Sustainable management of an exhaustible resource : a viable control model

Transcription

1 Susainable managemen of an exhausible resource : a viable conrol model V. Marine, L. Doyen Sepember 9, 2003 Absrac We examine susainabiliy condiions of an economic producion-consumpion sysem based on he use of an exhausible naural resource. Bu insead of sudying he environmenal and economic ineracions in erms of opimal conrol, we pay much aenion o he viabiliy of he sysem. The ex ane viabiliy, defined by a se of consrains combining a guaraneed consumpion and a sock of resource o preserve a any ime, refers o a Rawlsian inergeneraional equiy perspecive. Using he mahemaical concep of viabiliy kernel dealing wih he consisency beween he consrains and he conrolled dynamics, we reveal he susainable echnological configuraions and, whenever possible, policies opions and environmenal-economic saes o guaranee a perennial sysem. We poin ou he flexibiliy in exracionconsumpion choices. Numerical simulaions illusrae he general resuls. Key-words: Exhausible resource, Susainabiliy, Viabiliy kernel. JEL Classificaion: Q01, Q32, O13, C61. THEMA (THéorie Economique, Modélisaion e Applicaions). Universié Paris X-Nanerre, 200 Av. de la République, Nanerre Cedex, France. vincen.marine@u-paris10.fr CNRS, CIRED (Cenre Inernaional de Recherche pour l Environnemen e le Développemen), Campus Jardin Tropical de Paris, 45bis Avenue de la Belle Gabrielle, Nogen sur Marne, France. doyen@cenre-cired.fr 1

2 1 Inroducion In 1972, he repor of he Club of Rome The Limis o Growh [15] argued ha an unlimied economic growh is impossible because of he exhausibiliy of some resources. In response o his, economiss like Dasgupa, Harwick, Heal, Solow, Sigliz and ohers have developed models o deerminae wheher he presence of an exhausible resource was limiing he economy and, if so, how. Since 1987, he erm Susainable Developmen, defined in he Brundland repor Our Common Fuure [21], is used o describe his kind of concern. The quesion of susainable developmen is sill discussed nowadays. One basic concern is how o reconcile environmenal, social and economics requiremens. The issue of inergeneraional equiy plays anoher imporan role in he debae. In he presen paper, we are ineresed in he economic inerpreaion of susainabiliy and especially in he viable ineremporal use of exhausible resources. Firs, le us recall he main economic models and saemens referring o his concern, as in Heal [13]. The modelling on his opic are ofen derived from he classic cake-eaing economy firs sudied by Hoelling [14]. In such a model, one considers he allocaion of an exhausible resource which is he only good of he economy over an infinie horizon. The problem is o maximize he sum of discouned uiliy derived from he consumpion of he resource under he dynamic and he consrain max c(.) 0 U(c())e δ d Ṡ() = c() S() 0, where S() is he sock of resource, c() he consumpion and δ he discoun rae. Using he Maximum Principle, one deerminaes he opimal consumpion by he dynamic equaion ċ() c() = δ η(c()), where η(c) = cu (c)/u (c) > 0 is he elasiciy of subsiuion of marginal uiliy or Arrow-Pra relaive risk aversion. If he elasiciy η is a consan, he consumpion decreases exponenially a he rae δ/η, namely c() = c 0 e (δ/η), and he resource sock S() decreases o 0. Wih his approach, he only hing ha cares is he consumpion of resource, bu in several cases he socks of naural asses provide some uiliy. Thus, he model can be compleed by inroducing he sock of resource as an argumen of he uiliy funcion. Doing so, he problem becomes max c(.) 0 U(c(), S())e δ d under he same consrains as before. In his model, whenever he marginal uiliy of consumpion is infinie wihou consumpion, he sock is oally depleed. Oherwise, i is opimal o mainain forever a posiive sock of resource S which depends boh on he uiliy funcion and he discoun rae 1. 1 Such a susainable sock S saisfies δu c(0, S ) = U S(0, S ). 2

3 Neverheless, in hese wo models, he opimal consumpion c() declines owards zero. No posiive level of consumpion is susainable for an infinie horizon ime. A way o escape from an economy where he only susainable consumpion is zero consiss in considering ha he capial accumulaion and/or he echnical change can offse he diminuion of he resource exracion. In his form, imporan conribuions are [8, 9, 12, 19, 20]. In paricular, Dasgupa and Heal [8] expose an opimisaion problem where he economy is represened by he consumpion of a human-made reproducible capial K() produced by a echnology f(k, r) combining capial and exracion r() of he exhausible resource S(). The opimal dynamic conrol model becomes under he dynamics max c(.),h(.) 0 e δ U(c())d { K() = f(k(), r()) c(), Ṡ() = r(). For a resource essenial o producion 2, hey prove ha an opimal pah exiss if he acualisaion rae is high enough 3. In his case, he declining use of he resource is subsiued by he accumulaion of capial. Neverheless, in his model, if he acualisaion rae is posiive and he marginal produciviy of capial decreasing, he opimal pah displays a decreasing consumpion afer a while. Sigliz [20] adops a similar approach and inroduces an exogenous echnical change facor of he form e λ. The producion funcion is now f(k, r, L, ) = e λ K α r β L γ, where L is he labour force. If he populaion is growing a a consan rae n, a condiion o susain he growh is λ/n > β. In he case of a consan populaion wihou echnical change, here exiss an opimal pah if he share of capial in he producion is greaer han ha of resource i.e. α > β which is he same resul as in Solow [19]. Harwick [12] argues ha reinvesing he rens from he exploiaion of an exhausible resource in capial accumulaion leads o inergeneraional equiy wih consan oupu and consumpion along he ime. This resul, known as he Harwick-Solow rule, holds rue only if here is no capial depreciaion. All of hese aricles adop a discouned uiliarian approach. Ohers crierions can be proposed o cope wih he susainabiliy issues. For insance, Solow [19] examines he problem of he ineremporal allocaion of an exhausible resource hrough he Maximin or Rawls crierion [17]. The crierion is now o maximize he uiliy of he poores generaion, in he following sense max se of possible allocaions { min } U(c()). I urns ou ha he soluion is associaed wih he search of a consan consumpion for all generaions. Again, in he Cobb-Douglas case, a susainable posiive level of consumpion exiss if he share of capial in producion α is greaer han he share of resource β. The discouned uiliarian crierion is ofen considered as a dicaorship of he presen because he far fuure is no aking ino accoun. In he same way, he green 2 Namely if lim r 0 f r(k, r) =. 3 Namely if δ > ρ(1 η) wih ρ = lim x f(x)/(x) where x K r. 3

