Lee H. Endress James A. Roumasset Ting Zhou. Working Paper No January 2002

Transcription

1 SUSTAINABLE GOWTH WITH ENVIONMENTAL SPILLOVES: A AMSEY-KOOPMANS APPOAH by Lee H. Endress James A. oumasse Ting Zhou Working Paper No. 2-4 January 22

2 Susainable Growh wih Environmenal Spillovers: A amsey-koopmans Approach Lee H. Endress James A. oumasse Ting Zhou Absrac In his paper we apply he canonical approach of amsey Koopmans and Diamond o he problem of opimal and ineremporally-equiable growh wih a non-renewable resource consrain and show ha he soluion is susainable. The model is exended o cases involving environmenal ameniies and disameniies and renewable resources. The soluions equivalenly solve he problem of maximizing ne naional produc adjused for depreciaion in naural capial and environmenal effecs which urns ou o be boh susainable and consan even wihou echnical change. Universiy of Hawaii a Manoa Deparmen of Economics Social Science Building oom Maile Way Honolulu HI

3 1. Inro: To discoun or no o discoun? "Susainable growh" is commonly modeled as a problem of maximizing an ineremporal uiliarian welfare funcion subjec o he consrain ha consumpion or uiliy growh canno be negaive (e.g. Asheim 1995). The consrained-uiliarian approach applies posiive discouning of fuure uiliy consisen wih Koopmans' (196) demonsraion ha here does no exis a uiliy funcion defined on all consumpion sreams ha saisfies he usual axioms of raional choice and iming neuraliy (i.e. wihou discouning). In an alernaive approach (Belrai e. al and hichilnisky; 1996) 1 social welfare is modeled as a weighed average of convenional growh and a concern for susainabiliy. Boh schools of hough rejec he possibiliy of using a zero uiliy discoun rae o represen a social planner's concern for inergeneraional equiy. This is odd in ligh of he fac ha he pioneers of growh heory advocaed exacly ha. amsey warned ha he use of a posiive uiliy discoun rae is "ehically indefensible" and reveals "a weakness of he imaginaion." Koopmans himself (1965) noed ha "we welcome equally a uni increase in consumpion per worker in any one fuure decade Mere numbers canno give one generaion an edge over anoher " Moreover while Heal e. al. recognize Koopmans' (196) nonexisence heorem hey seem o overlook his subsequen soluion (1965) which relies precisely on he noion of "inergeneraional neuraliy" for a specific non-empy subse of feasible consumpion pahs and is capured in urn by he zero uiliy discoun rae. Accordingly he firs objecive of he presen paper is o reurn o he canonical approach of amsey and Koopmans and explore he exen o which concerns of he susainabiliy dialog can be capured by ha approach. 1 See also he discussion of his approach in Heal

4 Mos of he susainable growh lieraure focuses on he rade-off beween he accumulaion of produced capial and he depleion of naural capial (e.g. Toman e. al. 1995) o he neglec of he Brundland ommission s original emphasis on he imporance of managing iner-linkages beween povery populaion growh and environmenal degradaion. A second objecive of he presen paper herefore is o help develop hese linkages by exending he opimal growh framework o allow for environmenal ameniies and disameniies. A hird school of hough recommends maximizing green ne naional produc (e.g. World Bank 1997) wherein one subracs depreciaion of naural capial along wih ha of produced capial in reckoning ne naional produc. Weizman ( ) has indeed shown ha such a measure is formally equivalen o maximizing a uiliarian welfare funcion. Bu while his measure is commonly described as susainable income uiliy discouning permis is maximized value o evenually decline. In oher words maximizing susainable income is inconsisen wih opimal susainable growh as defined by eiher of he above-menioned schools. framework. We will show however ha his paradox does no arise in he amsey-koopmans Maximizing real naional income in he appropriaely adjused model yields an income sream ha is no only susainable bu also consan in he opimal program. Moreover he Weizman "hypoheical saionery equivalen income" is jus he golden rule uiliy level. In secion 2 below we presen he condiions for opimal and ehically neural growh in a model wih a non-renewable resource and a backsop echnology. The maxi-min soluion is shown o be a special case of his soluion albei one which is unlikely o be preferred. In secion 3 we exend he model o include environmenal ameniies and disameniies associaed wih he use of renewable and non-renewable resources. In secion 4 we invesigae he 3

