The relationship between welfare measures and indicators of sustainable development

Transcription

1 The relaionship beween welfare measures and indicaors of susainable developmen Geir B. Asheim May 12, 2008 Absrac This chaper invesigaes wheher measures of welfare improvemen indicaes susainabiliy. Firs i shows how he value of ne invesmens and real NNP growh (appropriaely exended if populaion growh is posiive) can be used o measure welfare improvemen, before urning o following wo quesions: (1) Does non-negaive value of ne invesmens imply susainable developmen? (2) Does susainable developmen imply non-negaive value of ne invesmens? The main conclusion is ha welfare improvemen is no a sufficien condiion for susainabiliy, bu under special condiions i is a necessary condiion for susainabiliy. Keywords and Phrases: Naional income accouning, dynamic welfare JEL Classificaion Numbers: C43, D60, O47, Q01 Address: Deparmen of Economics, Universiy of Oslo, P.O. Box 1095 Blindern, NO-0317 Oslo, Norway ( g.b.asheim@econ.uio.no)

2 1 Inroducion Wha is he relaionship beween welfare measures and indicaors of susainable developmen? This chaper sudies o wha exen measures of welfare improvemen can also be used as indicaors of susainabiliy. I builds on (and borrows freely from) published papers by myself and co-auhors (Asheim, 1994, 2003, 2004, 2007a; Asheim, Buchholz and Wihagen, 2003; Asheim and Weizman, 2001); mos of hese papers are included in Asheim (2007b). The relaionship beween welfare measures and indicaors of susainable developmen is paricularly ineresing a seing where here is populaion growh, and hus, in large par of his chaper I will allow for posiive populaion growh. Wha consiues welfare improvemen when populaion is changing? The answer depends on wheher a bigger fuure populaion for a given flow of per capia consumpion leads o a higher welfare weighs for people living a ha ime, or, alernaively, only per capia consumpion maes. When applying, e.g., discouned uiliarianism o a siuaion where populaion changes exogenously hrough ime, i seems reasonable o represen he insananeous well-being of each generaion by he produc of populaion size and he uiliy derived from per capia consumpion. This is he posiion of oal uiliarianism, which has been endorsed o by, e.g., Meade (1955) and Mirrlees (1967), and which is he basic assumpion in Arrow, Dasgupa and Mäler s (2003b) sudy of savings crieria wih a changing populaion. Wihin a uiliarian framework, he alernaive posiion of average uiliarianism, where he insananeous well-being of each generaion depends only on per capia consumpion, have been shown o yield implicaions ha are no ehically defensible. 1 Wha does susainabiliy mean when populaion is changing? If he economy cares abou susainabiliy (in he sense ha curren per capia uiliy should no exceed wha is poenially susainable), hen i becomes imporan o compare he level of individual uiliy for differen generaions, irrespecively of how populaion size develops. Therefore, uiliy derived from per capia consumpion seems more 1 See Dasgupa (2001b, Sec. 6.4) for a discussion of he deficiency of average uiliarianism. 1

3 relevan in a discussion of susainabiliy. Hence, wih posiive populaion growh, i is possible o have oal uiliy increasing hroughou so ha welfare improves, while a he same ime per capia uiliy is falling so ha developmen is no susainable. However, i urns ou ha he relaionship beween welfare improvemen and susainabiliy is no sraighforward even if populaion is consan. Since he major resuls of his chaper concern he problems of associaing measures of welfare improvemen wih susainabiliy, i is jusified o make raher sringen assumpions concerning he working of he economy, since he problems of such associaion will be even more serious in an economy wih a poorer performance. Hence, in he basic model presened in Secion 2, I assume ha he economy implemens a compeiive pah. In Secion 3 I show he welfare significance of he presen value of fuure consumpion changes even in he presence of populaion growh, while in Secion 4 I repor on how he presen value of fuure consumpion can be measured hrough naional accouning aggregaes (boh he value of ne invesmens and real NNP growh). On his basis, I presen in he following four secions a discussion of wheher measures of welfare improvemen can serve as indicaors of susainabiliy. The main conclusion (firs made by Pezzey, 2004) is ha welfare improvemen is no a sufficien condiion for susainabiliy, bu under special condiions i is a necessary condiion for susainabiliy. 2 Model Following Arrow, Dasgupa and Mäler (2003b) and Asheim (2004), assume ha populaion N develops exogenously over ime. The populaion rajecory {N()} =0 is deermined by he growh funcion Ṅ = φ(n) and he iniial condiion N(0) = N 0. Two special cases are exponenial growh, φ(n) = νn, 2

