Baumgaertner NSCERE

Baumgaertner NSCERE
Natural Science Constraints
in Environmental and
Resource Economics
Method and Problem
Stefan Baumgärtner
Alfred-Weber-Institute of Economics,
University of Heidelberg, Germany
University of Heidelberg Publications Online
http://archiv.ub.uni-heidelberg.de/Englisch/index.html
c Stefan Baumgärtner 2006
No part of this publication may be reproduced, stored in a retrieval system or
transmitted in any form or by any means, electronic, mechanical or photocopying, recording, or otherwise without explicit acknowledgement of authorship
and copyright.
Stefan Baumgärtner
University of Heidelberg
Alfred-Weber-Institute of Economics
Bergheimer Str. 20
D-69115 Heidelberg, Germany
[email protected]
This text is the revised version of a Habilitation Thesis that was submitted to
the Faculty of Economics and Social Studies, University of Heidelberg, Germany on 06 July 2005 and was accepted on 26 April 2006 (reviewers: Prof.
Malte Faber, Prof. Hans Gersbach).
Contents
Acknowledgements
v
1. Introduction
1
PART I THERMODYNAMICS
2.
3.
4.
5.
Thermodynamic Analysis: Rationale, Concepts, and Caveats
The Inada Conditions for Material Resource Inputs Reconsidered
Temporal and Thermodynamic Irreversibility in Production Theory
Necessity and Inefficiency in the Generation of Waste
with Jakob de Swaan Arons
6. Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
with Frank Jöst and Ralph Winkler
27
51
67
79
97
PART II BIODIVERSITY
7. Biodiversity as an Economic Good
123
8. Ecological and Economic Measures of Biodiversity
151
9. The Insurance Value of Biodiversity
173
10. Insurance and Sustainability through Ecosystem Management
201
with Christian Becker, Karin Frank, Birgit Müller and Martin Quaas
11. Optimal Investment in Multi-Species Protection
235
References
251
iii
iv
Natural Science Constraints in Environmental and Resource Economics
Acknowledgements
Interdisciplinary research means to bridge the gaps between traditional academic disciplines. This is hardly possible for a single researcher. As this study
aims to integrate perspectives, methods and insights from different natural and
social sciences as well as the humanities, I naturally owe a lot to many scholars
from different academic disciplines – such as economics, ecology, engineering,
physics, mathematics and philosophy – who have contributed in one way or
another to this research.
Chapters 5, 6 and 10 of this study originated from joint research with
Christian Becker, Karin Frank, Frank Jöst, Birgit Müller, Martin Quaas, Jakob
de Swaan Arons and Ralph Winkler. Working together with every single one
of them was a great opportunity, an even greater pleasure, and indispensable
for the kind of interdisciplinary research which essentially makes this study.
