Optimal policies to preserve tropical forests Preliminary draft


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1 Opimal policies o preserve ropical foress Preliminary draf Gilles Laorgue and Helene Ollivier February 1, 211 Absrac We develop a Norh/Souh growh model o invesigae he normaive quesion of inernaional ransfers o preserve ropical foress. The Souh convers fores lands ino agriculural lands o produce a nal consumpion good which is consumed by he Norh. We consider wo ways of incorporaing he exernaliy coming from deforesaion in he uiliy of he Norh: i) hrough an ameniy value, which reecs a coninuous willingness o preserve foress, or ii) hrough a minimum sock of ropical foress o be preserved. Firs, we solve he opimal program and we characerize he dynamic properies of he opimal deforesaion regime. Second, we derive he decenralized equilibrium oucome in which he exernaliy can be correced by a ransfer scheme from he Norh o he Souh. This ransfer funcion depends boh on he sock of fores (posiively) and on he deforesaion rae (negaively). Third, we characerize he dynamic implemenaion rule which mus be saised by his ransfer o resore he opimum. Finally, we analyze he specic cases of a pure fores subsidy policy and a pure deforesaion axaion policy and we illusrae hese ndings wih an analyical specied model. Keywords: REDD mechanism, deforesaion, dynamics, opimal regulaion, ax, subsidy. JEL classicaions: Q23, Q27, Q28. 1 Inroducion Reducing emissions from deforesaion and fores degradaion (REDD) in developing counries has been recenly a he cener of negoiaions on a renewed climae regime. Deforesaion in ropical counries is responsible for abou a quarer of oal world carbon emissions Toulouse School of Economics (INRALERNA). 21 allee de Brienne, 31 Toulouse, France. address: Deparmen of Agriculural and Resource Economics, Universiy of California, Berkeley. address: 1
2 and represen he main source of emissions in some developing counries. Reducing deforesaion could oer boh environmenal gains as a miigaion opion and an opporuniy o inegrae developing counries ino he inernaional negoiaions on climae change. An ocial agreemen has been recenly achieved on he REDD (or REDD+) mechanism during he UNFCCC conference of he Paries in Cancun, in December 21, as regards is scope, eligible aciviies, safeguards, and mehodological issues. 1 Agreeing on safeguards represen a major achievemen due o complex governance and insiuional issues, ha includes respec for he righs of Indigenous Peoples and local communiies, fores governance, and acions ha are consisen wih conservaion of naural foress and biodiversiy. To be par of he REDD mechanism, counries need o underake aciviies according o a phased approach. In he rs phase called readiness, counries develop naional sraegies, assess naional reference emission levels, and implemen a robus and ransparen naional fores monioring sysems. In he second phase, counries implemen heir naional policies aiming a preserving ropical foress. In he hird phase, counries can receive performancebased incenive paymens, ha is, paymens for veried emissions reducions. Designing he performancebased incenive scheme is one of he imporan challenges of he REDD mechanism, bu i also oers beer prospecs for realizing fores preservaion oucomes han mos previous fores policy inervenions (Pfa e al., 21). Since REDD projecs are mosly in heir infancy, resulbased incenive schemes have hardly been implemened, apar from he large bilaeral programs of Norway in Brazil, Indonesia, Guyana and Tanzania. As illusraed by he Amazon Fund, donaions (USD 1 billion) are linked direcly o resuls, i.e. o emission rends. 2 More precisely, paymens in a paricular year will depend on he dierence beween emissions from deforesaion in he previous year and a reference level, which is he average for he curren enyear calculaion period, and which is updaed every ve years. If emissions in a paricular year are higher han he reference level, no paymen will be made o he Amazon fund in he subsequen year. Hence, his scheme ress on expos evaluaion of he emissions due o deforesaion, and respecs naional sovereigny since he fund is managed by he Brazilian Developmen Bank (BNDES). However, he incenive scheme ha performs well for Brazil 1 See REDD sands for Reducing emissions from deforesaion and degradaion, whereas REDD+, in addiion o REDD, includes enhancing fores carbon socks hrough aciviies such as fores conservaion, fores resoraion and susainable fores managemen (Angelsen e al., 29; Kanowski e al., 21). 2 See Norwegian iniiaive on hp:// and more deails on he Amazon Fun on hp:// 2
