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Green National Accounting with a Changing PopulationAuthor(s): Geir B. AsheimReviewed work(s):Source: Economic Theory, Vol. 23, No. 3 (Apr., 2004), pp. 601-619Published by: SpringerStable URL: http://www.jstor.org/stable/25055775 .

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Economie Theory 23, 601-619 (2004) ?-;?DOI: 10.1007/s00199-003-0385-0 CCOIlOItlICTheory

? Springer-Verlag 2004

Green national accountingwith a changing population*Geir B. AsheimDepartment of Economics, University of Oslo, P.O. Box 1095 Blindem, 0317 Oslo, NORWAY(e-mail: [email protected])

Received: January 30, 2003; revised version: March 31, 2003

Summary. Following Arrow et al. (2003), this paper considers green national accounting when population is changing and instantaneous well-being depends bothon per capita consumption and population size. Welfare improvement is shownto be indicated by an expanded "genuine savings indicator", taking into accountthe value of population growth, or by an expanded measure of real NNP growth.Under CRS, themeasures can be related to the value of per capita stock changesand per capita NNP growth, using a result due toArrow et al. (2003). The resultsare compared to those arising when instantaneous well-being depends only on percapita consumption.

Keywords and Phrases: National accounting, Population, Dynamic welfare, Sustainability.JEL Classification Numbers: D60, D90,047, Q01.

1 IntroductionHow can welfare improvement be measured by national accounting aggregateswhen population is changing? The answer depends on whether a bigger futurepopulation for a given flow of per capita consumption leads to a higher welfareweights for people living at that time, or, alternatively, only per capita consumptionmattes.

* This paper is inspired by the recent investigation of the genuine savings criterion and the value ofpopulation by Arrow et al. (2003). I thank Kenneth Arrow, Partha Dasgupta, Lawrence Goulder, and areferee for helpful discussions and comments. I gratefully acknowledge the hospitality of the Stanford

University research initiative on the Environment, the Economy and Sustainable Welfare, and financialsupport from the Hewlett Foundation through this research initiative.

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602 G. B. Asheim

When applying discounted utilitarianism to a situation where population changes exogenously through time, it seems reasonable to represent the instantaneouswell-being of each generation by the product of population size and the utilityderived from per capita consumption. This is the position of 'total utilitarianism',which has been endorsed to by, e.g., Meade (1955) andMirrlees (1967), and whichis the basic assumption inArrow et al.' (2003) study of savings criteria with achanging population. Within a utilitarian framework, the alternative position of'average utilitarianism', where the instantaneous well-being of each generation

depends only on per capita consumption, have been shown to yield implicationsthat are not ethically defensible.1However, ifmaximin is applied

as a dynamic welfare criterion, then the instantaneous well-being of each generation will be represented by the utility derivedfrom per capita consumption. Moreover, if the welfare criterion cares about sustainability (in the sense that current per capita utility should not exceed what ispotentially sustainable), then it becomes important to compare the level of individual utility for different generations, irrespectively of how population size develops.Therefore, utility derived from per capita consumption seems more relevant in adiscussion of sustainability.

Following a suggestion by Samuelson (1961, p. 52), the welfare analysis inthe present paper does not presuppose a utilitarian framework, and allows for thepossibility that a requirement of sustainability is imposed. At this level of generality, one cannot give a definite answer to the question of whether only per capitaconsumption matters.In Sections 3-6 of this paper, I follow the basic assumption of Arrow et al.(2003) by letting - in the tradition of 'total utilitarianism' - instantaneous wellbeing depend not only on per capita consumption, but also population size. Withinamodel with multiple consumption and capital goods, I derive four ways for indicating welfare improvement, which are generalized or novel results. In Section 7,I compare the results to those arising when instantaneous well-being depends onlyon per capita consumption, and not on population size. In line with Arrow et al.(2003), I treat population as a form of capital. Section 2 introduces themodel, whileSection 8 concludes.