A number of people have spared some of their time and discussed drafts
or presentations of one or more of the papers, upon which this study draws. I
am grateful for their critical and constructive comments, as well as stimulating
discussions:
• Geir Asheim, Scott Barrett, Carl Beierkuhnlein, Lucas Bretschger, Erwin
Bulte, Jon Conrad, John Coulter, Salvatore di Falco, John Ehrenfeld,
Malte Faber, Karin Frank, John Harte, Sönke Hoffmann, Frank Jöst,
Giselher Kaule, Bernd Klauer, Andreas Lange, Karl Eduard Linsenmair,
Michel Loreau, Reiner Manstetten, Ulf Moslener, Klaus Nehring, Richard
Norgaard, Charles Perrings, Thomas Petersen, Rüdiger Pethig, Steve Polasky, John Proops, Clemens Puppe, Inge Røpke, Martin Quaas, Till
Requate, Joan Roughgarden, Johannes Schiller, Felix Schläpfer, Armin
Schmutzler, Thomas Schulz, Irmi Seidl, Thomas Sterner, John Tschirhart, Frank Wätzold, Hans-Peter Weikard, Gerhard Wiegleb, Bruce Wilcox, James Wilen, Ralph Winkler, Christian Wissel, Anastasios Xepapadeas, David Zilberman;
• Conference, workshop and seminar participants at Aachen, Ascona, Bad
Honnef, Berkeley, Berlin, Budapest, Cambridge, Castelvecchio, Chemnitz, Cottbus, Darmstadt, Delft, Geneva, Göttingen, Großbothen, Halle,
v
vi
Natural Science Constraints in Environmental and Resource Economics
Heidelberg, Karlsruhe, Leipzig, Lisbon, Mannheim, Monterey, Montreal,
Oberflockenbach, Paris, Rethymno, Santiago de Compostela, Schloss
Wendgräben, Sousse, St. Andreasberg, Stockholm, Strasbourg, Tenerife,
Tutzing, Venice, Versailles, Washington DC, Vienna and Zürich;
• Anonymous reviewers of the journals Ecological Economics, Economic
Theory, Ecosystem Health, Environmental and Resource Economics, Journal of Economic Dynamics and Control, Journal of Environmental Economics and Management, Journal of Industrial Ecology, Natural Resource
Modeling and Resource and Energy Economics;
The University of Heidelberg’s Interdisciplinary Institute for Environmental
Economics provided an ideal platform for conducting this research. I am grateful to the directors of the institute – Malte Faber and Till Requate in particular
– as well as to all researchers, students and visitors for creating an environment
of intellectual openness and scientific excellence. Eva Kiesele and Frank Schwegler have provided valuable research assistance; Dale Adams and Maximilian
Mihm have helped improve language and style; Simone Bauer has helped editing the references.
The initial drafts of Chapters 8, 9 and 11 were written while I was a Visiting
Scholar with the Energy and Resources Group at the University of California,
Berkeley, in the academic year 2001/2002. I am grateful for their hospitality – Dick Norgaard’s in particular – and for the most stimulating research
environment at Berkeley.
Last, not least, I am grateful for financial support from the Deutsche Forschungsgemeinschaft (DFG) under grant BA 2110/1-1 and from the Volkswagen
Foundation under grant II/79 628.
1. Introduction
Most, if not all, environmental problems of our time have their origin in human
economic activity. For example, the production of electricity from fossil fuels
depletes the Earth’s fossil fuel deposits and pollutes the atmosphere with greenhouse gas emissions; the use of water for industrial or agricultural production
and as a medium to dispose of wastes causes pollution of surface waters and
groundwater reservoirs; the consumption of a huge variety of products leaves behind enormous amounts of wastes, some of which are harmful to human health
and ecosystems; the satisfaction of mobility needs by modern traffic systems
and infrastructure destroys natural landscapes and habitat for many biological species. In all these cases, the economic benefits, which are the primary
justification for action, are intimately linked to environmental problems.1
In order to better understand how environmental problems arise from economic activity and how they may be solved, one needs to combine scientific
expertise from the natural sciences and from economics. For, it is the domain
of the natural sciences to analyze ‘nature’, while economics studies ‘the economy’. In this study, I contribute to this interdisciplinary task in a threefold
manner:
1. In Part I, I employ concepts and methods from thermodynamics in order
to study how this natural science puts constraints on the transformation
of energy and matter in the economic process of production.
2. In Part II, I analyze the problem of biodiversity loss and conservation by
combining concepts and methods from ecology and economics.
3. An underlying interest throughout this study is the methodological question of how to integrate concepts and methods from the natural sciences,
such as thermodynamics or ecology, and the social sciences, such as economics. This question is discussed in detail in this introductory chapter,
in order to provide a methodological basis for the actual analysis in Parts I
and II.
1
In conceptual terms, the structural cause behind many modern-day environmental problems is joint production (Baumgärtner et al. 2006). This captures the phenomenon whereby
human action always entails unintended side-consequences.