5 relaionship beween amsey-koopmans opimal growh and susainable income. Secion 5 provides a brief summary and concluding remarks. 2. Ineremporally-neural opimal growh wih a non-renewable resource 2.1 Koopmans impossibiliy heorem and his soluion An immediae obsacle o maximizing a uiliarian welfare funcion wihou discouning is ha he value of such a funcion is infinie for some feasible consumpion sreams. Koopmans (196) wen furher proving he impossibiliy of represening ineremporally-neural planner preferences over all consumpion vecors by any uiliy funcion. Building on he earlier work of amsey (1928) Koopmans (1965) argued ha one way ou of he dilemma of a non-exisen uiliy funcion of all consumpion pahs is o idenify a subse of all feasible pahs on which one can define a neural uiliy funcion. amsey's crierion for eligibiliy in he subse is a sufficienly rapid approach of he pah o a "bliss poin". Koopmans' crierion is less resricive: "We shall find ha in he presen case of a seady populaion growh he golden rule pah can ake he place of amsey's sae of bliss in defining eligibiliy" (Koopmans 1965 p.5). Specifically consumpion mus approach he golden rule consumpion level wih sufficien rapidiy ha he area of defici beween he "feliciy" of consumpion and ha of golden rule consumpion converges o a finie number i.e. V() < where V is he planner s uiliy defined as he area given in equaion 3. Using his crierion Koopmans (1965) demonsraes ha each eligible pah is superior o each pah ha is ineligible. Moreover one can rank eligible pahs and deermine one ha is opimal. 4

6 2.2 Susainabiliy wihou really rying Following Koopmans' mehod in his paper we propose an approach o susainabiliy ha is based on a resource managemen and capial accumulaion obained by seing he rae of ime preference ρ equal o zero. Insead of maximizing he social uiliy as a discouned sum he arge is now maximizing he uiliy funcion defined as an infinie sum of he difference beween he acual consumpion rajecory and he opimal consumpion. In order o inroduce naural capial ino he amsey-koopmans framework consider an economy ha uses a naural resource () in addiion o capial (K) and labor (L) o produce a single homogeneous good. Assume ha he producion echnology is consan reurns o scale so ha he producion funcions Q(KL) is homogeneous of degree one. In order o presen he argumen in is sarkes form we absrac from populaion growh and echnological change and normalize L = 1 such ha Q(KL) can be expressed as F(K). Following he sandard approach oupu of producion is divided among consumpion gross invesmen and he cos of providing he resource as an inpu o he producion process. Le è be he uni cos of exracing he naural resource and providing i as an inpu of producion. We assume ha his cos is a decreasing funcion of he resource sock (e.g. Heal 1976). Produced capial K depreciaes a he rae δ. The dynamic equaion governing capial accumulaion is = F(K ) äk K è( ) (1) In his secion he naural resource is assumed o be non-renewable and he dynamic equaion governing he resource sock becomes = (2) 5

7 We augmen his basic model by incorporaing a backsop echnology ha has a fixed uni exracion cos θ b. onsider for example he case of oil a non-renewable resource. Oil socks are drawn down as he economy grows unil uni cos θ has risen sufficienly o warran he swich o a superabundan bu high cos alernaive energy source (e.g. coal gasificaion nuclear fission/fusion solar energy). Once he swich has been made we assume ha he backsop echnology delivers energy a consan cos θ b. Therefore he locus of uni exracion cos is he lower envelope of curves θ () and θ b as shown in Figure 1. This case conrass wih he convenional Harwick-Solow model in which exracion coss are consan up o a specific quaniy afer which no amoun of he resource is obainable a any cos. 2 Social welfare over he feasible and eligible consumpion pahs makes use of he auxiliary feliciy funcion U( ). As in Koopmans (1965) we assume U > U < and lim U() = such ha periods of very low consumpion are avoided as much as possible. Following Koopmans he social planner's uiliy funcion is expressed as: 3 = [U( ) - V U ()]d (3) and he corresponding planner's opimizaion problem is: 2 The assumpion of rising exracion coss and a backsop echnology e.g. phoovolaics is considerably more realisic han he invered L-shaped exracion cos schedule ha is usually assumed (see hakravory e. al. 1997). An alernaive o he backsop assumpion would be o follow Hoelling (1931) wherein resource use is runcaed on he demand side. In our framework however his would require he complicaion of muliple consumpion goods. 3 To maximize H wih respec o he conrol variable implicily we require. orrespondingly he Kuhn- Tucker condiion is H / and he complemenary-slackness proviso ha ( H/ ) =. Bu in as much as we can rule ou he exreme case of = we posulae ha > implying an inerior soluion. 6