2Model

Following Arrow et al. (2003), I assume that population N develops exogenouslyover time. The population trajectory {N(t)}%Z0 is determined by the growth function

TV= (?)(N)1 See Dasgupta (2001b, Section 6.4) for a discussion of the deficiency of 'average utilitarianism'.

Dasgupta (2001b, p. 100) suggests 'dynamic average utilitarianism' as an alternative, where discountingseems to arise due to an exogenous and constant per-period probability of extinction. Since the presentpaper abstracts from any kind of uncertainty, this alternative criterion will not be discussed here.


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Green national accounting with a changing population 603

and the initial condition AT(0) = N?. Two special cases are exponential growth,(?)(N) = uN,

where v denotes the constant growth rate, and logistic growth,(N) 9N(l-?),

where v denotes the maximum growth rate, and AT* denotes the population sizethat is asymptotically approached. As mentioned by Arrow et al. (2003), the latterseems like themore acceptable formulation in a finite world. In general, denote byv(N) the rate of growth of population as a function of N, where

i/(N) = (f>(N)/N.Denote by C = (Ci,..., Cm) the non-negative vector of commodities thatare consumed. To concentrate on the issue of intertemporal distribution, I assumethat goods and services consumed at any time are distributed equally among the

population at that time. Thereby the instantaneous well-being for each individualmay be associated with the utility u(c) that is derived from the per capita vectorof consumption flows, c := C/N. Assume that u is a time-invariant, increasing,concave, and differentiable function. That u is time-invariant means that all variabledeterminants of current well-being are included in the vector of consumption flows.At any time, labor supply is assumed to be exogenously given and equal to thepopulation size at that time.Denote by K = (K\,..., Kn) the non-negative vector of capital goods. Thisvector includes not only the usual kinds of man-made capital stocks, but also stocksof natural resources, environmental assets, human capital, and other durable productive assets. Corresponding to the stock of capital of type j, Kj, there is a netinvestment flow: Ij :=Kj. Hence, I = (h,..., In) = K denotes the vector of netinvestments.

The quadruple (C,I,K, N) is attainable if (C,I,K,iV) G C, where C is aconvex and smooth set, with free disposal of consumption and investment flows.The set of attainable quadruples does not depend directly on time. I thus make anassumption of "green" or comprehensive accounting, meaning that current productive capacity depends solely on the vector of capital stocks and the population size.If C is a cone, then the technology exhibits constant returns to scale. An assumptionof constant returns to scale will be imposed only in Sections 6 and 7.

Society makes decisions according to a resource allocation mechanism that assigns to any vector of capital stocks K and any population size N a consumptioninvestmentpair (C(K, AT), (K, N)) satisfyingthat(C (K,N), I(K, JV), , N)is attainable.2 I assume that there exists a unique solution {K*(?)}^0 t0 the differential equations K*(t) ? I(K*(?), N(t)) that satisfies the initial condition

K*(0) = K?, where K? is given. Hence, {K*(t)} is the capital path that theresource allocation mechanism implements. Write C*(t) := C(K*(i), N(t)) andr(t):=I(K*(t),JV(t)).Say that the program {C* (t), I* (t),K* (t)}??0 is competitive if, at each t,

2 This is inspired by Dasgupta (2001a, p. C20) and Dasgupta and Maler (2000).


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604 G. B. Asheim

1. (C*(t),I*(t),K*(t),N(t)) is attainable,2. there exist present value prices of the flows of utility, consumption, labor input,andinvestment, p(t),p(t), w(t), q(t)), with p(t) > 0 andq(t) > 0, such thatCl C*(?) maximizes p(t)u(C/N(t)) - p(t)C/N(t) overallC,C2 (C*(t), I*(t),K*(?), N(t)) maximizes p(t)C - w(*)J\T q(t)I + q(t)Koverall (C,I,K,iV)

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p. 52) in the current setting, one can conclude that dynamic welfare is increasingat time t if and only if/oo

p(s)U*(s)ds>0. (5)To show that this welfare analysis includes discounted total utilitarianism, assumefor the rest of this paragraph only that society through its implemented programmaximizes the sum of total utilities discounted at a constant rate p. Hence, thedynamic welfare of the implemented program at time t is

/ooe-p(?-t)U*(s)d8.

