Scand.J. of Economics99(1), 1-13, 1997

Sustainability and Technical Progress MartinL. Weitzman HarvardUniversity,Cambridge,MA 01238, USA

Abstract A rigorousmodel connectstogetherthe followingthree basic concepts:(1) "sustainability" - meaningthe generalizedfuturepowerof an economyto consumeover time; (2) "Green NNP"- meaninga currentmeasureof nationalincomethatsubtractsoff fromGNP notjust depreciationof capitalbut also, more generally,depletionof environmentalassetsevaluated at currentefficiencyprices;(3) "technologicalprogress"- meaninga projectiononto the future of the so-called "Solowresidual".A simple general formulais derived.Some crude calculationssuggesta possiblystrongeffect of the residual,whichhintsthat our best present estimatesof long-termsustainabilitymay be largelydrivenby predictionsof future technologicalprogress.

I. Introduction "Sustainability"has become a popular catchwordin recent years. The word itself is subjectto variousinterpretations.In an oft-cited phrase,the Bruntland Commission' defined "sustainable development" to be "developmentthat meets the needs of the present without compromising the abilityof future generationsto meet their own needs." While it is not typically stated explicitly, the basic underlying concept behind most notions of sustainabilityin the literaturewould appearto be some implicit measureof the economy'sgeneralizedcapacityto produceeconomicwellbeing over time. In this paper "sustainability"is defined to be the annualizedequivalent of the present discounted value of consumption that the economy is of an economyis the capableof achieving.More precisely,"sustainability" hypotheticalconstant or "annuity-equivalent"level of consumptionthat would yield the same present discountedvalue as the actual consumption trajectorythe economy is able to deliver. In this context, "sustainable development"might refer to a time path whose "sustainability"over the future is never less than its currentconsumption. Now it turns out that there is a ratherremarkabletheoreticalrelationship between "sustainability",defined above, and what might be called

'WCED (1987); p. 43. ? The editors of the ScandinavianJournalof Economics1997. Publishedby BlackwellPublishers,108 Cowley Road, OxfordOX4 1JF, UK and 350 Main Street, Maiden, MA 02148, USA.

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"Green NNP" - the net aggregatethat subtractsoff from GNP not just depreciation of capital but also the value of depleted natural resources evaluated at competitivemarketprices. Under certain conditions,Green NNP exactlyequals sustainability.2 The force of this result is perhapsnot sufficientlyappreciated.A future "sustainabilitycrisis" caused by the exhaustion of a critical resource looming over the horizonshould manifestitself now. Sustainability,which is essentially a measure of future consumption, is, at least in principle, exactlyreflected in currentGreen NNP. While this resultcan serve as a powerfulconceptualguide for indicating how to think about the relationshipbetween sustainabilityand national income accounting,its practicalapplicabilityis somewhat limited by the assumptionsof the model. The most restrictiveassumption,by far, is the absence of technological progress.The existingresultthat a theoreticallycorrectmeasureof welfare just exactly equals a theoreticallycorrect measure of Green NNP relies completely on the time-autonomyof the system. But, to the extent that whateverendogenousand exogenousfactorsthoughtto underlietechnical change have been ignored, the situation can be interpreted"as if" there exists a time-dependentresidualshift factorthat increasesproductivitybut does not show up anywherein national income accounts, and hence is ignored by the existingframework. The consequences of technicalchange being absent from the standard time-autonomousmodel mightbe quite seriousfor the basicwelfareinterpretationof Green NNP. We know that futuregrowthis largelydrivenby the rate of technological progress, however it is conceptualized. Since Green NNP theoreticallyequals annuity-equivalentfuture consumption possibilitieswithoutthe "Solowresidual",the propermeasureof annuityequivalent future consumption possibilities with the "Solow residual" might conceivablycall for a sizable upwardadjustmentof Green NNP. This paper extends the existingstandardframeworkto include technological progress.Mathematically,I expandthe model of my 1976 paper to cover a situationwhere the technologydependson time. The resultsof the earlier paper can henceforth be viewed as a special case of the more general results obtained here. An exact expression is derived that indicates the appropriateupward correction of Green NNP required by the existence of technological progressor any other form of time dependency.A roughcalculationbased on reasonablevalues of the relevantparameterssuggeststhat the required corrections may be sizable - perhaps around 40 per cent or more of 2Thisis a rephrasingof the basic result in Weitzman(1976). ? The editors of the ScandinavianJournalof Economics1997.

