The Board of Regents of the University of Wisconsin System

What Is Sustainable Development?

Author(s): Graciela Chichilnisky
Source: Land Economics, Vol. 73, No. 4, Defining Sustainability (Nov., 1997), pp. 467-491
Published by: University of Wisconsin Press
Stable URL: http://www.jstor.org/stable/3147240
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 What Is Sustainable Development?

GracielaChichilnisky

ABSTRACT. This paper introducestwo axioms present without compromisingthe needs of

that capturethe idea of sustainabledevelopment, the future."2But what is sustainabledevel-
and characterizesthe welfare criterionthat they
opment?
imply.Theaxiomsrequirethatneitherthepresent Solow (1992) pointed out recently that
nor the futureshouldplay a dictatorial role in discussion of sustainabilityhas been mainly
society'schoicesovertime. TheoremsI and 2 show an occasion for the expressionof emotions
thereexist sustainablepreferenceswhichsatisfy
these axioms and provide a full characterization. and attitudes, with very little formalanalysis
Theorems3-5 studya standarddynamicalsystem of sustainabilityor of sustainablepaths for a
representing the growthof a renewable-resourcemodernindustrialeconomy.Formalanalysis
economy,give a "turnpike"theorem,and exhibit requiresvaluation, and the economic value
thedifferences between optimaandthe of a resource is usually derived from its
sustainable
ones accordingto discountedutilitarianism.(JEL contributionto utility.This suggeststhat the
013) crux of the matter is how to describevalue
so that it does not underestimatethe future's
I. A VORACIOUSUSE OF RESOURCES interests and utilities, so that the future is
given an equal treatment. This could be
For the first time in history,humanactiv- achieved in several ways. The challenge,
ity has reachedlevels at which it could alter however, is to develop economic theory
the planet's climate and its biological mix. which formalizesthis aim with the level of
Economicsis the drivingforce. Energyused clarityand substanceachievedby neoclassi-
for productionis obtained by burningfossil cal growth theory, and with the practical
fuels, and leads to carbon emissions. The scope of the currentapproachto cost bene-
emissions generated since the second world fit analysis that is based on neoclassical
war could alter the earth'sclimate,although growththeory.
there is as yet no scientific agreement on
the precise magnitudeof the effects. Biolo-
gists see the loss of biodiversityduring the
last fifty years as one of the four or five andThe author is UNESCO Professorof Mathematics
Economicsand Director,Programon Information
largest incidents of destructionof life on the and Resources,ColumbiaUniversity.Researchsupport
planet. Originatinglargely in the industrial from NSF grant No. 92-16028,the StanfordInstitute
countries, this voracious use of resources for TheoreticalEconomics(SITE),and from the Insti-
has been accompanied by increasing dis- tute for InternationalStudiesat StanfordUniversityis
gratefullyacknowledged,as are the comments of Y.
crepancies in resource consumption and Baryshnikov,P. Ehrlich,S. C. Kolm, D. Kennedy,D.
welfare between industrial and developing Kreps,L. Lauwers,D. Starrett,L. van Liedekierke,P.
countries.At the 1992United NationsEarth Milgrom,J. Roberts,R. Wilson,and H. M. Wu. Special
Summitin Rio de Janeiro,sustainabledevel- thanks are due to Kenneth Arrow, Peter Hammond,
Heal, MarkMachina,Robert Solow,Richard
opmentemerged as one of the most urgent Geoffrey Howarth,and an anonymousreferee.
subjects for international policy. One hun- This article was preparedfor an invited presenta-
dred and fiftyparticipatingnationsendorsed tion at a seminaron Reconsiderationof Values at the
UN Agenda 21, proposing aS part of its Stanford Institute for Theoretical Economics, orga-
nized by K. J. Arrowin July 1993.It was also an invited
policy agenda sustainable development presentation at the IntergovernmentalPanel on Cli-
based on the satisfactionof basic needs in mate Change (IPCC) Seminarin Montreux,Switzer-
developingcountries.This developmentcri- land, March1994,at a Seminaron Inconmensurability
terion was introducedby us in 1976 in the of Values at Chateauxdu Baffy,Normandy,April 1994,
BarilocheModel' and given further impetus and at the GraduateSchool of Business of Stanford
in 1987 when the BrundtlandCommission University in May 1994.
1 Chichilnisky(1977a, 1977b)and Herrera,Scolnik,
proposed that "sustainabledevelopment is and Chichilnisky(1976).
2 Brundtland
developmentthat satisfies the needs of the (1987,chap. 2, para. 1).

Land Economics * November 1997 * 73 (4): 467-91

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468 LandEconomics November1997

I.A. CostsToday,BenefitsTomorrow How to explain this experimental evi-

dence? How to make sense of our sensitivity
A well-knownproblem is that standard to time and integrate it into a criterion of
cost-benefitanalysisdiscountsthe future. It optimality?Several interesting alternatives
is thereforebiased againstpolicies designed to the discountedutility analysishave been
to providebenefits in the very long run. A proposed.So far none had reachedthe clar-
sharp example is the evaluationof projects ity and consistencyof the discountedutili-
for the safe disposalof waste from a nuclear tariancriterionused in cost-benefitanalysis,
power plant. Another is policies designed nor its analyticaltractability.Prominentex-
for the preventionof global warming.The amples are the "overtakingcriterion"and
benefits of both may be at least fifty to a Ramsey'scriterion.Both, however,are seri-
hundred years into the future. The costs, ously incomplete,failing to rank many rea-
however,are here today. In these cases, the sonable paths. The orderinginducedby the
inherent asymmetrybetween the treatment overtakingcriterion cannot be represented
of present and future makes it hard to jus- by a real valued function,makingit imprac-
tify investmentdecisionsthat large numbers tical to use. As a result, they lack the corre-
of individualsand organizationsclearly feel sponding"shadow"prices to evaluate costs
are well merited. Recent experimentalevi- and benefits in an impartialfashion. These
dence sheds new light on the matter. criteriathereforefail on practicalgrounds.

Evidence
I.B. Experimental I.C. A Criterion
for Sustainability

Several experimentshave measuredhow This paperproposessimple axiomswhich

people value the long run (see, e.g., Lowen- capture the concept of sustainability,and
stein and Thaler 1989, Cropper, Aydede, derivesthe welfare criterionwhich they im-
and Portney 1994, and the references in ply, see also Chichilnisky(1996). The crite-
Lowensteinand Elster 1992).Their findings rion that emerges is complete, analytically
clash with the traditional discounted ap- tractable,and representedby a real valued
proach.People are shown to value the pre- function.In optimizationit leads to well-de-
sent and the future differently,but not as fined shadowpriceswhich can be used for a
the standard analysis would predict. The "sustainablecost-benefit analysis."The ax-
experimental evidence indicates that the ioms provide internal consistencyand ethi-
present and the future are treated more cal clarity. They imply a more symmetric
even-handedly.Typicallywe do discountthe treatment of generations in the sense that
future,but the trade-offbetween today and neither the "present" nor the "future"
tomorrowblurs as we move into the future. should be favoredover the other. They nei-
Tomorrowacquires increasing relative im- ther acceptthe romanticviewwhichrelishes
portance as time progresses.It is as if we the future without regardsfor the present,
viewed the future through a curved lens. nor the consumeristview which ranks the
The relativeweight given to two subsequent present above all. The axioms lead to a
periods in the future is inverselyrelated to complete characterization of sustainable
their distance from today. The period-to- preferences,3 which are sensitive to the wel-
period rate of discount is inverselyrelated fare of all generations,and offer an equal
to the distance into the future. The experi- opportunityto the present and to the fu-
mental evidence shows that rate of discount ture. Trade-offsbetween present and future
between period t and period t + 1 de- consumptionare allowed.The existenceand
creases with t. Interestingly,studies of hu- characterizationof sustainable preferences
man responsesto sound summarizedin the appears in Theorems 1 and 2. Theorem 1
Weber-Fechner law, indicate similar re-
sponses to changes in sound intensity. The
human ear responds to sound stimuli in an 3 An alternativename suggestedby RobertSolowis
inverse relation to the initial stimulus. "intertemporallyequitablepreferences."