Then the change in dynamic welfare is given by?.dt v./* '.itj e-p^-Û*{s)ds)= -U*{t) + p ? e-^sÛt(s)ds/OO

e-ps?*(s)ds,where the second equality follows by integrating by parts. Hence, (5) follows bysetting{/*(*) ?0 = {e-'ôTurn now to the question of how to determine (5) by means of current pricesand quantities. Since u is concave and differentiable, and C is a convex and smoothset, with free disposal of consumption flows, it follows that, at each t,

U(t) := {(f/,I,K)| U = N(t)u(C/N(t)) and (C,I,K, N(t)) e C}is a convex and smooth set. Furthermore, it follows from Cl and C2 that, at each t,(U*(t),I*(t),K*(t)) maximizes

fl(t)U+ l(t)I + l(t)Kover all (?/, I,K) G 14(i). In particular,

-q(*) = ?(t)VKU(K*(t), N(t)) + q(t)VKI(K*(i), N(t)). (6)Denote by ip(t) themarginal value of population growth, measured in present valueterms. Since ip(t) is measured in present value terms, the decrease of the value ofpopulation growth, ?ip(t), equals themarginal productivity of the population stock:

It follows from (2) and the definition of U(K, N) that


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606 G. B. Asheim

where v(t) := p(t)(u(c*(t)) - Vcu(c*(t))c*(t)) denotes themarginal value ofconsumption spread, measured in present value terms. Hence, using (3), equation(7) can be rewritten as

-^(t) = v(t) + w(t) +WWW)), (8)and increasing N leads to three different kinds of marginal contributions:

1. Consumption is spread on more people: v(t),2. Output increases: w(t),3. Population growth increases: ip(t)(j)'(N(t)).By combining (6) and (7), one obtains

/iLr'= /i(VKtf-r + f?-WV)) (9)= -(ql* +q?* +MN) +iMN)))= -?(qr +WW)Assuming that

?m (q(f)r(f)+ tf(i)0(JV(t)))=Oholds as an investment value/population growth value transversality condition, onearrives at the following result by integrating (9) and using (5) as an indicator ofwelfare improvement.

Proposition 1. Dynamic welfare is increasing at time t if and only ifq{t)V{t)+m0.

This formally generalizes the "genuine savings indicator" to a case with population change by indicating welfare improvement by means of a positive value of netinvestments and population growth. However, while q can in principle be observedas market prices in a perfect market economy or calculated as efficiency pricesprovided the resource allocation mechanism implements an efficient program, oneneeds to consider how to calculate ip.This question is posed in the next section.

4 Value of population growthHow can themarginal value of population growth, ip(t), be calculated? Solving (8)and imposing lim ijj(t)= 0t-^ooas a terminal condition yields

/OO$$??l(v(s) + w(s))ds. (10)

Hence, the value of population growthis the integral of an expression that consistsof two factors,v(s) + w(s) and (f>(N(s))/(?(N(t)). Let us investigate(10) by

discussing these factors.


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The signofv(s) + w(s)If one assumes that the value of consumption, pC*, exceeds the total functionalshare of labor, wN, then it follows from (2) and the definition of v that u(c*) < 0is a sufficient condition for v + w to be negative. That u(c*) is negative, meansthat instantaneous well-being is reduced if an additional person is brought intosociety and offered the existing per capita consumption flows.4 Note that, since uis increasing, u(c) is negative for vectors with small consumption flows. E.g., if cis one-dimensional and u(c) ? In c, then u(c) < 0 if and only if c < 1.Since per capita consumption and, thus, utility derived from per capita consumption increases with development, one obtains the conclusion that v + w < 0ismore likely to hold for less developed societies.Since w > 0, it is sufficient for v + w to be positive that v and, thus,