4 golden rule crierion 4 inroduced by Belrai, Chichilnisky and Heal [6] and discussed in Heal [13], leads o preserve all he resource wih a zero consumpion pah. Thus i considers only he far fuure and is qualified of dicaorship of he fuure because i neglecs he presen needs. Chichilnisky [5] has developed a crierion ha allows o avoid boh he dicaorship of he presen and he dicaorship of he fuure 5. Wih his crierion, he consumpion decreases owards zero and a posiive sock of resource Ŝ is preserved 6. Thus, mos of he works dealing wih his problem relies on an opimaliy approach. Such a framework can be discussed because of he uniqueness of he associaed opimal decisions and he various crierions ha can be applied. Using one crierion or anoher, he resul is differen and, wihou choosing beween wo differen crierions, i is hard o make any non-conroversial recommendaion. Moreover, all hese approaches, excep Chichilnisky s one, are based on an a priori definiion of susainabiliy. The viabiliy approach and viable conrol framework [1] can be anoher way o explore his quesion. Insead of saying wha has o be, i aemps o say wha is able o be. Basically, his viabiliy approach focus on ineremporal feasible pahs. Thus i consiss in he definiion of a se consrains ha represens he good healh or by exension he effeciveness of he sysem a any momen, and in he sudy of condiions which allow hese consrains o be saisfied along ime including boh presen and fuure. So, i leaves he opimaliy framework and focus on he respec of consrains of conrolled dynamic sysems; in his way, we escape from he conroversy abou he crierion o opimise. More formally, he viabiliy approach deals wih dynamic sysems under sae and conrol consrains. The aim of he viabiliy mehod is o analyse compaibiliy beween he possibly uncerain dynamics of a sysem and sae or conrol consrains, and hen o deermine he se of conrols or decisions ha would preven his sysem from going ino crises i.e. from violaing hese consrains. We refer for insance o [2, 3, 11] for some sylised models in oher conexs. The Tolerable Windows Approach [4, 18] proposes a similar framework mainly concerned wih climae change issues. More specifically, in he environmenal conex, viabiliy may imply he saisfacion of boh economic and environmenal consrains. In his sense, i is a muli-crieria approach someimes known as co-viabiliy in reference o he co-evoluionary perspecive [16]. Moreover, since he viabiliy consrains are he same a any momen and he erm horizon is infinie, an inergeneraional equiy feaure is naurally inegraed wihin his framework. In his paper, we examine he quesion of he inergeneraional allocaion of an exhausible resource involved in producion wihin he viable conrol approach. Our purpose is o deermine wha are he feasible economics pahs (saes and conrols) respecing a se of consrains characerizing he susainabiliy of he sysem. Here, he consrains mainly include a guaraneed consumpion level and he scarciy of an 4 I consiss on maximising he uiliy a an infinie ime ; i.e. max c(.) lim U(c(), S()) 5 The crierion is W = θ 0 d()u(c(), S())d + (1 θ) lim U(c(), S()) where d() is he acualisaion facor ha can be exponenially decreasing or hyperbolic. 6 We can compare he opimal resource sock which is preserved wih he four crierions : S 0 > Ŝ > S > 0. The green golden rule is he mos proecive one. Chichilnisky s crierion leads o a posiive consumpion and preserves more resource han he Uiliarian crierion aking ino accoun he sock of resource. The classic Uiliarian crierion does no conserve any par of he sock. 4