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 469

shows that sustainablepreferences are dif- the neoclassical model. Each generation g
ferent from all other criteria used so far in has a preferencethat can be representedby
the analysisof optimal growth and of mar- a utility function ug for consumptionof n
kets. Theorem 3 studies a standarddynami- goods, some of which could be environmen-
cal system representingthe growthof a re- tal goods such as water, or soil, so that
newable resource. Sustainable preferences consumption vectors are in R", and ug:
are shown to be a natural extension of the Rn -- R. The availabilityof goods in the
"equaltreatmentcriterion"for finitelymany economyis constrainedin a numberof ways,
generations, in the sense that the optimal for exampleby a differentialequationwhich
solutions for such preferencesapproachthe represents the growth of the stock of a
"turnpike"of an equal-weightfinite horizon renewableresource,6and/or the accumula-
optimization problem as the horizon in- tion and depreciationof capital. Ignore for
creases. Theorem3 also shows that sustain- the moment population growth, although
able preferencesmatchthe experimentalev- this issue can be incorporatedat the cost of
idence in these cases, in the sense that they simplicity,but with little change in the re-
imply a rate of discount that decreases and sults.7The space of all feasible consumption
approaches zero as time goes to infinity. paths is indicatedby F.
Theorem 4 investigatesthe relationshipbe-
tween the optimal paths accordingto sus- F c ({x:x = E Rn). [1]
{Xg}g=x,2,...,
tainablepreferencesand discountedutilitar-
ianism in an extension of the classical
We chose a utility representationso that
Hotelling problem of the optimal depletion each generation'sutilityfunctionis bounded
of an exhaustibleresource.Theorem5 shows
that sustainableoptima can be quite differ- below and above: u: Rn-- R+, and
ent from discountedoptima,no matter how supx e R(U g(X)) < 00. Tfis choiceis not re-
small is the discountfactor. Subsequentex- strictive:it was shown by Arrow(1964) that
when rankinginfinite streamsof utilities as
amples show the implications for shadow done here one should work with bounded
prices.
utilityrepresentationssince doing otherwise
II. TWO AXIOMS FOR could lead to paradoxes.8Utility acrossgen-
SUSTAINABLEDEVELOPMENT erations is assumed to be comparable.In
orderto eliminatesome of the most obvious
The two following axioms are non-dic- problems of comparabilityI normalize the
tatorshipproperties.4Axiom 1 requiresthat
the present should not dictate the outcome
in disregardfor the future:it requiressensi- 4 See Arrow(1953),Chichilnisky (1994).In this case
we are concernedwith fairnessacrossgenerations,see
tivity to the welfare of generations in the also Solow(1992),Lauwers(1993, 1997).
distant future. Axiom 2 requires that the 5 Chichilnisky(1982, 1994, 1996).
welfare criterion should not be dictated by 6 See Chichilnisky(1993c) and Beltratti,Chichilni-
the long-runfuture,and thus requiressensi- sky,7andHeal (1995).
Populationgrowthand utilitariananalysisare well
tivity to the present.5 To offer a formal known to make an explosive mix, which is however
perspectivea few definitionsare required. outside the scope of this paper.
Each generationis representedby an in- 8A preferenceadmitsmore than one utilityrepre-
teger g, g = 1,2,...,oo. An infinitely lived sentation;amongthese one chooses a boundedrepre-
sentation.The need to workwithboundedutilityrepre-
world obviates the need to make decisions sentation in models with infinitelymany parameters
contingent on an unknown terminal date. was pointedout by Arrow(1964),who requiredbound-
Generations could overlap or not. Indeed edness to solve the problemthat originallygave rise to
agents could be infinitely long-lived and Daniel Bernouilli'sfamous paper on the "St. Peters-
evaluate development paths for their own burg paradox,"UtilityBoundednessTheorem(Arrow
futures. 1964, 27). If utilities are not bounded,one can find a
utilitystreamfor all generationswith as largea welfare
For ease of comparison,I adopt a formu- value as we wish, and this violates standardcontinuity
lation which is as close as possibleto that of axioms.

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470 LandEconomics November
1997

utility functions u so that they are non- DEFINITION2. The present consists of all
negative and all s'tare a common bound, feasible utilitystreamswhich have no future,
which I assumewithoutloss of generalityto i.e., it consists of the cutoffs of all utility
be 1: streams.

II.C. TheFuture
sup(Ug(Xg))xeRn ?R 1. [2]
g
By analogy, for any given utility stream
a, its "future"is representedby all infinite
The space of feasibleutilitystreamsfl is utility streams which are obtained as the
therefore "tail" resulting from cutting a off for any
finite numberof generations.
a: a =
0t =
=
{a}g=)l,2...,
tg
Ug(Xg)}g=
DEFINITION3. The K-th tail of y is the
1,2....
sequencewhosecoordinatesup to and includ-
and ing the K-thare zero and equalto those of y
x={Xg)}g=,2,.... F}. [3]
afterthe K-thgeneration.13
Each utilitystreamis a sequence of posi- For any two a, y E/4, let (aK,y ) be the
tive real numbersboundedby the number1. sequence defined by summingup or "past-
The space of all utility streams is therefore ing together"the K-th cutoff of a with the
containedin the spaceof all infinitebounded K-th tail of y.
sequences of real numbers, denoted /4.9
Our welfare criterion W should rank ele- II.D.No Dictatorial
Rolefor thePresent
ments of f, for all possible f c4.
DEFINITION 4. A welfarefunction W: 4-'
andCompleteness
II.A.Sensitivity R givesa dictatorialrole to thepresent,and is
called a dictatorshipof the present, if W is
The welfare criterionW is completeif it insensitiveto the utilitylevelsof all buta finite
is representedby an increasingreal valued numberof generations,i.e., if W is onlysensi-
function defined on all bounded utility tive to the "cutoffs"of utilitystreams,and it
streams1' W: / -- R'. It is sensitive, or disregardsthe utilitylevels of all generations
increasing,if whenevera utility stream a is from some generationon.
obtained from another P by increasingthe
welfareof some generation,then W ranksa
strictlyhigher than 1.11 9 Formally:f c4/ where 4 = {y: y = {y,g}= :
....
II.B. ThePresent yg
E R+, supglygl<
o}).Here I I denotes the absolute
value of yE R, which is used to endow 4. with a
standardBanachspace structure,definedby the norm
How to represent the present? Intu- -11in 4
11
itively,the present is representedby all the
utilitystreamswhichhave no future:for any Ilyll= supg=1,2,...lY. [4]
given utility stream a, its "present"is rep- The spaceof sequences/4 was firstused in economics
resented by all finite utility steams which
are obtained by cutting a off after any by Debreu
10The (1954).
representabilityof the order W by a real
numberof generations.Formally: valued function can be obtained from more primitive
assumptions,such as, for example, transitivity,com-
1. For any utilitystreama
DEFINITION pleteness,and continuityconditionson W.
and any integerKI let
aK
•4,
be the "K-cutoff"of
Formally: if a > P when W(ot) > W(3).
12In symbols: otK= ({g}g=1,2,... such that ag = ag
the sequencea, the sequencewhose coordi- if g K, and ag =0 if g >K.
nates up to and including the K-th are equal to < = 0 if
13In symbols: otr = {gg= such that
thoseof a, and zero afterthe K-th.12 K, and ag = ag if g > K. 1,2,... rg
g<

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 471

Formally,W is a dictatorshipof the present III. EXISTENCEAND

if for any two utility streams14a, CHARACTERIZATION OF
13 SUSTAINABLEPREFERENCES
W(a) > W(p)* Why is it difficult to rank infinite utility
streams?Ideallyone wouldgive equalweight
3N = N(a, p) to every generation. For example, with a
finite numberN of generations,each gener-
s.t. if K > N, W(aK, K)> W(K, K) ation can be assignedweight 1/N. But when
trying to extend this criterion to infinitely
foranyutilitystreamsy, ae . many generationsone encountersthe prob-
lem that, in the limit, every generation is
given zero weight.
The following axiom eliminates dictator- What is done usually to solve this prob-
ships of the present: lem is to attach more weight to the utilityof
near generations,and less weight to future
e Axiom 1: No dictatorshipof the ones. An example is of course the sum of
present. discountedutilities.Discountedutilities give
a bounded welfare level to every utility
II.E. No DictatorialRolefor the Future stream which assigns each generation the
same utility. Two numbers can always be
compared,so that the criterionso definedis
DEFINITION5. A welfarefunction W: /o -
R gives a dictatorial role to the future, and is clearly complete. However,the sum of dis-
called a dictatorship of the future, if W is
counted utilities is not even-handed:it dis-
insensitive to the utility levels of any finite regards the long-run future. I show below
that it is a dictatorshipof the present.
number of generations, or equivalently it is
Another solution is the criteriondefined
only sensitive to the utility levels of the "tails"
by the long-runaverageof a utilitystream,a
of utility streams. criterionused frequentlyin repeatedgames.
However, this criterion is not even-handed
Formally: for every two utility streams a, 3 either: it is biased in favorof the future and
against the present. It is insensitive to the
welfare of any finite number of gener-
W(0a)> W(P)
,** ations.'6
Here matters stood for some time. Ask-
3N = N(a, P.): if K > N,15
ing for the two axioms together, the non-
dictatorship of the present and the non-
W(yK,a K) > W((UK,PK), Vy, Eoo dictatorshipof the future, as I do there,
appears almost as if it would lead to an
The welfare criterion W is therefore only impossibilitytheorem. Not quite.
sensitiveto the utilities of "tails"of streams, Let us reason again by analogywith the
and in this sense the future alwaysdictates case of finite generations. To any finite
the outcome independentlyof the present.
The following axiom eliminates dictator-
14
ships of the future: Recall that all utility streams are in /. and they
are normalized so that sup, 1
and supg= 1,2,... ((g)) = IIPI 1.
1,2,..(a(g)) =- llo
* Axiom2: No dictatorshipof the future. 15An equivalentdefinitionfor our purposeswould
be obtainedby replacing"3N" by "for any K."
DEFINITION 6. A sustainable preference is a 16Other interesting incomplete intergenerational
criteriawhich have otherwisepoints in commonwith
complete sensitivepreferencesatisfyingAxioms sustainablepreferencesare foundin Asheim(1988,
1 and 2. 1991)

