Working Paper Green national accounting with a changing population. Memorandum, Department of Economics, University of Oslo, No.

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Asheim, Geir B. Working Paper Green naional accouning wih a changing populaion Memorandum, Deparmen of Economics, Universiy of Oslo, No. 2003,06 Provided in Cooperaion wih: Deparmen of Economics, Universiy of Oslo Suggesed Ciaion: Asheim, Geir B. 2003) : Green naional accouning wih a changing populaion, Memorandum, Deparmen of Economics, Universiy of Oslo, No. 2003,06 This Version is available a: hp://hdl.handle.ne/10419/63053 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 MEMORANDUM No 06/2003 Green naional accouning wih a changing populaion By Geir B. Asheim ISSN: Deparmen of Economics Universiy of Oslo

3 This series is published by he Universiy of Oslo Deparmen of Economics P. O.Box 1095 Blindern N-0317 OSLO Norway Telephone: Fax: Inerne: hp:// econdep@econ.uio.no In co-operaion wih The Frisch Cenre for Economic Research Gausadalleén 21 N-0371 OSLO Norway Telephone: Fax: Inerne: hp:// frisch@frisch.uio.no No 05 No 04 No 03 No 02 No 01 No 36 No 35 No 34 No 33 No 32 Lis of he las 10 Memoranda: Geir B. Asheim and Ylva Søvik The semanics of preference-based belief operaors. 30 pp.rolf Zhiyang Jia A Mixure Model of Household Reiremen Choice. 21 pp. Erling Eide Opimal Provision of Public Goods wih Rank Dependen Expeced. iliy. 21 pp. Hilde C. Bjørnland Esimaing he equilibrium real exchange rae in Venezuela. pp. Svenn-Erik Mamelund Can he Spanish Influenza pandemic of 1918 explain he baby-boom of 1920 in neural Norway?. 33 pp. Elin Halvorsen A Cohor Analysis of Household Saving in Norway. 39 pp. V. Bhaskar and Seinar Holden Wage Differeniaion via Subsidised General Training. 24 pp. Cahrine Hagem and Oar Mæsad Marke power in he marke for greenhouse gas emissions permis he inerplay wih he fossil fuel markes. 21pp. Cees Wihagen, Geir B. Asheim and Wolfgang Buchholz On he susainable program in Solow's model. 11 pp. Geir B. Asheim and Wolfgang Buchholz A General Approach o Welfare Measuremen hrough Naional Income Accouning. 21 pp. A complee lis of his memo-series is available in a PDF forma a: hp://

4 Green naional accouning wih a changing populaion Geir B. Asheim January 30, 2003 Absrac Following Arrow e al. 2003), his paper considers green naional accouning when populaion is changing and insananeous well-being depends no only on per capia consumpion, bu also populaion size. I is shown ha welfare improvemen can be indicaed by an expanded genuine savings indicaor, which also akes ino accoun he oal value of populaion growh, or by an expanded measure of real NNP growh. Pracical ways of approximaing hese measured are discussed. By assuming consan reurns o scale, he measures can be relaed o he value of per capia sock changes and per capia NNP growh, using a resul due o Arrow e al. 2003). The resul are compared o hose arising when insananeous well-being depends only on per capia consumpion, and no on populaion size. Keywords and Phrases: Naional accouning, Populaion, Dynamic welfare, Susainabiliy. JEL Classificaion Numbers: D60, D90, O47, Q01. This paper is inspired he recen invesigaion of he genuine savings crierion and he value of populaion by Arrow e al. 2003). I hank Kenneh Arrow, Parha Dasgupa, and Lawrence Goulder for helpful discussions and commens. I graefully acknowledge he hospialiy of he Sanford Universiy research iniiaive on he Environmen, he Economy and Susainable Welfare, and financial suppor from he Hewle Foundaion hrough his research iniiaive. Address: Deparmen of Economics, Universiy of Oslo, P.O. Box 1095 Blindern, N-0317 Oslo, Norway. Tel: Fax: g.b.asheim@econ.uio.no 1