5 exhausible resource. We use he mahemaical concep of viabiliy kernel o characerize he susainabiliy. Indeed, his kernel is he se of iniial resource and capial levels for which a feasible regime of exploiaion and a consumpion pah exis and saisfy he whole consrains along ime. Therefore he viabiliy kernel refers o some ex-pos viabiliy feaures and provides he rue consrains of he sysem. More specifically, by using he viabiliy kernel, we address he following quesions : - Wha are he echnology and producion configuraions allowing for susainable consumpion and exracion pahs? - Given a susainable echnology, wha are he iniial resource condiions for which such susainable policies exis? - Wha are he possible susainable conrols associaed wih hese susainable saes? The paper is organized as follows. Firs, in secion 2, we presen he model describing he dynamics of he sysem and we idenify he viabiliy consrains. We define he susainabiliy of he economy hrough he viabiliy kernel. Then, we give in secion 3 a axonomy of susainable and unsusainable economies depending on he echnology. We examine in more deails he case of a Cobb-Douglas producion funcion. Evenually, in secion 4 we provide some numerical simulaions and hen we conclude on he ineres and limis of he mehod. To resric he mahemaical conen wihin he core of he ex, he proofs of he formal resuls are presened in he appendix a he end of he paper. 2 The model 2.1 The dynamics Following [8, 19], we describe he economy wih he exhausible resource use by { Ṡ() = r(), K() = f(k(), r()) c() δk (1) where S() is he exhausible resource sock, r() sands for he exracion flow, K() represens he accumulaed capial, c() sands for he consumpion and he funcion f represens he echnology of he economy. Parameer δ is he rae of capial depreciaion, which can vanish. Thus he decisions or conrols of his economy are levels of consumpion c() and exracion r() respecively. 2.2 The susainabiliy consrains Now we consider he sae-conrol consrains. We firs assume ha he exracion r() is irreversible in he sense ha 0 r(). (2) We ake ino accoun scarciy of he resource by requiring 0 S(). More generally, we consider a sronger conservaion consrain for he resource as follows S S(). (3) 5

6 where S 0 sands for some guaraneed resource arge referring o a srong susainabiliy concern. We also assume he invesmen in he reproducible capial K o be irreversible in he sense ha 0 f(k(), r()) c(). (4) This ensures he growh of capial if here is no depreciaion. We also consider ha he capial is non negaive 0 K(). (5) The mos imporan requiremen we impose is relaed o some guaraneed consumpion level c along he generaions 0 < c c(). (6) This consrain refers o susainabiliy and inergeneraional equiy since i can be enounced in erms of uiliy in a form closed o Rawls crieria, namely U(c()) U. 2.3 The viabiliy kernel as an indicaor of susainabiliy Hereafer we shall say ha he evoluion (S(.), K(.), c(.), r(.)) of he economy is viable if he whole consrains (2), (3), (4), (5) and (6) hold rue for every period. A quesion ha arises now is wheher he dynamics of he resource and capial (1) are compaible and consisen wih hese consrains. In oher words, an objecive here is o idenify he iniial saes (S 0, K 0 ) ha are associaed wih viable decisions (c(.), r(.)) and viable rajecories (S(.), K(.)). This se of iniial sae is called he viabiliy kernel associaed wih he dynamics and he consrains. Since i depends on boh he producion funcion f and he consrains on basic needs c along wih he resource minimum sandard S, we denoe his kernel by Viab(f, c, S ): Viab(f, c, S ) = (S 0, K 0 ) here exiss decisions (c(.), r(.)) and saes (S(.), K(.)) saring from (S 0, K 0 ) saisfying condiions (1), (2), (3), (4), (5) and (6) for any ime R + In he general mahemaical framework, his se can alernaively be empy, or be he whole sae consrain se or even a sric par of he iniial sae consrain domain. Le us menion ha an inergeneraional equiy feaure is direcly induced by he viabiliy kernel. Indeed he required consrains mixing here economic and environmenal goals have o be saisfied for every generaion equivalenly. Moreover, he viabiliy kernel capures an irreversibiliy mechanism. Indeed, from he very definiion of his kernel, every sae ouside he viabiliy kernel will violae he consrain in finie ime whaever decisions applied. This means ha he crisis is irreversible. For insance, he exreme case where he viabiliy kernel is empy corresponds o a hopeless configuraion. 3 A axonomy of susainable and unsusainable economies In his secion, we analyse he susainabiliy of he economy wih respec o differen echnological srucure. To achieve his, we compue he viabiliy kernel Viab(f, c, S ) for several funcion f. We especially focus on he Cobb-Douglas case where he resuls are no sraighforward.. 6

7 3.1 Assumpions and noaions We firs assume ha he producion funcion f increases wih capial K and exracion r. We inroduce he exracion indicaor denoed by r (f, K) and defined by ) r (f, K) = inf (r 0 f(k, r) c. (7) along wih he minimal exracion indicaor r (f) defined by r (f) = inf K 0 r (f, K). (8) The value of his minimal exracion indicaor r (f) plays an imporan role for he susainabiliy concern. We disinguish some illusraive cases below. However, we will see ha he mos ineresing Cobb-Douglas case requires he inroducion of furher indicaors. 3.2 A non susainable economy : a srongly essenial resource We firs consider an economy where he resource is srongly essenial o he producion in he sense ha r (f) > 0: no producion is possible wihou a minimal quaniy of resource, sricly posiive for any sock of capial. For example, his is he case for a producion funcion as f(k, r) = min(ak, br) where r (f) = c b > 0. This case represens a complemenary poin of view as described in Daly [7] in he sense ha capial and naural resource are complemens raher han subsiues. For any iniial sae, when r (f) is sricly posiive, he quaniy of resource exraced a each ime is sricly posiive since r() r (f) and consequenly he sock is depleed in a finie ime. As he resource is necessary for producion, he producion sops in finie ime. Thus, he viabiliy kernel is empy. No surprisingly, he economy is no susainable. The proof of his proposiion is given in he appendix page 15. Proposiion 3.1 If he echnology f is such ha r (f) > 0 hen he economy is unsusainable in he sense ha he viabiliy kernel is empy Viab(f, c, S ) = for any posiive c and S. 3.3 A susainable economy : a non-essenial resource We now pay aenion o a non-essenial resource where r (f) = 0 = r (f, K + ): a sock of capial K + allows o produce c wihou using naural resource. If K K +, we can produce he quaniy c wihou using any par of he resource sock. So, any sae such ha K K + and S S is viable. This is he case for example if he producion funcion has an addiive form i.e. f(k, r) = ak α + br β where r (f) = 0 = r (f, (c /a) 1/α ). We hen obain he following proposiion which is proved in he appendix page 16. Noe ha, for all c and S, he viabiliy kernel is no empy. I is jus smaller whenever guaraneed hresholds c and/or S rise. Proposiion 3.2 Consider any posiive minimal consumpion c and a minimal resource sock S. If he echnology f is such ha here exiss a capial level K + wih r (f) = 0 = r (f, K + ) hen he economy is susainable in he sense ha he viabiliy kernel is no empy. 7