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472 LandEconomics November1997

U(2)

/
ir
L1
/ / i/V

ii

III

Lir in

FIGURE 1
The ThreeAxes Representthe Utilities of Finite Generations(U(1) and U(2)) and the
LimitingUtilityValue liminf U. Two Level Sets of the Rankingare Shown.One Dominates
OverFinite Generationsbut has a Lowerliminf. As the Weightson Finite Generationsare
Fixed,the Rankingin These Dimensionscan be Representedby the Intersectionof the
Level Set Restrictedto the U(1)-U(2)Planewith the VerticalAxis.The OverallRankingis
then Shownas the Sumof this Ranking(the CountablyAdditiveMeasure)with the Ranking
in the liminf Dimension(the PurelyFinitelyAdditiveMeasure).

number of generations one can assign The first part of Theorem 1 establishes
weights which decline into the future, and that the sum of a dictatorship of the present
then assign some extra weight to the last plus a dictatorship of the future is in fact
generation. This procedure, when extended neither. The first part is sensitive to the
naturally to infinitely many generations, is present, and the second is sensitive to the
neither dictatorial for the present nor for future. Furthermore such a sum admits
the future. It is similar to adding to a sum of trade-offs between the welfare of the pres-
discounted utilities, the long-run average of ent and of the future. It is represented
the whole utility stream. Neither part of the diagrammatically in Figure 1, which shows
sum is acceptable on its own, but together the trade-offs between the present's and the
they are. This is Theorem 1 below. Theorem future's utilities. The three axes represent
2 proves that under regularity conditions the utility levels of generations 1, 2, and,
this procedure gives a complete characteri- figuratively, co. The two triangular planes
zation of all continuous sustainable prefer- represent two indifference surfaces. One
ences. gives more utility to generations 1 and 2,

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73(4) WhatIs Sustainable
Chichilnisky: Development? 473

and undera dictatorshipof the presentthese (a) The sum of discounted utilities is a
choices would prevail;however the second dictatorshipof the present because for ev-
surfacegives more utilityto the long run, so ery E > 0, there exists a generation N so
that under certain conditions the second that the sum of discounted utilities of all
surface is chosen over the first. Theorem 1 other generationsbeyond N is lower than e
makes this reasoningrigorous. for all utility streams since all utilities are
The second part of Theorem 1 showsthat bounded by the number 1, [2]. Now, given
all known criteria of optimalityused until anytwoutilitystreamsa, 1, if W(a) > W(P)
now fail to satisfy the axioms postulated then W(a)> W(P) + E for some s > 0;
here. Therefore the sustainablepreferences therefore there exists a generation N be-
defined here performa role satisfiedby no yond which the utilities achieved by any
previouslyused criterion. generation beyond N do not count in the
What is perhaps more surprisingis that criterion W. This is true for any given dis-
the sustainablewelfare criteria constructed count factor.18
here, namely the sum of a dictatorshipof (b) The Appendix establishes that the
the present and one of the future, exhaust Ramsey's criterion is incomplete; this de-
all the continuousutilities which satisfy my rives from the fact that the distance to
two axioms.This means that any continuous Ramsey'sbliss path is ill-definedfor many
sustainablepreference must be of the form paths.
just indicated.This is Theorem 2, provedin (c) The Appendix establishes that the
the Appendix. overtakingcriterion is incomplete:see also
III.A.TheExistence Figure 2.
of Sustainable
Preferences (d) and (e) Lim inf and long-runaverages
1. Thereexistsa sustainablepref- are dictatorshipsof the future;furthermore
THEOREM the long-runaverageis also incomplete.19
erenceW: /. -- R, i.e., a preferencewhichis
sensitiveand does not assigna dictatorialrole Both (f) and (g), Rawlsianand basicneeds
to eitherthepresentor thefuture: criteria, are insensitive because they rank
equally any two paths which have the same
infimum even if one assigns much higher
W(a) = EXga, +4(a), [5]
g=1
utility to manyother generations.
III.B.A Complete Classification
of
where Vg, Xg> 0, ,hg < oo, and where SustainablePreferences
4(a) is thefunctionE•_
limg(, (sa g) extendedto
all of /, via Hahn-Banachtheorem."17 The following result characterizes sus-
Thefollowingwelfarecriteriaare not sus- tainable preferences.Additional conditions
tainable preferences:(a) the sum of dis-
countedutilities,for any discountfactor, (b) 17
Ramsey's criterion,(c) the overtakingcrite- The linear map "lim ,(a g)" is defined by using
the Hahn-Banachtheorem,as follows:define first the
rion, (d) liminf, (e) long-runaverages,(f) functionon the closed subset of 4. consistingof those
Rawlsianrules,and (g) basicneeds. sequences ag which have a limit, as that limit; the
functionis then extendedcontinuouslyto all sequences
PROOF.In the Appendix. in the space 4. by using the Hahn-Banachtheorem,
whichensuresthat such an extensionexists and can be
An intuitive explanation of this result constructedwhile preservingthe normof the function
follows. The preference defined in [5] is on the closed subspaceof convergentsubsequences.
18 InterestinglySolowhas suggestedmakingthe dis-
sustainablebecause it is complete, its first
term is sensitive to the present, in fact it countingfactor smallerthan one and decreasing:it is
possible to show that under certain conditions this is
increases with increases in the welfare of exactlywhat must be done to ensure the existenceof
every generation, and its second term is solutions.
sensitive to the long-run future. The next 19For examplethe two sequences (1,0, 0, 111,0, 0,
0,0,1,1,,1,1,0,10, 0, 0,0,0,0,...) and (0, 1,01,0, 0,
step is to show why all other criteriafail the 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1,1,1, 0...) are not compa-
sustainabilityaxiomsproposedhere. rable accordingto the long-runaveragescriterion.

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474 LandEconomics November
1997

Utility pendent and welfare criteria.The following

integral theorem decomposesa sustainablecriterion
from0 to T.
PathA i into the sum of two functions.The first is a
discounted utility with a variable discount
rate and the second is a generalizationof
/ long-runaveragesor a lim inf, and is called
a "purely finitely additive" measure. For
definitions and examples of purely finitely
additivemeasures the reader is referredto
the Appendix,Example 4: this latter mea-
Path B sure assigns all welfare weight to the very
long run. In particular,it assigns the value
zero to anysequencewhichhas finitelymany
non-zeroterms.22

THEOREM 2. Let W: 40 - R+ be a contin-

Time T
sustainablepreference.Then
uous independent
FIGURE2 Wis of theform Voaee:
NeitherPathOvertakes the Other,
Illustrating of the
the Incompleteness W(at)= E• gag+ 4()
Criterion. [6]
Overtaking g=l
whereVg X > 0, • lk, < o00,and # 0 is
a purelyfinitelyadditivemeasure.
on the welfare criterion W are now intro-
duced: W is continuouswhen it is defined PROOF.In the Appendix.
by a continuous function W: /4 -R.20
Continuityhas playeda useful role in social IV. ARE SUSTAINABLE
choice theory in the last ten years, in effect PREFERENCESREASONABLE?
replacingthe axiom of independenceof ir-
relevant alternatives and allowing a com- We saw that sustainable preferences
plete characterizationof domains in which emerge from well-definedand uncontrover-
social choice exists (Chichilnisky 1982,
1993a;Chichilniskyand Heal 1983).A simi- 20
lar role is found here for continuity:the A functionW whichis continuouswith respectto
the standardnorm of the space of sequences /. The
followingtheorem gives a full characteriza- norm is I11ll= supg= 1,2,... l1ag, and was defined above.
tion to all sustainable criteria which are Differentforms give rise to differentnotions of conti-
continuous. nuity but in the context of equitable treatment of
A standardpropertyof neoclassicalanal- generationsthe sup normis a naturalcandidate.
21See the Appendix.This simply means that the
ysis is that the rate of substitutionbetween indifferencesurfaces of the welfare criterion W are
two generations--whichis generallydepen-
dent on their level of consumption-is inde- hypersurfacesso that it is possibleto representit by a
linear function on utility streams:W(a + p) = W(a)
pendent of their levels of utility. This is a + W(p).Note that this does not restrictthe utilitiesof
widely used property: indeed the sum of the generation, ug,
in any way; in particular the u' s
discounted utilities, the most widely used need not be linearand so the marginalrate of substitu-
tion between consumptionat different dates can be
welfare criterion, certainly satisfies it. A non-constantand depend inter alia on consumption
welfare criterion W satisfyingthis property levels.
is called independent.21 The characterization 22A finitelyadditivemeasureon the integersZ is a
of a sustainablecriterionW in the following function?pdefinedon subsetsof the integers,satisfying
p(A)+ p(B) when AnB = ; if
theorem is simplest when W is continuous Ix(AUB)=
called purelyfinitely additivewhen it assignsmeasure
and independent.In consistencywith neo- zero to any finite subset of integers. See also the
classicalanalysis,we thereforeassumeinde- Appendixfor definitionsand examples.