3 is unlikely o sui Guyana, where deforesaion hardly occurs despie he large ropical fores. In his conex, Norway has oered up o USD 25 millions for preserving he sock of fores in Guyana. The design of he incenive scheme is crucial o avoid emissions from deforesaion, and i needs o ake ino consideraion boh he sock of fores and he rae of deforesaion depending on he characerisics of he recipien counries. To address he quesion of he design of he REDD mechanism, we develop a growh model wih deforesaion in a fores abundan economy ha will receive a ransfer from he inernaional communiy. Assuming ha deforesaion generaes a global exernaliy ha aecs he welfare of he inernaional communiy, we analyze he opimal policies ha resore he social opimum in a decenralized seing where he recipien economy does no immediaely suer from he exernaliy. We adop several simplifying assumpions. Firs, he analysis is held in parial equilibrium since we are only considering one producive secor in he recipien economy, and he secor is expororiened since all producs are consumed by he Norh. We make his assumpion wih he case of Brazilian expors of soybeans, Indonesian expors of palm oil, and expors of ropical wood and biofuels from ropical counries in mind. These producs are ofen linked o deforesaion paerns in ropical counries. This leads us o our second assumpion: we assume ha agriculure is he main driver for deforesaion, ha is, foresed lands are cleared for producive purpose, and agriculural expansion is fueling growh in ropical counries. In his conex, reducing deforesaion may have derimenal impacs on growh rends and requires ha he counry will be compensaed for he opporuniy coss of no clearing land for agriculure. The hird assumpion is ha only he inernaional communiy (rich indusrialized counries) or he Norh is willing o preserve ropical foress. I reecs he fac ha ropical counries are developing counries, hence heir prioriy is developmen even hough hey face a radeo beween environmenal preservaion and developmen. We consider wo ways of incorporaing he exernaliy in he uiliy of he Norh: hrough an ameniy value made explici in consumers' preferences, or hrough an environmenal hreshold. Norhern consumers are aeced by a loss in ameniy value when he sock of ropical foress is reduced. This reecs a coninuous willingness o preserve ropical foress. Since foress are sequesraing carbon and conribue o climae sabiliy, we can also relae he amospheric carbon cap ha climae scieniss recommend no o overshoo o avoid high climae disrupions wih a minimum sock of ropical foress o be preserved. The inuiion is ha, if he sock of ropical fores is reduced beyond his hreshold, he loss of carbon sequesraion sinks and he carbon released ino he amosphere by deforesaion would induce 3
4 some caasrophic damages. Ineresingly, he Norh faces a radeo beween preserving ropical foress and consuming goods ha are produced in deforesing counries. Finally, he fourh assumpion deermines how o implemen he REDD mechanism so ha he ropical counry would reduce is deforesaion rends. Since here is no consensus on he mos appropriae ransfer scheme in he lieraure, we consider ha he ransfer can depend boh on he sock of fores or on he rae of deforesaion. We obain ha... To he bes of our knowledge, he lieraure on paymens for preserving ropical foress has no proposed an opimal incenive scheme in he conex of a growh model wih deforesaion. I can serve as a benchmark for he negoiaions beween Norhern donors (e.g. Norway) and he recipien ropical counries. Using a renewable resource model, Sähler (1996) compares wo ransfer schemes aiming a preserving a fores sock, and concludes ha he scheme which oers a consan price per hecare of fores is more ecien han he scheme which oers a compensaion price ha rises wih fores scarciy, when foresabundan counries adop sraegic behaviors. van Soes and Lensink (2) use a ransfer scheme ha depends on boh he sock of fores and he rae of deforesaion o show ha penalizing he rise in he rae of deforesaion can be ecien for limiing and posponing carbon emissions from deforesing. The more recen debae on he evaluaion of he opporuniy coss of REDD projecs reveals ha he willingness o nance fores preservaion depends crucially on he expeced benes in erms of carbon emission reducions (Myers, 27; Eliasch, 28). Many