8 Max V s.. = F(K ) äk K = è( < T T ) K() = K () = (4) where U( ) is feliciy a he golden rule level of consumpion and T is he endogenous ime for which θ ) = θ T. ( b Since K = in he seady sae he golden-rule consumpion level can be found (generalizing from Solow 1956) by maximizing: = F(K ) äk è (5) b where is he amoun of he backsop resource consumed in he seady sae. The corresponding firs order condiions which comprise he golden rule for capial accumulaion and resource managemen are: K = F K δ = (6) and = F θb = (7) These condiions yield he golden rule seady sae levels K and. is now defined as b = F(K ) δ K θ (8) The Hamilonian for his problem is (for simpliciy he subscrips 's are dropped) H = [U() - U()] + ë[f( K ) äk è () ] + ø[ ] (9) Incorporaing he inequaliy consrains imposed on he problem we form he Lagrangian 7

9 L = H + φ {} + ô{è è()} (1) b such ha he complimenary slackness condiions associaed wih he inequaliies are L τ = τ[ θb θ()] = τ L φ = φ = φ (11) Applicaion of he maximum principle o his opimal conrol problem yields he following wo efficiency condiions (see Appendix I for deails): η( ) = F K δ (12) F F θ() = (FK δ) (13) ondiion (12) is he amsey condiion ha governs he opimal pah of consumpion leading o golden rule seady sae. In he analogous approach o he "modified golden rule" η( ) + ρ = F K δ here are wo parameers governing he savings and he rae of capial accumulaion. The firs is he absolue value of he consumpion elasiciy of he marginal uiliy η (). Lower η () implies a lower social opporuniy cos of savings (greaer olerance for inergeneraional inequaliy) more rapid capial accumulaion lower ineres raes and higher growh raes of consumpion. The second parameer is he social rae of ime preference ρ which reflecs he social valuaion of fuure feliciy in erm of oday's feliciy. However in our seing by reaing all generaions equally ρ =. 4 4 I would be unsound o derive equaion (12) by seing ρ= in he sandard amsey condiion. Noneheless he equaion ha one ges by doing exacly ha urns ou o be correc. 8

10 ondiion (13) is a generalizaion of Hoelling's ule. The LHS is he in siu marginal value of he resource and he HS is he marginal user cos. 5 Thus he opimal consumpion rajecory and he opimal moion of he sae variables K and are governed by wo inuiive condiions he amsey savings rule wih a zero uiliy-discoun rae and a general-equilibrium Hoelling rule for he case of rising exracion coss. 2.3 elaionship o oher approaches Harwick (1977) and Solow ( ) have shown ha for a obb-douglas producion funcion of capial and a non-renewable resource wih a consan exracion cos ha exracing he resource according o he Hoelling rule and hen saving exacly he resource rens hus generaed leads o a consumpion pah ha is susainable and consan over ime. The Harwick-Solow rule has been jusified ex pos as being he highes consumpion pah ha is inergeneraionally equiable in he sense of delivering equal consumpion o all generaions. This maxi-min consumpion pah may be generaed as a special case of our basic model. earranging equaion (12) we have: = FK δ η() (14) As he social aversion o ineremporal inequaliy η () approaches infiniy / generaing consan consumpion for all. As a special case suppose social planner's preferences are represened by = [U( ) - V U ()]d where U( ) akes he ES form: 5 Endress and oumasse (1994). For a parial equilibrium marke equivalen of equaion (13) see Hansen (198). 9