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 475

sial axioms. But how do they fit economic (1985) or Heal (1995). We are assumingthe
intuition and empirical evidence? The fol- stock of the resource to be an argumentof
lowing subsectionwill show that sustainable the utilityfunction,so that the resourceis in
preferences fit well our economic intuition the categoryof environmentalassets such as
about finite horizon optimization. Subsec- forests, landscapes,biodiversity,etc., which
tion IV.B will show that they also fit the provideservicesand value to humansociety
empiricalevidence ratherwell. via their stocks as well as via a flow of
consumption.The problemis therefore:
IV.A.A Turnpike
Theorem
maxaf dt + (1 - a) limu(ct, st)
Our economic intuition is grounded on u(ct, s)e-8(')'
finite horizons. Life on earth will certainly
s.t. st = r(st) - ct, Sogiven.
be of finite duration,although it is difficult
to determine its final date. It is therefore
[71
importantto determinewhether sustainable
preferencesare merelyan artifactof infinite To studythe asymptoticpropertiesof a max-
horizons, or are reasonable within a finite imum for this problem it is useful to intro-
world.
This section will show that sustainable duce the following definition,see also Bel-
tratti, Chichilnisky,and Heal (1995):
preferencescan be seen as a suitablegener-
alizationto infinitehorizonsof an intuitively
DEFINITION 7. Thegreengoldenruleg* is a
appealing criterion for finite horizons, one
which values all generations equally, thus stationarypath g* = {c*, s*} which achieves
the maximumutilitylevel whichis sustainable
providing"equal treatment."Indeed, for a
general class of dynamicoptimizationprob- forever,thatis,
lems, we will see that the limit of the opti-
mal solution accordingto a sustainablepref- g* = max
C
u(c, s)
erence has two interestingproperties:(i) it
is the "green golden rule" (see Beltratti, to c < R(s).
subject
Chichilnisky,and Heal 1995), that is, the
configuration of the economy giving the Equivalently:
maximumsustainableutilitylevel and (ii) as
the finite horizon increases,the optimal so- g* = max u(R(s), s)
S
lutions of equal treatment finite horizon
problems spend an increasing amount of so that it satisfies:
time progressivelycloser to the limit of the
path which is optimal accordingto sustain- au
able preferences. In other words, the opti-
mum according to sustainable preferences u aR
au au aR = as
or au . [8]
determinesa direction in which finite hori- ac as =--- as ' as au
zon equal treatment optima increasingly ac
move as the horizon increases.We refer to
this propertyas a "turnpike"property. DEFINITION 8. The equaltreatmentproblem
To see this we formalize a typical prob- for horizonT is:
lem of optimizinga sustainablepreference
over a constraintimposed by the dynamics
of a renewablestock. A renewable stock u(c, dt
st
grows over time t accordingto its own bio- maxfo St)
s.t. = r(s,) - c, so given.
logical dynamics,with growth rate r(s,); c, s,
of it is extractedfor consumption.The util-
ity dependson consumptionand the level of Its solution is called the equal treatment
the stock, as in for example Krautkraemer optimumover T generations.

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476 LandEconomics November1997

Green rule,theturnpike
golden
• offinite zerodiscount
horizon optima.
? ??

- -Optimalpaths,

infinitehorizon,
zero
isdeclining discount
rate
c = r(s)and
ds/dt=0---
so Environmentalstock s
FIGURE 3
The UtilitarianSolutionwith a RenewableResource,Stockan Argumentof the Utility
Functionand DiscountRate Fallingto Zero, Asymptotesto the Green GoldenRule. This is
also the Turnpikeof Finite HorizonEqualTreatment(Zero Discount)Optima.

DEFINITION9. The discounted utilitarian increases, the equal treatment optima for T
problem has the same constraint set as prob- generations are increasinglycloser to the green
lem [7] but the function to be maximized is, golden rule, a plan which is asymptotically
instead, the integralof utilities discounted by a approached by the optima of [7] with sustain-
constant positive discount rate: able preferences. Formally:
(1) The green golden rule g* has a "turn-
forafixed8 > 0. [10] pike property" for equal treatment optima.
fu(c,s)e"ddt, That is, as T--, o the distance between the
equal treatment optimum for T and g* e(T)g
THEOREM 3. The optimal path for problem goes to zero.
[7] with a sustainable preference exists if and (2) The optimal solution to problem [7]
only if the discount rate 8(t) approaches zero exists if and only if 8(t) - 0 as t - oo, in
in the limit, in which case the optimum is the which case it is a path (c,, s) which converges
"turnpike" of finite horizon problem [9] in asymptoticallyto the greengolden rule g*, and
which each generation is treated equally. This so to the turnpike of the equal treatment op-
means that as the number of generations T tima.

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 477

(3) By contrast, the discounted utilitarian to standardrates-perhaps 10 percent. As

optimum does not have the turnpikeproperty the horizon extends the implied discount
for equal treatment optima, for any positive rates drop, to in the region of 5 percent for
(fixed) discount rate 8: the optimal path for a thirtyto fiftyyears and downto of the order
positive discount rate 8 is uniformly bounded of 2 percent for one hundred years. The
away from g* and so from the turnpikeof the evidence for these statements is still tenta-
equal treatment optima. For a zero discount tive, and more research is needed to docu-
rate, the discounted utility problem has no ment fully how people trade off the future
solution. against the present. However, our frame-
work for intertemporaloptimizationhas an
PROOF. See Heal (1995). implicationfor discountingthat rationalizes
a behavior that hitherto has been found
Figure3 illustratesthe results.The green irrational.
golden rule is the pair of consumptionand
stock levels at which the marginal condi- IVC. Weber-Fechner
Law
tions for static optimality [8] are satisfied.
The solutions for the equal treatmentcrite- Well-knownresults from naturalsciences
rion for finite time horizonsT are indicated: establish that human responsesto a change
these solutions are found by noting that the in a stimulus are nonlinear, and are in-
stock at T should be zero, and then finding versely proportionalto the existing level of
the appropriateinitial conditionsby solving the stimulus. For example, the human re-
the Euler-Lagrange equations. As T - o00o, sponse to a change in the intensity of a
these paths come closer and closer to the sound is inverselyproportionalto the initial
green golden rule. This is the turnpikeprop- sound level: the louder the sound initially,
erty. the less we respondto a given increase.This
is an example of the Weber-Fechnerlaw,
IVB. EmpiricalEvidenceMatches which can be formalized in the statement
SustainablePreferences that humanresponseto a change in a stimu-
lus is inverselyproportionalto the preexist-
Sustainablepreferenceshelp explain the ing stimulus.In symbols.
empiricalevidence on time preferences.Re-
cent empiricalevidence on time preferences dr K
clashes with standard discounted utility = or r=Klogs
maximization.Howeverthis evidence is con- ds =- s
sistent with the solutions obtained from op-
timizingsustainablepreferencesin the con- where r is a response, s a stimulus,and K
text of a renewable resource, in the sense is constant. This law has been found to
that an optimum exists if and only if the applyto humanresponsesto the intensityof
discountrate falls over time, as in part (2) of both light and sound signals.The empirical
Theorem 3. results on discountingcited above suggest
There is a growing body of empirical that something similar is happeningin hu-
evidence that suggeststhat the discountrate man responsesto changes in the futurityof
which people apply to future projects de- an event: a given change in futurity (e.g.,
pends upon, and declines with, the futurity postponementby one year)leads to a smaller
of the project.See for example,Lowenstein responsein terms of the decrease in weight-
and Thaler (1989) or Cropper,Aydede, and ing, the further the event already is in the
Portney(1994).Over relativelyshort periods future. In this case, the Weber-Fechnerlaw
up to perhaps five years, they use discount can be applied to responses to distance in
rates which are higher even than commer- time, as well as to sound and light intensity,
cial rates-in the region of 15 percent or with the result that the discount rate is
more. For projects extending about ten inversely proportional to distance into the
years, the implied discount rates are closer future.

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478 LandEconomics November1997

ds/ldt~0

Green rule
golden
\

mn-n -Optimal
paths
...m

so Environmentalstock s
FIGURE4
TheGreenGoldenRuleis the Stationary System[7],andis a
Solutionof theAutonomous
SaddlePoint.