1
2
Natural Science Constraints in Environmental and Resource Economics
This chapter is organized as follows. Section 1.1 opens the methodological
discussion by characterizing the economic approach to studying economy-environment interactions. Section 1.2 discusses various concepts of nature in
economics, in order to shed more light on this defining element of environmental and resource economics. Section 1.3 then clarifies the role of the natural sciences for environmental and resource economics (Section 1.3.1) and
addresses the challenge of interdisciplinary integration of economics and the
natural sciences (Section 1.3.2). It introduces a fundamental distinction between two approaches to incorporating natural science constraints into environmental and resource economics – method-orientation and problem-orientation
(Section 1.3.3). Furthermore, it justifies the focus of this study on conceptual
analysis (Section 1.3.4).
After the methodological basis is thus prepared, the contents of this study is
introduced in the remainder of the chapter. Section 1.4 previews Part I, which
deals with thermodynamic analysis of economy-environment interactions and
is characterized by method-orientation. Section 1.5 previews Part II, which
deals with biodiversity loss and conservation and is characterized by problemorientation.
1.1
Economics and the Study of
Economy-Environment Interactions
According to a classic definition, economics is ‘the science which studies human behaviour as a relationship between ends and scarce means which have
alternative uses’ (Robbins 1932: 15). This definition has a wide scope and,
consequently, economics approaches a wide range of issues. One of these issues
is the relationship between human economic activity and the natural environment, which is the subject of the sub-discipline of environmental and resource
economics (e.g. Baumol and Oates 1988, Dasgupta and Heal 1979, Hanley et
al. 1997, Hartewick and Olewiler 1998, Kolstad 2000, Siebert 2004 and Tietenberg 2003). In line with Robbins’ definition, the approach of environmental
and resource economics to studying economy-environment interactions is characterized by
(i) a distinction between means – e.g. labor, capital, natural resources, ecosystem goods and services, or the environment’s absorptive capacity for pollutants and wastes – and ends – e.g. maximizing a firm’s profit or social
welfare;
(ii) the idea that means such as natural resources are scarce, which is usually
taken to mean that obtaining and utilizing them carries (opportunity)
Introduction
3
costs (e.g. Debreu 1959: 33, Eatwell et al. 1987);2 and
(iii) the existence of alternatives in using means to achieve ends, which implies
that there is scope for making choices and, at the same time, choices have
to be made about how to best use scarce means. Choice, thus, becomes
the true substance matter of economics.
This characterization has lead to the understanding that economics, including
its sub-field of environmental and resource economics, is essentially about optimization under constraints, with an objective function representing ends and
constraints as an expression of scarcity of means.
The aspect of scarcity allows defining more clearly the field of environmental
and resource economics as a sub-discipline of general economics: environmental
and resource economics studies those areas of optimizing human behavior subject to constraints where constraints are imposed by nature (Fisher 2000: 189).
Examples include the limited stock, concentration and spatial distribution of
mineral resources; the natural growth and interaction of biological resources;
the diffusion, transformation and decay of a pollutant in an environmental
medium; etc. In this view, the laws of nature captured by the environmental natural sciences, such as physics, hydrology, biology, ecology, geology, etc.,
are necessary for environmental and resource economics to gain an adequate
representation of relevant constraints.
This logic justifies an interdisciplinary cooperation between economics and
the natural sciences in the study of economy-environment interactions. But
there are at least three fundamental methodological problems for any such
cooperation (Becker and Baumgärtner 2005: Section 3):
1. The concepts and methods employed in different disciplines of the natural and social sciences – such as thermodynamics, ecology, or economics
– stem from, and are shaped by, very different disciplinary traditions,
cultures and self-images. It is not at all obvious that they are compatible
with each other.
2. Different disciplines have different research interests. What constitutes
an ‘interesting’ question for one discipline may be completely irrelevant
for another discipline, even within a common substantive domain, such
as ‘economy-environment interactions’.