11 ( η 1) U( ) = η > 1 (15) As η ges larger and larger he iniial level of consumpion increases and he consumpion rajecory becomes flaer. This is illusraed in Figure 2. No maer how high η he upper bound of consumpion remains a so long as η is no infinie. Once η becomes infinie however boh he upper and lower bound swich o _ in Figure 2 exacly he maxi-min level of ineremporal consumpion. An alernaive o maxi-min welfare is he concep of consrained uiliy maximizaion. The main idea due o Asheim (1988) is o apply a non-declining uiliy consrain ( U() ) o he maximizaion of uiliarian welfare. Bu as noed by Toman e. al. (1995) such an approach does no resolve how he social welfare funcion should direcly reflec concerns abou inergeneraional equiy. Besides he ad hoc naure of he uiliy consrain consrained opimizaion canno provide a full ranking of alernaives because alernaives ha violae he consrain canno be compared. In he Harwick-Solow economy for example if eiher he elasiciy of subsiuion beween naural capial and produced capial is less han one or he oupu elasiciy of naural capial is greaer han ha of produced capial (wih elasiciy of subsiuion equal o one) he susainabiliy consrain renders he maximizaion problem infeasible. In his case none of he feasible pahs can be ranked. 6 aher han adding a susainabiliy consrain or specifying axioms ha a "susainablycorrec" social planner's preferences mus saisfy (e.g. Belrai e. al. 1995) our approach follows amsey and Koopmans and finds an opimal and ineremporally neural growh pah. In 6 Even if consrained uiliy maximizaion is reformulaed as a lexicographic (vecor-valued) uiliy funcion (see Endress 1994) he model is sill characerized by he rejecion of radeoffs (Dasgupa 2). Tha is no negaive consumpion growh however close o zero can be jusified even if i affords higher susainable consumpion in he fuure. 1

12 he basic case above and in he cases below wherein he resource generaes an environmenal ameniy or disameniy we find ha he opimal pah is susainable even hough we do no require i o be so. 3. Exensions: Environmenal Effecs and enewable esources 3.1 Fund Polluion We now urn o he case wherein use of a non-renewable resource e.g. pero-chemically sourced energy generaes polluion. For simpliciy assume ha polluion ( E ) is emied as a consan proporion of resource use ( ) and ha emission unis are se such ha he proporion is one. Therefore E = T (16) < In he case of fund polluion emissions are assumed no o accumulae i.e. emissions bu no sock polluans ener ino he uiliy funcion. Now our maximand becomes = [U( E ) - U ( Max V E)] d (17) where U > U ; and U < U signifying increasing marginal disuiliy of polluion. E < EE < As in he basic case is associaed wih a oal swich o he backsop echnology (e.g. phoovolaics). Use of he primary resource and he corresponding emissions are hen boh zero so ha E =. Golden rule consumpion is herefore given by equaion (8) wih exacly he same as in he basic model. An opimum rajecory for consumpion and capial accumulaion saisfies: K and 11

13 Max s.. V = K [U( E = ) - = F(K ) δk U()]d θ( < T T ) K() = K () = (18) Applicaion of he maximum principle o his opimal conrol problem gives he following efficiency condiions (See Appendix II for deails): = F K δ (19) F F 1 UE UE θ() = + ( ) FK δ FK δ (2) Equaion (19) appears o be he familiar amsey condiion. However since U now has wo argumens he ime derivaive of Uc will involve a cross erm EE E ( )( ) + ( )( ) = FK δ (21) E If and E are separable argumens of he feliciy funcion equaion (21) collapses o he convenional amsey savings rule. Turning o equaion (2) he LHS and he firs erm on he HS consiue he generalized Hoelling condiion (cf. equaion (13)). The las erm on he HS is jus he marginal damage cos (MD = U U E ). The remaining erm U ( U E UE UE ) = ( ) E+ ( ) c E U U which is posiive or negaive respecively depending on wheher he firs or second erm dominaes. For example if he marginal damage cos is relaively fla and he income elasiciy 12

