o be ompeiive a an early dae while he dae of arrival of he full lean olar regime i delayed; or 3 < and hen he dae a whih olar energy begin o be ompeiive i delayed ogeher wih he dae of arrival of he purely lean energy regime. Wha an be aid abou he ae analyed in ub-eion 4.2 i lef a exerie, proeeding along he ame line of argumen. Whaever he kind of opimal prie pah, wo rong onluion may be drawn from he omparaive dynami riggered by more harmful manageable damage: 23

26 he ime a whih he eiling onrain begin o be effeive i delayed; he ime a whih he eonomy beome a purely lean renewable energy eonomy i alo delayed. 5.2 Comparaive dynami indued by a more ringen euriy onrain Le u onider now he effe of a more ringen euriy onrain, dz <, generaing a lower ap on he onumpion rae of he diry reoure when a he eiling, dx/dz >, hene a higher prie p when he onrain i effeive. A in he above ubeion, we examine fir he ae of an abundan bu oly renewable ubiue. Claim (a), (), (d) of Lemma 2 in Appendix C ae ha a more evere euriy level lower he reoure ren and p 3 (), hene induing a ime delay of he la phae of non-renewable exploiaion (Hoelling phae) before he omplee raniion oward lean energy: dλ dz >, dp 3 () dz Thee effe are illuraed in Figure 11. >, d dz < and d x dz >. < Figure 11 here > A uggeed by inuiion, an inreaed burden of he euriy onrain reul ino a lower hadow value of he reoure. The prie rajeory afer he eiling, p 3 (), i ranlaed down. Thu boh x and are delayed. Appendix C how in addiion ha beaue of he inreae of p implied by a lower ap on he onumpion rae of he reoure while a he eiling, he lengh of he la phae of reoure exraion i alo redued, d( x )/dz > : a rier euriy onrain delay farher in ime he inroduion of lean energy. Claim (b) of Lemma 2 in Appendix C ae ha dµ /dz <. A rier euriy level orrepond o a higher opporuniy o of he polluion ok, whih i alo a quie inuiive reul. A in he preeding ae of a higher level of manageable damage, a rier euriy onrain redue λ and inreae µ. Seondly, in boh ae, he raniion oward lean energy i delayed. Bu he overall effe of he variaion of eiher h or Z on he energy prie how harp differene. We have already een ha inead of a pivoal effe, a rier eiling implie a onanly dereaed level of p 3 (). Turning o 24

27 he variaion of he energy prie before he eiling, laim (e) and (f) of he Lemma 2 in Appendix C ae ha a rier eiling may indue eiher a global inreae of p 1 () above i referene level, dp 1 ()/dz <, [, 1 ), or eiher a dereae of p 1 () a he beginning of he pre-eiling phae followed by an inreae before : dp 1 ()/dz > and lim dp 1 ()/dz <. A rier euriy level may hu indue a higher rae of ue of he diry reoure a he beginning. Sine boh p and lim p 1 () are inreaed by a lower Z, we anno onlude onerning he effe on, a aed in laim (g) of Lemma 2 in Appendix C. If he dire effe of dz < on p dominae he indire effe on p 1 (), hen he eiling will be aained ooner in ime. Converely, if he dire effe on p i dominaed by he indire effe on he energy prie before he eiling, he eiling phae will be delayed. Noe ha he omparion beween he dire and indire effe i influened by he reoure demand elaiiy around p. For a given dx = αdz/ζ, a more elai demand will indue a higher inreae of p during he whole eiling phae, and hu a more imporan dire effe. Hene we ge he quie ounerinuiive reul ha a rier euriy onrain doe no neearily imply a quiker arrival a he eiling. Le u onider now he oher enarii. In he heap and abundan olar energy ae deribed in ub-eion 3.2, he ambiguiy of he variaion of p 1 () before remain. A more evere onrain on he ue of he diry reoure when a he eiling reul in an higher rae of ue of renewable energy during he eiling phae. Sine = x i deermined by he ame relaion a before, he dereae of λ indue a delay boh in he end of he eiling phae and in he omplee raniion oward lean energy. The ae of a rare and expenive renewable reoure deribed in ubeion 4.1 i imilar o he preeding ae. A lower Z ha an ambiguou effe on he energy prie p 1 () before he eiling and he ame applie o he variaion of. Sine p() = p 3 (), [, x ) and λ i dereaed, we onlude ha, y and x will be inreaed by a rier euriy onrain. Sine y i inreaed, he inroduion of he lean ubiue i delayed a already oberved for he preeding enarii. Noe ha while he lengh of he phae [, y ) i redued beaue of a higher level of p, he la phae [ y, x ) i only ranlaed laer in ime. The rare and heap renewable reoure ae i a bi more inriae. We mu ake are ha p = u (x + y ) i inreaed by a more evere euriy level ine x i dereaed. In he ae p < y, aume a uffiienly ligh inreae of p no revering he previou inequaliy. The iniial variaion of he energy prie remain ambiguou. Bu now ine p() = y while a he 25