u(c*)-Vcu(c*)c*are non-negative. That u(c*) ? Vci?(c*)c* is positive, means that instantaneouswell-being is increased if an additional person is brought into society even whenthe total consumption flows are kept fixed and must be spread on an additionalperson.5 Note that it follows from the concavity of u that i?(c) ? Vci?(c)c is nondecreasing as c increases along a ray where different commodities are consumedin fixed proportions. E.g., if c is one-dimensional and u(c) ? In c, then d(u(c) ?u'(c)c)/dc = 1/c, and u(c) ? u'(c)c > 0 if and only if c > e.Since it is reasonable to assume that per capita consumption as well as themarginal productivity of labor increases with development, one obtains the conclusion that v + w > 0 ismore likely to hold formore developed societies.Note that v and, thus, i? are not invariant under an additive shift in the utilityfunction (cf. Arrow et al. 2003, p. 224).

Thedevelopmentof (N(s))/(?>(N(t))If there is constant absolute population growth at all future times, then (N(s))/4>(N(t)) equals 1 throughout and (10) simplifies to

/oo(v(s) + w(s))ds.

If future absolute population growth is lower than the present - which occurson the decreasing part of a logistic growth function - then (?)(N(s))/ (j>(N(t)) issmaller than 1 throughout and it holds that/oo

(v(s) + w(s))ds,4 Observe that u has not been normalized to satisfy u(0) = 0. The analysis allows for the possibilitythat there are per capita consumption flows, c > 0, such that u(c) < 0. Cf. the concepts of 'well-being

subsistence', as discussed by Dasgupta (2001b, Ch. 14) in the tradition of Meade (1955) and Dasgupta(1969), and 'critical-level utilitarianism', as proposed by Blackorby and Donaldson (1984).5 As discussed by Dasgupta (2001b, Ch. 14), the condition u(c) ? Vci?(c)c = 0 is important inclassical utilitarian theories of optimal population: see also Meade (1955) and Dasgupta (1969).


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608 G. B. Asheim

provided that v(s) + w(s) > 0.If future absolute population growth is higher than the present - which occurswith exponential growth, entailing that the rate of growth of population, v(N)?

(j)(N)/N, is constant - then (j)(N(s))/(f)(N(t)) is greater than 1 throughout and itholds that /OO(v(s)+w(s))ds,

provided that v(s) + w(s) > 0.

Measuring the value of population growthby thepresent value of future wagesIf it holds that the total value of the current population N(t) valued by i?j(t) isapproximated the present value of future wages, then the total value of populationgrowth, ip(t)(j)(t), can be approximated through multiplying the present value offuture wages by the population growth rate, v(N) = cf)(N)/N.To investigate themerits of such an approximation, note that

= (v+ w +WiNfiN-ilH^N) (11)= wN + ?jl(u(c*) -Vcu(c*)c*)N + v'(N)NipN,where '(N) = d(u(N)N)/dN = vf(N)N + i/(N) has been used to establishthe last equality. Hence, the total value of the current population N(t) valued byip(t) can be expressed as follows:

/?OOOO

w(s)N(s)ds/OO/x(s)(?(c*(s)) - Vcu(c*(s))c*(s))iV(s)ds (12)

/OOvf(N(s))N(s)ip(s)N(s)ds,

provided that the following population value transversality condition holds:lim ip(t)N(t) = 0. (13)?-?oo

In the special case where u is homogeneous of degree 1, it follows that u(c*) ?Vct?(c*)c* = 0 throughout, so that in (12) the second term on the rhs. is equal tozero. In the special case where growth is exponential so that the growth rate v(N) isconstant, it follows that vf(N) = 0 throughout, so that in (12) the third term on therhs. is equal to zero. Hence, a linearly homogeneous u combined with exponentialpopulation growth are sufficient for the total value of population growth, ip(t)(j)(t),to be equal to

/OO w(s)N(s)ds. (14)


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In a developed society one would expect that/oo

H(s)(u{c*(s)) - Vcu(c*(s))c*(s))AT(s)ds > 0,while

/ vf(N(s))N(s)iJj(s)N(s)ds(t).