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conventionally-measurednational product. A possible implicationcould be that long-term sustainability,like so much else about the future, is largelydrivenby projectionsof technologicalprogress. I should make clear at the outset that the treatmentof time dependency per se is not original to this paper, since there already exists a sizable literature on the subject. Weitzman (1976) sketched the mathematical outlines of a correctiveexpression.Importantformal contributionswere made by Kemp and Long (1982), Lofgren (1992), Aronsson and Lofgren (1993), Asheim (1996), Nordhaus (1995), Hartwick (1995) and others. From this literatureit emerges that there are severalways to expressthe effects of time dependency,each one having a somewhatdifferent interpretation.The main contributionof the presentpaper is to derivea simple but quite general formulain whichthe Solow residualappearsas a natural link connectingsustainabilitywith nationalincome accountingwhen there is technologicalprogress. My hope is that this expressionmay be found useful because it is interpretable in terms of some already-familiar concepts from growththeory and other areas.

II. A Formulationof the Basic Problem To make the problem analyticallytractable,we abstractheroicallyin the spirit of Weitzman(1976). First of all, for simplicityit is assumed that (in effect) there is just one composite consumptiongood. It might be calculatedas an index number with given price weights,or as a multipleof some fixedbasketof goods, or more generallyas any cardinalutilityfunction.The importantthing is that the consumptionlevel in period t can be unambiguouslyregisteredby the single numberC(t). Thus, the paper assumesawayall of the problemsthat mightbe associatedwith constructingan "idealmeasure"of consumption akin to a utility function. Purging consumption of the index number problemwill allow us to focus more sharplyon the general meaning and significanceof combining it with investmentwhen there is technological progress. As in my earlierpaper, the notion of "capital"used here is meant to be quite a bit more general than the traditional"producedmeans of production" like equipment and structures.Most immediately,pools of natural resources are considered to be capital. Human capital should also be included,if we knew how to measureit. Under a verybroadinterpretation, environmentalassets generallymight be treated as a form of capital.3

3Maler (1991) includes a discussion of some of the relevant issues here. ? The editors of the ScandinavianJournalof Economics 1997.

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Suppose that altogether there are n capital goods, includingstocks of naturalresources.The stock of capital of typej (1 0 the conditions (C(t), K(t))eS(K(t); t) and K() = Ko0.

(4)

(5) where Ko is the original endowment of capital availableat startingtime t=0. ? The editors of the ScandinavianJournalof Economics1997.

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Generallyspeaking,there are an infinitenumberof feasibletrajectories. We can narrowfeasible trajectoriesdown to a unique familyby presuming a competitive-like economy with a fixed own rate of return on the consumptiongood equal to r. A competitivetrajectory{C*(t), K*(t)} with realinterestrater is anyfeasibletrajectoryfor whichthere existsan n-vector of investment prices {P(t)} such that, evaluated at all t>0 along the trajectory (6) G(K*(t), P(t), t) = C*(t)+P(t)K*(t) and, for eachj, dP 8G a = rPj(t)- dPj. OKj A/dt

(7)

Equation(6) just states thatwhatis actuallyproducedby the economyat any time maximizesits income - in other words,relativeprices are equal to marginalrates of transformation.Condition(7) is the well-knownintertemporalefficiencycondition of a competitivecapital market.4 Actually, equations (6) and (7) are necessary Pontryagin-typeconditions5for any solution to the optimal control problem of maximizingthe expression

C(t) e-rt dt

{

(8)

o

subjectto the constraint (C(t), K(t))eS(K(t); t) and obeying the initial condition K(O)= Ko.