3. It is not even obvious whether different disciplines could agree on the
exact substantive content of ‘economy-environment interactions’. This
requires an answer to the questions ‘What exactly is nature?’ and ‘What
2
For a more detailed discussion of the concept of scarcity of natural resources, goods and
services, see Baumgärtner et al. (in press).
4
Natural Science Constraints in Environmental and Resource Economics
exactly is the economy?’ Even worse, these questions may not have a
precise and unique answer even within each discipline.
As this study takes an economic approach to analyzing economy-environment
interactions, the second problem is obviously solved and the third problem
reduces to the question ‘What exactly is “nature” in the view of economics?’.
In the following section, I will address this question in detail. Section 1.3 then
addresses the first problem.
1.2
Concepts of Nature in Economics
As the concept of nature is crucial for defining the field of environmental and
resource economics as a sub-discipline of general economics, one needs to address the question: ‘What exactly is “nature” in the view of economics?’ As
economics deals primarily with the economy, not with nature, one should not
expect economics to have a clearly defined and encompassing notion of ‘nature’. Yet, there exist a number of different, mostly implicit, notions of nature
in economics (Becker 2005, Biervert and Held 1994, Schefold 2001). They are
embedded in different perspectives on the relationship between nature and the
economy, some of which are discussed in the following.3 Each of these perspectives highlights a particular aspect of this relationship and, thus, expresses a
particular concept of nature.
1.2.1
Nature as Part of the Economy
Natural resources and services have an obvious economic dimension insofar
as they may serve as production factors or directly as consumption goods.
Examples include the utilization of coal and iron ore for the production of
steel, or the appreciation of nature’s beauty by tourists during their vacation.
In their function as production factors or consumption goods, natural goods
share to some extent the general characteristics of any economic good: they are
relatively scarce, substitutable against other natural or manufactured goods,
subject to subjective valuation, and subject to individual or collective allocation
decisions.4
In conceptualizing nature as a set of goods and services which share the
essential characteristics of any other economic good, nature is seen as part
3
This classification follows Becker (2005) as far as the conceptualization of nature as part
of the subject matter of economics is concerned (‘what to explain’). In addition, Becker
(2005) also discusses concepts of nature as a model for understanding the subject matter of
economics (‘how to explain’).
4
One notable difference, which sets natural goods apart from other economic goods, is
that they are often public goods and that property rights are often not well defined.
Introduction
5
of the economy, like any other sector of the economy. It, thus, falls into the
domain of economic decision making. This is the traditional understanding of
environmental and resource economics, as it has emerged in the early twentieth
century as a sub-discipline of general economics (Gray 1913, 1914, Hotelling
1931, Pigou 1912, 1920), based on the methodological foundation of neoclassical
economic theory. It puts the economic decision maker at center stage, and this
procedure also defines ‘nature’: the economic perception of nature is reduced
to those objects and services, and the respective dimension of their physical
existence, that are of value to economic agents.
1.2.2
Nature as a Limit to Economic Activity
When people became aware of the existence of global and long-term environmental problems in the second half of the twentieth century – such as depletion
of the stocks of mineral resources and fossil fuels, land degradation, overfishing
of the oceans, climate change, rupture of the ozone layer, biodiversity loss, etc.
– this challenged the view that natural goods and services are essentially like
any other economic goods and services, and which stresses the manageability of, and human control over, nature. In contrast, it now became apparent
that nature may impose limits to economic activity and, to a considerable
extent, is beyond human control and management.5 The field of ecological economics emerged in the 1960s and 1970s from the insight that the traditional
approach of environmental and resource economics, which considers particular
environmental goods and services that share the general characteristics of any
economic good, was too narrow, and the treatment of nature in the analysis of
economy-environment interactions needed a systematic and more encompassing
approach (Costanza 1989, 1991, Costanza et al. 1997c, Røpke 2004).
A corner stone in the early arguments of ecological economics is the claim
that the laws of thermodynamics, which fundamentally govern the transformation of energy and matter, also govern economic action and economy-environment interactions insofar as these consist of energy/matter-transformations.