5 1 Inroducion How can welfare improvemen be measured by naional accouning aggregaes 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 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 e al. s 2003) 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 However, if maximin is applied as a dynamic welfare crierion, hen he insananeous well-being of each generaion will be represened by he uiliy derived from per capia consumpion. Moreover, if he welfare crierion 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 relevan in a discussion of susainabiliy. Following a suggesion by Samuelson 1961, p. 52), he welfare analysis in he presen paper does no presuppose a uiliarian framework, and allows for he possibiliy ha a requiremen of susainabiliy is imposed. A his level of generaliy, one canno give a definie answer o he quesion of wheher only per capia consumpion maers. In Secs. 3 6 of his paper, I follow he basic assumpion of Arrow e al. 2003) by leing in he radiion of oal uiliarianism insananeous well-being depend no only on per capia consumpion, bu also populaion size. Wihin a model wih muliple consumpion and capial goods, I derive four ways for indicaing welfare improvemen, which are generalized or novel resuls. In Sec. 7, I compare he resuls o hose arising when insananeous well-being depends only on per capia consumpion, and no on 1 See Dasgupa 2001b, Sec. 6.4) for a discussion of he deficiency of average uiliarianism. Dasgupa 2001b, p. 100) suggess dynamic average uiliarianism as an alernaive, where discouning seems o arise due o an exogenous and consan per-period probabiliy of exincion. Since he presen paper absracs from any kind of uncerainy, his alernaive crierion will no be discussed here. 2

6 populaion size. In line wih Arrow e al. 2003), I rea populaion as a form of capial. Secion 2 inroduces he model, while Sec. 8 concludes. 2 Model Following Arrow e al. 2003), I assume ha populaion N develops exogenously over ime. The populaion rajecory {N)} =0 is deermined by he growh funcion Ṅ = φn) and he iniial condiion N0) = N 0. Two special cases are exponenial growh, φn) = νn, where ν denoes he consan growh rae, and logisic growh, φn) = 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 e al. 2003), he laer seems like he more accepable formulaion in a finie world. In general, denoe by νn) he rae of growh of populaion as a funcion of N, where νn) = φn)/n. Denoe by C = C 1,..., C m ) he non-negaive vecor of commodiies ha are consumed. To concenrae on he issue of ineremporal disribuion, I assume ha goods and services consumed a any ime are disribued equally among he populaion a ha ime. Thereby he insananeous well-being for each individual may be associaed wih he uiliy uc) ha is derived from he per capia vecor of consumpion flows, c := C/N. Assume ha u is a ime-invarian, increasing, concave, and differeniable funcion. Tha u is ime-invarian means ha all variable deerminans of curren wellbeing are included in he vecor of consumpion flows. A any ime, labor supply is assumed o be exogenously given and equal o he populaion size a ha ime. Denoe by K = K 1,..., K n ) he non-negaive vecor of capial goods. This vecor includes no only he usual kinds of man-made capial socks, bu also socks of naural resources, environmenal asses, human capial, and oher durable producive asses. Corresponding o he sock of capial of ype j, K j, here is a ne invesmen flow: I j := K j. Hence, I = I 1,..., I n ) = K denoes he vecor of ne invesmens. 3

7 The quadruple C, I, K, N) is aainable if C, I, K, N) C, where C is a convex and smooh se. The se of aainable quadruples does no depend direcly on ime. I 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. If C is a cone, hen he echnology exhibis consan reurns o scale. An assumpion of consan reurns o scale will be imposed only in Secs. 6 and 7. Sociey makes decisions according o a resource allocaion mechanism ha assigns o any vecor of capial socks K and any populaion size N a consumpion-invesmen pair CK, N), IK, N)) saisfying ha CK, N), IK, N), K, N) is aainable. 2 assume ha here exiss a unique soluion {K )} =0 o he differenial equaions K ) = IK ), 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 ) := CK ), N)) and I ) := IK ), 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)uc/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. Assume ha he implemened program {C ), I ), K )} =0 is compeiive wih finie uiliy and consumpion values, 0 µ)n)uc )/N))d and and ha i saisfies a capial value ransversaliy condiion, 0 p)c )d exis, lim q)k ) = 0. 1) I follows ha he implemened program {C ), I ), K )} =0 maximizes 0 µ)n)uc/n))d 2 This is inspired by Dasgupa 2001a, p. C20) and Dasgupa and Mäler 2000). I 4

8 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) CK ), N)) N p) = µ) c uc )), 2) + q) IK ), N)) N, 3) q) = p) K CK ), N)) + q) K IK ), N)). 4) 3 Welfare analysis Wrie UK, N) := NuCK, N)/N) and U ) := UK ), N)) for he flow of oal uiliy. In line wih he basic analysis of Arrow e al. 2003), I assume for he nex four secions ha U ) measures he social level of insananeous well-being a ime. Assume ha, a ime, sociey 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 does no depend on. Moreover, assume ha, for a given iniial condiion, he opimal pah is ime-consisen, and ha sociey s resource allocaion mechanism implemens he opimal pah. If he welfare indifference surfaces in infiniedimensional uiliy space are smooh, hen, a ime, {µs} s= are local welfare weighs on oal uiliy flows a differen imes. 3 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 µs) U s)ds > 0. 5) To show ha his welfare analysis includes discouned oal uiliarianism, assume for he res of his paragraph only ha sociey 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 ρs ) U s)ds. Then he change in dynamic welfare is given by d ) e ρs ) U s)ds = U ) + ρ e ρs ) U s)ds = e ρ e ρs U s)ds, d 3 By idenifying he social level of insananeous well-being a ime wih U ), I 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. 5