4 where ν denoes he consan growh rae, and logisic growh, φ(n) = νn ( 1 N N ), where ν denoes he maximum growh rae, and N denoes he populaion size ha is asympoically approached. As menioned by Arrow, Dasgupa and Mäler (2003b), he laer seems like he more accepable formulaion in a finie world. In general, denoe by ν(n) he rae of growh of populaion as a funcion of N, where ν(n) = φ(n)/n. Le C represen a m-dimensional consumpion vecor ha includes also environmenal ameniies and oher exernaliies. Le u be a given concave and nondecreasing uiliy funcion wih coninuous parial derivaives ha associaes he insananeous well-being for each individual wih he uiliy u(c) ha is derived from he per capia vecor of consumpion flows, c := C/N. Assume an idealized world where c conains all variable deerminans of curren insananeous well-being, implying ha an individual s insananeous well-being is increased by moving from c o c if and only if u(c ) < u(c ). A any ime, labor supply is assumed o be exogenously given and equal o he populaion size a ha ime. Le K denoe a n-dimensional capial vecor ha includes no only he usual kinds of man-made capial socks, bu also socks of naural resources, environmenal asses, human capial (like educaion and knowledge capial accumulaed from R&Dlike aciviies), and oher durable producive asses. Moreover, le I (= K) sand for he corresponding n-vecor of ne invesmens. The ne invesmen flow of a naural capial asse is negaive if he overall exracion rae exceeds he replacemen rae. Assume again an idealized world where K and N conain all variable deerminans of curren producive capaciy, implying ha he quadruple (C, I, K, N) is aainable if (C, I, K, N) C, where C is a convex and smooh se, wih free disposal of consumpion and invesmen flows. Hence, he se of aainable quadruples does no depend direcly on ime. However, by leing ime be one of he capial 3

5 componens, his formulaion encompasses he case where echnology changes exogenously hrough ime. 2 We hus make an assumpion of green or comprehensive accouning, meaning ha curren producive capaciy depends solely on he vecor of capial socks and he populaion size. Sociey makes decisions according o a resource allocaion mechanism ha assigns o any vecor of capial socks K and any populaion size N a consumpioninvesmen pair (C(K, N), I(K, N)) saisfying ha (C(K, N), I(K, N), K, N) is aainable. 3 I assume ha here exiss a unique soluion {K ()} =0 o he differenial equaions K () = I(K (), N()) ha saisfies he iniial condiion K (0) = K 0, where K 0 is given. Hence, {K ()} is he capial pah ha he resource allocaion mechanism implemens. Wrie C () := C(K (), N()) and I () := I(K (), N()). Say ha he program {C (), I (), K ()} =0 is compeiive if, a each, 1. (C (), I (), K (), N()) is aainable, 2. here exis presen value prices of he flows of uiliy, consumpion, labor inpu, and invesmen, (µ(), p(), w(), q()), wih µ() > 0 and q() 0, such ha C1 C () maximizes µ()u(c/n()) p()c/n() over all C, C2 (C (), I (), K (), N()) maximizes p()c w()n + q()i + q()k over all (C, I, K, N) C. Here C1 corresponds o uiliy maximizaion, while C2 corresponds o ineremporal profi maximizaion. 4 The erm presen value reflecs ha discouning is aken care of by he prices. In paricular, if relaive consumpion prices are consan hroughou and here is consan real ineres rae R, hen i holds ha p() = 2 This leads o he problem of measuring he value of passage of ime using forward-looking erms. Mehods for such measuremen have been suggesed by e.g. by Aronsson e al. (1997), Kemp and Long (1982), Pezzey (2004), Sefon and Weale (1996), and Vellinga and Wihagen (1996). 3 This is inspired by Dasgupa and Mäler (2000), Dasgupa (2001a, p. C20) and Arrow, Dasgupa and Mäler (2003a). 4 To see ha p()c w()n + q()i + q()k is insananeous profi, noe ha p()c + q()i is he value of producion, w()n is he cos of labor and q()k is he cos of holding capial. 4

6 e R p(0). However, I will allow for non-consan relaive consumpion prices and will reurn o he quesion of how o deermine real ineres raes from {p()} =0 in his more general case. Assume ha he implemened program {C (), I (), K ()} =0 wih finie uiliy and consumpion values, is compeiive 0 µ()n()u(c ()/N())d and 0 p()c ()d exis, and ha i saisfies a capial value ransversaliy condiion, lim q()k () = 0. (1) I follows ha he implemened program {C (), I (), K ()} =0 maximizes 0 µ()n()u(c/n())d over all programs ha are aainable a all imes and saisfies he iniial condiion. Moreover, wriing c () := C ()/N(), i follows from C1 and C2 ha w() = p() C(K (), N()) N p() = µ() c u(c ()), (2) + q() I(K (), N()) N, (3) q() = p() K C(K (), N()) + q() K I(K (), N()). (4) 3 Welfare significance of he presen value of fuure consumpion changes Wrie U(K, N) := Nu(C(K, N)/N) and U () := U(K (), N()) for he flow of oal uiliy. In line wih he basic analysis of Arrow, Dasgupa and Mäler (2003b), assume ha U () measures he social level of insananeous well-being a ime. Assume ha, a ime, economy s dynamic welfare is given by a Samuelson- Bergson welfare funcion defined over pahs of oal uiliy from ime o infiniy, and ha his welfare funcion is ime-invarian (i.e., does no depend on ). Moreover, assume ha, for a given iniial condiion, he opimal pah is ime-consisen, 5