28 eiling, i redued ine p 1 () i hifed up before. During he eiling phae, he ue of lean olar energy i inreaed. λ being dereaed, he la phae of he diry reoure exploiaion, [, x ) i ranlaed laer in ime, delaying he full depleion of he diry reoure ok and hu he omplee raniion oward lean energy. If p > y, he variaion of p 1 () being ambiguou, y may be eiher redued or inreaed. A rier euriy level doe no neearily favor an earlier inroduion of lean renewable energy. The ame remark applie o ine p i inreaed. During he eiling phae, he lean energy upply i onrained o be equal o y while he ue of he diry reoure ha o be redued following a rier euriy onrain. The ariy ren of renewable energy i hu inreaed during he eiling phae, renewable energy produion being no more able o ompenae for a redued ue of he diry reoure. A in he preeding enario, boh and x are inreaed. We an hu onlude ha in all enarii, he omplee raniion oward lean energy i delayed by a more evere euriy level. 6 Conluion Sinn [28] emphaize forefully he poibiliy of green paradoxe. They are iuaion in whih limae poliy aelerae global warming inead of lowing i down, demonraing ha good inenion do no alway breed good deed. The main green paradox he refer o i he poibiliy ha an ill-hoen arbon ax, oo low a he beginning of he rajeory and inreaing oo fa laer, would make he reoure owner aelerae exraion now, in aniipaion of he inreae of he ax rae. Thi anno ou in our model where he arbon ax i opimal. Neverhele, ome oher form of green paradoxe may appear: we have proved ha a more ringen limae poliy in he form of a lower enenraion arge indue a delay in he peneraion of he lean renewable reoure and in he omplee raniion oward lean energy, and alo ha in ome ae hi more ringen limae poliy aelerae exraion now. Given heir grea relevane for he ondu of limae poliy, furher reearh on hee iue, and epeially on he robune of he green paradoxe o he peifiaion of he model, i needed. 26

29 Referene [26a] Chakravory, U., B. Magné and M. Moreaux, 26, Plafond de onenraion en arbone e ubiuion enre reoure énergéique, Annale d Eonomie e de Saiique, 81, [26b] Chakravory, U., B. Magné and M. Moreaux, 26, A Hoelling model wih a eiling on he ok of polluion, Journal of Eonomi Dynami and Conrol, 3(12), [1996] Farzin, Y. H. and O. Tahvonen, 1996, Global arbon yle and he opimal ime pah of arbon ax, Oxford Eonomi Paper, 48, [1996] Hoel, M. and S. Kverndokk, 1996, Depleion of foil fuel and he impa of global warming, Reoure and Energy Eonomi, 18, [28] Sinn, H.-W., 28, Publi poliie again global warming: a upply ide approah, Inernaional Tax and Publi Finane, 15, 36-Ű394. [26] Sern, N., 26, Sern Review: The Eonomi of Climae Change, HM Treaury, London. [1997] Tahvonen, O., 1997, Foil fuel, ok exernaliie and bakop ehnology, Canadian Journal of Eonomi, 3, [2] Toman, M. A. and C. Wihagen, 2, Aumulaive polluion, lean ehnology and poliy deign, Reoure and Energy Eonomi, 22, [1994] Ulph, A. and D. Ulph, 1994, The opimal ime pah of a arbon ax, Oxford Eonomi Paper, 46,