9 where he second equaliy follows by inegraing by pars. Hence, 5) follows by seing {µ)} =0 = {e ρ } =0. Turn now o he quesion of how o deermine 5) by means of curren prices and quaniies. Since u is concave and differeniable, i follows ha, a each, U) := {U, I, K) U = N)uC/N)) and C, I, K, N)) C} is a convex and smooh se. Furhermore, i follows from C1 and C2 ha, a each, U ), I ), K )) maximizes µ)u + q)i + q)k over all U, I, K) U). In paricular, q) = µ) K UK ), N)) + q) K IK ), N)). 6) Increasing N leads o hree differen kinds of marginal conribuions: 1. Consumpion is spread on more people: v) := µ) uc )) c uc ))c ) ), 2. Oupu increases: w), 3. Populaion growh increases: ψ)φ N)), where v) is he marginal value of consumpion spread, where w) is he wage rae, and where ψ) is he marginal value of populaion growh, all measured in presen value erms. Since ψ) is measured in presen value erms, he decrease of he value of populaion growh, ψ), equals he marginal produciviy of he populaion sock: ψ) = v) + w) + ψ)φ N)) = µ) UK ), N)) N + q) IK ), N)) N + ψ)φ N)), where he second equaliy follows from 2), 3), and he definiion of UK, N). By combining 6) and 7), one obains 7) µ U = µ K U I + U N φn)) = qi + qi + ψφn) + ψ d d φn) )) = d d qi + ψφn) ). Assuming ha lim q)i ) + ψ)φn)) ) = 0 holds as an invesmen value/populaion growh value ransversaliy condiion, one arrives a he following resul by inegraing 8) and using 5) as an indicaor of welfare improvemen. 6 8)

10 Proposiion 1 Dynamic welfare is increasing a ime if and only if q)i ) + ψ)φn)) > 0. This formally generalizes he genuine savings indicaor o a case wih populaion change by indicaing welfare improvemen by means of a posiive value of ne invesmens and populaion growh. However, while q can in principle be observed as marke prices in a perfec marke economy or calculaed as efficiency prices provided he resource allocaion mechanism implemens an efficien program, one needs o consider how o calculae ψ. This quesion is posed in he nex secion. 4 Value of populaion growh How can he marginal value of populaion growh, ψ), be calculaed? Solving 7) and imposing as a erminal condiion yields ψ) = lim ψ) = 0 φns)) φn)) vs) + ws) ) ds. 9) Hence, he value of populaion growh is he inegral of an expression ha consiss of wo facors, vs) + ws) and φns))/φn)). Le us invesigae 9) by discussing hese facors. The sign of vs) + ws). If one assumes ha he value of consumpion, pc, exceeds he oal funcional share of labor, wn, hen i follows from 2) and he definiion of v ha uc ) 0 is a sufficien condiion for v + w o be negaive. Tha uc ) is negaive, means ha insananeous well-being is reduced if an addiional person is brough ino sociey and offered he exising per capia consumpion flows. 4 Noe ha, since u is increasing, uc) is negaive for vecors wih small consumpion flows. E.g., if c is one-dimensional and uc) = ln c, hen uc) 0 if and only if c 1. 4 Observe ha u has no been normalized o saisfy u0) = 0. The analysis allows for he possibiliy ha here are per capia consumpion flows, c 0, such ha uc) < 0. Cf. he conceps of well-being subsisence, as discussed by Dasgupa 2001b, Ch. 14) in he radiion of Meade 1955) and Dasgupa 1969), and criical-level uiliarianism, as proposed by Blackorby and Donaldson 1984). 7