8 3.4 The Cobb-Douglas case : a weakly essenial resource For sake of simpliciy and o mach wih he lieraure referred before, we now focus on wha Dasgupa and Heal call essenial resources in [8] namely resources needed in producion bu wih an unbounded poenial producion. For such a purpose, we now use a Cobb-Douglas producion funcion. This case is ineresing because a fine subsiuion effec occurs beween naural resource and capial and an infinie capial accumulaion can offse a decreasing use of he resource even if i is sill essenial o producion No capial depreciaion, no echnical progress We consider he usual case of a Cobb-Douglas producion funcion f(k, r) = K α r β. In his case he echnology f is such ha r (f) = 0 = r (f, + ). I urns ou ha he economy is susainable if and only if he resource elasiciy β is smaller han he capial elasiciy α. In ha case he capial accumulaion allows o compensae in he producion sysem he decline of he exhausible resource. Thus we recover Dasgupa-Heal [8] and Solow [19] resuls. A susainabiliy condiion linking he iniial resource and capial levels is also required o achieve acual susainabiliy. Le us menion ha he proof of proposiion (3.3), given in he appendix p.16 for α > β, combines angenial and Hamilonian properies. Proposiion 3.3 Consider any posiive minimal consumpion c and resource level S and a echnology f(k, r) = K α r β wih β < 1. We have { if β > α Viab(f, c, S ) = { } (S, K) such ha S V (K, c, S ) if α > β. where V is a funcion defined by V (K, c, S ) = 1 ( ) 1 β c β K β α β + S α β 1 β. (9) An irreversibiliy concern : Hence, he viabiliy kernel Viab(f, c, S ) represens he se of all saes for which a leas a viable policy (choice of conrols) exiss. If he iniial sae lies ouside his viabiliy kernel, or if he sae runs ou of i, hen he iniial viabiliy consrains will be violaed in a finie ime and he economy is no more susainable whaever admissible decisions of consumpion and exracion applied; he crisis can no be avoided: If S S() < V (K(), c, S ) hen T >, S(T ) < S or c(t ) < c. Consequenly, some fuure generaions will no mee heir needs. In fac, every generaion greaer han T will no saisfied he whole consrains in an irreversible way. Flexibiliy concerns : Whenever hey are possible, he relevan policies consis in mainaining he sae wihin he viabiliy kernel, avoiding he resource crisis menioned before. Given a viable curren sae (K, S) Viab(f, c, S ), relevan viable conrols 8

9 (r, c) ensure ha he velociies ( K, Ṡ) are angen or inward o he viabiliy kernel.7 Therefore here is no uniciy of he susainable conrols in he viabiliy kernel excep on is boundary S = V (K, c, S ). In his sense, his mehod does no provide an unique policy bu he se of all viable policies. Numerical examples illusraing his flexibiliy resul are given in secion 4. More specifically, for any sae (K, S) wihin he inerior of he viabiliy kernel Viab(f, c, S ), appropriae viable conrols belong o he regulaion se { C(K, S) = (r, c) such ha c } c f(k, r). (10) r (f, K) r On he boundary, viable feedbacks are reduced o ( c r (K, S) = 1 β c (K, S) = c. ) 1 β K α β, (11) Moreover, le us remark ha susainable conrols of (10) or (11) are depiced in erms of feedback decisions which depend on he sae of he sysem. I is well-known ha such closed-loop conrols are beer suied for adapaive properies han openloop conrols, depending only on ime. This means ha applying such decisions rule may yield o relevan saes even wih errors, shocks or perurbaions occurring along ime since he feedback decisions ake ino accoun he curren sae of he sysem. In he presen conex, his implies ha some evens modifying in an unforeseen way he resource or he capial socks could be compensaed and assimilaed by he exracion and consumpion decisions and generaes a relevan susainable evoluion. A sensibiliy and robusness concern : If we examine he sensibiliy of he viabiliy kernel wih respec o he consrains c and S, and he parameers α and β, we obviously deduce from he definiion of V (K, c, S ) ha : The greaer α is, he greaer he viabiliy kernel is. This represens he fac ha for a given capial level, he greaer he share of capial in producion is, he smaller he resource needed o produce a arge level of oupu will be. This sensibiliy saemen is illusraed on figure 1. The susainabiliy decreases wih c : The size of he viabiliy kernel diminishes when c increases i.e. c 1 > c2 = Viab(K, c 1, S ) Viab(K, c 2, S ) The susainabiliy decreases wih S : The size of he viabiliy kernel diminishes when S increases i.e. S 1 > S2 = Viab(K, c, S 1 ) Viab(K, c, S 2 ) 7 In a more formal way, relevan conrols saisfy he generalized angenial condiion (f(k, r), r) T Viab(f,c,S )(K, S) where T V (x) refers o a angen cone o he se V a sae x. Thus, in he inerior of he viabiliy kernel, every admissible velociy is available. 9