30 APPENDIX Appendix A. Equaion yem deermining he endogenou variable Sub-eion 3.2. Only four variable have o be deermined here ine = x, ha i λ,µ, and. They olve he following yem of four equaion: q d (p 1 ())d + ( ) x = X, Z e α + ζ q d (p 1 ())e α( ) d = Z, (A.1) (A.2) p 1 ( ) = p, p 3 ( ) = y. (A.3) Sub-eion 4.1. In hi ae ix endogenou variable mu be deermined: λ,µ,,, y and x. They are given a he oluion of he following yem of ix equaion: q d (p 1 ())d+ ( ) y x [ x+ q d (p 3 ())d+ q d (p 3 ()) y ] d = X, y (A.4) Z e α + ζ q d (p 1 ())e α( ) d = Z, (A.5) p 1 ( ) = p, p 3 ( ) = p, p 3 ( y ) = y, p 3 ( x ) = p. (A.6) Sub-eion 4.2. Paragraph Here he five endogenou variable are λ,µ,, and x, and hey are olving he following yem of five equaion: q d (p 1 ())d+ ( ) x [ x+ q d (p 3 ()) y ] d = X, (A.7) Z e α + ζ q d (p 1 ())e α( ) d = Z, (A.8) p 1 ( ) = y, p 3 ( ) = y, p 3 ( x ) = p. (A.9) 28

31 Paragraph The ix endogenou equaion are now: λ,µ, y,, and z whih olve he following yem of ix equaion: y [ q d (p 1 ())d+ q d (p 1 ()) y ] d+ ( ) x [ x+ q d (p 3 ()) y ] d = X y (A.1) y [ Z e α +ζ q d (p 1 ())e α( ) d+ζ q d (p 1 ()) y ] e α( ) d = Z (A.11) p 1 ( y ) = y, p 1 ( ) = p, p 3 ( ) = p, p 3 ( x ) = p. (A.12) y 29

32 Appendix B. Comparaive dynami aben he euriy onrain Abundan olar energy ae Toally differeniaing (25) and noing ha q d (p 3 ( x )) = ỹ, we ge: x qd 3 ()e ρ d ỹ dλ x = ζ qd 3 ()d dh ρ + α e ρx ρλ e ρx d x 1 where q d 3 () denoe dq d (p 3 ())/dp 3 (). Le M be he marix of he lef-hand ide of hi equaion and he deerminan of M. We ge: and Hene wih Hene: Noe ha: λ = ρλ e ρx x M 1 = 1 ρλ e ρx x = dλ d x = ρλ e ρx x x e ρx = ζ 1 ρ + α q d 3 ()e ρ e ρx ỹ < ỹ x qd 3 ()e ρ d λ x q d 3 ()d ỹ <, q d 3 ()e ρ d e ρx x dλ = ζ λ dh ρ + α <, d x dh = ζ x ρ + α >. dh,. q d 3 ()d >. < λ = ρλ e ρx x qd 3 ()d ỹ x ρλ e ρx qd 1 ()e ρ d e ρx ỹ < 1, 3

33 hene: dλ dh < ζ ρ + α. Rare olar energy ae Wihou he eiling onrain and wih a rare olar energy flow endowmen y, hree endogenou variable have o be deermined: λ, he oal mining ren, y, he ime a whih olar beome ompeiive, and x, here x > y, he ime a whih he oal ok i exhaued and he eonomy wihe o a lean regime. They olve he following hree equaion yem: y x [ q d (p 3 ())d + q d (p 3 ()) ỹ ] d = X y p 3 ( y ) = y and p 3 ( x ) = p. Toally differeniaing he above yem and noing ha q d (p 3 ( x )) = ỹ, we obain: x qd 3 ()e ρ d ỹ dλ x e ρy ρλ e ρy d y = ζ qd 3 ()d ρ + α 1 dh e ρx ρλ e ρx d x 1 Le be he deerminan of he marix of he lef-hand ide member of hi equaion: [ x ] = ρλ e ρy e ρx ρλ q3 d ()e ρ d ỹ <. We have: dλ d y = ζ 1 ρ + α λ y dh, d x x 31

