Table Of ContentTHE ECONOMICS OF NON-CONVEX ECOSYSTEMS
THE ECONOMICS OF NON-MARKET GOODS AND RESOURCES
VOLUME 4
Series Editor: Dr. Ian J. Bateman
Dr. Ian J. Bateman is Professor of Environmental Economics at the School of
Environmental Sciences, University of East Anglia (UEA) and directs the research
theme Innovation in Decision Support (Tools and Methods) within the Programme on
Environmental Decision Making (PEDM) at the Centre for Social and Economic
Research on the Global Environment (CSERGE), UEA. The PEDM is funded by the UK
Economic and Social Research Council. Professor Bateman is also a member of the
Centre for the Economic and Behavioural Analysis of Risk and Decision (CEBARD) at
UEA and Executive Editor of Environmental and Resource Economics, an international
journal published in cooperation with the European Association of Environmental and
Resource Economists (EAERE).
Aims and Scope
The volumes which comprise The Economics of Non-Market Goods and Resources
series have been specially commissioned to bring a new perspective to the greatest
st
economic challenge facing society in the 21 Century; the successful incorporation of
non-market goods within economic decision making. Only by addressing the
complexity of the underlying issues raised by such a task can society hope to redirect
global economies onto paths of sustainable development. To this end the series
combines and contrasts perspectives from environmental, ecological and resource
economics and contains a variety of volumes which will appeal to students, researchers,
and decision makers at a range of expertise levels. The series will initially address two
themes, the first examining the ways in which economists assess the value of non-
market goods, the second looking at approaches to the sustainable use and management
of such goods. These will be supplemented with further texts examining the
fundamental theoretical and applied problems raised by public good decision making.
For further information about the series and how to order, please visit our Website
http:/www.wkap.nl/series.htm/ENGO
The Economics of
Non-Convex Ecosystems
Editedby
Partha Dasgupta
CambridgeUniversity,FacultyofEconomics,Cambridge,UK
and
Karl-GöranMäler
BeijerInternationalInstituteofEcologicalEconomics,
RoyalSwedishAcademyofSciences,Stockholm,Sweden
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-2515-7
Print ISBN: 1-4020-1864-9
©2004 Kluwer Academic Publishers
NewYork, Boston, Dordrecht, London, Moscow
Print ©2004 Kluwer Academic Publishers
Dordrecht
All rights reserved
No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Visit Kluwer Online at: http://kluweronline.com
and Kluwer's eBookstoreat: http://ebooks.kluweronline.com
Contents
TheEconomicsofNon-ConvexEcosystems:Introduction
P.Dasgupta,K.-G. Mäler 1
Scale andScaling in EcologicalandEconomicSystems
J. Chave, S. Levin 29
ConvexRelationshipsinEcosystemsContainingMixtures
ofTrees andGrass
R.J.Scholes 61
ManagingSystemswithNon-convexPositiveFeedback
W.A.Brock, D.Starrett 77
TheEconomicsofShallowLakes
K.-G. Mäler, A.Xepapadeas,A. deZeeuw 105
_
Multiple Species Boreal Forests W hat Faustmann
Missed
A.-S. Crépin 127
Evaluating Projects and Assessing Sustainable Develop-
mentin Imperfect Economies
K.J. Arrow,P.Dasgupta,K.-G. Mäler 149
The Economics of Non-Convex Ecosystems:
Introduction
PARTHADASGUPTA1andKARL-GÖRANMÄLER2
1CambridgeUniversity,FacultyofEconomics,SidgwickAvenue,CambridgeCB39DD,UK
(E-mail:[email protected]);2BeijerInternationalInstituteofEcological
Economics,Stockholm,Sweden
Abstract. The word “convexity” is ubiquitous in economics, but absent from economics. In this
paper we explain why, and show what difference it makes to economic analysis if ecosystem
non-convexities are taken seriously. A simple proof is provided of the connection between “self-
similarity” and “power laws”. We also provide an introduction to each of the papers in the
Symposiumanddrawoutthewayinwhichtheyformalinkedsetofcontributions.