10 In oher words, all he iniial saes ha belong o he viabiliy kernel for a minimal consumpion c (respecively a guaraneed resource level S ), are also in he viabiliy kernel for all smaller level of consumpion (respecively level of proeced resource). Therefore, sronger ex-ane susainabiliy requiremens induce sronger ex-pos susainabiliy condiions. An equiy concern : Le us now discuss he equiy quesion. We have a glance a such a concern from he Maximin crierion poin of view as in Solow [19]. In he se of consrains, we rea every generaions equally wih respec o he minimal level of consumpion bu we do no cope wih he curren consumpion. We now search for he maximal level of consumpion susainable forever given an iniial sae. For his purpose, we compue he maximal guaraneed consumpion c + for which a given iniial sae (K 0, S 0 ) belongs o he viabiliy kernel. This consumpion is he maximum susainable level of consumpion and solves he equaion V (K 0, c, S ) = S 0 wih respec o c, where V (K 0, c, S ) is given by he equaion (9). We obain c + = (1 β) ( (S 0 S )(α β) ) β α β 1 β 1 β K 0 (12) which is exacly he resul given by Solow [19, p 39] for S = 0. Furhermore, Solow noes [ibid. p 35] ha maximizing he consan level of consumpion is an unusual sor of maximum problem and [he] do[esn ] no see any obvious direc approach.. In his aricle, he uses a âonnemen approach. Le us sress he fac ha he use of he viabiliy kernel is a sraighforward way o deal wih his kind of maximin concern. A sep owards srong susainabiliy? I is ofen argued in he debae on susainabiliy ha a par of he resource have o be proeced. The quesion is generally How much of he resource have o be conserved? Our concern was How much of he resource could be conserved? We have seen ha he viabiliy kernel is smaller when he guaraneed level of resource S growhs. We can search he maximal quaniy of resource ha can be preserved forever keeping he iniial sae (K 0, S 0 ) in he viabiliy kernel. This quaniy is given by S + = S 0 V (K 0, c, 0). Thus, we can mainain a quaniy S such ha 0 S S +. Indeed, he acual siuaion is susainable if S + S. Moreover, if c represens he needs of each generaion, he exra consumpion may represen i wans. There exiss an arbirage beween consumpion and conservaion for he quaniy S + S. The resource can be oally consumed or conserved according o one or anoher crierion of inergeneraional equiy. I depends on he inerpreaion of he susainable developmen (weak or srong). Dobson [10] suggess a classificaion of susainabiliies in accordance wih he prioriy in susainabiliy (presen or fuure generaions, needs or wans, humans or naure) Wha does happen wih capial depreciaion and echnological change? We inroduce a depreciaion erm in he dynamics of capial, i.e. K = K α r β δk c. 10

11 No susainabiliy of an economy wih capial depreciaion : I urns ou ha his depreciaion erm δk condemns he susainabiliy of he economy as proves in anoher conex by [19]. This also means ha aking ino accoun a populaion or labour growing a a consan rae jeopardizes he viabiliy of his economy in a per capia reasoning. Proposiion 3.4 If α < 1 and δ > 0, he viabiliy kernel Viab(f, c, S ) is empy. Technological change as a soluion: Thus, in presence of a capial depreciaion erm, here is no susainable pah for a Cobb-Douglas producion funcion. We now wonder wheher a echnological progress erm inegraing he producion funcion allows o resore viable pahs. To achieve his, we now consider he model: { K() = A(K, )K α r β c() δk() Ṡ() = r(), where A(K, ) sands for he echnological progress depending on accumulaed capial K and ime. I urns ou ha a sufficien condiion for susainabiliy o occur refers o a echnological progress greaer han he form K 1 α. We focus on an endogenous form A(K) = K 1 α+ɛ alhough he resul migh also encompass he sronger case of an exogenous echnical change facor of he form e λ adoped for insance by Sigliz [20]. The proof is given in he Appendix page 18. Proposiion 3.5 If here is a capial depreciaion erm a a posiive consan rae δ and if he producion funcion has he form f(a, K, r, ) = A(K, )K α r β wih β < α, β < 1 and A(K, ) = K 1 α+ɛ wih ɛ > 0, hen he viabiliy kernel is no empy. We exend he resul of proposiion (3.3) and express he viabiliy kernel as he epigraph of he funcion V (K) defined by K V (K) = V (c ) + c wih V (c ) = S ( ) 1 β c +δx β c A(x)(1 β) of he viabiliy kernel are given by 4 Numerical simulaions 1 ( c + δx ) 1 β β A(x)β A(x)(1 β) x α β dx x α β dx. Similarly, viable conrols on he boundary r (K, S) ( c + δk = A(K)(1 β) c (K, S) = c. ) 1 β K α β, We presen here some simulaions and illusraing resuls for he Cobb-Douglas case. Shape of he viabiliy kernel: Figure 1 represens he sensibiliy of he viabiliy kernel wih respec o α and β. We compue numerical illusraions for he funcion 11