7 and ha economy s resource allocaion mechanism implemens he opimal pah. If he welfare indifference surfaces in infinie-dimensional uiliy space are smooh, hen, a ime, {µ(s} s= are local welfare weighs on oal uiliy flows a differen imes. 5 Following a sandard argumen in welfare economics, as suggesed by Samuelson (1961, p. 52) in he curren seing, one can conclude ha dynamic welfare is increasing a ime if and only if µ(s) U (s)ds > 0. (5) To show ha his welfare analysis includes discouned oal uiliarianism, assume for he res of his paragraph only ha economy hrough is implemened program maximizes he sum of oal uiliies discouned a a consan rae ρ. Hence, he dynamic welfare of he implemened program a ime is e ρ(s ) U (s)ds. Then he change in dynamic welfare is given by d d ( ) e ρ(s ) U (s)ds = U () + ρ = e ρ e ρs U (s)ds, e ρ(s ) U (s)ds (6) where he second equaliy follows by inegraing by pars. Hence, (5) follows by seing {µ()} =0 = {e ρ } =0. The following resul provides a connecion beween welfare improvemen and he presen value of fuure changes in consumpion. Proposiion 1 Under he assumpions of Secion 2, µ(s) U (s)ds = p(s)ċ (s)ds + v(s)φ(n(s))ds. 5 By idenifying he social level of insananeous well-being a ime wih U (), we assume ha here are sable welfare indifference surfaces in infinie-dimensional space when he well-being of each generaion is measured by oal uiliy, irrespecively of how consumpion flows and populaion size develop. Discouned oal uiliarianism leads o linear indifference surfaces in his space. 6

8 where v() := µ() ( u(c ()) c u(c ())c () ) denoes he marginal value of consumpion spread, measured in presen value erms. 6 Proof. Since U () = N()u(C ()/N()), we have ha U = c u(c )Ċ + φ(n)u(c ) N c u(c )c ν(n), The resul follows from (2) and he definiion of v(), since ν(n)n = φ(n). Corollary 1 The presen value of fuure consumpion changes, p(s)ċ (s)ds, indicaes welfare improvemen in each of he following wo siuions: (1) There is a consan populaion. (2) The uiliy funcion u is linearly homogeneous. Proof. Par (1) follows direcly from (5) and Proposiion 1. Par (2) follows from (5) and Proposiion 1 hrough he applicaion of Euler s heorem. These resuls can be generalized o he case where he economy s resource allocaion mechanism does no implemen an opimal pah. In paricular, (6) depends solely on he properies of discouned uiliarianism, and does no rely on he resource allocaion mechanism implemening an opimal or even an efficien pah. Furhermore, if (2) is used o define consumpion shadow prices, hen Proposiion 1 and Corollary 1 remain rue. In he case wihou populaion growh, I have hrough Asheim (2007a, Proposiion 2(b)) generalized he resuls of his secion o he case of any ime-invarian Samuelson-Bergson welfare funcion saisfying a condiion of independen fuure (so ha he ranking of wo pahs ha coincide from he curren ime o a fuure ime is he same a any ime beween and ), wihou making any assumpions abou he working of economy s resource allocaion mechanism. 6 Tha v() is posiive means ha insananeous well-being is increased if an addiional individual is brough ino economy even when he oal consumpion flows are kep fixed and mus be spread on an addiional person. See Asheim (2004, Sec. 4) for a discussion of he erm v(). 7

9 4 Measuring he presen value of fuure consumpion changes hrough naional accouning aggregaes To ie he curren chaper o conribuions on he heory of welfare accouning, i is worhwhile o recapiulae how he presen value of fuure consumpion changes, which welfare significance was invesigaed in Secion 3, can be measured hrough naional accouning aggregaes. Wihin he seing of he model of Secion 2, here are wo ways o measure he presen value of fuure consumpion changes, p(s)ċ (s)ds: Through (i) he value of ne invesmens and hrough (ii) real NNP growh, where each measure has been exended o ake care of populaion growh. Proposiion 2 Under he assumpions of Secion 2, p(s)ċ (s)ds = q()i() + Proof. By combining (3) and (4), one obains w(s)φ(n(s))ds. pċ = p ( K C I + C N φ(n)) = ( qi + qi + wφ(n) ) = d d( qi ) + wφ(n). (7) Assuming ha lim q()i () = 0 holds as an invesmen value ransversaliy condiion, and w(s)φ(n(s))ds exiss, he resul is obained by inegraing (7). Turn nex o he quesion of how exended real NNP growh can measure he presen value of fuure consumpion changes. For his purpose, follow Asheim and Weizman (2001) and Sefon and Weale (2006) by using a Divisia consumer price index when expressing comprehensive NNP in real prices. The applicaion of a price index {π()} urns he presen value prices {p(), q()} ino real prices {P(), Q()}, P() = p()/π() Q() = q()/π(), 8