Key words: bifurcation points, environmental Kuznets curve, hysteresis, irreversibility, non-
convexity, Pontryagin Principle, power laws, property rights, separatrix, structural stability,
thresholds
1. Economists’Convexities andNature’sNon-Convexities
Theword“convexity” isubiquitous ineconomics, butabsent fromecology. There
isareasonforeach.Aspricesareprominentinmoderntransactions,itisbutnatural
that we would wish to uncover the ways in which the price system is capable of
functioning as a resource allocation mechanism. In recent years economists have
identi ed notonlythewayinwhichpricesaggregate dispersed pieces ofinforma-
tion, but also the sense in which they re ect the relative scarcities of goods and
services. Wenowknowthatthepricesystemcanbeanef cientallocation mecha-
nismiftransformation possibilities amonggoodsandservices–inandovertime–
constitute aconvexset.1 However, except in the case ofpartial economic systems,
as in models of industrial organization (Tirole 1993), or of systems harbouring
very speci c forms of non-convexities, as in modern growth models (Jones 1998)
or in models of poverty traps based on the connection between nutritional status
and human productivity (Dasgupta and Ray 1986, 1987; Dasgupta 1997) or in
models of spatial economies (Fugita, Krugman and Venables 1999), we still do
nothaveaclearunderstandingofthemechanismsbywhichresourcesareallocated
in non-convex environments. So we economists continue to rely on the convexity
assumption, alwayshopingthatitisnotanembarrassing simpli cation.
Ecologists have no comparable need to explore the structure of convex
sets. They are interested in identifying pathways by which the constituents of
1
P. Dasgupta& K.-G.Mäler (eds.), The Economics of Non-Convex Ecosystems,1 _ 27.
©2004 Kluwer Academic Publishers. Printed in the Netherlands.
2 PARTHADASGUPTAANDKARL-GÖRANMÄLER
an ecosystem interact with one another and with the external environment. A
large body of empirical work has revealed that those pathways in many cases
involve transformation possibilities amongenvironmental goods andservices that,
together, constitute non-convex sets. Often the non-convexities re ect positive
feedbacks in Human-Nature interactions. Mathematical ecologists therefore study
the structural stability of ecosystems and the sizes and shapes of their basins
of attraction for given sets of environmental parameters.2 Such notions as the
“resilience” ofecosystems areexpressions ofthisresearchinterest.3
The price mechanism is especially problematic in economic systems charac-
terised by positive feedback processes. We now know that in such environments
it may prove impossible to decentralise an ef cient allocation of resources by
means exclusively of prices. Ef cient mechanisms would typically involve addi-
tional social contrivances, such as (Pigouvian) taxes and subsidies, quantity
controls, social norms of behaviour, and so forth. This was proved formally in
a justly famous article by Starrett (1972), who showed that for certain types
of non-convexities associated with environmental pollution, a competitive price
equilibriumsimplydoesnotexist:marketsforpollutionwouldbeunabletoequate
demandstosupplies.Starrett’snon-convexitiesarepresentwhenlossestraceableto
environmentalpollutionarebounded.Ifthemarketpriceforpollutioninsuchsitua-
tions were negative (i.e., the pollutor has to pay the pollutee), pollutees’ demand
would be unbounded, while supply would be bounded. On the other hand, if the
pricewerenon-negative, demandwouldbezero,whilesupply, presumably, would
bepositive.4
Starrett’s ndingimpliedthatprivatepropertyrightstoenvironmentalpollution
would not be capable of sustaining an ef cient allocation of resources by means
of the price system. In a subsequent note, Starrett (1973) demonstrated by means
of an example that if property rights are awarded to polluters, even such a non-
price resource allocation mechanism as the core may not exist. But he showed
(Starrett1972)thatasuitablychosen setofpollution taxes,together withasystem
ofcompetitivemarketsforothergoodsandservices(assumingthatthelatterconsti-
tute a convex sector), would be capable of supporting an ef cient allocation of
resources. Asthere are no markets forpollution insuch anallocation mechanism,
the problem of equating supply to demand in pollution activities is bypassed. The
moral would appear to be that social dif culties arising from the non-convexities
can be overcome if the State were to assign property rights in a suitable way –
permitting private rights to the convex sector, but reserving for itself the right to
controlemissions anddischarges.5
2. EcologicalThresholdsandtheEnvironmentalKuznetsCurve
Despitetheecologist’sstrictures,weeconomistshaveremainedambivalenttoward
Nature’snon-convexities. Oftenenough,thatambivalencerevealsitselfonlyindir-
ectly. For example, it is even today commonly thought that, to quote an editorial
THE ECONOMICS OF NON-CONVEX ECOSYSTEMS: INTRODUCTION 3
intheUK’sTheIndependent (4December 1999), “... (economic) growthisgood
fortheenvironment because countries needtoputpovertybehind theminorderto
care”; or, to quote The Economist (4 December 1999, p. 17), “... trade improves
the environment, because it raises incomes, and the richer people are, the more
willingtheyaretodevoteresources tocleaning uptheirlivingspace”.