drawbacks limi he expeced carbon emission reducions: carbon sequesraion in fores is no permanen; here is a risk ha deforesaion will move oward non paricipaing counries, leading o carbon leakage; and he governance issue is ofen problemaic in ropical developing counries, implying a high risk ha he ransfer will be divered o corrupion and renseeking (Myers, 27). We absrac from hese issues by considering only one recipien economy characerized wih neiher marke nor insiuional failures. The remainder of he paper is organized as follows. Secion 2 presens he simple model of growh wih deforesaion. Secion 3 presens he opimum, ha is, wha would be he opimal deforesaion rends for he consumers who are consuming he good ha is responsible for he exernaliy. Secion 4 presens he opimal behaviors of he producing Souh and of he consuming Norh, and secion 5 describes he incenive mechanism ha is required o resore he opimum in his conex. Secion 6 illusraes wih a specied model, and secion 7 provides concluding remarks. 4
5 2 The model Consider a fores abundan economy wih an inniely lived represenaive agen. economy is composed of one secor, whose produc is consumed. Only one facor of producion, land, is required, bu we disinguish beween old agriculural land and newly deforesed land, he laer being more producive han he former. And deforesing generaes an exernaliy suered by he represenaive consumer. In our economy, deforesaion is enirely driven by agriculural expansion. We focus on he issue of land allocaion beween a producive land use and an "idle" land use, hence we neglec sources of revenue from fores such as imber producion van Soes and Lensink (2). We assume ha deforesaion is irreversible, ha is, once a hecare of fores has been cleared, i is used for agriculure and hus fores canno regrow. Following Harwick e al. (21), he clearing process is represened as a sicky adjusmen of endowmens, where he sock of "unproducive" fores F decreases while he sock of agriculural land A rises, since he economy has iniially "oo much" foresed land (large F ). The dynamics is The F = h = A, (1) where h denoes he rae of deforesaion or he amoun of newly cleared land a period, and F and A are given. Given he oal amoun of land L in he economy, we have for any : L = F + A. (2) The producion funcion of he agriculural secor, whose produc x is homogeneous, akes he following form: x = f(a, h ) (3) where boh he sock of agriculural land, A, and he newly deforesed land, h, are facors of producion. We assume ha heir produciviy dier since he clearing and burning of biomass creae a boos in produciviy in newly deforesed land, due o he release of nuriens from ashes. The feriliy gap only lass one period, and afer one period, he land falls ino he sock of agriculural land whose produciviy is normalized o one. The parial derivaives of he producion funcion wih respec o agriculural land and o newly cleared land are denoed by f A and f h, respecively. We assume ha echnology f(.) exhibis decreasing reurns o scale, since here is only one inpu considered (land). The use of inpus is no cosly, since he resource is in open access and once deforesed he land is owned by he producer. 5
6 The enire producion of he homogeneous good is consumed by he represenaive household. Since his preferences exhibi ameniy values, his insananeous uiliy funcion U(.) depends boh on his consumpion x and on he sock of ropical fores F. Denoe by W he ineremporal uiliy funcion, which is dened by: W = U(x, F )e ρ d (4) where ρ is he social discoun rae, which is assumed o be consan. The parial derivaives of he insananeous uiliy funcion wih respec o consumpion and o fores preservaion are denoed by U x and U F, and hey are boh posiive. We also assume ha he funcion U(.) is sricly concave and saises he Inada condiions. We do no need o make any assumpion on he sign of he cross derivaive U xf. 3 To avoid any environmenal caasrophe, we assume ha he sock of fores, F, needs o say above a minimum F or, equivalenly, ha he sock of deforesed land A needs o remain inferior o a hreshold Ā = L F. Hence, we have F F A Ā,, (5) wih F < F and Ā > A. Assuming < F < L implies ha < Ā < L. Combining his consrain wih he ameniy value of he fores implies ha he following disconinuous welfare funcion V (x, F ) capures he wofold represenaion of he exernaliy from deforesing: { U(x, F ), if F F x, V (x, F ) = (6), if F < F 3 The opimum The social planner chooses he deforesaion pah h ha maximizes he ineremporal