14 of environmenal qualiy is high enough he enire erm is negaive. This would imply ha he opimal polluion ax would be less han he marginal damage cos. 3.2 A Non-enewable esource wih Environmenal Disameniies Now consider he case of sock polluion such as greenhouse gases wherein emissions conribue o he sock of polluion M which depreciaes a rae ξ. 7 Our model now becomes: Max s.. V = K [U( M ) - U ( M)]d = F(K ) äk = ξ M M = ξ M è( < T T < T T ) K() = K () = (22) where U > U ; and U < U corresponding o he previous secion and M < MM < where M = and is he Solow golden rule consumpion as before. 8 Applicaion of he maximum principle leads o a amsey condiion ha is idenical o (19) and he expansion of U will be analogous o equaion (21). If and M are separable we have he convenional amsey savings rule once more. The generalized Hoelling condiion for his case is: F F 1 UM µξ θ() = ( )( ) (23) FK δ FK δ 7 This model is similar o ha of Nordhaus (e.g. 1991) albei wih explici consideraion of resource depleion bu wihou he inervening climae model. 8 In he golden rule seady sae he economy has swiched o he backsop resource and he sock of polluion has depreciaed o zero. Analogous o consumpion in he Koopmans model he sock of polluion in he opimal rajecory asympoically approaches is golden rule level bu never acually reaches i. 13

15 where µ is he shadow price of he polluion sock (see Appendix III). The las erm on he HS U is he marginal exernaliy cos for sock polluion. I is smaller han MD ( U M ) because he shadow price of polluion µ is negaive. We now consider he special case of he above for which ξ = i.e. he sock polluan does no depreciae. In his case he backsop echnology is immediaely employed and he primary resource is never used so ha he golden rule sock of polluion is zero ( M = ). This resul provides a case in which he sraegy of srong susainabiliy is opimal. In is usual jusificaion srong susainabiliy is associaed wih he preservaion of naural capial and is defended as an ecological imperaive no derived (see e.g. Pearce and Barbier 2). Srong susainabiliy criics have derided he sraegy as being a caegory misake i.e. no derived from more fundamenal objecives and for denying resource-rich economies a major source of savings and capial formaion (Dasgupa and Mahler 1995). The zero polluion resul exemplifies a differen approach o srong and weak susainabiliy han is usually found in he lieraure. Insead of proposing he sraegy as boh he objecive and he means of opimal growh our approach separaes ends and means. Bu while ecologically-oriened proponens of preservaion sugges ha srong susainabiliy is especially imporan when naural capial is essenial and irreplaceable our resul suggess ha srong susainabiliy is an opimal sraegy when naural capial has an abundan and perfec subsiue. 3.3 A enewable esource wih Environmenal Ameniies 14

16 Nex consider he case of a renewable resource such as a fores ha generaes an environmenal benefi and eners he uiliy funcion as a sock. epresening he growh funcion as G () he dynamic equaion governing he resource sock becomes = G() (24) In he secion below we show ha using a amsey-koopmans approach soluion o his problem is well defined and he opimal rajecory leading o such seady sae can be idenified. The problem a hand is Max s.. V = K = F( K [U( ) - ) äk = G( ) U ( )]d è( ) K() = K () = (25) where and are now defined as heir BH green golden rule values. Applicaion of he maximum principle yields he following efficiency condiions (See Appendix IV for deails): = F K δ (26) [F θ()](fk δ) = {F + [F θ()]g θg() + U U } (27) As expeced consumpion pah is governed by a amsey condiion (26). The LHS of equaion (27) measures he benefi of exracing a marginal uni oday [ F θ()](fk δ) and he HS is he marginal cos of such exracion associaed wih: 1) he poenial resource appreciaion F ; 15