10 implying ha he real ineres rae, R(), a ime is given by R() = π() π(). A Divisia consumpion price index saisfies implying ha ṖC = 0: ṖC = d d π() π() = ṗ()c () p()c (), ( p π ) C = πṗc πpc π 2 = 0. Define comprehensive NNP in real Divisia prices, Y (), as he sum of he real value of consumpion and he real value of ne invesmens: Y () := P()C () + Q()I (). Proposiion 3 Under he assumpions of Secion 2, ( ) p(s) R() π() Ċ (s)ds Proof. Since d d d d (Q()I ()) = 1 d π() = Ẏ () + d d ( ) w(s) π() φ(n(s))ds. d (q()i ()) + R()Q()I () ( ) ( ) w(s) π() φ(n(s))ds = w() π() φ(n()) + R() w(s) π() φ(n(s))ds, i follows from ṖC = 0 and expression (7) ha ( ) 0 = 1 π() p()ċ () + d d (q()i ()) w()φ(n()) ( ) = d d (P()C ()) + d d (Q()I ()) + d w(s) d π() φ(n(s))ds R() ( Q()I () + ) w(s) π() φ(n(s))ds. (8) Hence, he resul is obained by using Proposiion 2 and he definiions above. 9

11 5 Susained developmen implies welfare improvemen Proposiion 1 shows ha he presen value of fuure consumpion changes is an indicaor of welfare improvemen, also under exogenous populaion growh, while Proposiions 2 and 3 show how he presen value of fuure consumpion changes can be measured by means of naional accouning aggregaes. The remaining four secions of his chaper consider he relaionship beween he presen value of fuure consumpion changes, and hus he naional accouning aggregaes of Proposiions 2 and 3, on he one hand, and he susainabiliy of he pah, on he oher hand. To concenrae aenion on inergeneraional issues, absrac hroughou from inraemporal disribuion by assuming ha all individuals living a ime obains he average uiliy level, u(c ()), where c () = C ()/N() is he per capia consumpion along he implemened pah. Following Pezzey (1997), one can disinguish beween susainable and susained developmen. A pah consiues susainable developmen if, a each ime, he per capia uiliy level a ime can poenially be shared by all individuals of fuure generaions. A pah consiues susained developmen if, a each ime, u(c ()) is non-decreasing. Any susained developmen is also susainable. On he oher hand, he converse does no hold, since a developmen can be susainable even if a generaion makes a sacrifice for he benefi of successors ha lowers is own per capia uiliy below hose of is predecessors. If populaion growh is non-negaive and developmen is susained, hen i is a sraighforward conclusion ha ha he presen value of fuure consumpion changes is non-negaive and ha welfare improves. Proposiion 4 If ν(n(s)) 0 and du(c (s))/d 0 for all s >, hen If, in addiion, u(c ()) 0, 7 hen p(s)ċ (s)ds 0. µ(s) U (s)ds 0. 7 Tha u(c ()) is posiive means ha insananeous well-being is increased if an addiional individual is brough ino economy and offered he exising per capia consumpion flows. 10

12 Proof. Since c = C /N, he following holds a each s > : du(c )/d = c u(c (Ċ ) ) /N c ν(n). By (2) and he premisses of he proposiion, p(s)ċ (s) 0 a each s >, hereby esablishing he firs par of he proposiion. Since U = Nu(c ), we have hereby esablishing he second par. U = Ndu(c )/d + ν(n)nu(c ), I is an equally obvious resul ha welfare improvemen, measured by a posiive presen value of fuure consumpion changes, or a posiive presen value of fuure changes in oal uiliy, canno serve as an indicaor of susainabiliy if here is posiive populaion growh. The reason is ha declining per capia uiliy hroughou (i.e., du(c (s))/d < 0 for all s > ) is consisen wih a posiive presen value of fuure consumpion changes and a posiive presen value of fuure changes in oal uiliy if populaion growh is sufficienly large. Therefore, an invesigaion of converse versions of Proposiion 4 is of ineres only in he case of consan populaion. In his case i follows from Proposiions 1 and 2 ha boh he presen value of fuure consumpion changes, p(s)ċ (s)ds, and he value of ne invesmens, q()i (), are exac indicaors welfare improvemen independenly of he properies of he funcion u (cf. foonoe 6). The nex secion repors on he negaive resul ha a posiive value of ne invesmens a ime does no imply ha developmen a ime is susainable. The subsequen Secion 7 presens an invesigaion of he quesion of wheher susainable developmen, raher han he sronger premise of susained developmen used in Proposiion 4, is sufficien for non-negaive value of ne invesmens, when here is no populaion growh ha faciliaes an expansion of he economy. I follows from he analysis of Pezzey (2004) ha i is indeed he case ha developmen is susainable only if he value of ne invesmens is non-negaive, in he special case where he economy implemens a discouned uiliarian opimum. resul does no hold in general. 11 However his