uiliy W subjec o (3)(2), (5) and o he nonnegaiviy consrain h. The exended Hamilonian (in curren value) of his onesaevariable problem is: H = U(x, F ) + λ h + µ h + η (Ā A ) (7) where λ is he cosae variable associaed wih agriculural land accumulaion, and µ and η are he Lagrangian mulipliers associaed wih he nonnegaiviy consrain on he 3 The erm U xf measures he degree of subsiuion or complemenariy beween consumpion and naural capial preservaion in welfare. In he Michel and Roillon (1996) erminology, he uiliy funcion exhibis a "compensaion eec" if U xf < (i.e. a decrease in naural capial implies an increase in he marginal uiliy of consumpion ha leads sociey o consume more) and a "disgus eec" if U xf >. 6
7 deforesaion rae and wih he fores hreshold consrain, respecively. The saic and dynamic rsorder condiions are: and he ransversaliy condiion is given by and he complemenary slackness condiions are U x f h = λ µ (8) λ = ρλ U x f A + U F + η, (9) lim e ρ λ A =, (1) µ h =, µ, h (11) η (Ā A) =, η, A Ā. (12) Condiion (8) implies ha he marginal gain of deforesaion in erms of uiliy, which is relaed o an increase in consumpion, mus be equal o he social marginal cos of deforesaion, λ, as long as h >. Denoe by φ(f, h ) U x f h he marginal gain of deforesaion. Inegraing he dierenial equaion (9) from up o yields: [ ] λ = λ + (U F + η s U x f A )e ρs ds e ρ (13) From he ransversaliy condiion (1), i implies ha λ = (U F + η s U x f A )e ρs ds, which leads o λ = (U F + η s U x f A )e ρ(s ) ds. (14) A any period along he opimal rajecory, he social marginal cos of deforesaion mus be equal o he discouned sum from up o of he ow of he marginal ameniy value of preserving he fores plus he marginal social value of he hreshold consrain, minus he marginal uiliy gain from an increase in agriculural land, U x f A. Reducing he sock of fores by one uni provides a direc marginal decrease in uiliy, U F, and an indirec marginal loss since we are approaching he hreshold by one more uni, which is capured by η. We denoe by ψ(f, h ) U F + η U x f A he ne marginal gain of preserving he fores. Proposiion 1 Characerizaion of he opimal soluion: 1. Inerior soluion: a each ime, an opimal deforesaion pah is characerized by he following equivalen equaions: φ(f, h ) = ψ(f s, h s )e ρ(s ) ds (15) ρ = φ(f, h ) φ(f, h ) + ψ(f, h ) φ(f, h ) 7 (16)
8 2. If he economy converges oward a seadysae, i mus be characerized by: ρ = 1 ( ) UF + η f A. (17) f h U x Proof: Equaion (15) resuls from combining (8) and (14), and (16) is obained by diereniaing (15) wih respec o ime. Equaion (17) is derived by seing φ equal o zero in (16). η corresponds o he value of η a seady sae. In our economy, no deforesaion occurs a he seadysae, hence he sock of fores remains consan (if posiive). Condiion (17) can be inerpreed as a modied Clark and Munroe Golden Rule (Swallow, 199) ha equalizes he social discoun rae o he social marginal rae of reurn of undeforesed land. Deforesaion ends when he marginal gain of deforesaion muliplied by he social discoun rae equals he social ne marginal gain of preserving he fores. Denoing by A he opimal saionary sock of agriculural land, we observe wo cases: if A = Ā, hen η, he hreshold consrain is binding, which impedes more deforesaion; whereas if A < Ā, hen η =, and he hreshold plays no role in he ending of deforesaion. Given ha Ā < L, some fores will remain a seady sae, even for negligible ameniy values. 4 The decenralized economy Now, consider ha here is no inernaional social planner who balances he benes and he damages from deforesaion. Insead, we represen a simplied world economy where Souh is he producer of a single expored good and he landowner of ropical foress, and where Norh consumes he good and suers from he exernaliy of deforesaion. The price of he expored good is normalized o one. Since he environmenal exernaliy is no considered by Souh, Norh mus oer o Souh a condiional ransfer o avoid ha mos of he fores is cu down. Denoe by S(F, h ) he funcional class of ransfers ha depends on boh he remaining sock of fores F and he insananeous deforesaion rae h. Inuiively, an ecien ransfer scheme should be designed such ha S F and S h. 4.1 Deforesing Souh We assume ha fores is an open access resource in Souh, hence here is no marke for land. Clearing he land is sucien o obain he propery righs. Assuming ha deforesaion is irreversible implies ha Souh will ake ino accoun he shadow cos of deforesaion. 8