34 wih λ = ρλ e ρx [ ρλ e ρy x y x = ρλ e ρx [ x = ρλ e ρy [ x ] q3 d ()d ỹ q d 3 ()e ρ d e ρy x q d 3 ()e ρ d e ρx x ỹ <, ] q3 d ()d whih anno be igned, ] q3 d ()d + [ e ρx e ρy] ỹ. Hene: Noe ha < λ λ dλ = ζ dh ρ + α <, d y dh = ζ y ρ + α indeerminae, d x dh = ζ x ρ + α >. < 1, from whih we onlude ha: dλ dh < ζ ρ + α = dµ dh. 32

35 Appendix C. Comparaive dynami under he euriy onrain Differeniaing he yem (2)-(22) wih repe o (λ,µ,,, x ) and (h,z) reul in he following linearized yem in marix form: Iλ X Iµ X ỹ dλ Iλ Z Iµ Z dµ 1 ζe α π( ) d 1 ρλ d 1 ρλ d x where we denoe: Noe ha: I X λ I X µ ζ I Z µ ζ 2 π( ) I X h I Z h ζ k( ) x q1 d () e ρ d + ρλ + ζ α I X h = IX µ ζ = ζ α + ρ I X h I Z h e α e ρ e ρ e ρx dh 3 () e ρ d > (A.13) 1 () e (α+ρ) d = I Z λ > (A.14) I Z h = IZ µ ζ ζ 1 () e (2α+ρ) d > (A.15) ( µ h ) (α + ρ)e α > α + ρ (A.16) 1 () ( e (α+ρ) 1 ) d x 3 () d > / < (A.17) 1 () ( e (α+ρ) 1 ) e α d > (A.18) [ 1 [ ] 1 ζ q1 d ( ) e ] ρ, k( ) α ζ q3 d ( ) e ρ. (A.19) [ x ] q1 d () d + q3 d () d 1 () e α d IZ µ ζ JZ h. IX µ ζ JX h (A.2) (A.21) We ae he following Lemma regarding he onequene of an inreae dh > of he marginal manageable damage. α ( ζ ) e α k( ) k( ) dz, 33

36 Lemma 1 In he expenive and abundan olar energy ae, an inreae dh > of he marginal manageable damage reul in: (a) A dereae of he reoure hadow value, dλ /dh < ; (b) An inreae of he polluion ok opporuniy o, dµ /dh > ; () A pivoal effe on p 1 () before, dp 1 ()/dh > and lim dp 1 ()/dh < ; (d) A delay in he arrival o he eiling, d /dh > ; (e) A pivoal effe on p 3 () around (, x ), dp 3 ()/dh > and lim x dp 3 ()/dh < ; (f) A delay in he omplee raniion oward lean energy, d x /dh > ; (g) An ambiguou effe on he end of he eiling phae, d /dh > / <. (h) An inreae of he duraion of he po-eiling phae, d( x )/dh >. Proof: Le be he deerminan of he fir marix of he linearized yem. = (ρλ ) 2 π( ) [ δ + ỹiµ Z (ρλ ) 1] where δ Iλ XIZ µ Iλ ZIX µ. We now hek ha δ > implying ha >. Making ue of (A.13), (A.14) and (A.15): x [ ][ ] δ = Iµ Z q3 d () e ρ d + ζ 2 q1 d () e ρ d q1 d () e (2α+ρ) d Le: ζ 2 [ 1 () e (α+ρ) d] 2. a() 1 () 1/2 e ρ 2, b() 1 () 1/2 e (α+ ρ 2 ), hen: a()b() = 1 () e (α+ρ). Thu δ i equal o: δ = I Z µ x { [ q3 d () e ρ d + ζ 2 ][ ] [ ] 2 } a 2 ()d b 2 ()d a()b()d Applying he Cauhy-Shwarz inequaliy for inegral, he la differene i poiive, Iµ Z > hene implie ha δ > and >. Nex dλ /dh = λ h / where: { λ ζ h = α + ρ (ρλ ) 2 π( ) δh λ ỹ } Iµ Z e ρx, ρλ and δ λ h IX h IZ µ I X µ I Z h. We hek ha δλ h < implying ha λ h < ine I Z µ >. Making ue of (A.2) and (A.21): δ λ h = I Z µ x 3 () e ρ d + [ I X µ ζ JX h 34 ] I Z µ I X µ [ I Z µ ζ JZ h ]..