17 2) he forgone uiliy from using one uni of he resource U / U ; 3) he marginal user cos (he remaining wo erms). Addiionally seady sae is defined by seing F = and F = δ K U ( ) = [F (K ) θ( )]G x θ G( ) (28) U ( ) The LHS of (28) is he slope of he indifference curve beween & as shown in Figure 3. Following Heal (1998) he oher curve in Figure 3 is he resource feasibiliy fronier represening he maximum level of for each level of when opimizing over produced capial K. This curve is hump-shaped because is slope (he HS of (28)) is posiive for smaller values of (since F θ > θ < and G > ). I coninues o be posiive unil G becomes sufficienly negaive o reverse he sign. Equaion (28) hus represens he angency beween he indifference curve and he resource feasibiliy fronier. Noe ha he seady sae levels of * and * are precisely he green golden rule level of consumpion and resource sock &. Finally noe ha he sock polluion case from secion 3.2 wihou depreciaion of he polluion sock can be wrien a special case of he model here where G() =. Since M = we can rewrie he uiliy funcion such ha U is a posiive funcion of insead of a negaive funcion of M. 4. Ne Naional Produc An alernaive o he convenional approaches of opimizing susainably-weighed or susainably-consrained growh is o maximize green or "susainable" income. Weizman ( ) has shown ha green naional produc defined as ne naional produc minus he depreciaion of naural capial is indeed a linear approximaion of he Hamilonian of a 16

18 uiliarian welfare funcion i.e. maximizing uiliarian welfare and green naional produc are roughly equivalen. Green naional produc is also used inerchangeably wih susainable income (Pearce and Barbier 2; World Bank 1997). Since Weizman assumes posiive discouning however his leads o a paradox. The Hamilonian for uiliarian welfare maximizaion wih posiive discouning can decline over ime in he ransiion o he modified golden rule seady sae (Dasgupa and Heal 1979 chaper 13). Tha is maximizing susainable income leads o non-susainable income and consumpion pahs. 9 In his secion we show ha his paradox disappears in he case of iming neuraliy. onsider a general case ha combines he models of secions 3.2 and 3.3 such ha boh he disameniy M and he ameniy of he sock are presen. The Hamilonian along he opimum rajecory remains consan; ha is dh / d = 1 where H = [U( M ) - U ( M)] + λ K + ψ +µ M (29) A he seady sae K = = M = and U( M ) = U( M) implying a zero value for H. onsequenly H = for all ime. Thus U( M) = U( M ) + λ K + ψ +µ M (3) The golden rule level of uiliy U( M) is he consan green ne naional produc in uiliy unis for all ime periods. Thus maximizing green ne naional produc is equivalen o opimizing he amsey-koopmans welfare funcion and resuls in susainable income. 9 In he ransiion o he modified golden rule seady sae oal capial sock consumpion and green naional produc ypically rise and hen fall albei capial and green naional produc fall before consumpion. 1 See Appendix V for mahemaical derivaion. For simple presenaion capial leers wih subscrip sands for variables along opimal rajecory. Indeed his also implies ha our soluion saisfies he ransversaliy condiion for he infinie-horizon non-discouning problem as examined by hiang (1992). 17

19 This proposiion is illusraed using a much simpler case where oal capial is aken only o include produced capial K and insananeous uiliy is defined solely on consumpion good U(). In Figure 4 11 curve (aa) represens he feasibiliy fronier of he economy a ime =. onsumpion level is he maximum aainable level of consumpion a ime = if no invesmen were o ake place and U () is he associaed level of uiliy. The uiliy-invesmen pair ( ) U K lies on he opimal rajecory o he seady sae. As capial is accumulaed he feasibiliy fronier moves ouward and owards he righ unil maximum aainable consumpion reaches he golden rule level and K = as depiced by curve (bb). The shadow price of capial illusraed by he slope of he angency line decreases monoonically. For he general case considered above wih hree ypes of capial NNP sill ransiions o U( ) he shadow price of capial decreases monoonically and he oher shadow prices ransiion o consan values albei no monoonically. 5. Summary and concluding remarks The lieraure on susainable growh has foundered on he quesion of wheher o represen susainabiliy as an ad hoc consrain on he objecive funcion or by resricing he social planner s preferences. Insead of searching for wha is opimal and susainable we follow he canonical approach of amsey Koopmans and Diamond and solve for wha is opimal and 11 Figure 4 generalizes Weizman s (1976) illusraion of he hypoheical saionary equivalen consumpion of NNP by illusraing he ransiion o he seady sae. Unlike he Weizman case however NNP remains consan under iming neuraliy and equal o he seady sae consumpion level. We also illusrae NNP in uiliy unis o avoid he linear approximaion problem. 18