13 In boh he nex wo secion I adop he raher sringen assumpions on he working of he economy imposed in Secion 2 and follow he analysis presened in Asheim, Buchholz and Wihagen (2003). The populaion is assumed o be consan and normalized o 1, implying ha c = C. As he resuls on he relaionship beween susainabiliy and he value of ne invesmens are negaive, such sringen assumpions make he resuls sronger. In he concluding remark I discuss he reliabiliy of welfare improvemen, as measured by he value of ne invesmens, as an indicaor of susainable developmen in an economy ha works less perfecly. 6 Does non-negaive value of ne invesmens imply susainable developmen? Consider he following claim: If he value of ne invesmens q()i () is non-negaive for (0, T ), hen, for any (0, T ), u(c ()) can be susained forever given K (). This claim is no rue in he Dasgupa-Heal-Solow model (see, e.g., Dasgupa and Heal, 1974, and Solow, 1974). In his model here are wo capial socks: manmade capial, denoed by K M, and a non-renewable naural resource, he sock of which is denoed by K N. So, K = (K M, K N ). The iniial socks are given by K 0 = (KM 0, K0 N ). The echnology is described by a Cobb-Douglas producion funcion F (K M, I N ) = K a M ( I N) b depending on wo inpus, man-made capial K M and he raw maerial I N ha can be exraced wihou cos from he nonrenewable resource. The oupu from he producion process is used for consumpion and for invesmens in man-made capial I M. Hence, (c(), I(), K(), 1) is aainable a ime if and only if c() + I M () K M () a ( I N ()) b where a > 0, b > 0 and a + b 1, and c() 0, K M () 0, K N () 0, and I N () 0. Wih r() := I N () denoing he flow of raw maerial, hese assumpions enail 0 r()d K 0 N and r() 0 for all 0. 12

14 Wriing i() := I M (), he compeiiveness condiion C2 requires ha c () + i () = KM() a r () b (9) p() = q M () (10) q M () b KM() a r () b 1 = 1 (11) q M () a KM() a 1 r () b = q M (), (12) where (11) follows from q M () b KM ()a r () b 1 = q N () and 0 = q N () by choosing exraced raw maerial as numeraire: q N () 1. Noe ha (11) and (12) enail ha he growh rae of he marginal produc of raw maerial equals he marginal produc of man-made capial; hus, he Hoelling rule is saisfied. Assume ha a > b > 0. Then here is a sricly posiive maximum consan rae of consumpion c ha can be susained forever given K 0 (see, e.g., Dasgupa and Heal, 1974, p. 203). I is well known ha his consan consumpion level can be implemened along a compeiive pah where ne invesmen in man-made capial is a a consan level ī = b c/(1 b). To give a counerexample o he claim above, fix a consumpion level c > c. Se i = bc /(1 b) and define T by T 0 (i /b) 1 b (K 0 M + i ) a b d = K 0 N. (13) For (0, T ), consider he pah described by K (0) = K 0 and c () = c i () = i r () = (i /b) 1 b (KM 0 + i ) a b, which by (13) implies ha he resource sock is exhaused a ime T. This feasible pah is compeiive during (0, T ) a prices p() = q M () = r ()/i and q N () = 1, implying ha he value of ne invesmens q M ()i q N ()r () is zero. Hence, even hough he compeiiveness condiion C2 is saisfied (while C1 does no apply) and he value of ne invesmens is non-negaive during he inerval (0, T ), he consan rae of consumpion during his inerval is no susainable forever. 13

15 The pah described above for he Dasgupa-Heal-Solow model is in fac no efficien, since he capial value ransversaliy condiion (1) is no saisfied: A ime T a cerain sock of man-made capial, K M (T ) = K0 M + i T, has been accumulaed. A he same ime he flow of exraced raw maerial falls abruply o zero due he exhausion of he resource. Wih a Cobb-Douglas producion funcion, he marginal produciviy of r is a sricly decreasing funcion of he flow of raw maerial for a given posiive sock of man-made capial. This implies ha profiable arbirage opporuniies can be exploied by shifing resource exracion from righ before T o righ afer T, implying ha he Hoelling rule is no saisfied a ha ime. As he pah in his counerexample is inefficien, i migh be possible ha he value of ne invesmens does no indicae susainabiliy in he example due o his lack of efficiency. However, his is no rue eiher. The claim above does no become valid even if we consider pahs for which compeiiveness holds hroughou and he capial value ransversaliy condiion is saisfied. Again, counerexamples can be provided in he framework of he Dasgupa-Heal- Solow model. Asheim (1994) and Pezzey (1994) independenly gave a counerexample by considering pahs where he sum of uiliies discouned a a consan uiliy discoun rae is maximized. If, for some discoun rae, he iniial consumpion level along such a discouned uiliarian opimum exacly equals he maximum susainable consumpion level given KM 0 and K0 N, hen here exiss an iniial inerval during which he value of ne invesmens is sricly posiive, while consumpion is unsusainable given he curren capial socks KM () and K N (). I is no quie obvious, however, ha he premise of his saemen can be fulfilled, i.e., ha here exiss some discoun rae such ha iniial consumpion along he opimal pah is barely susainable. This was subsequenly esablished for he Cobb-Douglas case by Pezzey and Wihagen (1998). The fac ha heir proof is quie inricae indicaes, however, ha his is no a rivial exercise. Consequenly, anoher ype of counerexample is provided here. This example is also wihin he Dasgupa-Heal-Solow model and resembles he one given above. 14