9 However, only an inernaional ransfer from Norh can compel Souh o consider he social opporuniy cos of no preserving he fores. The program of Souh is: max {h, } [f(a, h ) + S(F, h )]e rsds d (18) subjec o (1), (2) and o h. The exended Hamilonian of his program is: where λ S H S = f(a, h ) + S(F, h ) + λ S h + µ S h (19) is he cosae variable associaed wih he accumulaion of he sock of agriculural lands, and where µ S is he Lagrangian muliplier associaed wih he nonnegaiviy consrain on he deforesaion rae. The saic and dynamic rsorder condiions, he ransversaliy condiion, and he complemenary slackness condiion are, respecively: f h + S h = λ S µ S (2) λ S = r λ S + S F f A (21) lim e rsds λ S A = (22) Using (21) and (22), we can express λ S λ S = As long as h >, we have µ S µ S h =, µ S, h (23) respecs he following arbirage condiion: as follows: (S F f A )e s rudu ds (24) =, hence given (2) and (21), he inerior soluion r = φ S (F, h ) φ S (F, h ) + ψs (F, h ) φ S (F, h ) (25) where φ S (F, h ) f h + S h is he ne marginal pro from deforesing and ψ S (F, h ) S F f A is he ne marginal pro of preserving one uni of fores. Comparing (25) wih (16) shows ha he insananeous pro ows are now discouned a he ineres rae r, insead of ρ, and ha he gains or losses from deforesing are no evaluaed in erms of uiliy. 4.2 Environmenallyfriendly Norh Norh is he only consumer of he good whose producion generaes deforesaion and is willing o preserve ropical fores due o is ameniy value. The program of he donor is: max {x, } U(x, F )e ρ d (26) 9
10 subjec o Ḃ = r B x S(F, h ), (27) where B denoes Norhern wealh, which can be used for consumpion or for invesmen a he ineres rae r. I can be easily shown ha solving he Norhern consumer's program leads o he sandard KeynesRamsey condiion: ρ U x U x = r (28) 4.3 Characerizaion of he decenralized equilibrium A paricular equilibrium, associaed wih a paricular ransfer funcion S(F, h ), is a vecor of rajecories {h, A ; r } such ha: i) Souhern producer maximizes is ineremporal pro funcion; ii) Norhern consumer maximizes is uiliy; iii) he marke of he nal good is perfecly compeiive and cleared:, x = f(a, h ); and iv) Souh receives a ransfer S(F, h ) from Norh. From he previous analysis of individual behaviors, we obain Proposiion 2 For a given class of ransfer funcions S(F, h ), he decenralized equilibrium is characerized as follows: 1. During any sricly posiive deforesaion regime, he wo following equivalen equaion mus hold a he equilibrium: φ S (F, h ) = 1 U x, ρ U x = φ S (F, h ) U x φ S (F, h ) + ψs (F, h ) φ S (F, h ) ψ S (F s, h s )U x,s e ρ(s ) ds (29) 2. If he economy converges oward a seadysae equilibrium, i mus be such ha: Proof: (3) ρ = ψs (F, h) φ S (F, h) = S F f A S h + f h (31) Inegraion of (28) from up o s yields: e s rudu = (U x,s /U x, ) e ρ(s ). Combining his las resul wih (2) and (24) implies (29). (3) is a direc implicaion of (25) and (28). 5 Opimal ransfer analysis 5.1 Firsbes opimum implemenaion rule The opimal ransfer funcion is such ha he characerizing condiions of he opimum in Proposiion 1 coincide wih he characerizing condiions of he decenralized equilibrium 1
11 in Proposiion 2. We obain Proposiion 3 During any sricly posiive deforesaion regime a he decenralized equilibrium, he ransfer funcion S(F, h ) mus saisfy he wo following equivalen condiions in order o resore he rsbes opimaliy: S h U x = (S F U x U F η s )e ρ(s ) ds U F + η U x = S F + Ṡh S h ( ρ U x U x ) (32) The condiion saes ha when he hreshold F is no reached, he marginal rae of subsiuion beween consumpion and fores preservaion is equal o he marginal ransfer per uni of fores, plus he change of he marginal ransfer per uni of deforesaion, minus he marginal ransfer per uni of deforesaion muliplied by he rae of ineres. A priori, here exiss an inniy of ransfer funcions among he funcional class S(F, h ) ha saisfy he condiion (32). In he nex subsecion, we will resric he analysis o a se of sandard poliical ools, namely axes and subsidies. 5.2 Example of ransfers Subsidy on he remaining sock of fores Consider ha he ransfer akes he form of a uniary subsidy on he remaining sock of fores, i.e. S(F, h ) = σ F, wih σ for any. From (32), he opimal subsidy rae mus be equal o: σ = U F + η U x (33) When he hreshold F has no been reached (η = ), he opimal subsidy rae mus be equal o he marginal rae of subsiuion beween consumpion and preservaion of he fores. Once he hreshold is reached, he opimal subsidy rae is larger since η. Is dynamic properies canno be characerized wihou a deeper invesigaion on he properies of he uiliy funcion U(x, F ). An analyic example will be examined laer. If he damage due o deforesaion was only capured by he hreshold consrain, he opimal subsidy rae mus be equal o zero as long as he hreshold of fores is no reached and o he marginal social cos of he consrain in moneary erms (i.e. divided by he marginal uiliy) hereafer. Norh would hen perceive a subsidy only once his fores sock has been driven down o he allowed minimal level, o sop furher deforesaion. 11