12 S = V (k) defining he viabiliy kernel Viab(f, c, S ) as he epigraph 8 of V as enounced analyically in proposiion 3.3. We use differen shares α of capial o represen such a sensibiliy. Oher parameers are se o c = 1, S = 0 while β = 1 α. Upper curves correspond o smaller α (greaer β). We show ha he greaer α is, he lower he value funcion V is and he greaer he viabiliy kernel Viab is. 70 K V(K) alpha=0.52 alpha=0.55 alpha=0.60 K Figure 1: Sensibiliy of he viabiliy kernel Viab(f, c, S ) and he funcion V (K) wih respec o he capial share α. The viabiliy kernel is he epigraph (he se above he funcion) of he funcion V (.) defined in proposiion 3.3. The viabiliy kernel growhs wih α. Viable pahs: Figure 2 represens some emporal pahs associaed wih several susainable policies. I especially illusraes he flexibiliy in he decision choices wihin he viabiliy kernel. We sar from he inerior of he viabiliy kernel which means ha iniial resource S(0) is greaer han V (K(0)). We define he following exracion rule r(c, K) = ( ) 1 c β K α β. (13) 1 β Such an exracion, associaed wih a consan consumpion pah leads o he opimal use of he resource as defined by he Harwick-Solow rule. We also define he maximal 8 The epigraph of a funcion V : X R is defined by Epi(V ) = {(x, y), V (x) y}. 12

13 susainable consumpion afer ime by c + () = (1 β) ( (S() S )(α β) ) β 1 β K() α β 1 β. (14) This is he maximal consumpion ha could be susained for an endless ime, saring from socks K() and S() a ime. Sock S() Resource 28 S() Sock Resource K() Capia l K() 200 K() Capial l (a) Resource sock S() (b) Capial sock k() c() Consumpio n c() 2. 6 c() Consommaion Consumpio n h() Exracio n r() r() 16 h() Exracion n (c) Consumpion c() (d) Exracion r() Figure 2: Differen viable evoluions (saes and conrols) over ime [0, 100]. Oher parameers are c = 1, S = 0 while α = 2/3 and β = 1 α. Iniial condiions are K 0 = 1 and S 0 = We focus on consumpion pahs c(). On figure 2 (c), we see ha here is a diversiy in he susainable ineremporal consumpions. We disinguish he following viable evoluions: A firs susainable rajecory is obained wih he consumpion fixed o c() = c and he viable exracion se o r() = r(c(), K()) = r(c, K()). I appears ha a par S + > S of he resource sock is preserved providing a srong susainabiliy perspecive. I corresponds o he lower consan consumpion pah on figure 2 (c) and o he upper curve of resource sock S() on figure 2 (a). Anoher viable pah is ploed from he consumpion c() = c + () and r() = r(c + (), K()). I urns ou ha he consumpion remains consan c() = c + > c as displayed by upper consan pah on figure 2 (c) and only he par S of he 13

14 resource sock is preserved. This consumpion-exracion pah is he resul of he maximin program as given in Solow [19]. Anoher susainable consumpion is defined as a convex combinaion of he previous policies c() = λc + + (1 λ)c wih λ [0, 1] while he exracion pah is defined as in (13). Again he consumpion pah is a consan one c() = c(0) and a par of he resource S is preserved (S + > S > S ). A viable rajecory wih a decreasing consumpion: he consumpion is again c() = λc + () + (1 λ)c bu he exracion is lower han r(c(), K()) defined in (13). We ake r() = r(c, K()). In his configuraion, he accumulaion K() is no high enough o allow for a consan consumpion pah and he consumpion decreases owards he guaraneed hreshold c. In his case, he preserved sock of resource is also lower han in he previous consan consumpion case even if he exracion is lower a he beginning. As oo few capial is accumulaed o offse he decreasing use of he resource, more resource has o be exraced afer some ime. A viable rajecory wih an increasing consumpion: here he consumpion again is se o c() = λc + () + (1 λ)c bu he exracion is greaer han r(c(), K()). We ake for insance r() = r(c + (), K()). More capial is accumulaed han in he Maximin pah, so c + () increases and c() rises as i is defined as a par of c + (). Doing so, less resource is preserved han in he consan consumpion case saring from c(0) since only a par S + of he sock is preserved. This las case illusraes an ineresing siuaion. Suppose ha he reference pah is given by he maximin crierion: consumpion is consan and equals c + (0). If he firs generaion consumes less han his level and accumulaes he difference, hen all fuure generaions are able o have a consan consumpion c + (1) > c + (0). This corresponds o he sacrifice of he presen generaion. Moreover, if every generaion consumes only a fracion of he maximun susainable c + (), a growing susainable consumpion pah is possible despie he exhausibiliy of he resource. Noe ha for a consan consumpion pah c() = c(0), capial accumulaion K() is also consan. I is increasing if he exracion is greaer han he reference one defined by equaion (13), and decreasing oherwise (see figure 2 (b)). We also see on figure 2 (d) ha he exracion r() quickly declines owards zero in every case. 5 Conclusion In his sudy, we address he problem of he ineremporal allocaion of an exhausible resource wih concerns of inergeneraional equiy and susainabiliy. Mos of he formal researches and analyses on his subjec have adoped he opimal conrol heory approach. We use here anoher framework based on he viable conrol heory. Insead of maximizing an objecive funcion, he main purpose of his approach is o focus on he role of consrains and o characerize admissible pahs and decisions. The use of he viabiliy kernel as an indicaor of susainabiliy allows o characerize he se of policies and saes ha do no drive he sysem ino crisis siuaions. We mach his approach wih he definiion of susainable developmen as a developmen ha mees he needs of he presen wihou compromising he abiliy of fuures generaions 14