(9)

(10) Thus, {C*(t), K*(t)} can be considereda solutionof the optimalcontrol problem (8)-(10), and what we have been calling "inclusive"or "Green" Net National Product- the expression(6) - is the currentvalue Hamiltonian maximizedover the controlvariables.Therefore,an alternativebut equivalentapproachto the one taken in this paper would be to ask what, if anything,the Hamiltonianalong an optimalgrowthpath measureswhen the productionpossibilitiesset exhibitstime dependence. In the optimalcontrol context,r standsfor the rate of pure time preference or, alternatively,a fixed probabilitythat the world ends in any given period. By either interpretation,the discount factor {e-r'} serves as a 4 See Weitzman (1976, footnote 5). 5See Weitzman (1976, footnote 6). ? The editors of the ScandinavianJournalof Economics1997.

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naturalsystemof weightsfor aggregatingor averagingvariablesover time. The next section makes extensive use of this philosophy and this mechanism.

III. Sustainabilityand Green NNP Let X(t) represent any time-dependentvariable.The annuity-equivalent value of X, which henceforthwill be denoted by [X], is the hypothetical constant level of X that would yield the same present discountedvalue at time zero as the time series {X(t)}. The discount rate to be used here is "naturally"the same value of r that is inherent in the entire analytical framework. By this definition,[X] satisfiesthe equation [X]

e-r

0

dt=

X(t)

e-r'

dt

(11)

0

which can be rewrittenas [X] =r

X(t) e-rdt.

(12)

0

It will be seen at once that [X] is interpretableas a weighted timeaverageof {X(t)}, with weights {e-rt} applyingat time t. To be able to make any formal statements about "sustainability",the concept must first be defined formally.The basic motivatingidea behind the concept of "sustainability"underlyingmost of the literaturewould appear to be some "good" aggregatemeasure of the economy'sgeneralized power to sustainfuture consumption. In this paper, sustainabilityis defined formallyto be the hypothetical constant "annuity-equivalent" level of consumptionthat would yield the same present discounted value as the optimal consumption trajectory {C*(t)} that the economy is able to deliver. To me, there is something intuitivelyappealingabout defining"sustainability"to be the time-weighted averageof futureconsumptionpossibilities,where the "weight"is the equilibriumdiscountrate on consumptionthat the actors in the economy themselvesare displaying.6Note that the annualizedequivalentconsumption flow used here to define sustainabilityis a hypotheticallevel that may not actuallybe attainable. 61 realize that there is an extensiveliteratureon the appropriatesocial discountrate,which representsa set of issues I am sidesteppinghere. If the appropriatediscountrate is different than the competitiveown rate of returnon consumption,then in principlethe formulasof this paper could all be redone usingcorrespondingshadowor efficiencyprices- although I would hate to be the one who has to make such recalculationsin practice. ? The editors of the Scandinavian Journal of Economics 1997.

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Fromthe above definition,sustainabilityat time t, denoted T(t), is given by the formula: T(t)-=r

C*(s) e-r(-') ds.

(13)

In this context, the phrase "sustainabledevelopment"might be interpreted to mean a trajectoryalong which P(t) > C*(t) (14) for all t. In conformitywith the symbolicnotation previouslyintroduced,[C*], defined by (12) above for X(t) = C*(t), denotes the economy's sustainabilityat the currenttime zero, or: T(0) = [C*]. Let Y*(t) denote inclusiveor Green NNP at time t. Then: Y*(t) = C*(t) + P(t)K*(t) = G(K*(t), P(t), t).