20 ineremporally neural. This frees us o explore he condiions under which such a program resuls in susainable uiliy We find ha opimal and ineremporally neural growh is susainable even in he presence of non-renewable resources. Adding a consrain ha resrics growh of consumpion or uiliy o be nonnegaive would be no only ad hoc bu also redundan. The necessary condiions for opimal growh require ha he economy save a he rae given by he familiar amsey condiion and ha resource use and conservaion conform o a generalized Hoelling condiion. The consrain of weak susainabiliy which requires ha he depleion of naural capial no exceed he accumulaion of produced capial is similarly redundan. Toal capial increases along he opimal growh pah albei a a declining rae. The model is exended o accommodae environmenal ameniies and disameniies resuling in modificaions of he amsey and Hoelling condiions. In he cases of fund and sock polluion he amsey condiion is expanded o include a disameniy erm. The Hoelling condiion conains an addiional erm he "marginal exernaliy cos" which is however less han he marginal damage cos for he sock polluion case and ambiguously so for he fund polluion case. This means ha he opimal polluion ax may be less han is Pigouvian level even wihou second-bes consideraions of public finance. 12 Anoher ineresing resul concerns he case wherein he sock of polluion does no depreciae. In his case he opimal sraegy urns ou o be no o ouch he non-renewable resource and immediaely exploi he more cosly bu non-polluing backsop. This resul shows ha he sraegy of srong susainabiliy which is ofen advocaed on he grounds ha naural 12 See e.g. Bovenberg and Goulder (1996) for a discussion of second-bes emission axes and he double-dividend debae. 19

21 capial is essenial and irreplaceable urns ou o be correc in he opposie case i.e. where i has a perfec subsiue. The model is furher generalized o he case of a renewable resource wih a renewable resource. Again he ransiion o he seady sae is governed by amsey and Hoelling condiions and he laer conains a marginal exernaliy cos erm ha is similar o he sock polluion case. The amsey-koopmans approach also resolves he paradox beween opimizing growh wih inergeneraional equiy and maximizing green naional produc. Moreover when inergeneraional equiy is aken o mean amsey-koopmans inergeneraional neuraliy real naional income is susainable and consan in he opimal program and he Weizman "hypoheical saionery equivalen income" is acually aained a he golden rule seady sae. 2

22 Appendix I The problem is o minimize he difference beween acual consumpion and opimal consumpion subjec o dynamic consrains on capial accumulaion and resource use: Max s.. V = K = [U( ) - U ()]d F( K = ) äk è( < T T ) K() = K () = The Hamilonian expression for his problem is (for simpliciy he subscrips 's are dropped) H = [U() - U()] + ë[f( K ) äk è () ] + ø[ ] The sandard firs order condiions for his opimal conrol problem are: (1) (2) (3) (4) K = U = λ[f λ = = λ = λ[fk = ψ = λθ θ()] ψ = δ] From he firs and he hird condiions (5) = λ and ( 6) = λ[fk δ] λ Equaing expressions for λ and rearranging yields 21

23 (7) = FK δ or η( ) = F K δ where η () = >. From he second necessary condiion: (8) ψ = λ[f θ()] + λ[f = λ(f K δ)[f θ ] θ()] + λ[f + θ ] From he fourh necessary condiion ( 9) = λθ ψ Equaing expressions for ψ and rearranging yields or K = ( 1) (F δ)[f θ()] + F (11) F F θ() = (FK δ) 22