15 o mee heir own needs. If he needs of each generaion are defined equally hrough he viabiliy consrains in an inergeneraional equiy perspecive, hen all he susainable pahs lie wihin he viabiliy kernel. Furhermore, if he sae variables of he economy leave he viabiliy kernel, he consrains will be violae in a finie ime and fuure generaions will no be able o mee heir needs. Such crisis siuaions are irreversible. We deal wih he quesion of susainabiliy when an exhausible resource is involved in producion. This analysis highlighs he role of he echnology funcion in he susainabiliy of an economy. We disinguish non-essenial and essenial resources o reveal he susainabiliy of he economy. We focus on he Cobb-Douglas case in which he economy is susainable if he share of capial in producion is greaer han ha of resource. We find resuls close o Solow [19] and Dasgupa and Heal [9] wihin he Maximin framework because we impose he same consrain on minimal consumpion level for each generaion. We iniially adop a weak susainabiliy poin of view in he sense ha he main consrain is a minimal level of consumpion for each generaion and ha a subsiuion beween capial and naural resource is possible. Neverheless we also have a srong susainabiliy concern, inroducing a consrain abou a guaraneed sock of resource o be preserved. Insead of defining is level ex-ane, or deermining is opimal level linked o a crierion of susainabiliy, we reveal he sock of resource ha can be mainained forever wihou compromising he whole susainabiliy of he economy. We also show ha an economy wih depreciaion of capial is no susainable unless a significan echnological change occurs. One may poin ou ha he viabiliy approach induces flexibiliy in he choices. We do no say wha is he opimal allocaion bu wha are all he susainable choices. We have flexibiliy in he decisions hrough all possible consumpions and exracions ha allow o remain wihin he viabiliy kernel. Given he iniial condiions, we also propose how o adap in a relevan way he guaraneed hresholds associaed wih he susainable requiremens. Of course here remains a lo of works o be done. In paricular, we aim a including uncerainies in dynamics, needs and preferences which may raise quesions relaed o he value of informaion and he resoluion of uncerainies. Exensions o climae change concerns is anoher exciing perspecive. More generally, we hink ha he viable conrol approach provides an ineresing analyical framework o cope wih susainable developmen issues. A Appendix : he proofs Proof of proposiion (3.1) Assume for a while ha he viabiliy kernel is no empy namely Viab and consider any iniial sae (S 0, K 0 ) Viab. Then, from he very definiion of he viabiliy kernel, here exiss saes (S(), K()) saring a (S 0, K 0 ) and conrol (r(), c()) such ha he consrains and dynamics are saisfied along ime. In paricular, we can wrie Consequenly, we have r() r (K(), f) r (f) > 0. S() = S 0 0 r(s)ds S 0.r (f). 15

16 Therefore, here exiss a ime T < + such ha S(T ) < 0. This does no respec he scarciy resource consrain and we derive a conradicion. Consequenly, he viabiliy kernel is empy or Viab =. Proof of proposiion (3.2) Le us consider capial K + such ha r (f) = 0 = r (f, K + ) or equivalenly f(k +, 0) = c. We prove ha all he saes {K K +, S S } are viable 9. We choose conrols r = 0 and c [c, f(k, 0)]. For such conrols, we have Ṡ() = 0 and K() 0. Consequenly he dynamics and consrains are saisfied. Proof of proposiion (3.3) for α > β Following noaions of [1], we need o prove ha he se Viab(f, c, S ) is indeed he viabiliy kernel of he se R + R + for he se-valued dynamics G defined (Ṡ, K) G(S, K) = {( r, f(k, r) c) (r, c) C(K, S)} wih he se-valued map C previously defined by equaion (10). Now assume ha 1 > β and α > β and consider V defined by (9). We need o prove ha he viabiliy kernel Viab(f, c, S ) coincides wih he epigraph Epi(V ) of he funcion V. To achieve his, we proceed in four seps as follows 1. Epi(V ) R + R +, 2. Epi(V ) is a viable domain for he dynamics G, 3. The hypograph 10 Hyp(V ) is an invarian domain for he dynamics G. 4. Viab(f, c, S )\Epi(V ) =. 1) The domain of V is reduced o R +. Since α β > 0, for any K posiive, we obviously have 0 V (K). Therefore Epi(V ) is conained in he se R + R +. 2) Consider he specific feedback conrols defined by (11). Firs i is clear ha r (K, S) and c (K, S) are admissible conrols since f(k, r (K, S)) = K α c 1 β K α = c 1 β = c (K, S) c. 9 The Viabiliy kernel conains every sae wih a sock of resource large enough o allow for a sufficien capial accumulaion o reach K + wihou violaing he consrain on he resource sock. The expression of he whole viabiliy kernel is no given here for sake of simpliciy. I implies o solve a Hamilon-Jacobi-Bellman equaion close o he one given below in (15) ogeher wih he boundary condiion V (K + ) = S. One can refer o he mehod and proof used below in he Cobb-Douglas case. 10 The hypograph of a funcion V : X R is defined by Hyp(V ) = {(x, y), V (x) y}. 16