11 Since per capia consumpion and, hus, uiliy derived from per capia consumpion increases wih developmen, one obains he conclusion ha v + w < 0 is more likely o hold for less developed socieies. Since w > 0, i is sufficien for v + w o be posiive ha v and, hus, uc ) c uc )c are non-negaive. Tha uc ) c uc )c is posiive, means ha insananeous wellbeing is increased if an addiional person is brough ino sociey even when he oal consumpion flows are kep fixed and mus be spread on an addiional person. 5 Noe ha i follows from he concaviy of u ha uc) c uc)c is non-decreasing as c increases along a ray where differen commodiies are consumed in fixed proporions. E.g., if c is one-dimensional and uc) = ln c, hen d uc) u c)c ) /dc = 1/2c), and uc) u c)c 0 if and only if c e. Since i is reasonable o assume ha per capia consumpion as well as he marginal produciviy of labor increases wih developmen, one obains he conclusion ha v + w > 0 is more likely o hold for more developed socieies. Noe ha v and, hus, ψ are no invarian under an addiive shif in he uiliy funcion cf. Arrow e al., 2003, p. 224). The developmen of φns))/φn)). If here is consan absolue populaion growh a all fuure imes, hen φns))/φn)) equals 1 hroughou and 9) simplifies o ψ) = vs) + ws) ) ds. If fuure absolue populaion growh is lower han he presen which occurs on he decreasing par of a logisic growh funcion hen φns))/φn)) is smaller han 1 hroughou and i holds ha ψ) < provided ha vs) + ws) > 0. vs) + ws) ) ds, If fuure absolue populaion growh is higher han he presen which occurs wih exponenial growh, enailing ha he rae of growh of populaion, νn) = φn)/n, is consan hen φns))/φn)) is greaer han 1 hroughou and i holds ha ψ) > vs) + ws) ) ds, 5 As discussed by Dasgupa 2001b, Ch. 14), he condiion uc) c uc)c = 0 is imporan in classical uiliarian heories of opimal populaion; see also Meade 1955) and Dasgupa 1969). 8

12 provided ha vs) + ws) > 0. Measuring he value of populaion growh by he presen value of fuure wages. If i holds ha he oal value of he curren populaion N) valued by ψ) is approximaed he presen value of fuure wages, hen he oal value of populaion growh, ψ)φ), can be approximaed hrough muliplying he presen value of fuure wages by he populaion growh rae, νn) = φn)/n. To invesigae he meris of such an approximaion, noe ha dψn) d = ψn + ψṅ) = v + w + ψφ N) ) N ψφn) = wn + µ uc ) c uc )c ) N + ν N)NψN, where φ N) = dνn)n)/dn = ν N)N + νn) has been used o esablish he las equaliy. Hence, he oal value of he curren populaion N) valued by ψ) can be expressed as follows: ψ)n) = ws)ns)ds + + µs) uc )) c uc ))c ) ) Ns)ds ν Ns))Ns)ψs)Ns)ds, provided ha he following populaion value ransversaliy condiion holds: 10) 11) lim ψ)n) = 0. 12) In he special case where u is homogeneous of degree 1, i follows ha uc ) c uc )c = 0 hroughou, so ha in 11) he second erm on he rhs. is equal o zero. In he special case where growh is exponenial so ha he growh rae νn) is consan, i follows ha ν N) = 0 hroughou, so ha in 11) he hird erm on he rhs. is equal o zero. Hence, a linearly homogeneous u combined wih exponenial populaion growh are sufficien for he oal value of populaion growh, ψ)φ), o be equal o νn) In a developed sociey one would expec ha while ws)ns)ds. 13) µs) uc )) c uc ))c ) ) Ns)ds > 0, ν Ns))Ns)ψs)Ns)ds < 0, 9

13 since exponenial growh canno be mainained indefiniely. Hence, in a developed sociey, he wo addiional erms in 11) go in differen direcions, implying ha i canno easily be deermined wheher 13) over- or underesimaes he oal value of populaion growh, ψ)φ). 5 Real NNP growh as a welfare indicaor To invesigae o wha exen real NNP growh indicaes welfare improvemen in he presence of a changing populaion, I follow Asheim and Weizman 2001) and Sefon and Weale 2000) by using a Divisia consumpion 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), 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 ). Define likewise real prices for uiliy, consumpion spread, and populaion growh: M) = µ)/π) V ) = v)/π) Ψ) = ψ)/π). 10

14 Since Q) = q)/π) + R)Q) Ψ) = ψ)/π) + R)Ψ), i follows from 8) ha M U + d d QI + ΨφN) ) = R QI + ΨφN) ). 14) Moreover, keeping in mind ha U = Nuc ), V = M uc ) c uc )c ), P = M c uc ), and ṖC = 0, one obains M U uc = M ) c uc )c ) φn) + c uc )Ċ ) = V φn) + d d PC ). 15) Hence, by combining 14) and 15), i follows ha Ẏ + V φn) + d d ) ΨφN) = R QI + ΨφN) ). In view of Prop. 1, his leads o he following resul. Proposiion 2 Dynamic welfare is increasing a ime if and only if Ẏ ) + V )φn)) + d d ) Ψ)φN)) > 0, provided ha he real ineres rae, R), is posiive. I end his secion by discussing he following quesion: If naional accounans can esimae he genuine savings indicaor, QI, and real growh in comprehensive NNP, Ẏ, bu no he erms ha capure he welfare effecs of populaion change, which of QI and Ẏ is he beer indicaor of welfare improvemen? If one assumes ha, in a more developed sociey, absolue populaion growh, φn), is posiive bu decreasing owards zero, and he marginal value of consumpion spreading, V, is posiive, enailing ha also he marginal value of populaion growh, Ψ, is posiive, hen i follows ha boh V φn) and ΨφN) are posiive, bu evenually decreasing. As he sociey is geing near o having populaion sauraed a N if a logisic growh funcion is followed), hen V φn) > 0 due o a posiive value of consumpion spread, while d d ) ΨφN) < 0 since populaion growh is decreasing owards zero. Hence, 11