12 5.2.2 Tax on he deforesaion ow Consider ha he ransfer akes he form of a ax on he deforesaion ow, hence S(F, h ) = τ h, wih τ for any. From (32), we have τ = 1 (U F + η s )e ρ(s ) ds (34) U x A each period, he opimal ax rae mus be equal o he discouned sum of he insananeous ow of social marginal gain from preserving he fores, divided by he marginal uiliy of consumpion o obain he expression in moneary erms (given ha he consumpion good is he numeraire). Since η = for any < T, where T denoes he dae a which he sock of fores reaches he minimal hreshold, he expression of τ rewrien as: τ = [ e ρ T U x U F e ρs ds + ] T (U F + η s )e ρs ds, [, T ) e ρ U x (U F + η s )e ρs ds, [T, + ) can be If he deforesaion exernaliy was capured only by he minimal hreshold, he opimal ax would be sricly posiive even when < T. This is in conras wih he subsidy on he sock of fores in he case where fores has no ameniy value. Logdiereniaing (34) gives he growh rae of he opimal ax on deforesaion a each ime : τ = ρ U x U F + η τ U x (U F + η s )e ρ(s ) (36) ds The dynamics of he opimal ax resuls from he dierence of wo componens. The rs one, ρ U x /U x, is simply he ineres rae, i.e. he rae a which fuure gains or losses are discouned, assumed here o be posiive. 4 The second erm is he raio of he insananeous social marginal uiliy gain of preserving he fores over he discouned sum of he same marginal gain, which is also posiive. We hen have wo opposie eecs ha drive he dynamics of he ax, which implies ha he ax can eiher rise or fall over ime Policymix Since Souhern paricipaion ino he ransfer program is volunary, he ax scheme is unlikely o be acceped. Hence, he paricipaion consrain requires ha S(F, h ). 4 Noe ha, since social preferences exhibi a conservaion moive for ropical fores, his discoun rae can be expanded as r = ρ+ɛ(x )g x,+h U xf /U x, where ɛ(x ) is he inverse of he elasiciy of ineremporal subsiuion of consumpion and g x, is he insananeous growh rae of consumpion. As compared wih he sandard decomposiion of he discoun rae, we mus ake ino accoun he ameniy eec capured by h U xf /U x, whose sign is no specied a priori. An exensive discussion on his poin can be found in Schuber (26). (35) 12
13 To respec his condiion, he scheme needs o combine a subsidy for fores preservaion a rae σ and a ax on he deforesaion process a rae τ : S(F, h ) = σ F τ h. If one of hese wo policy insrumens is se equal o is rsbes opimal level dened above, hen obviously, he second one mus be equal o zero. Since he model considers a single source of exernaliy, a unique policy insrumen is required o resore opimaliy. However, he enforcemen of he rsbes soluion hrough one insrumen may no be achievable due o budgeary, socioeconomic or poliical addiional consrains, hence he use of boh insrumens may be required. To illusrae his, consider ha Norh has no se he subsidy rae a is opimal level, hence σ < σ,. In his paricular case, Norh could also implemen a secondbes opimal ax τ sb τ sb o resore he opimum, such as = 1 U x (σs σ s )e ρ(s ) ds > (37) U x Since he donor was no able o oer he opimal subsidy rae, he incenive for preserving fores is no sucienly high, hence a ax is required o limi deforesaion. However, we need o have σ F τ sb h for all o ensure Souhern paricipaion. Conversely, consider ha he ax rae τ eecively levied by Norh is such ha τ < τ,. From (32), he secondbes subsidy rae σ sb, which is necessary o resore he opimum, is given by: U x σ sb s e ρ(s ) ds = U x (τ τ ) σ sb = ( ρ U x U x ) (τ τ ) ( τ τ ) > (38) Since Norh has no se he ax a is opimal level, Souh is emped o defores more, hence a subsidy on he fores sock is necessary o curb deforesaion. However, we need o have σ sb F τ h o allow Souh o paricipae. 6 Illusraion 6.1 Analyical specicaions and idenicaion of he possible cases In order o furher develop he analysis, we adop he following specicaions. Firs, assume a separable insananeous uiliy funcion: U(x, F ) = x1 ɛ 1 ɛ + γf, wih ɛ >, γ > (39) where 1/ɛ is he consan ineremporal elasiciy of subsiuion in consumpion. posiive o ensure ha he higher he sock of preserved ropical fores, he higher he uiliy. Second, consider a producion funcion wih decreasing reurns o scale, where wo γ is 13