The view’s origin can be traced to World Bank (1992), which observed an
empirical relationship between GNP per head and atmospheric concentrations of
industrial pollutants. Based on the historical experience of OECD countries, the
authors of the document suggested that, when GNP per head is low, concentra-
tions ofsuch pollutants asthesulphur oxides increase asGNPperhead increases,
but that when GNP per head is high, concentrations decrease as GNP per head
increases further.6 Among economists this relationship has been christened the
“environmental Kuznetscurve”.7 Themoralthatwouldappeartohavebeendrawn
from the nding is that resource degradation is reversible: degrade all you want
now,Earthcanbereliedupontorejuvenate itlaterifyourequire it.
The presumption is false. Nature’s non-convexities are frequently a manifes-
tation of positive feedback processes, which in turn often means the presence of
ecological thresholds. But if a large damage were to be in icted on an ecosystem
whose ability to function is conditional on it being above some threshold level
(in size, composition, or whatever), the consequence would be irreversible.8 The
environmental Kuznets curve was detected for mobile pollutants. Mobility means
that, so long as emissions decline, the stock at the site of the emissions declines.
However,reversalisthelastthingthatwouldspringtomindshouldagrassland ip
to become covered by shrubs, or should the Atlantic gulf stream shift direction or
come to a halt, or should a source of water disappear, or should an ocean shery
become a dead zone owing to over shing. As a metaphor for the possibilities of
substituting manufactured and human capital for natural capital, the relationship
embodiedintheenvironmental Kuznetscurvehastoberejected.9
Althoughnon-convexities areprevalentinglobalecosystems(e.g.,oceancircu-
lation, global climate), it is as well to emphasise the spatial character of many
positive feedback processes. The latter have a direct bearing on the rural poor
in the world’s poorest regions. Eutrophication of ponds, or salinization of soil,
or biodiversity loss in a forest patch involve crossing ecological thresholds at
a spatially localised level. Similarly, the metabolic pathways between an indi-
vidual’s nutritional status and his or her capacity to work, and those between
a person’s nutritional and disease status involve positive feedback.10 Studies of
extremepovertybasedonaggregationattheregionalornationallevelcantherefore
mislead greatly. The spatial con nement of many of the non-convexities inherent
inHuman-Natureinteractions needsalwaystobekeptinmind.
The connection between rural poverty in the world’s poorest regions and the
state of the local ecosystems should be self-evident. When wetlands, inland and
coastal sheries, woodlands, forests, ponds and lakes, and grazing elds are
damaged (owing, say, to agricultural encroachment, or urban extensions, or the
4 PARTHA DASGUPTA AND KARL-GÖRAN MÄLER
constructionoflargedams,ororganizationalfailureatthevillagelevel),traditional
dwellers suffer. For them – and they are among the poorest in society – there
are frequently no alternative source oflivelihood. In contrast, for rich eco-tourists
or importers of primary products, there is something else, often somewhere else,
which means that there are alternatives. Whether or not there are substitutes for a
particular resource is therefore not only a technological matter, nor a mere matter
of consumer taste: among poor people location can matter too. Thepoorest of the
poorexperience non-convexities inawaytherichdonot. Eventherange between
a need and a luxury is context-ridden. Macroeconomic reasoning glosses over the
heterogeneity ofEarth’s resources and the diverse uses to which they areput – by
peopleresiding atthesiteandbythoseelsewhere.11
3. Motivation BehindthisSymposium
Despitetheirprevalence,weeconomistscontinuetoshowlittleinterestinNature’s
non-convexities. Forexample, thenowenormous literature on“greenaccounting”
has been built largely on the backs of economic models in which non-convexities
aresafelyoutofsight.12 Inviewofthis,theBeijerInternational InstituteofEcolo-
gical Economics in Stockholm invited a group of ecologists and economists to
studytheeconomicsofnon-convex ecosystems. Beginningin1998,thegroupmet
regularly.13 Weknewthatmuchfundamentalworkhadalreadybeenaccomplished
on non-linear dynamical systems. Some of it had already been used to study the
economics of non-convex social environments.14 Our idea therefore was to use
these techniques so as to study proto-typical Human-Nature interactions. Speci c
applications wereverymuchinourmind.