24 Appendix II The problem is: Max s.. V = K [U( E = ) - = F(K ) δk U()]d θ( < T T ) K() = K () = The Hamilonian for his problem is (for simpliciy he subscrips is dropped): H = [U( E) - U()] + λ[f( K ) δk θ() ] + ψ[ ] The sandard necessary firs order condiions for his opimal conrol problem are: (1) (2) (3) (4) K = U = U E λ = + λ[f = λ = λ(fk = ψ = λθ θ()] ψ = δ) From he firs equaion (5) = λ U and from (3) ( 6) = [FK λ δ] Equaing hese wo expressions for λ and rearranging gives ( 7) = F K δ 23

25 Differeniaing equaion (2) wih respec o ime ( 8) ψ = UE + [F θ()] + [F + θ] Equaing wih (4) o obain: Meanwhile ( 9) UE (FK δ)[f θ()] + F = U U E UE UE UE UE ( 1) = ( ) + = ( ) (FK δ) Plugging back ino (9) F 1 UE UE ( 11) F θ() = + ( ) F δ F δ U U K K 24

26 Appendix III The problem is: Max s.. V = K = [U( M ) - U ()]d F( K = ξ M M = ξ M ) äk è( ) < T T < T T K() = K () = The Hamilonian for his problem is (for simpliciy he subscrips 's are dropped) H = [U( M) - U ()] + ë[f( K ) äk è () ] + ø[ ] +µ ( - ξm) The sandard necessary condiions for his opimal conrol problem are: (1) (2) (3) (4) (5) = λ = = λ[f θ()] ψ + µ = = λ = λ[fk δ] K = ψ = λθ = µ = U M µξ M From he firs and he hird condiions (6) = λ and ( 7) λ = λ[fk δ ] Equaing expressions for λ and rearranging yields 25

27 ( 8) = F K δ From he second necessary condiion: ( 9) ψ µ = λ[f θ ()] + λ[f + θ ] From he fourh and fifh necessary condiions ( 1) ψ µ = λθ + U M µξ Equaing expressions on he HS of (9) & (1) plugging in he expression for λ and rearranging yields ( 11) F F θ () = F δ K U µξ U M 1 * F δ K 26

28 Appendix IV The problem is: Max s.. V = K = F( K [U( ) - ) äk = G( ) U ( )]d è( ) K() = K () = The Hamilonian for his problem is (for simpliciy he subscrips 's are dropped) H = [U( ) - U ( )] + ë[f( K ) äk è () ] + ø[g() ] The sandard necessary condiions for his opimal conrol problem are: (1) (2) (3) (4) K = U = λ[f λ = = λ = λ[fk = ψ = U θ()] ψ = δ] λθ + ψg From he firs and he hird condiions (5) = λ and ( 6) = λ[fk δ] λ Equaing expressions for λ and rearranging yields ( 7) = F K δ From he second necessary condiion: 27

29 ( 8) ψ = [F θ()] + [F θ G() + θ ] From he fourh necessary condiion ( 9) = U + U θ U [F θ()] G ψ Equaing expressions for ψ and rearranging yields ( 1) F θ()(fk δ) = F + [F θ()]g θ G() + U U 28

30 Appendix V For a general case ha combines he models of secions 3.2 and 3.3 consider he following problem: Max s.. V = K M M = F( K = [U( M ξ M ) äk = G( ) ) - è( U ( M )]d ) K() = K () = M() = M The Hamilonian is (for simpliciy he subscrips 's are dropped) H = [U( M) - U ( M)] + ë[f( K) äk è() ] + ψ[g() ] + µ ( - ξm) wih firs order condiions: (1) (2) (3) (4) (5) = λ = = λ[f θ()] ψ + µ = = λ = λ[fk δ] K = ψ = U λθ + ψg = µ = UM µξ M Now on he opimal rajecory leading o seady sae consider (6) H = [U( M ) - U( M)] + λ K + ψ +µ M We ake he ime differeniaion for simple presenaion ignoring he 's 29

31 dh d = U = [U = + U + UM M+ λk+ λ[fk K+ F δ K θ θ() ] + ψ + ψ[g ] + µ M+ µ [ ξ M] λ]+ [UM + µ µξ]m+ [ λ+ λ(fk δ)]k+ [U λθ + ψ+ ψg ] + { λ[f θ()] ψ + µ } where he las sep uilizes he above five firs order condiions. 3

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