According to the laws of thermodynamics, energy and matter cannot be created or destroyed (First Law), and in any transformation of energy and matter
a non-negative amount of entropy is created (Second Law). This should fundamentally constrain the set of feasible economic actions (Ayres 1978, Ayres and
Kneese 1969, Boulding 1966, Faber et al. 1995[1983], Georgescu-Roegen 1971,
Kneese et al. 1972). More recently, a similar line of argument emphasizes the
role of ecological relationships for the functioning and resilience of ecosystems
and, thus, their ‘carrying capacity’ in terms of economic use and impact that
5
This view led to the report on The Limits to Growth by the Club of Rome (Meadows et
al. 1972). As a matter of history, the view that nature is beyond human control and acts as
a limit for human action, governs all pre-modern thinking. It also shows up in the writings
of the classical economists, e.g. Malthus (1798).
6
Natural Science Constraints in Environmental and Resource Economics
ecosystems can withstand before they loose their ability to generate ecosystem
goods and services (Arrow et al. 1995, Daily 1997b, Gunderson and Holling
2002, Perrings 1995b, 2001, Perrings et al. 1995a).
The conceptualization of nature implicit in these arguments is that of a set
of laws of nature which systematically determine what is possible and what is
not – not only in the realm of the natural phenomena, but also as far as economic action and economy-environment interactions are concerned. Thereby,
nature limits the potential scope of economic action. It is not only relatively
scarce, as in the view of environmental and resource economics, but imposes
an absolute scarcity on the economy (Baumgärtner et al., in press). This conception of nature also leads to a modified view on the human economy, which
appears to be limited by, and contingent upon, nature.
1.2.3
The Economy as Part of Nature
One intellectual consequence of recognizing nature as a limit to economic action
is a fundamental change in perspective: nature is no longer seen as part of the
economy but the economy is seen as part of nature.6 Accordingly, the ‘vision’
(in the sense of Schumpeter)7 of ecological economics is that the human economy
is an open subsystem of the larger, but finite, closed, and non-growing system
of non-human nature (e.g. Ayres 1978, Boulding 1966, Daly 1991[1977], Faber
and Proops 1998, Georgescu-Roegen 1971). In this perspective, the dynamics
of economy-environment interactions appears as a co-evolution of two systems
that both have their internal structure and dynamics, and mutually influence
each other’s development (Norgaard 1981, 1984, 1985, 1994).
This view of the economy as part of nature is very encompassing. It includes different aspects of the economy-environment-relationship under a unifying perspective, which appear isolated in the nature-as-part-of-the-economyperspective and the nature-as-a-limit-to-economic-activities-perspective respectively:
• First, nature provides a number of goods and services that may be of
value for, and utilized by, optimizing economic agents. This is the aspect
of the economy-environment-relationship which has been stressed by the
nature-as-part-of-the-economy-perspective (Section 1.2.1) and which is
the focus of traditional environmental and resource economics.
• Second, while these natural goods and services share essential characteristics of other economic goods, they are crucially distinct from the latter
6
Brown (2001: 5) argues that the change of perspective from nature-as-part-of-theeconomy to the-economy-as-part-of-nature amounts to a scientific revolution not unlike the
transition from the geocentric to the heliocentric world view in the Copernicanean revolution.
7
Schumpeter (1954: 42) defines a vision as the ‘preanalytic cognitive act that necessarily
precedes any scientific analysis’.
Introduction
7
in that they come from a geobiophysical environment governed by laws
of nature which are beyond human control and, thus, impose exogenous
(and, in general, not constant) limits to human economic activity. This
is the aspect of the economy-environment-relationship which has been
stressed by the nature-as-a-limit-to-economic-activities-perspective (Section 1.2.2) and which was the focus of early ecological economics.
Thus, nature is conceptualized as a set of goods and services plus a set of laws
of nature governing the provision of these natural goods and services.