14 ypes of land are perfec subsiues: x = f(a, h ) = (A + νh ) β, wih < β < 1, ν > 1, (4) where ν is he feriliy boos of he newly deforesed land, h, compared o he old agriculural land, A. We rs look a he opimal saionary regime. There are wo possible saionary socks of agriculural lands: he "naural" seadysae level A owards which he economy would end in he absence of he maximal hreshold consrain on A and he imposed seadysae level Ā. In boh cases, he deforesaion rae h equals zero and he resuling consumpion level simply becomes x = x = (A ) β or x = x = Āβ, depending on he saionary amoun of agriculural land. Assume ha level Ā is high enough so ha A < Ā L. In his case, he economy converges owards a seadysae which is characerized by (17) wih η =, saring from A < A. Using he specied analyical forms (39) and (4), U x f h = νu x f A = νβx 1 ɛ 1/β so ha (17) implies: [ β(ρν + 1) A = γ ] 1 1 β(1 ɛ) (41) where [1 β(1 ɛ)] > for any < β < 1 and ɛ >. Conversely, assume ha Ā is low enough so ha he hreshold consrain becomes binding a some dae T >. Hence, from (12), we have η = for any < T and η for any T. Equaion (17) implies η = β(ρν + 1)Āβ(1 ɛ) 1 γ for any T. Replacing γ by β(ρν + 1)(A ) β(1 ɛ) 1 in his las expression, using (41), we obain: {, [, T ) η = β(ρν + 1)Ψ, [T, + ) (42) where Ψ is dened as: Ψ = Āβ(1 ɛ) 1 (A ) β(1 ɛ) 1. (43) wih β(1 ɛ) 1 < for any β (, 1) and ɛ >. The muliplier η is nonnegaive if and only if Ψ, ha is, if Ā A. Consequenly, we have wo cases: eiher Ā > A and he opimal saionary regime is unconsrained, or Ā A and he saionary regime is driven by he hreshold consrain. We invesigae hese wo cases in he following subsecions. 6.2 The unconsrained case: A < Ā Opimal rajecories If he hreshold consrain is no binding, η = for any, and he opimal rajecories are given by he following proposiion: 14
15 Proposiion 4 If A < Ā, he opimal rajecories are (he upperscrip u refers o opimaliy in he unconsrained case), for any : A u = A (A A )e ν = L F u (44) ( A h u ) A = e ν (45) ν x u = x = (A ) β. (46) Proof: See he proof in appendix. Given ha he uiliy funcion is addiively separable in x and F, and is linear in F, he economy insananeously jumps from he beginning of he planning horizon o a saionary consumpion regime. However, he opimal deforesaion rae decreases exponenially over ime from is iniial level h = (A A )/ν, and ends asympoically o zero. The sock of agriculural lands coninuously increases over ime, from A a he iniial period o is asympoic saionary value A. Observe ha a sricly posiive sock of fores F will remain in he seady sae, where F = L A, due o he ameniy value of fores. In he absence of such a conservaion moive, he opimal deforesaion pah could exhaus enirely he fores sock in nie ime 5. These opimal rajecories are depiced in Figure 1. Table 1 shows he sign of he parial derivaive of he variables of ineres wih respec o he main parameers. We nd noably ha he opimal saionary levels of agriculural land and consumpion are increasing in he pure rae of ime preference ρ, as well as in he new deforesed land produciviy boos ν, and are decreasing in he marginal ameniy value γ. Table 1: Comparaive dynamics analysis The unconsrained case X X/ ρ X/ γ X/ ν A u + depends on A, β, ɛ, ρ, ν h u + depends on A, β, ɛ, ρ, ν x u + + A + + F + 5 Proof available upon reques. 15
16 h h u h * * A A = F F A, L L u F * F * A F u A A Figure 1: Opimal rajecories of h, A and F The unconsrained case 16
17 6.2.2 Opimal policies In he unconsrained case, consider rs a pure subsidy on he remaining sock of fores: S(F, h ) = σ F. From (33), he opimal subsidy rae is σ u = γ(a ) ɛβ,. Since he marginal rae of subsiuion beween consumpion and preservaion of he fores is consan hrough ime in our specied framework, he opimal rae of subsidy mus also be consan. Obviously, i is increasing in he marginal ameniy value, γ. I is also increasing in ρ and ν, since a rise in hese wo parameers resuls in a larger saionary level of agriculural land. τ u Second, consider a pure axaion policy: S(F, h ) = τ h. From (34), i implies: = γ(a ) ɛβ /ρ,. The opimal deforesaion ax rae corresponds o he discouned sum from up o of he ows of he opimal