5 Real NNP growth as a welfare indicatorTo investigate to what extent real NNP growth indicates welfare improvement inthe presence of a changing population, I follow Asheim and Weitzman (2001)and Sefton andWeale (2000) by using a Divisia consumption price index whenexpressing comprehensive NNP in real prices. The application of a price index{ir(t)} turns hepresentvalueprices {p(?), q(t)} into realprices {P(?), Q(?)},

P(?) = p(t)/7T(t)Q(?) = q(?)/7r(?),

implying that the real interest rate, R(t), at time t is given by*(*) = -*$

A Divisia consumption price index satisfiesTT(t) p(t)C*(t)7T(?) p(?)C*(?) '

implying that PC* = 0:

Define comprehensive NNP in realDivisia prices, Y(t), as the sum of the real valueof consumption and the real value of net investments:y(t):=p(t)c*(t) + Q(t)r(t).

Define likewise real prices for utility, consumption spread, and population growth:M(t) = p(t)Mt)V(t) = v(t)/v(t)${t)=m/*(t)


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610 G. B. Asheim

SinceQ(f) = q(t)/?r(t) ?(t)Q(t)$(t)=i>(t)/n(t) + RW(t),

it follows from (9) thatM?* + ft (QI*+ ?ty(JV))= ?(QI* + #4>(N)). (15)

Moreover, keeping inmind thatU* = Nu(c*), V = M(u{c*) - Vctt(c*)c*),

P = MVc?(c*), and PC* = 0, one obtainsM?* = M((?(c*) - Vcu(c*)c*)4>{N) + Vcu(c*)C*) v (16)= l^(JV)+ ?(PC*).

Hence, by combining (15) and (16), it follows thatY + V{N) ft(V(N)) .

In view of Proposition 1, this leads to the following result.Proposition 2. Dynamic welfare is increasing at time t if and only if

Y(t)+ V(t)4>(N(t)) &{V(t) 0 due to a positivevalue of consumption spread, while ^(^(/)(N)) < 0 since population growthis decreasing towards zero. Hence, when using Proposition 2 and approximating Y + V(?)(N) -h ^(^(?)(N)) by Y, one would be missing two terms withopposite signs. On the other hand, when using Proposition 1 and approximating

QI* + &(f)(N) by QI*, one would be missing one term with a positive sign. Thus,if it is impracticalor

impossible to calculate terms involving the value of populationchange, then real NNP growth, Y, may be an interesting alternative to the "genuinesavings indicator", QI*, as an approximate indicator of welfare improvement.


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In the present section I have considered real growth in totalNNP, not real growthin per capita NNP. To be able to analyze per capita measures - as Iwill do in thenext section - one needs to impose an assumption of constant returns to scale. Onecan, however, use (16) tomake the following observation:M?* = Mu(c*)0(JV) + PC'-(d(ppy - v(N)).

Hence, if u(c*) is non-negative - so that instantaneous well-being is not decreasedif an additional person is brought into society and offered the existing per capitaconsumption flows - and real per capita consumption increases throughout, then itfollows from (5) that dynamic welfare is improving.

6 Per capita measuresIn the present section I consider two per capita measures: 'value of net changes inper capita stocks' and 'real growth in per capita NNP'. These will be consideredin turn in each of the two following subsections. In both subsections I impose theadditional assumption of constant returns to scale.

6.1 Value of net changes inper capita stocksDenote by k*(t) :=K*(t)/N(t) the vector of per capita capital stocks. Since

K ? N N N ? N ^VJK

and (f)(N) = v(N)N, it follows that

(cf. Arrow et al., 2003, p. 222). Hence, by Proposition 1,welfare is increasing attime t if and only ifqk* + v(N) (qk* + i?) > 0. This means that dynamic welfarecan be improving even if the value net changes inper capita stocks, qk*, isnegative,provided that the term v(N) (qk* + i?) is sufficiently positive. How can qk* -f i?be calculated?To allow analysis of this question - in particular, to derive a generalized versionof Arrow et al.'s (2003) Theorem 2 - one must impose constant returns to scale byassuming that C is a convex cone. Then it follows directly from C2 that, at each t,

p(t)C*(t) - w(t)N(t) + q(t)I*(t) + q(*)K*(t) = 0 , (17)or, equivalently,

-^f^Pitiqt)-^). (is)


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612 G. B. Asheim

This means that the value of capital equals the present value of the differencebetween the value of consumption and the functional share of labor,/OO {p(s)C'(s)-w(s)N{s))ds,

provided that (1) holds as a capital value transversality condition.It now follows from (2), (11), and (18) thatd(i?jN)= _^pC* _^ + ^(c*)7v + vf(N)Ni}jN

+ N(nu(c*) + i/'{N)rl>N).