16 consumpion c 2 c 1 max. sus. cons. given K 0 T 1 T 2 ime Figure 1: Non-negaive value of ne invesmens does no imply susainabiliy In paricular, a pah idenical o ha described in he firs counerexample during an iniial phase can always be exended o an efficien pah. Moreover, his second counerexample can be used o show ha here exis regular pahs wih non-negaive value of ne invesmens during an iniial phase even if a b, enailing ha a posiive and consan rae of consumpion canno be susained indefiniely. The example, illusraed in Figure 1, consiss of hree separae phases wih consan consumpion, consruced so ha here are no profiable arbirage opporuniies a any ime, no even a he wo poins in ime, T 1 and T 2, where consumpion is no coninuous. Boh capial socks are exhaused a T 2, implying ha consumpion equals zero for (T 2, ). In he consrucion of he example, KM 0 is given, while K0 N is reaed as a parameer. Fix some consumpion level c 1 > 0 and some erminal ime T 1 of he firs phase of he pah. In he inerval (0, T 1 ) he pah is as in he firs example described by K (0) = K 0 and c () = c 1 i () = i 1 r () = (i 1 /b) 1 b (KM 0 + i 1) a b, where i 1 = bc 1 /(1 b), bu wih he difference ha he resource sock will no be exhaused a ime T 1. As in he firs example, he value of ne invesmens equals zero during his phase. 15

17 The second phase sars a ime T 1. Consumpion jumps upward disconinuously o c 2 > c 1, bu we ensure ha he flow of raw maerial is coninuous o remove profiable arbirage opporuniies. Consumpion is consan a he new and higher level c 2, and, by he generalized Harwick rule firs esablished by Dixi, Hammond and Hoel (1980), he value of ne invesmens measured in presen value prices mus be consan. I.e., here exiss ν 2 < 0 such ha, for all (T 1, T 2 ), q M ()i () = r () + ν 2. By (9) and (11), his equaliy may (for any c and ν) be wrien as K M () a r() b c = b K M () a r() b 1 (r() + ν). (14) As KM a rb b KM a rb 1 r = (1 b) KM a rb, his implies ( c = (1 b) K M () a r() b 1 b ) 1 b ν. (15) r() Since boh K M () and r () are coninuous a ime T 1, we can now use (15) o deermine ν 2 as follows: ( c 2 = (1 b) KM(T 1 ) a r (T 1 ) b 1 b ) 1 b ν 2 r. (16) (T 1 ) By choosing c 2 > K M (T 1) a r (T 1 ) b (> c 1 ) and fixing ν 2 according o (16), q M ()i () = r () + ν 2 combined wih (9) deermines a compeiive pah along which invesmen in man-made capial becomes increasingly negaive. Deermine T 2 as he ime a which he sock of man-made capial reaches 0, and deermine K 0 N such ha he resource sock is exhaused simulaneously. Wih boh socks exhaused, consumpion equals 0 during he hird phase (T 2, ). The Hoelling rule holds for (0, T 1 ) and (T 1, T 2 ), and by he consrucion of ν 2, a jump of he marginal produciviy of he naural resource a T 1 is avoided such ha he Hoelling rule obains even a T 1. Thus, he pah is compeiive hroughou. By leing u(c) = c and, for all (0, T 2 ), µ() = p(), i follows ha he pah saisfies all assumpions of Secion 2. Noe ha he above consrucion is independen of wheher a > b. If a b, so ha no posiive and consan rae of consumpion can be susained indefiniely, we have hus shown ha having non-negaive value of ne invesmens during an iniial 16

18 phase of a regular pah is compaible wih consumpion exceeding he susainable level. However, even if a > b, so ha he producion funcion allows for a posiive level of susainable consumpion, a counerexample can be obained. For his purpose, increase c 2 beyond all bounds o ha ν 2 becomes more negaive. Then T 2 decreases and converges o T 1, and he aggregae inpu of raw maerial in he inerval (T 1, T 2 ) being bounded above by r(t 1 ) (T 2 T 1 ) since r() is decreasing converges o 0. This in urn means ha, for large enough c 2, c 1 canno be susained forever given he choice of KN 0 needed o achieve exhausion of he resource a ime T 2. 7 Does susainable developmen imply non-negaive value of ne invesmens? The counerexample of Figure 1 shows ha non-negaive value of ne invesmens on an open inerval is no a sufficien condiion for having consumpion be susainable. Consider in his secion wheher his is a necessary condiion: Does a negaive value of ne invesmens during a ime inerval imply ha consumpion exceeds he susainable level? The following resul due o Pezzey (2004) shows ha such a converse implicaion holds under discouned uiliarianism. Proposiion 5 Le T > 0 be given. Consider a pah {c (), I (), K ()} =0 saisfying he assumpions of Secion 2 in a consan-populaion economy, wih {µ()} =0 = {e ρ } =0. If he value of ne invesmens q()i () is negaive for (0, T ), hen, for any (0, T ), u(c ()) canno be susained forever given K (). Proof. I follows from (2) and (7) ha µ()du(c ())/d + d(q()i ())/d = 0, implying d(µ()u(c ()))/d+d(q()i ())/d = µ()u(c ()). By combining his wih µ() = e ρ and lim q()i () = 0, so ha µ(s)ds = µ()/ρ and µ()u(c ()) + q()i () = µ(s)u(c (s))ds = ρ µ(s)u(c (s))ds, 17