1.3
1.3.1
Methodological Position of this Study
The Role of the Natural Sciences
In this study, I adopt the perspective of the economy as part of nature (Section 1.2.3), since this is an encompassing economic perspective on the relationship between human economic activity and nature. As described above, in this
perspective nature provides a number of goods and services which share essential characteristics of other economic goods, but are crucially distinct from the
latter in that they come from a geobiophysical environment governed by laws
of nature.8
Within this perspective, the role of the natural sciences now becomes clearer.
The natural sciences, such as physics, chemistry, biology, hydrology, geology
etc., are necessary for the field of ecological, environmental and resource economics to the extent that their concepts and laws give a clear, systematic
and encompassing description of the characteristics of the goods and services
provided by nature, and of the relationships that govern their generation and
provision. Thereby, they describe the natural world as offering a potential for,
but also as setting limits to, economic action.
The relationship between the natural sciences and economics in the study
of economy-environment interactions then is, in principle, as follows. Concepts
and laws of the natural sciences are essential ingredients in the characterization
and delimitation of the ‘commodity space’ (as far as natural goods and services
are concerned) and the ‘set of feasible economic actions’. That is, they serve
to conceptualize the objects of economic action (as far as natural goods and
services are concerned) and to formulate the constraints on economic action
(as far as they are imposed by laws of nature). Their role is limited to this
particular task. The ranking of feasible actions and the explanation of which
action is chosen by an optimizing economic agent (homo economicus) do not
8
Being based on such a perspective, this study cannot be classified as belonging either to
environmental and resource economics or to ecological economics. Instead, it displays typical
characteristics of both approaches.
8
Natural Science Constraints in Environmental and Resource Economics
require any input from the natural sciences, but are subject to economic analysis proper.9 Thereby, ecological, environmental and resource economics is,
first of all, economics in that it centrally studies optimizing human behavior
under constraints based on the standard concepts and tools of economics; it is
informed by the natural sciences insofar as the formulation of the commodity
space and constraints is concerned.
1.3.2
The Challenge of Interdisciplinary Integration
Insofar as ecological, environmental and resource economics is defined as a
subfield of general economics by the integration of laws and concepts from the
natural sciences, it is inherently interdisciplinary. Hence, the methodological
challenge arises of how exactly to integrate concepts and laws from the natural sciences into an economic analysis. Different procedures and degrees of
interdisciplinary integration are imaginable and have been distinguished (e.g.
Becker and Baumgärtner 2005: Section 3.2):
(i) One potential approach of how to study a subject matter from different
disciplinary perspectives is a multidisciplinary analysis in which different disciplines make statements about the same subject matter, but
they do so in isolation. That is, each discipline addresses the aspects that
it considers relevant, and it does so in its own terminology and based on
its own set of concepts, methods and theories. For example, in a multidisciplinary analysis of greenhouse gas emissions by economists, legal
scientists and atmospheric scientists, the economists would study the optimal allocation of emissions based on their costs and benefits; the legal
scientists would study the restrictions on emissions imposed by existing, national or international, regulations; and the atmospheric scientists
would study the physical or chemical impact of emissions on the state of
the atmosphere. Their results would typically be reported as an additive
compilation of independent disciplinary sub-reports, each written by one
disciplinary sub-group. In such a multidisciplinary analysis, the different
disciplinary contributions are not integrated in any substantial manner;
this task is left to the reader. The question remains open whether the
different disciplines have really studied the same subject matter.
(ii) A method may be transferred from one discipline into another one, where
it is then applied to the substantive domain of the importing discipline
according to the scientific criteria and organizational structures of that
discipline. Such a transfer of method is a unidirectional relation and
9
In contrast, some have argued that seeing the economy, and human action in general,
as fundamentally constrained by nature should make a difference for how to describe and
analyze human behavior (e.g. Costanza et al. 1997a, Daly 1991, 1992a, Faber et al. 1996).