15 when using Prop. 2 and approximaing Ẏ + V φn) + d d ΨφN) ) by Ẏ, one would be missing wo erms wih opposie signs. On he oher hand, when using Prop. 1 and approximaing QI + ΨφN) by QI, one would be missing one erm wih a posiive sign. Thus, if i is impracical or impossible o calculae erms involving he value of populaion change, hen real NNP growh, Ẏ, may be an ineresing alernaive o he genuine savings indicaor, QI, as an approximae indicaor of welfare improvemen. In he presen secion I have considered real growh in oal NNP, no real growh in per capia NNP. To be able o analyze per capia measures as I will do in he nex secion one needs o impose an assumpion of consan reurns o scale. One can, however, use 15) o make he following observaion: M U = Muc )φn) + PC dpc )/d PC νn) ). Hence, if uc ) is non-negaive so ha insananeous well-being is no decreased if an addiional person is brough ino sociey and offered he exising per capia consumpion flows and real per capia consumpion increases hroughou, hen i follows from 5) ha dynamic welfare is improving. 6 Per capia measures In he presen secion I consider wo per capia measures: value of ne changes in per capia socks and real growh in per capia NNP. These will be considered in urn in each of he wo following subsecions. In boh subsecions I impose he addiional assumpion of consan reurns o scale. 6.1 Value of ne changes in per capia socks Denoe by k ) := K )/N) he vecor of per capia capial socks. Since k = K N Ṅ K N N = I N νn)k and φn) = νn)n, i follows ha qi + ψφn) N = q k + νn) qk + ψ ) cf. Arrow e al., 2003, p. 222). Hence, by Prop. 1, welfare is increasing a ime if and only if q k + νn) qk + ψ ) > 0. This means ha dynamic welfare can be improving even if he value ne changes in per capia socks, q k, is negaive, provided ha he erm νn) qk + ψ ) is sufficienly posiive. How can qk + ψ be calculaed? 12

16 To allow analysis of his quesion in paricular, o derive a generalized version of Arrow e al. s 2003) Theorem 2 one mus impose consan reurns o scale by assuming ha C is a convex cone. Then i follows direcly from C2 ha, a each, or, equivalenly, p)c ) w)n) + q)i ) + q)k ) = 0, 16) d q)k ) ) = p)c ) w)n). 17) d This means ha he value of capial equals he presen value of he difference beween he value of consumpion and he funcional share of labor, q)k ) = ps)c s) w)ns) ) ds, provided ha 1) holds as a capial value ransversaliy condiion. I now follows from 2), 10), and 17) ha or, equivalenly, dψn) d = pc wn ) + µuc )N + ν N)NψN = dqk ) d + N µuc ) + ν N)ψN ). d qk + ψn ) = N µuc ) + ν N)ψN ). d If 1) and 12) hold as ransversaliy condiions, hen q)k ) + ψ)n) = and, by dividing by N), q)k ) + ψ) = Ns) µs)uc s)) + ν Ns))ψs)Ns) ) ds, Ns) N) µs)uc s)) + ν Ns))ψs)Ns) ) ds. 18) By differeniaing boh sides of 18) w.r.. ime, one obains d qk + ψ ) = µuc ) + ν N)ψN + νn)qk + νn)ψ d = µuc ) ν N)qK + φ N) qk + ψ ), where I have followed Arrow e al. 2003) by using φ N) = ν N)N +νn) o esablish he las equaliy. Inegraing his yields q)k ) + ψ) = φns)) φn)) µs)uc s)) ν Ns))qs)K s) ) ds. 19) In ligh of Prop. 1, 18) and 19) lead o he following resul: 13