37 Developing and implifying, we obain: δ λ h = I X µ J Z h I Z µ 1 () d I Z µ [ x Developing ank o (A.14), (A.21), and (A.15): I X µ J Z h I Z µ 1 () d = ζ 2 {[ [ 3 () e ρ d + ] q3 d () d. x ][ 1 () e (2α+ρ) d ][ 1 () e (α+ρ) d ] 1 () d ]} 1 () e α d Le f() 1 () e (α+ρ) and g() 1 (), [, ). f() > g() implie ha: [ ][ ] [ ][ ] f()e α d g()d f()d g()e α d = A λ h >. Hene δh λ <, reuling in λ h > whih implie dλ /dh <, whih prove laim (a) of Lemma 1. Nex, dµ /dh = µ h /, where: { µ ζ h = α + ρ (ρλ ) 2 π( ) δ µ h + ỹ ( ) } I X ρλ µ e ρx + Ih Z, and δ µ h IX λ IZ h IX µ I X h. We hek ha δµ h > implying ha µ h > ine I X µ > and I Z h >. ζ 2 A λ h δ µ h = I Z h +ζ x q3 d () e ρ d + Iµ X { [ {[ +ζ x ][ q1 d () e ρ d 3 () d ][ q1 d () e (α+ρ) d q1 d () d ] [ q1 d () e (2α+ρ) d ] [ ] 2 } q1 d () e (α+ρ) d ][ q1 d () e ρ d The fir erm ino brake i poiive hank o he Cauhy-Shwarz inequaliy. The eond erm ino brake i of he form ( f()e α d )( g()d ) ( ) f()d (g()e α d) and f() q1 d () e ρ > g() q1 d () implie ha hi erm i alo poiive. We onlude ha δ µ h >, hene µ h > and dµ /dh > whih prove laim (b) of Lemma 1. ]} q1 d () e α d 35

38 Nex: dp 1 () dh = dλ dh + ζ dµ = ζ(ρλ ) 2 π( ) (α + ρ) ζ(ρλ ) 2 π( ) (α + ρ) { λ h + ζ µ h} dh = 1 { δh λ + ζδ µ h + ỹ { δ p1 h + ỹ δ p11 h ρλ [ ζi Z ρλ h + e ρx (ζiµ X Iµ Z ) ] } }. We now hek ha δ p1 h δ p1 > and δ p11 h >. Making ue of (A.2) and (A.21): h = I X λ (I Z µ ζj Z h ) J X h (I Z µ ζi X µ ) + I X µ (J Z h I X µ ) x = ζ 2 q3 d () 1 (τ) e { ατ e (α+ρ)τ (e ρ 1) (e ρ e ρτ ) } dτd [ ][ ] +ζ 2 q1 d () e ρ d q1 d () e α (e (α+ρ) 1)d [ ][ ] ζ 2 q1 d () d q1 d () e α (e (α+ρ) e ρ )d [ ][ ] +ζ 2 q1 d () e (α+ρ) d q1 d () e α (1 e ρ )d { x ζ 2 q3 d () 1 (τ) e ατ h(,τ)dτd [ ] [ ] + q1 d () e α (e (α+ρ) 1)dτ q1 d () (e ρ 1)d [ ][ ]} q1 d () (e (α+ρ) 1)dτ q1 d () e α (e ρ 1)d, where h(,τ) e (α+ρ)τ (e ρ 1) (e ρ e ρτ ), τ <. Sine e ρ e ρτ e ρ 1, h(,τ) (e (α+ρ)τ 1)(e ρ 1), implying ha he fir inegral i poiive. Nex, f() e (α+ρ) 1 > g() e ρ 1 implie ha he la differene of inegral produ i alo poiive. Hene δ p1 h >. Then: δ p11 h = I Z µ (1 e ρx ) + ζ[e ρx I X µ J Z h ] = ζ 2 e ρx = ζ 2 e ρx 1 () e α { e (α+ρ) (e ρx 1) (e ρx e ρ ) } d 1 () e α h( x,)d, implie ha δ p11 h >. Thu δ p1 h > and δ p11 h > imply ogeher ha dp 1 ()/dh >. A permanen inreae of he energy prie p 1 () over he ime 36