17 This means ha (r (K, S), c (K, S)) C(K, S). Furhermore, we have ( ) V (K) f(k, r (K, S)) c (K, S) = V β (K)c 1 β = 1 ( c ) 1 β β β α α β 1 β β ( c ) 1 β β K α β = 1 β = r (K, S). Consequenly, we have he Hamilonian formulaion { V (K)(f(K, r) c) + r } 0 inf (r,c) C(K,S) c 1 β α K In angenial erms, his also means ha here exiss admissible (r, c) such ha (f(k, r) c), r) T Epi(V ) (K, V (K)) Therefore he se Epi(V ) is a viabiliy domain for he dynamics (1). β c β 1 β 3) Now we prove ha he hypograph Hyp(V ) of V is an invarian domain or equivalenly ha for any K R + we have { V (K)(f(K, r) c) + r } 0 inf (r,c) C(K,S) Since V (K) is negaive, we obain inf (r,c) C(K,S) { V (K)(f(K, r) c) + r } = inf r r (K) Now le us denoe by g he funcion g(r) = V (K)(f(K, r) c ) + r A sraighforward compuaion provides he opimaliy condiions 0 g (r) and g (r (K, S)) = 0. { V (K)(f(K, r) c ) + r }. Thus r (K, S) is he minimum of g and consequenly for any K R + inf (r,c) C(K,S) {V (K)(f(K, r) c) + r} = g (r (K, S)) = 0. (15) 4) I remains o prove ha he viabiliy kernel and he epigraph of V coincide. By virue of 1) and 2) and Viabiliy heorem in [1], we deduce ha Epi(V ) is conained in Viab(f, c, S ) he union of viable domain for he dynamics G conained in R + R + : Epi(V ) Viab(f, c, S ). Now assume for a while ha here exiss (K 0, S 0 ) Viab(f, c, S ) and (K 0, S 0 ) / Epi(V ). Firs, he condiion (K 0, S 0 ) Viab means ha here exiss viable conrols (r(.), c(.)) and saes (S(.), K(.)) saring from (K 0, S 0 ). 17

18 Now consider he difference e() = V (K()) S(). By assumpion, 0 < e(0) = e 0 = V (K 0 ) S 0. On he oher hand, using Hamilonian formulaion (15) in asserion 3), we wrie for almos every posiive ime ha ė() = V (K())(f(K(), { r()) c()) + r() V (K())(f(K(), r) c) + r } inf (r,c) C(K(),S()) 0. Thus e(.) is decreasing. Using he assumpion 0 e 0 = V (K 0 ) S 0, we infer ha e() e(0) = e 0 or equivalenly V (K()) S() e 0. Since (S(.), K(.)) is a viable rajecory, we have S() 0 which implies ha V (K()) e 0 > 0. This allows o infer ha K() is bounded namely K() K. Since r() is admissible, his implies ha r() is bounded from below namely r() r (K()) r (K ) > 0, where r (K) = ( c K α ) 1 β. Consequenly, he sock of resource saisfies S() = S 0 0 r(s)ds S 0 r (K ). Therefore here exiss a ime T where S(T ) < 0 and we derive a conradicion. We conclude ha Viab(f, c, S ) = Epi(V ). Proof of proposiion (3.3) for α β If he viabiliy kernel was no empy, here would exis a leas one exracion-consumpion pah for which he se of consrains is respeced. Then, he same exracion pah wih consan consumpion limied o c would also be in he viabiliy kernel. Ye, Solow gives he proof in [19, Appendix B] ha no consan posiive consumpion pah exiss for such a sysem if α β. Then he viabiliy kernel is empy. Proof of proposiion (3.5) We presen he proof for he smalles endogenous echnological change i.e. A(K) = K 1 α+ε where ε > 0. We show ha, in his case, he viabiliy kernel is he epigraph of a funcion V (K) wih V (K) = V (c ) + K c 1 β ( δ + c /x ) (1 β)/βx (ε+β)/β dx. 1 β Furhermore, we require lim K V (K) = S, which yields 1 ( δ + c /x ) (1 β)/βx V (c ) = S + (ε+β)/β dx. c β 1 β 18

19 Le us prove ha V (c ) is finie. Firs, we wrie We deduce ha (V (c ) S )β(1 β) (1 β)/β = (V (c ) S )β ( 1 β δ+1 c ( δ + c ) 1 β β x x (ε+β)/β dx c (δ + 1) 1 β β x (ε+β)/β dx. ) (1 β)/β c x (ε+β)/β dx [ ] β ε x ε/β c = β ε c ε/β. Thus V (c ) is bounded. I remains o prove ha he viabiliy kernel is he epigraph of he funcion V (K): we use he same seps han in proposiion (3.3): V (K) is defined on R +. V (K) is posiive and V (K) S. The main sep is o demonsrae he Hamilonian condiions { V (K) ( A(K)K α r β δk c ) } + r = 0 Wih V (K) = 1 β c = c and r = resul. inf (r,c) C(K,S) ( δ+c /K 1 β ( δ+c /K 1 β ) (1 β)/βk (ε+β)/β, aking specific feedback conrols ) 1/βK ε/β, making some algebra, we obain he desired References [1] Aubin J.P. (1991) Viabiliy Theory. Birkhäuser, Springer Verlag. [2] Bene C., Doyen L. and Gabay D. (2001) A Viabiliy Analysis for a Bioeconomic model. Ecological Economics, 36, pp [3] Bonneuil N. (1994) Capial accumulaion, ineria of consumpion and norms of reproducion. Journal of Populaion Economics, 7, pp [4] Bruckner T., Peschell-Held G., Toh F.L., Fussel H.-M., Helm C., Leimbach, M. and Schellnhuber H.J. (1999), Climae change decisionsuppor and he olerable windows approach, Environmenal Modeling and Assessmen, Earh sysem analysis and managemen, Ediors: H.-J. Schellnhuber and F.L. Toh, 4, pp [5] Chichilnisky G. (1996) An axiomaic approach o susainable developmen. Social Choice and Welfare, 13(2), pp

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