(15)

(16) The principaltask of this paper is to elucidate the relationshipbetween T(t) and Y*(t). Strictlyfor notationalease and without loss of generality we choose to performthe evaluationat the present time t = 0, so that we will be comparingT(0) with Y*(O). To this end, we must define the following two summarystatistics of relevantgrowthrates: [?']

=[y*

(17)

and

-= [8Y*/8t] [Y*I

(18)

where aY*/at denotes aG/Ot. The summarygrowthstatisticg is interpretableas an expressionof the averagefuturegrowthrate of Green NNP. The summarygrowthstatisticA is an expressionof the averagefuturegrowthrate of the "residual",which captures the pure effect of time alone on enhancement of productive capacitynot otherwiseattributableto capital accumulation.7 7 It is theoretically possible that A could be negative for some situations. For example, an exporting country facing declining terms of trade over time might have a negative value of A if the time-deteriorating terms-of-trade effect has a strong enough economic impact. ? The editors of the ScandinavianJournalof Economics 1997.

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To understandbetter the meaning of definitions(17) and (18), rewrite them as: (y*(t)-gY*(t))

e-r dt = 0

(19)

, 0

and CC' iay*

J

t (t)-Y*(t)

e-rdt = 0.

(20)

Conditions(19) and (20) make clear the sense in whichg and Acan be interpretedas weighted average growth rates of, respectively,aggregate output and the "residual".Consider, for example, (19). The weighted amountbywhich {Y*(t)} exceeds {gY*(t)} over time is exactlyequal to the weighted amountby which {gY*(t)} exceeds {Y*(t)} over time, where the weighting factor at time t is, naturally,chosen to be {ert}. Likewisefor aY*/at and 2Y* in (17).

The main use of (17) and (18) will ultimatelybe to treat g and 2 as parametricallygiven projectionsof averagefuturegrowthrates in orderto analyze the relationshipbetween sustainabilityand currentGreen NNP, and to understand better what critical features of the growth path it depends upon. For this purpose of performingsensitivity analysis, the proposed definitions (17) and (18) are more than adequate representations of the underlyingconcepts behind projectedvalues of g and A.Note that in the special case of steady exponentialgrowth,definitions(17) and (18) reduce to the underlyingconstantrates of growthexactly. The primaryaim of the present paper is to determine the degree to whichcurrentinclusiveor Green NNP, Y*(0), reflectsfuturesustainability, T(0), in the presence of technological progress. Let ? stand for the appropriate"technologicalprogress premium"needed to convert Y*(0) accuratelyinto 'P(0).By definition, the correction factor 0 satisfies the condition T(0) = Y*(0) [1 + 0].

(21)

The main result of the paper is the followingformula(22). Theorem.Underthe assumptionsof the model,the appropriate technological progresspremiumis 0 =

r-g ? The editors of the Scandinavian Journal of Economics 1997.

(22)

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IV. Proof of the Main Proposition In the proof, algebraicmanipulationsare compressed to save space. To keep the proof simple, I assume that any relevant variables are timedifferentiablealmost everywhereon te [0, oo). Taking the total time derivativeof Y*(t) = G(K*(t), P(t), t) along an optimal trajectory,

aG '0t

n aG . n aG. Y*(t)Z = KE K -PKj

j=l aK

iI

(23)

From (3), (6) and the theory of cost functions8it follows that along an optimal trajectory the following duality conditions must be satisfied for allj: aG

.

-=K*.

(24)

aPi

Substitutingfrom (24) and (7) into (23), and cancelingout terms of the form EPjKj*,yields along an optimal trajectorythe equation: n

y*(t) = r

.

+ PjK*+ j=!

aG

.

(25)

at

By substitutingfrom (16), expression(25) becomes equivalentto

y*(t) = r(Y*(t)-C*(t))+aG at

(26)

Applyingthe stationary-equivalenceoperator(12) to (26) transformsit into the equation [y*] = r([Y*]- [C*]) +

-ay*-

.