marginal rae of subsiuion (MRS) beween x and F. Given ha he opimal MRS is consan over ime in our specied example, he uniary ax simplies o MRS/ρ. The opimal ax rae is consan hrough ime, increasing in γ and ν and decreasing in ρ since shor run consumpion, and deforesaion, are more valuable when ρ is high. 6 σ u Imagine ha Norh is no willing o oer more ha half of he opimal subsidy: we have = σ u /2. To resore he opimum, he ransfer scheme mus combine he subsidy wih a secondbes ax, which is se as follows: τ sb he scheme mus respec he following condiion: = τ u /2. To ensure Souhern paricipaion, L A (A A )e /ν [1/(νρ) 1]. (47) If ρν 1, he inequaliy is saised for all, whereas, if ρν < 1, he condiion is saised if ρν is high, and if he saionary sock of fores is large compared o he rise in agriculural land from is iniial level, given ha e /ν < The consrained case: Ā A Opimal rajecories Consider now ha he hreshold consrain becomes binding a some dae T. In his case, he Lagrange muliplier associaed wih he hreshold consrain is given by (42). 6 A simple calculus leads o: τ u ρ = β ɛβ 1 β(1 ɛ) ρ 2 ( ρν + 1 which is negaive for any ɛ and β 1. γ ) β 1 1 β(1 ɛ) [ (β 1)(ρν + 1) ɛβ 1 β(1 ɛ) ], 17
18 Proposiion 5 If Ā A hen, he opimal rajecories are (he upperscrip c refers o opimaliy in he consrained case), for any : [ A + 1 ] 1 A c ν (zc β(1 ɛ) 1 s) e s ν ds e ν, [, T ) = Ā, [T, + ) (48) F c = L A c (49) h c = 1 [ ] (z c 1 ) β(1 ɛ) 1 A c (5) ν β x c = (z c ) β(1 ɛ) 1, (51) where z c is dened by: ] Ā β(1 ɛ) 1 ρν+1 + Ψ [e ( z c ν )(T ) 1, [, T ) = Ā β(1 ɛ) 1, [T, + ) (52) Proof: See he proof in appendix. Le us analyze he dynamic properies of hese opimal rajecories. Using (52), he rajecory of z c is increasing and convex for any [, T ) and consan for any T. I is depiced by he Norh Eas quadran of Figure 2. By consrucion, x c is obained as a decreasing and convex ransformaion of z c of Figure 2. as shown in he Souh Eas quadran The opimal consumpion pah is declining hrough ime as long as he deforesaion process goes on and i reaches he imposed sabilizaion level x = Āβ a ime T. Consumpion decreases because producion is resriced, due o he consrain on agriculural land. The rajecory of A c mus be increasing since A c = h c is nonnegaive by assumpion. Using (5) and (51), Ä c = ḣc = (1/ν)[ẋ c (x c ) 1/β 1 /β A c ] which is clearly nonposiive since ẋ c and A c. Finally, diereniaing again his las expression wih respec o ime leads o ḧc since ẍ c and Äc. We obain an increasing and concave opimal rajecory of agriculural land accumulaion and a decreasing and convex opimal deforesaion pah for any [, T ) as represened in Figure 3. The opimal dae T a which he sock of deforesed land reaches he imposed maximal hreshold Ā is endogenously deermined by he coninuiy condiion on he opimal rajecory A c. I is hus soluion of he following equaion: 1 T ν 1 (zs) c β(1 ɛ) 1 e s ν ds = Āe T ν A. (53) 18
19 A z β ( 1 ε ) 1 c z x = z β β ( 1 ε ) 1 c z x β A T c x 45 c x x Figure 2: Opimal rajecory of x The consrained case Performing a comparaive dynamics analysis on he opimal rajecories amouns o performing a comparaive saics analysis on T. We underake such an exercise by sudying he impac of he pure rae of ime preference and of he marginal ameniy value on T. Parially diereniaing (53) wih respec o ρ leads o: [ 1 T ν ρ (zc T ) 1 β(1 ɛ) 1 e T ν + T z c s ρ ( ) ] 1 (z c 1 β(1 ɛ) 1 β(1 ɛ) 1 s) 1 e s ν ds = T Āe T ν ρ ν Given z c T = Āβ(1 ɛ) 1, his expression simply reduces o z c / ρ =, for any [, T ). From (41), (43) and (52), we obain afer simplicaions: z c [( ) ] ρν + 1 T γν [ ρ = Ψ + (T ) = [,T ) ν ρ β(ρν + 1) 2 1 e ( ] ρν+1 ν )(T ) (55) For any [, T ), he RHS of his equaliy is negaive. Since Ψ is posiive, a necessary condiion for geing a negaive LHS is T/ ρ. (54) Inuiively, he opimal dae a which he sock of fores reaches is hreshold level occurs earlier when ρ increases. This resul implies ha he iniial deforesaion level h c mus increase, since he area below he rajecory of h c beween and T is consan: T hc d = Ā A. Then, an increase in ρ resuls in faser deforesaion: F c is reduced and A c is augmened. The eec of an increase in ρ on he opimal rajecories of he deforesaion rae and he sock of agriculural land is represened in Figure 4 by a shif from he solid line o he dashed line. 19
20 h h c h A A T A, L L c F F F A c A A T Figure 3: Opimal rajecories of h, A and F The consrained case 2
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