Asourunderstandingoftheissuesimproved,wefeltitwouldbedesirabletooif
thearticleswepreparedwerewrittenexpansively.Soweencouragedourselvesalso
toreviewwhathasalreadybeenachievedinthesubject. Wefeltthatthecollection
could then serve as a self-contained body of work, useable in graduate courses in
environmental andresource economics, perhaps useful aswelltoresearchers who
wish to work in this eld. In order to increase accessibility to the collection, we
feltitwouldbeappropriate topublishitinajournal, ratherthanasabook. Weare
thereforemostgratefultothemembersoftheEditorialBoardofEnvironmental and
ResourceEconomics,especiallyIanBatemanandKerryTurner,fortheenthusiasm
theyhaveshowntowardourenterprisefromthetimeweapproached them,andfor
theencouragement they havegiven ustoproduce thecollection intheform that it
nowappears.
THE ECONOMICS OF NON-CONVEX ECOSYSTEMS: INTRODUCTION 5
4. MacroRegeneration FunctionsofEcosystems
4.1. PRODUCTION FUNCTIONS AND COMMODITY POSSIBILITY SETS
Incontrasttotheproductionscale-economiesstudiedintraditionalpricetheory,the
non-convexities associated with ecological thresholds manifest themselves across
time. Nevertheless, there is a formal connection between the two types of non-
convexities.
To see this, consider the economist’s old stand-by, a one-commodity, constant
population economy, where the commodity in question is durable and non-
deteriorating. Timeiscontinuous andtheeconomyisdeterministic.15 LetK (≥0)
t
and C (≥ 0) be the “capital” stock and the flow of consumption, respectively, at
t
t (≥ 0). Output is assumed to be given by the production function F(K), where F
is differentiable everywhere, F(0) = 0 and F(K) > 0 for some K > 0. An extreme
assumption,muchusedingeneralequilibriumtheory,isthatoutputisfreelydispos-
able.Subsequentlywewillseewhattheassumptioninvolvesandwhyitsviolation
issignificant forthesubject ofthisSymposium.16
LetI denoteinvestment att.Sinceoutput isfreely disposable, wemayexpress
t
thebalanceofflowsateachdateintheeconomybytheinequality,
I ≤ F(K)−C, fort ≥ 0,andK (> 0)isgiven, (1a)
t t t 0
which,withoutrigorous justification, wewriteas,
dK/dt ≤F(K)−C, fort≥ 0,andK (> 0)isgiven. (1b)
t t t 0
A programme is a complete forecast of the economy, from the present (t = 0)
toinfinity.Aprogrammecanbeexpressedas{K,C,dK/dt} ∞.Forsimplicityof
t t t 0
exposition, wesuppose thatanyprogrammesatisfying (1b)isfeasible; whichisto
saythattheeconomyisnotsubjecttoanyotherconstraint.
Thequestion iswhetherthesetoffeasibleprogrammes isconvex.
TheanswerdependsonwhetherFisconcaveeverywhere(henceforth,concave).
Toconfirmthis,assumethatFisconcave. Consideranytwofeasible programmes,
which we write as {K∗, C∗, dK∗/dt} ∞ and {K(cid:5), C(cid:5), dK(cid:5)/dt} ∞. By definition,
t t t 0 t t t 0
bothsatisfy(1b).Hence,
dK∗/dt≤ F(K∗)−C∗, fort≥ 0,andK ∗ = K , (2)
t t t 0 0
and dK(cid:5)/dt≤ F(K(cid:5))−C(cid:5), fort≥ 0,andK (cid:5) = K . (3)
t t t 0 0
Nowchooseanumberγ,where0 ≤ γ ≤ 1.Define
C¯ = γC∗ +(1−γ)C(cid:5) andK¯ = γK∗+(1−γ)K(cid:5). (4)
t t t t t t
Wewishtoconfirmthat{K¯ ,C¯ ,dK¯ /dt} ∞ alsosatisfies(1b).
t t t 0
From(2)and(3),wehave,
γdK∗/dt ≤ γF(K∗)−γC∗, fort≥ 0,andK ∗ = K ,(5)
t t t 0 0
and (1−γ)dK(cid:5)/dt ≤ (1−γ)F(K(cid:5))−(1−γ)C(cid:5), fort ≥ 0,andK (cid:5) = K . (6)
t t t 0 0