(27)

Substitutingfrom (17), (18) into (27) turns it into the condition

g[Y*]= r([Y*]- [C*)])+ [Y*],

(28)

which can be rewrittenas [C*] = [Y*])

.

(29)

See footnote 9 in Weitzman (1976). ? The editors of the ScandinavianJournalof Economics 1997.

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Equation(29) relates [C*] to [Y*].Now, essentially,what remainsto be done is to determine the relationbetween [Y*] and Y*(O). Towardsthat end, a mechanicalintegrationby partsof the integralform Jre-r'Y*dt (= Ju dv, where u =_re-r, dv=Y* dt) yields the expression (30) [y*] = -rY*(O)+r[Y*]. Substitutingfrom (17) into (30) to eliminate [Y*] gives the desired relation:

[Y*]= Y*(0)(, ).

(31)

Now, finally,using (31), substitutefor [Y*] in (29) and rearrangeterms to get [C*] = Y*(0)

1+ . \ r-g

(32)

The above expression (32) is equivalent to (22), thus concludingthe D proof. Note that the basic result of Weitzman (1976) - when technology is independentof time, T(0) = Y*(0) - correspondshere to the specialcase of (32) where i = 0.

V. Conclusions If we think generally of 0 as a parameter quantifyingthe appropriate "technologicalprogresspremium",then how mightthis parameterbest be estimated?Howeverimperfectit mightbe, as a practicalmatterwe have a better intuitivefeeling for projectingfuture rates of the "Solowresidual" than for forecasting the relevant future parametervalues or functional forms of any existing model of endogenous growth theory. If we go the route of this paper, then we have a methodologyfor estimating0 and can at least hope that it could be a decent approximationfor what might also be derived from the "right"form of a more-fully-specifiedmodel where innovationand externalitiesare endogenouslydetermined.9 Suppose that defensive environmentalspending in an advancedindustrialeconomysuch as the United States can serve as a very roughmeasure of the welfare loss of the negative environmental externalities it is 9 For

an example of a model in the spirit of endogenous growth theory being used to address issues of social accounting and welfare measurement, see Aronsson and Lofgren (forthcoming). ? The editors of the ScandinavianJournalof Economics1997.

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intended, in part, to offset.10The total "cost of a clean environment"is currentlyestimated to be about 2% of GNP."lAs for the depletion of naturalcapital like subsoil minerals,forests, or topsoil, this is currentlya negligible fraction of national product in the United States.'2To a tolerable degree of approximation,then, it becomes difficult to argue that making all the proper adjustmentsfor depleted natural resources and deteriorated environmentalassets might bring conventionallymeasured NNP down by more than about a couple of percentage points when it is convertedover to Green NNP.'3 What about the upwardadjustmentof NNP indicatedby the "technological change premium"?In this case, formulas (21), (22) give some handle on the theoretically appropriate relationship between (future) sustainability and (present) inclusive NNP. Here, it seems that the correction factor is considerablylarger, perhaps an order of magnitude greater. As very rough estimates, suppose the following numbers are chosen'4:

r = 5% = annual after-taxreal returnon capital g = 2.5% = annual real growthrate of NNP i = 1% = annualgrowthrate of total factor productivity With these numbers,'5the technologicalchange premiumis 0 = 40%. A warning is in order about this kind of exercise. First of all, it is undertakenin what might be called a "commandoptimum",presuming that negative and positive externalities are internalized in an optimal manner. Second, some very rough ballparknumbersthat, at best, apply only to the present situation are being extrapolated far forward as presumptiveforecasts. No one should feel fully at ease projectingsuch crude estimatesas have been made above onto an indefinitefuture.And, of course, 0 will change with differentassumedvalues of the underlyingparameters.Yet, when all is said and done, I believe it is fair to say that a reasonable parametric "'I realize that several importantissues are being glossed over here, but it is only the approximatemagnitudeof this numberthat mattersin the presentcontext. " See EPA (1991). 12See U.S. CommerceDept (1994, p. 15). t3See the appropriatesection of Weitzmanand Lofgren (forthcoming)for a more exact descriptionof the relevantcalculation.For the purposesof this paper,it does not matterif the correctionin the text above is off by a factor of two. 14 Such numberscould be justifiedby reference to e.g. Jorgenson(1994), BLS (1994) and Nordhaus(1995). '5Note that it makes no differencewhetherthe calculationis done on a per capitabasisor, as here, on a total basis,becausethe numerator/ and the denominator(r-g) of the formula (22) for 0 are invariantto such an alteration. ? The editors of the ScandinavianJournalof Economics1997.