In our economicproblem zero in the limit, the discount factor A(t)

goes to zero, and the integral flA(t) dt =
1JeKlog t dt = fltK dt convergesfor K nega-
maxf u(c, s)A(t) dt + limu(c, s),
oo
tive, as it alwaysis.
A discount factor A(t)= eK logt has an
the discountrate is q(t) = W(t)/A(t),where interesting interpretation:the replacement
of t by log t implies that we are measuring
A(t) is the discountfactor. One can formal- time differently:by equal proportionalin-
ize the Weber-Fechnerinterpretationas fol-
lows: crements rather than by equal absolute in-
crements. This is consistent with the ap-
1 dA K proach taken in, for example, acoustics,
q(t) = = - or A(t) = eKlOgt tK where in response to the Weber-Fechner
-
A dt t law sound intensityis measuredin decibals
which respond to the logarithmof the en-
where K is a negativeconstant. ergy content of the soundwaves.In general,
Such a discount factor meets all of the non-constant discount rates can be inter-
conditionsrequiredfor the existence of sus- preted as a nonlineartransformationof the
tainableoptima:the discountrate q goes to time axis.

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 479

V. DIFFERENCESBETWEEN
STANDARDAND
SUSTAINABLEOPTIMA dC/dt=O

VA. RenewableResources

The next step in exploringthe properties

of sustainable preferences is to study the Utilitarian
stadtionary
difference between sustainable and dis- soludan

counted utilitarianoptima. We do this first ds/dt=O

for the case of optimizationwith renewable roe
pwlen
GrCm
resources, comparingthe optimal path for
sustainable preferences, which solves [7]
above, with that which is optimal for the
integral of discountedutilities, namely [10]
above. In fact most of the elements needed
for this comparisonare in place. Figure 4
shows the optimalpath for sustainablepref-
erences, which asymptotes to the green
golden rule, namely the point of tangency S
between an indifferencecurve of the utility lEnvironmental stock s
function u(c, s) and the growth function FIGURE5
r(s). A stationarysolution to the utilitarian Dynamicsof the Utilitarian The
Solution:
case is characterizedby the followingcondi- GreenGoldenRule-the Highest
tions: UtilityLevel-is the Pointof
Sustainable
TangencyBetweenan Indifference Curve
andthe GrowthCurve.
r(s,) = Ct
u'2(St)
=
2

- s'(s,) [11]
both cases, because the stock of the re-
u',(ct) source is an argument of the utility func-
tion, the stationarystock exceeds that giving
The first equationin [11]just tells us that the maximumsustainableyield. This is an
a stationarysolution must lie on the curve obvious consequence of the utility of the
on which consumption of the resource stock in its own right.
equals its renewal rate: this is obviouslya
prerequisitefor a stationarystock. The sec- VB. ExhaustibleResources
ond is derivedfrom the standardconditions
for dynamicoptimizationand it implies that Considernow the rather simpler case of
the indifference curve cuts the renewal exhaustibleresources,in essence an exten-
function from above as shown in Figure 5. sion of the familiarHotelling case. For sus-
This involves a lower long-runstock and a tainable preference we consider the prob-
higher long-run level of consumptionthan lem
the sustainableoptimum,shown in Figure 4
and in this sense is less conservative:it also
involvesa lower long-runutilitylevel. In the
utilitariansolution for a constant discount
afu(c,, s)e-e'dt
max o0

+(1 - a) lim s,), a > 0, [12]

rate, as the discount rate falls to zero, the u(ct,
stationary solution moves to the green subject to St = -ct, st > 0 Vt
golden rule defined in [8]. However, for a
constant discount rate of zero, the utilitar-
ian problem has no solution. Note that in and we contrast this with the equivalent

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480 LandEconomics November
1997

Utility

Slopedui()/d

Slopesatisfiesu2(S*ads=
u
u2(S)

Stocks andconsumption
c
s*

FIGURE 6
the
Determining Stationary Stock of the EnvironmentalAsset: The MarginalUtilityof the
StockEqualsthe MarginalUtilityof Consumptionat Zero Timesthe DiscountRate. The
StockRises as the DiscountRate Falls.

discounted utilitarian formulation A sufficientconditionfor the optimumto in-

volve the preservationof a positivestockfor-
everis thatthe marginalutilityof consumption
maxf0 u(ct, st)e-st dt at zero is finite, < oo, and that there
[13] u'l(O)
existsa finite stock level s*, the optimalsta-
subjectto s, = -ct, st >0 Vt. tionary stock, such that = u'2(s*). In
so > s*, thentotal
u'l(O)8
this case, if the initialstock
In both cases the stock of the resource is an consumptionover time will equal so - s*: if
argument of the utility function, so that the so < s*, thenconsumptionwillalwaysbe zero
resource is in the category of environmental and the entirestock will be conservedon an
assets such as forests, landscapes, biodiver- optimalpath. If on theotherhandthemarginal
sity, etc., which provide services and value to utilityof consumptionat c = 0 is infinite,then
human society via their stocks as well as via it will not be optimalto conserveanypositive
a flow of consumption. The solution to the stocklevel indefinitely.
utilitarian case is summarized in the follow-
ing theorem, which is illustrated in Figures The determination of the stock s* which
6 and 7: is conserved forever is illustrated in Figure
6. Figure 7 shows in the space of consump-
THEOREM4. Consider an optimal solution tion and the remaining stock the phase dia-
to problem [12] when the utilityfunction is gram for the differential equations which
additively separable, u(c, s) = Ul(c) + u2(s). define the conditions necessary for optimal-

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 481

8uc
(c,s)Fcr-=uc,s)

10
SS '0 '0 SO

#
Utilitarianpath -
S"Greengoldenrule
Chichilnisky path
FIGURE 7
The Dynamicsof DepletionPaths OptimalAccordingto AlternativeOptimalityCriteria.

Initial
stock:maintained
bygreengoldenrule
so ---------------------------------------------------------------------
so

StockwithChichilnisky
criterion

s* -------------------- - - - ------------------------- --- --

ockwith
u(c,s)
----- -----
u(c):stockgoesto zero.
problem
Hotelling

Time.

FIGURE 8
The Time Pathsof the EnvironmentalStockunderAlternativeOptimalityConcepts.

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482 LandEconomics 1997
November

ity in the utilitarian case [13]. The initial always approximatea sustainable optimum
stock so is given: the optimal initial con- by a sequence of paths which approximates
sumptionis that level which leads asymptot- the solutions of a discountedproblem?The
ically to s = s* and c = 0 along a path following result gives a negative answer to
satisfyingthe necessaryconditionsfor opti- these two questions.It is not alwayspossible
mality. to approximatesustainableoptima by paths
When do sustainablepreferenceslead to which approachdiscountedoptima.Sustain-
more conservation, to a larger long-run able optima and discounted optima can be
stock, than the discountedutilitarianframe- far apart.24
work with the same discountrate? This oc-
curs when the weight (1 - a) on the limit- THEOREM5. Consider a sustainable prob-
ingvalue of utilityin the maximandis strictly lem:
positive. Figure 8 compares the optimal
paths of s in this problem for the alter- max W(a ), [14]
native optimality criteria that we have ({aa:fla )
examined.For the case of sustainablepref-
erences, one can show that the same dif- where W(a) = Eg=kXg g + P(a) is a sus-
ferential equationsas in the utilitariancase
are necessary for optimality,but that the tainablepreferenceand f c/ is the set of
initial and final conditionswill be different all feasibleutilitystreams.Let ac* be a unique
from those in the utilitariancase. In particu- sustainable optimum of [14] and W* =
lar, the final stock will be higher,the initial W(a*). Assumealso thatthereexistsa unique
consumption level lower, and the initial discountedoptimumP*, for the problemof
shadowprice higher(see Heal 1995). maximizingover the same set f the dis-
counted utility Then in general
g=lXgotg.25
VI. SUSTAINABLEOPTIMATHAT ARE the sustainableoptimaa* cannotbe approxi-
FAR FROM DISCOUNTEDOPTIMA matedby a sequenceof feasibleutilitystreams

We saw that sustainablepreferences are

substantiallydifferent from other welfare 23 Formally:let F be a convexand closed subsetof
criteriawhich have been used in the litera- feasiblepathsin a linearspace X such as, for example,
ture. It remains however to study how dif- X =/., or X = RN. A vectorp3 F is calledoptimalin
F if it maximizesthe value of a function U: X -- R.
ferent they are in practice, for example, The vector P is called a discountedoptimum if U:
whether the optimal solutions of problems / -- R is a discountedsum of utilities as defined in
which maximizesustainablepreferencesare [19];p is called a sustainableoptimumif U: X -- R is a
sustainablepreference.
substantiallydifferent from the optimal so- 24 This result has also implicationsfor the support
lutions to discountedproblems. and decentralizationof sustainableoptima(Corollary6
To answerthis question I compareprob- and ExampleVII.A).It maynot be possibleto approxi-
lems which face the same constraints,but mate sustainableoptimaby pathswhich approximately
each of which maximizesdifferent welfare maximizethe presentvalue of profitsunder any stan-
dard sequence of prices. However,level independent
criteria.The purposeis to explorewhat dif- sustainablepreferencescan be used in certaincases to
ference this makes in practice. A problem define shadowprices at which the optimalpaths maxi-
which maximizes a sustainable preference mize value. These are "generalized"prices, but they
will be called a sustainableproblem.If the can be givena precisemathematicalformulationand a
welfare criterion is a discounted sum of precise economic interpretation.They measure the
utilities as defined in [19], I call this a dis- value both of the shortand of the long run,and can be
called "sustainableprices."
countedproblem. The correspondingsolu- 2 This result aims at showing that the result of
tions are called sustainableoptima and dis- optimizinga sustainablepreferencemay be quite dif-
counted optima.23 ferent from that of optimizingits "discounted"part.
This is whywe comparethe optimumof W with that of
Can one always approximate a sustain- EX ot~. One could allow the case where the discount
able optimum by paths which optimize dis- factor varies over all possible discount factors; this
counted problems? Or even better: can one latter case is consideredin ExampleVII.A below.