39 inerval [, ] would imply boh a earlier arrival a he eiling and a redued ue of he reoure during he ime inerval, and hene a lower polluion aumulaion, a onradiion wih repe o he polluion aumulaion ondiion beween Z and Z. We onlude ha he modified prie rajeory mu u one he referene prie pah, reuling in a pivoal effe on p 1 () ombined wih a delayed arrival a he eiling. Hene i inreaed wih h. Thi prove he laim () and (d) of Lemma 1. Nex, making ue of he expreion of and λ h : dp 3 () dh = dλ dh + = ζ(ρλ ) 2 π( (α + ρ) ζ(ρλ ) 2 π( ) (α + ρ) We now hek ha δ p3 h δ p3 h = I Z µ ζ α + ρ = 1 [ λ h + ζ ] α + ρ { Ih X Iµ Z Iµ X Ih Z + Iλ X Iµ Z (Iµ X ) 2 + ỹ } ( 1 e ρ x ) I Z ρλ µ { δ p3 h + ỹ } ( 1 e ρ x ) I Z ρλ µ. >. Making ue of (A.2) and (A.21): [ ] [ ] I X µ I Z ζ JX h Iµ X µ ζ JZ h + Iλ X Iµ Z (Iµ X ) 2 ( ) ( ) = Iµ Z I X λ Jh X I X µ I X µ Jh Z. Thank o (A.13), (A.14) and (A.15): x δ p3 h = Iµ Z q3 d () ( e ρ 1 ) d {[ ][ +ζ 2 q1 d () e (2α+ρ) d [ ][ q1 d () e (α+ρ) d 1 () ( e ρ 1 ) ] d ]} 1 () e α ( e ρ 1 ) d. Le f() q1 d () e (α+ρ) and g() q1 d () (e ρ 1). Sine: e (α+ρ) > e ρ > e ρ 1, we ge f() > g(). Thu ( f()e α d)( g()d) ( f()d)( g()e α d) > implie ha he erm ino brake i poiive. We onlude ha δ p3 h > implying ha dp 3 ()/dh > and proving he fir par of laim (e) in Lemma 1. 37

40 Then leing dp 3 ( x )/dh lim x dp 3 ()/dh by a ligh abue of noaion: dp 3 ( x ) dh = dλ dh e ρx + ζ α + ρ = ζ(ρλ ) 2 π( )e ρx { ( I Z (α + ρ) µ I X λ e ρx Jh X ζ(ρλ ) 2 π( )e ρx (α + ρ) δp3x h, ) I X µ ( I X µ e ρx J Z h )} uing he ame mehod a before. We now hek ha δ p3x h <. x δ p3x h = e ρx Iµ Z q3 d () ( e ρ e ρx) d {[ ][ ζ 2 q1 d () e (2α+ρ) d [ ][ q1 d () e (α+ρ) d 1 () e ( ρx e ρx e ρ) ] d ]} 1 () e α e ρx ( e ρx e ρ) d. < x implie ha he fir erm of he above um i negaive. Le f() q1 d () e (α+ρ) and g() q1 d () (1 e ρ( x) ). f() > g() i equivalen o e ρx > e ρ e α m(). We hek immediaely ha m() = and m() <, hu f() > g() implie ha he la erm ino brake i poiive and hene ha δ p3x h < whih implie in urn ha dp 3 ( x )/dh <. Thi prove he la par of laim (e) in Lemma 1. Sine dp 3 ()/dh i a dereaing funion of ime, dλ /dh being negaive, we onlude from dp 3 ()/dh > and lim x dp 3 ()/dh < ha i p 3 () hould pivo around ome ime 3 beween and x. Thi implie ha d x /dh >, ha i laim (f) of Lemma 1. Bu may be loaed anywhere inide he inerval hu depending upon < / >, may eiher inreae of dereae following an inreae of h, whih i laim (g) of Lemma 1. La, making ue of he differeniaion of he gro urplu ime oninuiy requiremen a and x, p 3 ( ) = p, p 3 ( x ) = y, i i immediaely heked ha: d x d dh = ζ ) (e ρ e ρx >, ρλ (α + ρ) ha i laim (h) of Lemma 1. The proof of Lemma 1 i now omplee. Conerning he effe of a rier euriy level on he opimal pah, we ae he following Lemma: 38