17 Proposiion 3 Arrow e al., 2003, Theorem 2) Assuming consan reurns o scale, dynamic welfare is increasing a ime if and only if where q)k ) + ψ) = q) k ) + νn)) q)k ) + ψ) ) > 0, = Ns) N) µs)uc s)) + ν Ns))ψs)Ns) ) ds φns)) φn)) µs)uc s)) ν Ns))qs)K s) ) ds. Hence, if uc ) is non-negaive so ha insananeous well-being is no decreased if an addiional person is brough ino sociey and offered he exising per capia consumpion flows and ν N) < 0 so ha he rae of growh of populaion decreases as populaion increases, hen qk + ψ > 0, meaning ha welfare improvemen is possible even if he value of ne changes in per capia socks is negaive. In he case of exponenial populaion growh so ha he rae of growh of populaion is consan i follows ha ν N) = 0, Ns)/N) = φns))/φn)), and q)k ) + ψ) = 6.2 Real growh in per capia NNP Ns) N) µs)uc s))ds. In he previous subsecion I have ranslaed he resul on he generalized genuine savings indicaor in a seing where here is populaion growh, Prop. 1, ino a finding, Prop. 3, saed in per capia erms. In his subsecion I do he same for Prop. 2 by ranslaing a resul on real growh in oal NNP ino a finding saed in erms of real growh in per capia NNP. The obained resul will be repored below as Prop. 4. Also in his subsecion I impose he addiional assumpion of consan reurns o scale. Wrie y) := Y )/N) for real per capia NNP and i ) := I )/N) for he vecor of per capia invesmen flows. I follows from 16) ha y = Pc + Qi = W + RQ Q ) k since q/π = RQ Q, wih W ) = w)/π) denoing he real wage rae. Moreover, ẏ = Ẏ N Ṅ Y N N = Ẏ N νn)y Hence, by combining hese wo observaions i follows ha Ẏ N = ẏ + νn)w + νn) RQ Q ) k. 20) 14

18 Likewise, since ψ/π = V + W + Ψφ N) and ψ/π = RΨ Ψ, one obains V + W + Ψφ N) + Ψ = RΨ. 21) The sage is now se for expressing Ẏ + V φn) + d d ΨφN) ) in per capia erms and hus, derive a fourh expression ha indicaes welfare improvemen: Using 20) and 21) i follows hrough edious bu sraighforward calculaions ha Ẏ 1 N + V φn) + d d ΨφN))) = ẏ νn) Qk + νn)r Qk + Ψ ). On he basis of Props. 2 and 3, his leads o he following resul. Proposiion 4 Assuming consan reurns o scale, dynamic welfare is increasing a ime if and only if ẏ) νn)) Q)k ) + νn))r) Q)k ) + Ψ) ) > 0, provided ha he real ineres rae, R), is posiive, where Q)k ) + Ψ) = = Ns) N) Ms)uc s)) + ν Ns))Ψs)Ns) ) ds φns)) φn)) Ms)uc s)) ν Ns))Qs)K s) ) ds. In he seing of a one-secor model like he one considered by Arrow e al. 2003) where consumpion, invesmen, and capial are all one-dimensional, and oupu is spli beween consumpion and invesmen he anicipaed capial gains are zero: Q = 0. Under his simplifying assumpion one can draw he following conclusion from Prop. 4: If a posiive ineres rae R, a non-negaive uc ), and a posiive and decreasing νn) lead o νn)r Qk + Ψ ) being posiive, hen welfare improvemen is possible even if real per capia NNP is decreasing. 7 Welfare when only per capia consumpion maers In his secion I invesigae how he welfare analysis will change if I insead of leing oal uiliy, N)uc )), consiue he insananeous well-being a ime le insananeous well-being a ime depend only on he uiliy derived from he vecor of per capia consumpion flows, uc )), bu no on he size of he populaion, N). This is he underlying assumpion made by, e.g., Hamilon 2002). The following is a sraighforward adapaion of he analysis of Sec. 3. Wrie ŨK, N) := uck, N)/N) and Ũ ) := ŨK ), N)) for he flow of per capia 15