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73(4) WhatIs Sustainable
Chichilnisky: Development? 483

1,2,... which approximates the dis- maximizationof presentvalue.26A standard

counted
{}nn= optimum 3*. This is true for any price p is a sequence of prices, p =
sequenceof discountfactors{Xg}. pg, ...) which assignsa well-defined
present value to every stream, p(y)=
(P,.".,
PROOF. In the Appendix. Eg= Pg-g, ffor eo all y • 27 There exist con-
tinuous linear functions on /e which can-
The intuitionfor this result is the following not be representedin this manner.An ex-
example: ample is the long-runaverageof a sequence
at: this assignsa presentvalue zero to every
EXAMPLE 1. Consideran economy which sequence with finitely many terms. If the
uses treesas a necessaryinputto productionor long-runaveragewas representableas a se-
consumption;withoutthisinputtheeconomy's quence, in the limit, it would assign zero
utilityis zero. The dynamicsof treereproduc- value to every sequence, which is obviously
tion requiresthat unlessthefirstN periodsthe untrue.28The Corollarybelow and the fol-
economy preservesa minimum number of lowing Example 4 construct two specific
trees,the speciesbecomesextinctafterK + N economic examples.
periods, in which case thereis zero utilityat
everyperiod from there on. The economy's COROLLARY6. Thereexistsa problem29
feasible set of utilitystreamsif is described
then as follows: a minimuminvestmentde- max, [15]
notedE > 0 is requiredduringeach of thefirst •an(W(a))
N periodsto ensurethatthe utilitylevelsin all withthepropertythat its sustainableoptimum
periods from the (N + K)-th on, is above does not maximizepresentvalue withinif at
zero. Once this thresholdis reached,then all
any standard price p = ( p, P2 ...). There-
utilitylevelsin eachperiodafterthe (N + K)- fore for any standardprice systemp thereare
th exceed E. Thenfor everydiscountfactor,
suboptimalpaths which have strictlylarger
there is an N, K for which the sum of dis- value than the sustainableoptimum
present
countedutilitiesis maximizedat a path which that is, for all p, 33p*E fl:
leads to the eventualeliminationof theforest. ta*,
Instead, for a sustainablepreferencewhich
gives sufficient weight to the long run the gE1P*g< SupP IE Pgg)i-
optimumwill keeptheforestalive andyielding g= resultgistruefor any discount
a minimumutilitylevel E forever. Therefore
the two optima are apart by at least E; any This result is truefor any discount factor.
sequenceof paths which approachesthe dis-
countedoptimumwill not approachthe sus-
tainablesolution. 26
Here I considera specialcase where the utilities
are linear; the problem can then be formulated
AND VALUE
VII. SUSTAINABILITY readilywithout introducingany furthernotation.The
Ur
generalcases can be analyzedalongsimilarlines, at the
cost of more notation. General formulationsof the
The followingcorollaryand exampleshow problemof optimaand intertemporalprofitmaximiza-
that the notion of value derived from sus- tion can be found in the literature(Debreu 1954);a
tainable preferences is rather distinctive. simple formulationin infinitedimensionalspaces that
Paths which are optimal under sustainable fits well our purposesis in Chichilniskyand Kalman
(1980).
preferences may not maximizepresent dis- 27 In some cases, the present value coincideswith
counted value according to any standard intertemporalutilitymaximization, see Chichilniskyand
price system. Therfore, environmentalre- Kalman (1980). Note that if p is a standardprice
sources with a large value in the long run, system, represented by the sequence (Pi, P2,...),
then by definitionthis sequence satisfiesE•~
may not appear valuable under a standard pg < oo
notion of present value profit maximization. 2 In fact, all purelyfinitelyadditivemeasureson Z
definereal valuedfunctionson sequenceswhichcannot
The following corollary explores the con- be representedby sequences.
nection between sustainable optima and the 29Vg, Ug
9 0.

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484 LandEconomics November1997

PROOF.Without loss of generality, con- 0, that is, p -y?p -1 Vy E U. By the

sider the case where the utilityfunctions u usual marginal rate of substitution argu-
are the identitymap. In this case, the set of ments,
feasible utilitystreamsfl coincideswith the
set of feasible consumption streams. Fur- [16]
thermore,since the welfare function W de- Pt = P2t-1.
fines an independent sustainable prefer-
ence, the firstterm of W defines a sequence I shall show that p, must be zero, so that
of standardpriceswith the desirableproper- the whole sequence {pt}t= 1,2,... must be
ties, namely p = {Dg\i= 1,2,.. defines the zero. Assume to the contrarythat p, Z 0.
Define z
presentvalue EXogag or each sequence of •e • by
,.30 Theorem3 showed that in general the
value maximizingsequence p* is different zt
= 1/pt
from a*. In particular,this implies that the
sustainableoptimuma* does not generally and z'n E+
maximizevalue for the standardprice sys- by
tem p. Since the results of Theorem 5 are
true for any sequence of "discountfactors" zt = X* if t n, and 0 otherwise.
V , hg > 0 and
{X}g=,1,2... satisfying
1X9g
< m, the corollary
lollows. Then Vn, z 2 zn so that
Eg=
VII.A.A SustainableOptimumWhichDoes
Not MaximizeExpectedValueat Any Standard p(z) > p(zn),
[17]
PriceSystem
but
The results of Corollary 6 can be
strengthenedfurther by means of another
example. Consider a feasible path Pe13 ptzn = n > p(z)
which maximizesa continuousconcave util- t= 1

ity function U within a convex set F c /, for some n sufficiently large, contradicting [17].
but such that at no standardprice system p
does 1 maximizepresent value.31 For c E
[0,oc)let The contradiction arises from the as-
sumption that p, is not zero. Therefore
u,(c)= 2'c for
c<
1/22t and P, = 0 and by [16]the entire price sequence
P= {Pt}t=1,2... is identically zero. It is
u,(c) = 1/2' for c > 1/2'. therefore not possible to support the con-

Now, for any sequence c 8' let U(c)=

u,(ct), which is well defined, continu- 30 It can be shown that the value EgXgag
E7= concave, and can be interpretedas the intertemporalp(a)-=
profit of the
ous, increasing of /.. Let
p Ef= be definedby "plan"a.
31This is from Example1 in Chichilniskyand Heal
(1993, 369), which is reproducedhere for the reader's
Pt = 1/22t+ 1 convenience.This exampledeals with the minimization
ratherthan the optimizationof a functionover a set,
and let32 but the results are of course equivalent.The example
constructsa feasible set F C:/ which is non-empty,
closed and concave,and a continuousconcavefunction
F = UP = {k U(y) 2 U(3)};; U: 4-. R whichattainsa non-zeroinfimumU(P) at B
•e/: in F, such that the only sequence of prices p =
which can support p in F is identically
F is a closed convex subset of /4. Now (Pn=1,2,...
zero.
assume that p = {pt}t is a standard 32 We call this set UOin sympathywith the notation
1,2,... of Chichilniskyand Heal (1993).
supportingprice systemfor the set U , P, 2