Introduction
9
does not aim at a bidirectional relationship between two disciplines. For
example, economics has adopted the so-called ‘Le Chatelier Principle’
from classical thermodynamics and uses it (under the name of ‘comparative statics’) to study the properties of economic equilibria (Samuelson
1947).
(iii) In an interdisciplinary division of labor different disciplines address
the same subject matter in such a manner that they each base their investigation on their own disciplinary set of concepts, methods and theories,
and exchange results via clearly defined data interfaces. This may be a
recursive procedure. In this approach the interdisciplinary coordination
and cooperation pertains to the input and output of data and results;
it does not cover the internal elements and structure of the disciplinary
analyses. An example is the interdisciplinary analysis of global anthropogenic climate change by coupled simulation models, where demographic
and economic models produce projections about future emission paths;
these serve as input into climate models, which predict climate change;
and the climate data thus obtained are then, again, fed into the economic
models of optimal emission choice.
(iv) While in an interdisciplinary division of labor each discipline retains autonomy over how to set up and carry out its analysis, a closer coordination and cooperation is possible. In an interdisciplinary integrated
analysis the concepts, methods and theories of different disciplines are
closely related and adjusted to each other with regard to the joint interdisciplinary scientific aims. This happens in a discussion process among
scientists that clarifies what disciplinary concepts, methods and theories
are adequate to the joint interdisciplinary endeavor, how they relate to
each other, and how they need to be adjusted to each other with regard
to the interdisciplinary scientific aims. An example is the ecologicaleconomic model analysis of biodiversity management policies, where the
different disciplinary sub-models are adjusted to each other within a joint
research perspective, so that they are formulated in the same functional
terms and operate on similar spatial and temporal scales (Wätzold et al.,
forthcoming).
(v) In the extreme, interdisciplinary integrated analysis may lead to transdisciplinary science, that is, the modification of disciplinary, and the
emergence of new interdisciplinary, concepts, methods or theories. They
are firmly based on established concepts, methods or theories of one particular discipline, which are modified so as to fit with concepts, methods
or theories from another discipline. If the original disciplines do not accept this modification, it may lead to the emergence of a new scientific
10
Natural Science Constraints in Environmental and Resource Economics
discipline which is defined by its subject matter and its concepts, methods
and theories. The reference to the disciplines from which it has emerged
is then purely historical. An example is the field of molecular biology,
which has emerged as an independent discipline from the integration of
concepts, methods and theories from biology, chemistry and physics. But
it is also possible that the interdisciplinary modifications act back on the
original disciplines and leave a permanent impact on them.
(vi) The notion of transdisciplinary problem solving is sometimes used
in an even wider meaning to denote cooperation beyond the boundaries of
science, e.g. with stakeholders or practitioners disposing of non-scientific
knowledge. The discourse with, and participation of, such social actors
and groups should help to identify relevant research questions and conceptual structures of some problem under study, which later on facilitates the
adoption and implementation of solutions. An example is the search for,
and sustainable management of, pharmaceutical substances embedded
in the naturally occurring biodiversity.10 This endeavor brings together
academic scientists – such as biologists, chemists and physiologists – and
indigenous people with their traditional knowledge about the medicinal
impact of local plants.
One cannot generally say that one of these approaches is superior to the others.
All of them have merit in some respect and shortcomings in some other respect.
Which approach to follow when combining insights from different disciplines
depends on the scientific aims and the subject matter to be studied.
For the purpose of ecological, environmental and resource economics, and
the purpose of this study, approach (i) will clearly not do because it cannot
guarantee that exactly those natural science insights which are relevant for
economics are taken up. This would require at least some minimal exchange
with economics and, thus, go beyond a multidisciplinary analysis. Also, a
multidisciplinary approach cannot guarantee that natural science insights are
put forward in a manner compatible with the conceptual structure and terminology of economics. This requires an interdisciplinary division of labor or
integrated analysis of economics and the natural sciences. On the other hand,
the approaches (v) and (vi) are very ambitious and go far beyond standard
science. So, the analysis in this study is interdisciplinary in the sense of approaches (iii) and (iv). While the procedure in Part I (Thermodynamics) is
predominantly characterized by an interdisciplinary division of labor between
economics and thermodynamics, the procedure in Part II (Biodiversity) is predominantly characterized by an interdisciplinary integrated analysis between
economics and ecology. This will be explained in more detail in the following.