19 Weizman s (1976) main resul can be esablished: ( ) µ(s) u(c ()) + q() µ() I () ds = µ(s)u(c (s))ds. (17) Since he pah saisfies he condiion of Secion 2, i maximizes µ(s)u(c(s))ds over all feasible pahs. This combined wih (17) implies ha he maximum susainable uiliy level given K () canno exceed u(c ()) + q()i ()/µ(). Suppose q()i () < 0 for (0, T ). Then u(c ()) > u(c ()) + q()i ()/µ(). Hence, u(c ()) exceeds he maximum susainable uiliy level and canno be susained forever given K (). I is no, however, a general resul ha susainabiliy implies non-negaive value of ne invesmens. This will be esablished nex by showing ha he following claim is no rue, even under he condiions of Secion 2: If he value of ne invesmens q()i () is negaive for (0, T ), hen, for any (0, T ), u(c ()) canno be susained forever given K (). Also in his case a counerexample will be provided in he framework of he Dasgupa-Heal-Solow model. Assume ha a > b so ha he producion funcion allows for a posiive level of susainable consumpion. Again, he example (which is illusraed in Figure 2) consiss of hree separae phases wih consan consumpion, consruced so ha here are no profiable arbirage opporuniies a any ime, no even a he wo poins in ime, T 1 and T 2, where consumpion is no coninuous. As before, K 0 M is given, while K0 N is reaed as a parameer. Fix some consumpion level c 1 > 0 and some erminal ime T 1 of he firs phase of he pah. Consruc a pah ha has consan consumpion c 1 and obeys he generalized Harwick rule by having a negaive and consan value of ne invesmen. I.e., q M ()i () = r () + ν 1 wih ν 1 < 0 in he inerval (0, T 1 ), where T 1 is small enough o ensure ha K M (T 1) > 0. Le he pah have, as is second phase, consan consumpion c 2 > 0 and obeying he generalized Harwick rule wih ν 2 > 0 in he inerval (T 1, T 2 ). To saisfy he Hoelling rule a ime T 1, c 2 and ν 2 mus fulfill (16); hence, by choosing c 2 < (1 b) K M (T 1) a r (T 1 ) b i follows ha ν 2 > 0. Le 18

20 consumpion c 3 c 1 max. sus. cons. given K 0 c 2 T 1 T 2 ime Figure 2: Susainabiliy does no imply non-negaive value of ne invesmens KM (T 2) and r (T 2 ) be he sock of man-made capial and he flow of raw maerial, respecively, a ime T 2. A his poin in ime he pah swiches over o he hird phase wih zero value of ne invesmens, where he consan level of consumpion is deermined by c 3 = (1 b) KM (T 2) a r (T 2 ) b. Since a > b, he producion funcion allows for a posiive level of susainable consumpion, and here exiss an appropriae choice of KN 0 ha ensures resource exhausion as so ha he capial value ransversaliy condiion (1) is saisfied. This iniial resource sock depends on T 1 and T 2, bu i is finie in any case. Keep T 1 fixed and increase T 2. As T 2 goes o infiniy, hen he sock KN 0 needed will also end o infiniy. 8 The same holds rue for he maximum susainable consumpion level c ha is feasible given KM 0 and he iniial resource sock K0 N deermined in his way. Hence, by shifing T 2 far enough ino he fuure, i follows ha c 1 < c. Thus, a regular pah can be consruced which has a firs phase wih a negaive value of ne invesmens even hough he rae of consumpion during his phase is susainable given he iniial socks. Boh our counerexamples are consisen wih he resul of Proposiion 2 (in he case wih no populaion growh) ha he value of ne invesmens measures he presen value of all fuure changes in of uiliy. I follows direcly from ha resul ha if along an efficien pah uiliy is monoonically decreasing/increasing 8 I follows from (15) and c 2 > 0 ha r () > bν 2/(1 b) (> 0) for all (T 1, T 2). 19

21 indefiniely, hen he value of ne invesmens will be negaive/posiive, while uiliy will exceed/fall shor of he susainable level. The value of ne invesmens hus indicaes susainabiliy correcly along such monoone uiliy pahs. Hence, he counerexamples of Figures 1 and 2 are minimal by having consumpion (and hus uiliy) consan excep a wo poins in ime. I is worh emphasizing he poin made in Asheim (1994) and elsewhere ha he relaive equilibrium prices of differen capial socks oday depend on he properies of he whole fuure pah. The counerexamples of Figures 1 and 2 show how he relaive price of naural capial depends posiively on he consumpion level of he generaions in he disan fuure. Thus, he fuure developmen in paricular, he disribuion of consumpion beween he inermediae and he disan fuure affecs he value of ne invesmens oday and, hereby, he usefulness of his measure as an indicaor of susainabiliy oday. 8 Concluding remark As shown in Secions 3 and 4, he value of ne invesmens measures welfare improvemen in an economy wih no populaion growh, given ha he assumpions of Secion 2 are saisfied and he economy s dynamic welfare is given by a ime-invarian Samuelson-Bergson welfare funcion ha leads o a ime-consisen opimal pah. In Secions 6 and 7 we have shown ha he value of ne invesmens canno serve as a reliable indicaor of susainabiliy, even in a consan-populaion economy ha saisfies he assumpions of Secion 2. I is worh o noice ha he reliabiliy of he value of ne invesmens as an indicaor of susainabiliy is furher undermined if he resource allocaion mechanism implemens neiher an opimal nor an efficien pah. Consider, e.g., an economy where radiional growh is promoed hrough high invesmen in reproducible capial goods, bu where incorrec (or lack of) pricing of naural capial leads o depleion of naural and environmenal resources ha is excessive boh from he perspecive of shor-run efficiency and long-run susainabiliy. Then uiliy growh in he shor 20