= N(pu(c*) + v'(N)ipN)

dt= 0,


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where

q(t)k*(t)+ V(t)/OO #$(A*(a)?(c*(5)) u'(N(s)Ms)N(S))ds/OO

$$$(M?)?(c*(*)) - ^'(iV(S))q(S)K*(S))dS.Hence, if w(c*) is non-negative - so that instantaneous well-being is not decreasedif an additional person isbrought into society and offered the existing per capita consumption flows -and v'(N) < 0- so that the rate of growth of population decreasesas population increases, then qk* + i? > 0, meaning that welfare improvement ispossible even if the value of net changes in per capita stocks is negative.In the case of exponential population growth - so that the rate of growth ofpopulation is constant - it follows that i/(N) = 0, N(s)/N(t) = (N(s))/(?(N(t)),and

/oo

"$n(s)u(c*(s))ds.

6.2 Real growth inper capita NNPIn the previous subsection I have translated the result on the generalized "genuinesavings indicator" in a setting where there is population growth, Proposition 1,into a finding, Proposition 3, stated in per capita terms. In this subsection I do thesame for Proposition 2 by translating a result on real growth in total NNP into afinding stated in terms of real growth inper capita NNP. The obtained result willbe reported below as Proposition 4. Also in this subsection I impose the additionalassumption of constant returns to scale.Write y(t) :=Y(t)/N(t) forrealpercapitaNNPand i*(t) := l*(t)/ N(t) forthe vector of per capita investment flows. It follows from (17) that

y = Pc* + Qi* = W + (RQ - Q)k*since ? q/7r

=RQ

?Q, with W(t)

=w(t)/7r(t) denoting the real wage rate.Moreover,

Hence, by combining these two observations it follows that

%=y + v(N)W + v(N)(RQ-Q)k\ (21)Likewise, since ?ip/ir = V + W + \P'(N) and -tp/ir = R&

-4', one obtains

V + W + V

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614 G. B. Asheim

The stage is now set for expressing Y + V(?(N) + ^ (& 0,provided that the real interest rate, R(t), ispositive, whereQ(?)k*(t)+ #(?)/OO

?g(M(sMc?) + i/(JV(s))#(s)JV(s))

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In analogy to the demonstration in Section 3, it can be shown that this welfareanalysis includes discounted utilitarianism, where society through its implementedprogram maximizes the sum of per capita utilities discounted at a constant rate p.However, as pointed out by Dasgupta (2001b, Section 6.4), it is hard to offer anethical defense for such discounted average utilitarianism.The framework of Asheim and Buchholz (2002) can, however, be used show howthewelfare analysis of Section 3 applies also to cases likemaximin, where societythrough its implemented program maximizes infimum of per capita utilities, andwelfare criteria that impose sustainability as a constraint. In such cases, it seemsappropriate to adopt the assumption of the present section and let instantaneouswell-being depend only on per capita consumption, and not on population size (seealso Pezzey 2003, Section 4).Since u is concave and differentiate, and C is a convex and smooth set, withfree disposal of consumption flows, it follows that, at each t,

U(t) := {(17,1,K)| ? = u(C/N(t)) and (C, I,K, N(t)) e C}is a convex and smooth set. Furthermore, it follows from Cl and C2 that, at each t,(i/*(t),I*(t),K*(t)) maximizes

/i(t)?7+ q(t)I + q(t)Kover all (?7,1, K) e ?(t). In particular,

-q(t) = ?(t)VK?(K*(t), N(t)) + q(?)VKI(K*(?), N(t)). (24)Denote by ip(t) is themarginal value of population growth, measured in presentvalue terms, under the alternative welfare analysis. It follows that

_??. ?^""?'^(o^y'.^w)) (25)= v(t)+w(t) + j>(t)'(N{t)),where the second equality follows from (2), (3), and the definition of U (K, N),with v(t) :? ? p(t)c*(t) denoting the marginal value of consumption spread,measured in present value terms, under the alternative welfare analysis. Hence,increasing N leads to three different kinds of marginal contributions:1. Consumption is spread on more people: v(t),2. Output increases: w(t),3. Population growth increases: ^(t)(f)f(N(t)).