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analysis done on (21) seems to make the following conclusion hard to resist: wouldappearto dependmore criticallyon futureprojec"Sustainability" tions of the residualthan on the typicalcorrectionsnow beingundertakenin the name of green accounting.Because it omits the role of technological progress,NNP, whetherconventionallymeasuredor green-inclusive,likely understatesan economy'ssustainability. The ultimate origins of the "residual"are still poorly understood by economists.But if future growthrates of technologicalprogressresemble those of the past, we are probablyunderestimatingsignificantlyan economy's future power to consume when we identify it with current NNP, howeverinclusivethat measure is made.

References Aronsson, T. and L6fgren, K.-G.: Welfare measurement of technological and environmental externalities in the Ramsey growth model. Natural Resources Modeling 7(1), pp. 1-14, 1993. Aronsson, T. and Lofgren, K.-G.: Social accounting and welfare measurement in a growth model with human capital. Scandinavian Journal of Economics 98(2), 185-201, 1996. Asheim, G. B.: Capital gains and net national product in open economies. Journal of Public Economics 59, 419-34, 1996. Bureau of Labor Statistics: Estimates of multifactor productivity for private business sector, 1948-92. Washington, DC, July 1994. Hartwick, J. M.: Constant consumption paths in open economies with exhaustible resources, Review of International Economics 3(3), 275-83, 1995. Jorgenson, D. W.: Investment and economic growth. The Simon Kuznets Lectures, Yale University, Nov. 11-13, 1994. Kemp, M. C. and Long, N. V.: On the evaluation of social income in a dynamic economy. Ch. 12. In G. R. Feiwel (ed.), Samuelson and Neoclassical Economics, Kluwer-Nijhoff, Dordrecht, 1982. Lofgren, K.-G.: Comment on C. R. Hulten, Accounting for the wealth of nations: The net versus gross output controversy and its ramifications. Scandinavian Journal of Economics 94 Supplement, S25-S28, 1992. Maler, K.-G.: National accounts and environmental resources. Environmental and Resource Economics 1, 1-15, 1991. Nordhaus, W. D.: How should we measure sustainable income? Mimeo, Yale University, Oct. 1995. Pearce, D. and Atkinson, G.: Capital theory and the measurement of sustainable development. Ecological Economics 8, 103-8, 1993. Solow, R. M.: On the intergenerational allocation of resources. Scandinavian Journal of Economis 88(1), 141-9, 1986. U.S. Commerce Department: Integrated economic and environmental satellite accounts. Survey of CurrentBusiness, 33-49, Apr. 1994. U.S. EPA: Environmental Investments: The Cost of a Clean Environment. Island Press, Washington, DC, 1991. ? The editors of the ScandinavianJournalof Economics 1997.

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Weitzman, M. L.: On the welfare significance of national product in a dynamic economy. QuarterlyJournal of Economics 90, 156-62, 1976. Weitzman, M. L. and Lofgren, K.-G.: On the welfare significance of green accounting as taught by parable, forthcoming in Journal of Environmental Economics and Management. World Commission on Environment and Development: Our Common Future. Oxford University Press, London, 1987. First version submitted January 1996; final version received April 1996.

? The editors of the ScandinavianJournalof Economics 1997.