22 o inermediae run will, if he uiliy discoun rae ρ is large enough, lead o curren growh in discouned uiliarian dynamic welfare. Hence, boh he value of ne invesmens and real NNP growh will be posiive. 9 A he same ime, he resource depleion may seriously undermine he long-run livelihood of fuure generaions, so ha curren uiliy far exceeds he level ha can be susained forever. References Aronsson, T., Johansson, P.O., and Löfgren, K.-G. (1997), Welfare Measuremen, Susainabiliy and Green Naional Accouning. Edward Elgar, Chelenham. Arrow, K., Dasgupa, P.S., and Mäler, K.-G. (2003a), Evaluaing projecs and assessing susainable developmen in imperfec economies. Environmenal and Resource Economics 26, Arrow, K.J., Dasgupa, P.S., and Mäler, K.-G. (2003b), The genuine savings crierion and he value of populaion. Economic Theory 21, Asheim, G.B. (1994), Ne Naional Produc as an Indicaor of Susainabiliy. Scandinavian Journal of Economics 96, Asheim, G.B. (2003), Green naional accouning for welfare and susainabiliy: A axonomy of assumpions and resuls, Scoish Journal of Poliical Economy 50, Asheim, G.B. (2004), Green naional accouning wih a changing populaion. Economic Theory 23, Asheim, G.B. (2007a), Can NNP be use for welfare comparisons? Environmen and Developmen Economics 12, Asheim, G.B. (2007b), Jusifying, Characerizing and Indicaing Susainabiliy. Springer, Dordrech. 9 I follows from Asheim (2007a, Proposiions 1(b) and 2(b)) ha he value of ne invesmens and real NNP growh measure welfare improvemen even if he resource allocaion mechanism is imperfec, provided ha appropriae shadow prices are applied. 21

23 Asheim, G.B., Buchholz, W., and Wihagen, C. (2003), The Harwick Rule: Myhs and Facs, Evironmenal and Resource Economics 25, Asheim, G.B. and Weizman, M.L. (2001), Does NNP growh indicae welfare improvemen? Economics Leers 73, Dasgupa, P.S. (2001a), Valuing objecs and evaluaing policies in imperfec economies, Economic Journal 111, C1 C29. Dasgupa, P.S. (2001b), Human Well-Being and he Naural Environmen. Oxford Universiy Press, Oxford. Dasgupa, P.S. and Heal, G.M. (1974), The opimal depleion of exhausible resources, Review of Economic Sudies (Symposium), Dasgupa, P.S. and Mäler, K.-G. (2000), Ne naional produc, wealh, and social well-being, Environmen and Developmen Economics 5, Dixi, A., Hammond, P., and Hoel, M. (1980), On Harwick s rule for regular maximin pahs of capial accumulaion and resource depleion, Review of Economic Sudies 47, Kemp, M.C. and Long, N.V. (1982), On he evaluaion of social income in a dynamic economy: Variaions on a Samuelsonian heme, in G.R. Feiwel (ed.), Samuelson and Neoclassical Economics, Kluwer Academic Press, Dordrech, Meade, J.E. (1955), Trade and Welfare. Oxford Universiy Press, Oxford. Mirrlees, J.A. (1967), Opimal growh when he echnology is changing. Review of Economic Sudies (Symposium Issue) 34, Pezzey, J.C.V. (1994), Theoreical Essays on Susainabiliy and Environmenal Policy. Ph.D. Thesis, Universiy of Brisol. Pezzey, J.C.V. (1997), Susainabiliy consrains versus opimaliy versus ineremporal concern, and axioms versus daa, Land Economics 73, Pezzey, J.C.V. (2004), One-sided unsusainabiliy ess wih ameniies and shifs in echnology, rade and populaion, Journal of Environmenal Economics and Managemen 48,

24 Pezzey, J.C.V. and Wihagen, C.A. (1998). The rise, fall and susainabiliy of capialresource economies, Scandinavian Journal of Economics 100, Samuelson, P. (1961), The evaluaion of social income : Capial formaion and wealh, in F.A. Luz and D.C. Hague (eds.), The Theory of Capial, S. Marin s Press, New York, Sefon, J.A. and Weale, M.R. (1996), The ne naional produc and exhausible resources: The effecs of foreign rade, Journal of Public Economics 61, Sefon, J.A. and Weale, M.R. (2006). The concep of income in a general equilibrium, Review of Economic Sudies 73, Solow, R.M. (1974), Inergeneraional equiy and exhausible resources, Review of Economic Sudies (Symposium), Vellinga, N. and Wihagen, C. (1996), On he Concep of Green Naional Income, Oxford Economic Papers 48, Weizman, M.L. (1976), On he welfare significance of naional produc in a dynamic economy, Quarerly Journal of Economics 90,