41 Lemma 2 In he expenive and abundan olar energy ae, a dereae dz < of he euriy level reul in: (a) A dereae of he reoure hadow value, dλ /dz > ; (b) An inreae of he polluion ok opporuniy o, dµ /dz < ; () A delay in he raniion oward lean energy, d x /dz < ; (d) A delay in he end of he eiling phae, d /dz < ; (e) An ambiguou effe on he iniial energy prie p 1 (); (f) An inreae of he energy prie before he eiling lim dp 1 ()/dz < ; (g) An ambiguou effe on he beginning of he eiling phae, d /dz > / <. Proof: I i immediaely heked ha: dλ dz = (ρλ { ) 2 π( ) e α I X µ + α } ζ ( )Iµ Z > (A.22) { [ dµ dz = (ρλ ) 2 π( ) e α Iλ X + ỹ ] + α } ρλ ζ ( )Iµ X <,(A.23) ha i he laim (a) and (b) of Lemma 2 if dz <. Sine dp 3 ()/dz = (dλ /dz)e ρ >, (, x ), he prie pah i hifed down for a dz < during he la phae before he raniion oward lean energy. Thi implie ha x ha o inreae. Sine p i alo hifed up by a rier eiling onrain, i alo inreaed. Noe ha: αe ρ d x dz d dz = 1 ρλ ζ q3 d ( ) >. The lengh of he po eiling phae i redued by a rier euriy onrain. We have heked laim () and (d) of Lemma 2. Making ue of he expreion (A.22) and (A.23) of dλ /dz and dµ /dz: dp 1 () dz = e ρ [ dλ dz + ζ dµ dz eα = (ρλ ) 2 π( )e α e [ζ ρ ] ( Iλ X + ỹ ρλ ) ] + α ζ ( ( )e α ζi X µ e α Iµ Z (ρλ ) 2 π( )e α e ρ δ p Z. Making ue of (A.13) and (A.14): ) e α I X µ δ p Z > < x 1 (τ) e ρτ [1 + α ζ ( )e α(τ+ ) ] (e α(τ ) 1 ) dτ 3 () e ρ d + ỹ ρλ K p Z. 39

42 Le I p Z () be he inegral in he LHS of hi inequaliy. Ip Z () i a dereaing funion of, and I p Z () > while Ip Z ( ) <. Thu eiher I p Z () < Kp Z and dp 1 ()/dz <, [, ), eiher I p Z () > Kp Z and here exi a unique Z (, ) uh ha dp 1 ()/dz >, < Z and dp 1 ()/dz <, ( Z, ). We onlude ha he iniial energy prie p 1 () may eiher be hifed up or down following a rier Z level, whih i laim (e) of Lemma 2 while he energy prie i in all ae hifed up ju before, whih i laim (f). La, ine he eiling prie p and he energy prie before boh inreae, nohing an be aid abou he direion of hange of, hi i laim (g) of Lemma 2. The proof of Lemma 2 i now omplee. 4

43 u', p y p + λ + ζμ x ζμ ( ) + ζ h x λ + + ρ α ζμ () ζμ () λ ( ) ζ h = ρ + α x + λ λ () x x Phae a he eiling qxy,, () = x( ) q x q() = x( ) = x ( ) = x( ) q y q() = y( ) = y y( ) = x( ) = Phae a he eiling x Figure 1: Opimal pah: Cae of abundan renewable reoure and y > p

44 u', p p x y + λ + ζμ ζμ ( ) ζμ () ζ h = ρ + α x + ζ h λ + ζμ () ρ + α x + λ λ () x Phae a he eiling qxy,, () = x () q y q ( ) = x+ y ( ) = y q () = y ( ) = y ( ) y = y x x x( ) = x y() = Phae a he eiling x ( ) = Figure 2: Opimal pah: Cae of abundan renewable reoure and y < p NB: The figure i drawn auming ha x < y x, bu he revere ould hold