By combining (24) and (25), one obtains

?U =?(VK?'!* + ??-{N)) (26)= -(ql* +ql* +U(N) + $&{

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616 G. B. Asheim

Assuming that

tlim(q(?)r(t)+

^(t)0(JV(?)))=Oholds as an investment value/population growth value transversality condition, onearrives at the following result by integrating (26), and using (23) as an indicator ofwelfare improvement.Proposition 5. Dynamic welfare is increasing at time t ifand only if

q(t)V(t) + i?(t)ooas a terminal condition yields

/oo

$$?{P(8)C'(8)-W(8))d8.It follows that themarginal value of population growth is negative, provided thatthe value of consumption, pC*, exceeds the total functional share of labor, wN.This reflects that population growth means that society's assets must be sharedamong more people, while - under this alternative welfare analysis - there is nocountervailing effect. Note that i? is invariant under an additive shift in the utilityfunction.

By repeating the analysis of Section 5, it follows that dynamic welfare is increasing at time t if and only if

Y(t)+ V(t) 0,provided that the real interest rate, R(t), is positive, where, under the alternativewelfare analysis, V(t) = v(t)/it(t) is themarginal value of consumption spread,and &(t) = i?(t)/7r(t) is themarginal value of population growth, measured in realterms.

Adapting the analysis of Section 6.1 to the welfare criterion considered in thepresent section, yields the following result:Proposition 6. Assuming constant returns to scale, dynamic welfare is increasingat time t ifand only if

q(?)k*(?) i/(tf(t))(q(t)k*(t) ?,(t))> 0 ,where

/oo%$v'(N(s))?,(s)N(s)ds

/OO^^u'(N(s)Us)K*(s)ds.


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If a positive and decreasing v(N) leads to v(N) (qk* + -0) being positive (sincev'(N) < 0, and keeping inmind that t/i< 0), then welfare improvement is possibleeven if the value of net changes in per capita stocks is negative. This result has aclear intuitive interpretation: When the rate of population growth is decreasing, it isnot necessary for the current generation to compensate fully for current populationgrowth in order for dynamic welfare to be non-decreasing.Since i/'(N) = 0 if population growth is exponential, one obtains as a corollarythe following result, shown by Hamilton (2002) under discounted average utilitarianism and by Dasgupta (2001b, p. 258) under 'dynamic average utilitarianism'.6Proposition 7. Assuming constant returns to scale and exponential populationgrowth, dynamic welfare is increasing at time t if and only if

q(t)k*(t)>0.Note that this result does not entail that dynamic welfare is increasing if and only ifreal per capita wealth is increasing, since time-differentiating real per capita wealth,QK*/AT, yields

d(QK*/N) = qk* |dt 7Twhere it does not follow from our assumptions thatQk* = 0. However, since

sal - 9*1 _ KQ?- - QKV^IL _ \7T ~ N N N ~ N VQK* "J >it follows that qk* can be signed by comparing the ratio of the value of net investments and wealth with the rate of growth of population.7

By repeating the analysis of Section 6.2, it follows thaty - i/(JV)Qk*+ i/(JV)?(Qk*+ #) = ?(Qk* -h i/(AT)(Qk* ?)). (27)

Hence, Proposition 6 implies that dynamic welfare is increasing at time t if andonly ify(t) - i/(^(?))Q(?)k*(*) v(N(t))R(t)(Q(t)k*(t) 9(t))>0,

provided that the real interest rate, R(t), is positive. Define z(t) by?(*):= P(t)c*(t) + Q(t)k*(t).