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 485

cave set UO with a non-zero standardprice The notion of value derivedfrom sustain-
system.33 able preferences is distinctive.Paths which
are optimal under sustainable preferences
VIII. CONCLUSIONS may not maximizevalue accordingto any
standardprice system. Therefore, environ-
mental resources with a large value in the
I have defined a set of axioms which
long run, may not appearvaluable under a
capture the idea of sustainability,and char- standardnotion of profitmaximization.
acterized the sustainable preferences that These resultsmay help to disentanglethe
they imply. I also analyzed other criteria apparentcontradictionsin valueswhichwere
used in the literature,and found that they discussedin the
do not satisfymy axioms.Discounted utility noted that beginningof this paper.We
governments and international
fails to satisfy the non-dictatorshipof the
organizations appear seriously concerned
present. This agrees with the viewpoint of about global environmentalproblemswhich
many practitioners,who have pointed out lie so far into the future that with current
the inadequacyof discountedutilityfor ana- discounted
utilitymeasuresthey do not lead
lyzing sustainable growth.34Rawlsian and to substantialeconomicloss. The axiomsfor
basic needs criteria are insensitive, since sustainable
preferences proposed here may
they only regardthe welfare of the genera- help resolve this contradiction.Discounted
tion which is least well-off. The overtaking maximizationand sustainabilitylead
criterion and its relative the catching up profit
to different value systems. Some trade-offs
criterionare incomplete as orders and can- are
not be representedby real valued functions. same. possible, but the two values are not the
The empiricalevidencewe have today
They fail to comparemanyreasonablealter- is more in favor of sustainablepreferences
natives. This decreases their value as tools than discounted utility. Solow (1992) has
for decision making.Ramsey'scriterionhas
a similardrawback:it is defined as the inte- proposedthat sustainabilityshould allow in-
tergenerationaltrade-offs, but no genera-
gral of the distance to a "bliss"utility level, tion should be favoredover any other. This
but this integralis often ill-defined.3V standardis met by sustainablepreferences
The sustainable preferences proposed
when applied to the "present"and to "fu-
here and characterizedabove circumventall
ture" generations.The long run does matter
of these problems.From the practicalpoint
and so does the short run. Indeed, indepen-
of view, they satisfy two desirable criteria:
dent sustainable preferences can define
they fit our intuition of finite horizon prob- shadow for sustainableoptima,which
lems, because in important examples they can be prices used for project evaluation and for
have a turnpike property with respect to
the characterizationof optimal solutions.
equal treatmentfinite horizon problems.In Severalof the aims of this paperhave there-
addition, they fit rather well empirical ob-
servations that indicate that people's per-
ceptions of the future imply lower discount
rates as time progresses.Importantclasses
33 Further examplesof phenomenarelated to the
by dynamic problems have a solution ac- results in Theorem2 and Corollary1 can be found in
cording to sustainable preferences only if Dutta (1991).
the implied discount rates are decreasing 34For example,Dasguptaand Heal (1979),Broome
time. (1992), Cline (1992).
through 35Hammond(1993) has definedagreeablepaths as
I also showed that sustainable prefer- those which are
approximatelyoptimal for any suffi-
ences give rise to optimal solutions which ciently long horizon, in the sense that the welfare
are different from those obtained by dis- losses inflictedby consideringonlyfinite horizonsgo to
counted optimizationcriteria.A path which zero as the length of the finitehorizongoes to infinity.
criterionis not designedas a complete order but
is optimal under a sustainable preference The rather as a way of identifyingacceptable paths. A
may not be approximatedby paths which similarissue ariseswith the overtakingcriterion,which
approximatediscountedoptima. is ill-definedin manycases.

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486 LandEconomics November1997

fore been reached,and several of the ques- standard topology of 4: the norm defined by
tions that we posed have been answered. I111= UPg1,2,... Ia(g)I.
But perhapsthe results open up at least
as manynew questions.It remainsto under- IX.B.Independence
stand the concern for the long-run future
The welfare criterionW: 4~- R will be said
which is observed in practice, and which to give independenttrade-offsbetweengenera-
appearsformalizedin the axioms proposed tions,andcalledindependent, whenthe marginal
here and their impliedpreferences.Nobody rate of substitution
betweenthe utilitiesof two
alive today,not even their heirs, has a stake generationsg, andg2 dependsonlyon the iden-
on the welfare of fifty generationsinto the titiesof the generations,
thatis, on the numbers
future. Yet many humans care about the g, andg2, andnoton the utilitylevelsof the two
long-run future of the planet, and the re- generations.Independence of the welfarecrite-
sultsof this paperindicatethat axiomswhich rionis notneededin Theorem1. It is usedsolely
formalize this concern are not altogether in the characterization
of Theorem2, to allowus
to obtaina simplerepresentation of all sustain-
unacceptable. One may then ask: whose able preferences.Formally:let 4* be the space
welfare do sustainable preferences repre- of all continuousrealvaluedlinearfunctionson
sent?
Perhapsan answerfor this riddle may be
found in a wider understanding of hu- DEFINITION10. Thewelfare W:4~- R
criterion
mankindas an organismwho seeks its over- if Va, P3E
is independent e ,
all welfare over time. Such proposals have
been advancedin the concepts of a "selfish W(a) = W(P)-*
gene," or, more practically,in Eastern reli-
gions whichview the unity of humankindas 3 X e/*,X = X(W), such that X(a) = X(P).
a natural phenomenon. If such unity ex-
isted, humankindwould make up an un- Thispropertyhas a simplegeometricinterpreta-
usual organism,one whose parts are widely tion,whichis perhapseasierto visualizein finite
distributedin space and time and who is dimensions.For example:consideran economy
lackinga nervoussystem on which the con- with n goodsandtwo periods.Let oa= (la2t2),
12) e aR2and
sciousness of its existence can be based. P = (131, denote two feasible utility
Perhapsthe recent advancesin information streams. Then 1 areequivalentaccording
technology, with their global communica- to the welfarecriterionW: R2 -* R, that is,
tions and processingreach, are a glimmerof = W(P),if andonlyif thereexistsa num-
W(oa)
ber ji = ji(W), > 0, suchthat
the emergence of a nervous system from 7i
whicha global consciousnessfor humankind
could emerge. Oa2- 12
131
- P1 . [18]
Otl
IX. APPENDIX The geometric interpretationof [18] is that the
indifferencesurfacesof W are affine linear sub-
IX.A. Continuity spaces of R2. Level independenceimplies that
the indifferencesurfacesof the welfare function
In practicalterms the continuityof W is the W are affine hyperplanesin 4/. In particular,W
requirementthat there should exist a sufficient can be representedby a linearfunctionon utility
statisticfor inferringthe welfarecriterionfrom streams,that is, W(a + p) = W(o) + W(P).Ex-
actualdata.Thisis an expressionof the condi- amples of welfare criteriawhich satisfy this ax-
tion that it should be possible to approximateas iom are all time-separable discounted utility
closely as desired the welfare criterion W by functions, any linear real valued non-negative
sampling over large enough finite samples of functionon 4e, and the welfarecriteriain Theo-
utilitystreams.Continuityof a sustainablecrite- rem 2. As alreadymentioned,this axiomis used
rion functionW: 4 -* R is not needed in Theo- to provide a tight representationof sustainable
rem 1; it is used solely for the characterizationin preferences,but is not strictlynecessaryfor the
Theorem2. Continuityis definedin terms of the main results.

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 487

DEFINITION 11. A continuousindependentsus- The long-runaveragecriterioncan be defined in

tainablepreferenceis a completesensitiveprefer- our context as follows: a utility stream ae is
ence satisfyingAxioms 1 and 2 and whichis con- preferredto another P if in average terms, the
tinuousand independent. long-run aggregate utility37 achieved by ae is
largerthan achievedby 1. Formally:3N, K > 0:
IX.C. Definitionof PreviousWelfareCriteria
To facilitate comparison,this section defines 1 T+M 1 T+M
some of the more widelyused welfare criteria.A F t > -
F P,
g=M g=M
functionW: /, -* R is called a discountedsum of
utilitiesif it is of the form: VT > N and M > K. [24]

W(a) = EahXg, e
Va E [19]
IX.D. Countableand FinitelyAdditiveMeasures
7,
g= 1
DEFINITION12. Let (S, E) denotethefield of all
where Vg, Xg 2 0 and X is called
<c; subsetsof a set S withthe operationsof unionsand
E'lkg
the discount factor. Ramsey's welfare criterion intersectionsof sets. A real valued,boundedaddi-
(Ramsey 1928) ranks a utility stream ao= tive set function on (S, E) is one which assignsa
= 1, 2 e above another = {Pgg=1, 2... real value to each elementof (S, E), and assigns
{agO}g 3 the sum of the valuesto the union of two disjoint
e- if ....
the utility stream a is "closer"to the
bliss path, namely to the sequence = sets.
{1,1,...,1,...), than is the sequence p. For-
DEFINITION 13. A real valuedboundedadditive
mally: setfunctionis calledcountablyadditiveif it assigns
the countablesum of the values to a countable
E (1- pg). [20] unionof disjointsets.
g=1 (1-ta,)_< g=1
EXAMPLE 2. Probabilitymeasureson the real
A Rawlsianrule (Rawls 1971)rankstwo utility numbers,R, or on the integersZ, are typicalexam-
streamsaccordingto whichhas a higherinfimum ples of such countablyadditivefunctions.Any se-
value of utility for all generations.36This is a quenceof positivereal numbers{X}g= 1, such
natural extension of the criterion proposed ini- 2...
that Eg= <00 defines a countably additive mea-
tially by Rawls (1971). Formally:a utility stream sure Lon••the integersZ, by the rule
a is preferredto another P if
1?(A)= E Xg,VACZ.
2,... > inf{gg=1,2,...
inf{ag}g=1, [21] geA

The criterion of satisfactionof basic needs DEFINITION 14. A real valuedboundedadditive

introducedin Chichilnisky(1977a)ranksa utility set function (9 on (S, E) is called purelyfinitely
stream a over another 3 if the time requiredto additive(see Yosidaand Hewitt1952) if whenever
meet basic needs is shorter in a than in p. a countablyadditivefunctionv satisfies:
Formally:
VA e (S,E), v(A) q(p(A), then
T(ac) T(P3), [22]
_ v(A)= 0 VA e (S, E).
where T(a) = min{t:aOg b Vg t), for a given
b which representsbasic needs. The overtaking This meansthat the only countablyadditivemea-
criterion(von Weizaicker1967) ranks a utility sure which is absolutelycontinuouswith respect
stream at over another 1 if a eventuallyleads to
a permanentlyhigher level of aggregate utility
than does 13.Formally:a is preferredto 13if 3N: 36 RelatedRawlsian rulesare discussedin Asheim
M M (1988).
VM > N, [23] 37 This is only one of the possibledefinitions
of
Eag > Ei ~g. long-runaverages.For otherrelateddefinitions with
g=1 g=1
similarpropertiessee Dutta(1991).