45 u', p p y ( ) γ = y p y p + λ + ζμ x ζμ ( ) ζμ () ζ h = ρ + α x + ζ h λ + ζμ () ρ + α λ ( ) x + λ λ () x y x Phae a he eiling qxy,, () = x () q x q () = x ( ) = x ( ) = x ( ) q y ( ) = ( ) + ( ) q x y y ( ) = y y q () = y () = y y() = x () = Phae a he eiling y ( ) x x Figure 3: Opimal pah: Cae of rare renewable reoure and p < y

46 u', p p ( ) γ y () γ = y p y y + λ + ζμ x λ ζ h ρ α + + x + x + λ x x Phae a he eiling qxy,, y + x () = x () q y x q () = x+ y ( ) = y x() = x q ( ) = x ( ) + y y ( ) = y y q () = y () = y () y = y x x( ) y() = Phae a he eiling x x () = Figure 4: Opimal pah: Cae of rare renewable reoure, y < p and p < y

47 u', p p p () γ y ( ) γ = y p y y () γ y ( ) γ y + λ + ζμ x λ ζ h ρ α + + x + x + λ x y x Phae a he eiling qxy,, () = x ( ) q q () = x () + y x + y x x() q ( ) = x+ y x( ) = x () = () + ( ) q x y y ( ) = y y q () = y ( ) = y x() y() = x ( ) = y Phae a he eiling x Figure 5: Opimal pah: Cae of rare renewable reoure, y < p and y < p

48 μ μ ( ) e ρ + α μ Addiional arbon hadow o indued by he euriy onrain ( + ) h / ρ α Manageable damage marginal hadow o of arbon x Phae a he eiling Figure 6: Time profile of he marginal o of he amopheri arbon ok

49 μ (1) (2) y + λ + ζμ x x = y + d x x Figure 7: Effe of an inreae dh > of he manageable damage, aben he eiling onrain, when he lean primary reoure i abundan NB: (1) pah (2) pah x e ρ ζμ ρ + λ + + λ + dλ e + ζ μ + dμ x ( ) ( )

50 μ (1) (2) p y2 p3 ( ) y1 2 y1 y x d < d 2 > d > y1 y x Figure 8: Conraed effe of an inreae dh > of he manageable damage, aben he eiling onrain, when he lean renewable primary reoure i rare NB: (1) pah x e ρ + λ + ζμ, (2) pah + ( λ + dλ ) e ρ + ζ ( μ + dμ ) x Cae Cae ( ) ( ) < p :, and d y 1 y1 y1 y1 > p :, and d y 1 y2 y2 y2

51 p3 () (1) y p1 () (2) p 1 3 d > d < d > x x Figure 9: Effe of an inreae of he manageable damage, dh > Cae of an abundan bu oly renewable reoure, an aive euriy onrain and 3 >. NB: (1) () () ( λ λ ) ζ p3 + dp3 = x + + d e ρ + (2) () () () () ( h+ dh) ρ + α h+ dh p1 + dp1 = p3 + dp3 + ζ μ + dμ e ρ α ρ+ α ( + )

52 p3 () (1) y p1 () (2) p 1 3 d > d > d > x x Figure 1: Effe of an inreae of he manageable damage, dh > Cae of an abundan bu oly renewable reoure, an aive euriy onrain and 3 <. NB: (1) () () ( λ λ ) ζ p3 + dp3 = x + + d e ρ + (2) () () () () ( h+ dh) ρ + α h+ dh p1 + dp1 = p3 + dp3 + ζ μ + dμ e ρ α ρ+ α ( + )

53 p3 () (1) y (2) p1 ( ) p + dp dp p x d > d > d > x Figure 11: Effe of a lower euriy polluion ok, dz<. Cae of an abundan bu oly renewable reoure. NB: The Figure illurae a ae where ( ) () + ( ) p ( ) + dp ( ) (1) p dp (2) dp / 1 dz > and d / dz <