19 uiliy. In he alernaive welfare analysis, Ũ ) measures he social level of insananeous well-being a ime, and dynamic welfare is increasing a ime if and only if µs) Ũ s)ds > 0, 22) where, for each, µ) = µ)n). In analogy o he demonsraion in Sec. 3, i can be shown ha his welfare analysis includes discouned uiliarianism, where sociey hrough is implemened program maximizes he sum of per capia uiliies discouned a a consan rae ρ. However, as poined ou by Dasgupa 2001b, Sec. 6.4), i is hard o offer an ehical defense for such discouned average uiliarianism. The framework of Asheim and Buchholz 2002) can, however, be used show how he welfare analysis of Sec. 3 applies also o cases like maximin, where sociey hrough is implemened program maximizes infimum of per capia uiliies, and welfare crieria ha impose susainabiliy as a consrain. In such cases, i seems appropriae o adop he assumpion of he presen secion and le insananeous well-being depend only on per capia consumpion, and no on populaion size see also Pezzey, 2003, Sec. 4). Since u is concave and differeniable, i follows ha, a each, Ũ) := {Ũ, I, K) Ũ = uc/n)) and C, I, K, N)) C} is a convex and smooh se. Furhermore, i follows from C1 and C2 ha, a each, Ũ ), I ), K )) maximizes µ)ũ + q)i + q)k over all Ũ, I, K) Ũ). In paricular, q) = µ) K ŨK ), N)) + q) K IK ), N)). 23) Increasing N leads o hree differen kinds of marginal conribuions: 1. Consumpion is spread on more people: ṽ) := p)c ), 2. Oupu increases: w), 3. Populaion growh increases: ψ)φ N)), where, under he alernaive welfare analysis, ṽ) is he marginal value of consumpion spread, and ψ) is he marginal value of populaion growh, all measured in presen 16

20 value erms. I follows ha ψ) = ṽ) + w) + ψ)φ N)) = µ) ŨK ), N)) N + q) IK ), N)) N + ψ)φ N)), 24) where he second equaliy follows from 2), 3), and he definiion of ŨK, N). By combining 23) and 24), one obains µ Ũ = µ K Ũ I + Ũ N φn)) = qi + qi + ψφn) + ψ d d φn) )) = d d qi + ψφn) ). 25) Assuming ha lim q)i ) + ψ)φn)) ) = 0 holds as an invesmen value/populaion growh value ransversaliy condiion, one arrives a he following resul by inegraing 25), and using 22) as an indicaor of welfare improvemen. Proposiion 5 Dynamic welfare is increasing a ime if and only if q)i ) + ψ)φn)) > 0. Solving 24) and imposing as a erminal condiion yields lim ψ) = 0 ψ) = φns)) φn)) ps)c s) ws) ) ds. I follows ha he marginal value of populaion growh is negaive, provided ha he value of consumpion, pc, exceeds he oal funcional share of labor, wn. This reflecs ha populaion growh means ha sociey s asses mus be shared among more people, while under his alernaive welfare analysis here is no counervailing effec. Noe ha ψ is invarian under an addiive shif in he uiliy funcion. By repeaing he analysis of Sec. 5, i follows ha dynamic welfare is increasing a ime if and only if Ẏ ) + Ṽ )φn)) + d d Ψ)φN)) ) > 0, 17

21 provided ha he real ineres rae, R), is posiive, where, under he alernaive welfare analysis, Ṽ ) = ṽ)/π) is he marginal value of consumpion spread, and Ψ) = ψ)/π) is he marginal value of populaion growh, measured in real erms. Adaping he analysis of Sec. 6.1 o he welfare crierion considered in he presen secion, yields he following resul: Proposiion 6 Assuming consan reurns o scale, dynamic welfare is increasing a ime if and only if q) k ) + νn)) q)k ) + ψ) ) > 0, where q)k ) + ψ) = Ns) = N) ν Ns)) ψs)ns)ds φns)) φn)) ν Ns))qs)K s)ds. If a posiive and decreasing νn) leads o νn) qk + ψ ) being posiive since ν N) < 0, and keeping in mind ha ψ < 0), hen welfare improvemen is possible even if he value of ne changes in per capia socks is negaive. This resul has a clear inuiive inerpreaion: When he rae of populaion growh is decreasing, i is no necessary for he curren generaion o compensae fully for curren populaion growh in order for dynamic welfare o be non-decreasing. Since ν N) = 0 if populaion growh is exponenial, one obains as a corollary he following resul, shown by Hamilon 2002) under discouned average uiliarianism and by Dasgupa 2001b, p. 258) under dynamic average uiliarianism. 6 Proposiion 7 Assuming consan reurns o scale and exponenial populaion growh, dynamic welfare is increasing a ime if and only if q) k ) > 0. Noe ha his resul does no enail ha dynamic welfare is increasing if and only if real per capia wealh is increasing, since ime-differeniaing real per capia wealh, QK /N, yields dqk /N) d = q k π + Qk, 6 Cf. foonoe 1. Dynamic average uiliarianism as inroduced by Dasgupa, 2001b, pp. 100 and 258) coincides wih discouned average uiliarianism when populaion growh is exponenial. 18