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488 LandEconomics November1997

to a purelyfinitelyadditivemeasure,is the mea- IX.E. Proofof Theorem1

sure whichis identicallyzero.
PROOF. To establish the existence of a sustain-
EXAMPLE3. Any real valued linear function V:
R definesa boundedadditivefunction V on able preferenceW: 4, -- R, it sufficesto exhibit
the field (Z, E) of subsetsof the integersZ as a function W:/. -- R satisfyingthe two axioms.
For any a E- consider
follows:

VA c Z, V(A)= V(aA) [25] = +

W(ao) Eg=~ [liminf{aog}g=i1,2,... ,
8gg
whereaa is the "characteristic
function"of the set with 0 < 8 < 1.
A, namelythe sequencedefinedby

W satisfies the axioms because it is a well-de-

SA = ({ such that
g=1,2,..
fined, non-negative,increasingfunctionon /4; it
is not a dictatorshipof the present (Axiom 1)
a = 1 if geA and aA = 0 otherwise. because its second term makes it sensitive to
changes in the "tails"of sequences; it is not a
[26] dictatorshipof the future (Axiom 2) because its
first term makes it sensitive to changes in
"cutoffs"of sequences.
EXAMPLE 4. Typicalpurelyfinitely additiveset The next task is to show that the following
functionson thefield of all subsetsof the integers, welfare criteriado not define sustainableprefer-
(Z, E), are the liminf functionon 4, definedfor ences: (a) Ramsey'scriterion,(b) the overtaking
each a /, by criterion,(c) the sum of discountedutilities, (d)
•
liminf, (e) long-runaverages(f) Rawlsiancrite-
liminf(a) = liminf {etg}. [27] ria, and (g) basic needs. The Ramsey'scriterion
g=1,2,... defined in [20] fails because it is not a well-de-
fined real valued function on all of 4, and
cannot thereforedefine a complete order on 4.
Recallthatthe liminf of a sequenceis the infimum To see this it suffices to consider any sequence
of theset of pointsof accumulationof thesequence. a e=- for which the sum in [20] does not con-
The"long-runaverages"functionis anotherexam- verge. For example,let
ple: it is definedfor each a E- by

=
1 K+N O (aOLgg=1,2,... where Vg, = (g - 1)/g.
lim - Oag
K,N-oo
EO . [28]
[281
Kg=N
Then ag -- 1 so that the sequence approaches
It is worth noting that a purelyfinitely additive the "bliss" consumption path P = (1, 1,...,
set function ( on the field of subsets of the 1,...). The rankingof a is obtainedby the sum
integers (Z, E) cannot be representedby a se- of the distancebetween a and the bliss path 3.
quence of real numbersin the sense that there Since lim ooE-N (1 - ag) = lim ll/g EN1=
exists no sequence of positivereal numbers,X = does not converge, Ramsey'swelfare criterion
which defines 4, that is, there is for no X does not define a sustainablepreference.
{X,,}
such that The overtakingcriteriondefinedin [23]is not
a well-definedfunction of 4, since it cannot
rank those pairs of utility streams a, e3 =,, in
VA cZ, whichneither a overtakesP, nor P overtakes a.
4,(A)= neA
h,.-
Figure 2 exhibitsa typicalpair of utilitystreams
which the overtakingcriterionfails to rank.
For examplethe liminf: 4 -* R, definesa purely The long-run averages criterion defined in
finitelyadditiveset functionon the integerswhich [24] and the liminf criteriondefined in [27] fail
is not representableby a sequence of real num- on the grounds that neither satisfies Axiom 2;
bers. both are dictatorshipsof the future. Finallyany

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73(4) WhatIs SustainableDevelopment?
Chichilnisky: 489

discountedutilitycriterionof the form the welfare criterion W: e4 -~ R, defining a

non-negative,continuouslinearfunctionalon e.
As seen abovein Example3, [25]and [26],such a
W(a) = E ag Xg function defines a non-negative,bounded, addi-
g=1
tive set function denoted W on the field of sub-
sets of the integers Z, (Z, E).
>0
whereVg,Xkg and 0 The representationtheorem of Yosida and
EXg<
g= 1 Hewitt (Yosida 1974; Yosida and Hewitt 1952)
establishesthat everynon-negative,bounded,ad-
is a dictatorshipof the present, and therefore ditive set function on (S, E), the field of subsets
fails to satisfyAxiom 1. This is because E of a set S, can be decomposedinto the sum of
a non-negativemeasure?Ljand a purelyfinitely
Vy s.t. sup (yg) 1, and VE > 0, additive,non-negativeset function PL2on (S, E).
•E/ g= 1,2,... <
It follows from this theorem that W can be
representedas the sum of a countablyadditive
:N>0,
N=N(E): 1 Yg < E, [29] measure and a purely finitely additivemea-
g=N sure on the141,
integers Z. It is immediateto verify
that this is the representationin [5].To complete
and therefore,since the characterizationof an independentsustain-
able preferenceit sufficesnow to show that nei-
W(a) > W(3) =* 3E > 0: W(a) - W(P3)> 3E, ther X nor 4 are identicallyzero in [5]. This
followsfrom Axioms 1 and 2: we saw above that
then by [29] discountedutilityis a dictatorshipof the present,
3N > 0 such that so that if 4= 0, then W wouldbe a dictatorship
ye- f, of the present,contradictingAxiom 1. If on the
Vr,
V K > N. other hand X 0, then W would be a dictator-
W(K,I rK) > W(K,yIK),
ship of the future because all purely finitely
The functionW thus satisfiesthe firstpart of the additivemeasuresare, by definition,dictatorships
definitionof a dictatorshipof the present,that is, of the future, contradictingAxiom 2. Therefore
neither X nor 4 can be identicallyzero. This
W(a) > W(P3) completesthe proof of the theorem.
3N, N = N(a,3p): Vy, /
IX.F. Proofof Theorem5
E•=
with ? 1 and |lall
ll•ll < 1, The statement of Theorem 5 is: Considera
sustainableoptimumgrowthproblem
W(aK, YK)> W(K, UK), VK > N.

The reciprocalpart of the definitionof dictator- maxW(otg),

ship of the presentis immediatelysatisfied,since
if Va, -y / such that iall? 1, 11P13l ? 1, where atg = {ug(Xg))g -
e
1,2,... C/c, [30]
W(aK, UK)> W(aK, YK), and obviouslythis im-
plies W(a)> W(P). Therefore W is a dictator-
ship of the present and violatesAxiom 1. where 0Ois the set of all feasible utilitystreams
Finallythe Rawlsianwelfarecriterionand the and W is an independentsustainablepreference.
criterion of satisfaction of basic needs do not By Theorem2 W must be of the form:
define independentsustainablepreferences:the
Rawlsiancriteriondefinedin [21]fails because it
is not sensitive to the welfare of many genera- W(Xa) E agOtg
+ (aO), VOa
e , [31]
tions: only to that of the less favoredgeneration. g=1
Basic needs has the same drawback.
where g, kg > O, :=,g < oo, and $4#0 is a
IX.E. Proofof Theorem2 purely finitely additiveindependentmeasure on
Z. Assume that there exists a unique solutionto
PROOF. Consider a continuous independent problem [30], denoted a* and called a sustain-
sustainablepreference.It must satisfyAxioms 1 able optimum,with welfare value W* = W(a*).
and 2. There exists a utility representationfor Assume also that there exists a unique solution,

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490 LandEconomics November1997

denoted 13*and called a discountedoptimum,for finitelyindependentmeasure 4 has most of the

the problemof maximizingover the same set fl "weight,"that is, when 0 ~ 0. Indeed,for 0 ~ 0,
the discountedutility the sustainableoptimuma* satisfies

max(U(a)) where U(a) = , [32] Ee* forg>K+N.

ae g=1 gag,
Since both a* and P* are unique,and
whichis the firsttermdefiningthe preferenceW
in [31].Thenin generalthe sustainableoptima
IIa*- P"*11E > 0,
a* cannotbe approximated by a sequenceof
feasibleutilitystreams{p3N}=1,2,"...which ap- it is clear that a sequence {P")which approaches
proximates the discountedoptimumP*, that is,
for all suchsequences p* cannot approachalso a*. This completesthe
proof of the theorem.

limn g=1
F, Xg1 max
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