Since y = Pc* + Qi* and k* = i* - i/(JV)k*, it follows that?= ?(!/-KW)Qk*) (28)= y- u{N)Qk* - i/(JV)Qk* - v''{N)

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618 G. B. Asheim

Proposition 8. Assuming constant returns to scale and exponential populationgrowth, dynamic welfare is increasing at time t ifand only if

z(t) >0,provided that the real interest rate net of population growth, R(t) ? v(N(t)), ispositive.

It is important to observe that z is not real growth in per capita NNP; rather, zis real growth in the sum of the value of per capita consumption and the value ofnet changes in per capita stocks.

8 Concluding remarksIn this paper, I have followed a standard argument in welfare economics - whichwas suggested by Samuelson (1961, p. 52) in the current setting - and identifiedwelfare improvement with

/oo?(s)?(N(S)u(c*(S)))ds>0,

if both population size and per capita consumption contribute to instantaneouswell-being, or, /oo

?(8)?(u(c*{8)))d8>0.if only per capita consumption matters. In each case, the analysis encompassesdiscounted utilitarianism (which, however, seems more d?fendable in the formercase).

Through Propositions 1-81 have established eight ways to indicate welfare improvement, depending on which welfare criterion is adopted, on whether populationgrowth is exponential, and on whether the technology exhibits constant returns toscale. Thereby, this paper offers a substantial generalization and extension of Arrowet al.' (2003) analysis and results.These two cases are of interest for different reasons: Following Arrow et al.(2003, p. 221), there are strong arguments in favor of associating instantaneous wellbeing with total utility, Nu(c*), when investigating whether utilitarian welfare isincreasing over time. However, in line with Pezzey (2003), one might argue thatpercapita utility, u(c*), ismore relevant in a discussion of sustainability. This wouldmean that development is said to be sustainable at the current time, if the currentlevel of individual utility derived from per capita consumption flows, u(c*), can

potentially be sustained indefinitely.If per capita utility, u(c*), is sustained throughout, so that d(u(c*))/dt > 0at all times, then it follows directly from the criterion for welfare improvementthat welfare is non-decreasing, in the case where only per capita consumptionmatters for instantaneous well-being. However, the converse implication does nothold: Non-decreasing welfare does not imply that current instantaneous well-beingcan potentially be maintained forever. In fact, it can be shown (cf. Pezzey 2003,


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Section 4) - under the provision that {?(t)}^0 is an exponentially decreasingfunction - that non-decreasing welfare is a necessary, but not sufficient, conditionfor sustainable per capita utility.

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accounting. Memorandum 32/2002, Department of Economics, University of Oslo (2002)Asheim, G. B., Weitzman, M. L.: Does NNP growth indicate welfare improvement? Economics Letters73, 233-239 (2001)Blackorby, C, Donaldson, D.: Social criteria for evaluating population change. Journal of Public Eco

nomics 25, 13-33 (1984)Dasgupta, P.S.: On the concept of optimum population. Review of Economic Studies 36, 294-318(1969)Dasgupta, P. S.: Valuing objects and evaluating policies in imperfect economies. Economic Journal 111,C1-C29 (2001a)Dasgupta, P. S.: Human well-being and the natural environment. Oxford: Oxford University Press 2001bDasgupta, P. S., Maler, K.-G.: Net national product, wealth, and social well-being. Environment and

Development Economics 5, 69-93 (2000)Hamilton, K.: Saving effort and population growth: theory and measurement. The World Bank (2002)Meade, J.E.: Trade and welfare. Oxford: Oxford University Press 1955Mirrlees, J.A.: Optimal growth when the technology is changing. Review of Economic Studies (Sym

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population. CRES, Australian National University, Canberra (2003)Samuelson, P. A.: The evaluation of 'social income' :capital formation and wealth. In: Lutz, FA., Hague,

D.C. (eds.) The theory of capital, pp. 32-57. New York: St. Martin's Press 1961Sefton, J.A., Weale, M. R.: Real national income. NIESR, London (2000)

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