22 where i does no follows from our assumpions ha Qk = 0. However, since q k π = Q K N Ṅ QK N N = QK QI N QK ν ), i follows ha q k can be signed by comparing he raio of he value of ne invesmens and wealh wih he rae of growh of populaion. 7 By repeaing he analysis of Sec. 6.2, i follows ha ẏ νn) Qk + νn)r Qk + Ψ ) = R Q k + νn)qk + Ψ) ). 26) Hence, Prop. 6 implies ha dynamic welfare is increasing a ime if and only if ẏ) νn)) Q)k ) + νn))r) Q)k ) + Ψ) ) > 0, provided ha he real ineres rae, R), is posiive. Define z) by z) := P)c ) + Q) k ). Since y = Pc + Qi and k = i νn)k, i follows ha ż = d d y νn)qk ) = ẏ νn) Qk νn)q k ν N)φN)Qk. 27) By combining 26) and 27) wih an assumpion of exponenial populaion growh so ha ν N) = 0 and Prop. 6 implies Qk + Ψ = 0), i follows ha ż) = R) νn)) ) Q) k ), which by Prop. 7 means ha he following resul is obained. Proposiion 8 Assuming consan reurns o scale and exponenial populaion growh, dynamic welfare is increasing a ime if and only if ż) > 0, provided ha he real ineres rae ne of populaion growh, R) νn)), is posiive. I is imporan o observe ha ż is no real growh in per capia NNP; raher, ż is real growh in he sum of he value of per capia consumpion and he value of ne changes in per capia socks. 7 I am graeful o Parha Dasgupa for making his observaion. 19

23 8 Concluding remarks In his paper, I have followed a sandard argumen in welfare economics which was suggesed by Samuelson 1961, p. 52) in he curren seing and idenified welfare improvemen wih µs) d ds Ns)uc s)) ) ds > 0, if boh populaion size and per capia consumpion conribue o insananeous wellbeing, or, µs) d ds uc s)) ) ds > 0. if only per capia consumpion maers. In each case, he analysis encompasses discouned uiliarianism which, however, seems more defendable in he former case). Through Props. 1 8 I have esablished eigh ways o indicae welfare improvemen, depending on which welfare crierion is adoped, on wheher populaion growh is exponenial, and on wheher he echnology exhibis consan reurns o scale. Thereby, his paper offers a subsanial generalizaion and exension of Arrow e al. s 2003) analysis and resuls. These wo cases are of ineres for differen reasons: Following Arrow e al. 2003, p. 221), here are srong argumens in favor of associaing insananeous well-being wih oal uiliy, Nuc ), when invesigaing wheher uiliarian welfare is increasing over ime. However, in line wih Pezzey 2003), one migh argue ha per capia uiliy, uc ), is more relevan in a discussion of susainabiliy. This would mean ha developmen is said o be susainable a he curren ime, if he curren level of individual uiliy derived from per capia consumpion flows, uc ), can poenially be susained indefiniely. If per capia uiliy, uc ), is susained hroughou, so ha duc ))/d 0 a all imes, hen i follows direcly from he crierion for welfare improvemen ha welfare is non-decreasing, in he case where only per capia consumpion maers for insananeous well-being. However, he converse implicaion does no hold: Non-decreasing welfare does no imply ha curren insananeous well-being can poenially be mainained forever. In fac, i can be shown cf. Pezzey, 2003, Sec. 4) under he provision ha { µ)} =0 is an exponenially decreasing funcion ha non-decreasing welfare is a necessary, bu no sufficien, condiion for susainable per capia uiliy. 20

24 References Arrow, K., Dasgupa, P.S., and Mäler, K.-G. 2003), The genuine savings crierion and he value of populaion. Economic Theory 21, Asheim, G.B. and Buchholz, W. 2002), A general approach o welfare measuremen hrough naional income accouning. Memorandum 32/2002, Deparmen of Economics, Universiy of Oslo. Asheim, G.B. and Weizman, M.L. 2001), Does NNP growh indicae welfare improvemen? Economics Leers 73, Blackorby, C. and Donaldson, D. 1984), Social crieria for evaluaing populaion change. Journal of Public Economics 25, Dasgupa, P.S. 1969), On he concep of opimum populaion. Review of Economic Sudies 36, 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 Mäler, K.-G. 2000), Ne naional produc, wealh, and social well-being. Environmen and Developmen Economics 5, Hamilon, K. 2002), Saving effor and populaion growh: heory and measuremen. The World Bank. 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. 2003), One-sided unsusainabiliy ess wih ameniies and shifs in echnology, rade and populaion. CRES, Ausralian Naional Universiy, Canberra. Samuelson, P. 1961), The evaluaion of Social income : Capial formaion and wealh. In The Theory of Capial eds. Luz and Hague), pp S. Marin s Press, New York. Sefon, J.A. and Weale, M.R. 2000), Real Naional Income. NIESR, London. 21