10
See the detailed discussion of this approach to ‘bio-prospecting’ on page 127.
Introduction
1.3.3
11
Method Orientation and Problem Orientation
Any scientific analysis studies a certain subject matter (problem) with a certain
toolbox of concepts, methods and theories (method ). Accordingly, any scientific research program may be driven either by the primary purpose of better
understanding a certain subject matter (solving a problem) or by the primary
purpose of further elaborating and advancing the methodological toolbox of
science (enhancing a method ).11 The same goes for interdisciplinary science:
the purpose of interdisciplinary science, or the challenge arising from interdisciplinary cooperation, may be primarily related either to solving a particular
problem or to enhancing a particular scientific method.
In this study, I follow both approaches to the interdisciplinary integration
of natural science constraints into ecological, environmental and resource economics: Part I of this study is primarily motivated by enhancing a particular
method – thermodynamic analysis in ecological, environmental and resource
economics; Part II is primarily motivated by contributing to the solution of a
particular problem – biodiversity loss and conservation. Both the particular
method and the particular problem under study are typical examples for the
respective approach. In each approach, the interdisciplinary challenge of how
to integrate natural science constraints into ecological, environmental and resource economics has a different character and, therefore, entails a different kind
of solution. Hence, Parts I II of this study yield complementary insights into
how to conceptualize natural science constraints in ecological, environmental
and resource economics.
The method-oriented approach
In the method-oriented approach of Part I of this study, the method of thermodynamics serves to conceptualize constraints on economic action, in particular
on production processes which can be described as a transformation of energy
and matter. The two most elementary constraints stem from the first and
second laws of thermodynamics: conservation of mass (First Law) and irreversibility (Second Law). These constraints are exogenous to economic action
and fixed. Therefore, in order to capture thermodynamic constraints in ecological, environmental and resource economics, an interdisciplinary division of
labor between thermodynamics and economics is possible. The task of thermodynamics in this division of labor is to conceptualize and formalize these
constraints so that they are compatible with the conceptual structure and terminology of economics. For instance, the laws of thermodynamics can impose
restrictions on the set of feasible economic allocations. Once this first step
has been completed, it is the task of economics to then study which alloca11
Of course, the motivation of a scientific research program may also be a combination of
these two polar cases.
12
Natural Science Constraints in Environmental and Resource Economics
tions are, and should be, chosen individually and collectively. Since this second
step of the analysis does not have any repercussions on the first step, a clear
division of labor between thermodynamics and economics is possible. As a result, the method-oriented approach is much closer to the categorically distinct
approaches of the two disciplines of thermodynamics and economics than the
problem-oriented approach.
The problem-oriented approach
Part II of this study follows a problem-oriented approach to studying biodiversity loss and conservation. This is a complex and multifarious real-world
problem at the intersection of economies and ecosystems. Being a real-world
phenomenon, it first has to be translated into scientific terms before it can
be studied by scientific means. This is a challenge that does not occur in the
method-oriented approach, as the starting point of the method-oriented approach is already – by definition – within the realm of science. Any real-world
problem can be translated such that it falls into the domain of one discipline
or the other, or a certain set of disciplines. While there is considerable freedom (even arbitrariness) as to how to translate a real-world phenomenon into
scientific terms, it then is up to the different disciplines involved in an interdisciplinary problem-oriented analysis to make sure that their respective contributions really deal with the same phenomenon and fit with each other. In
most cases, an interdisciplinary division of labor will not do for that purpose.
Rather, a problem-oriented approach most often requires an interdisciplinary
integrated analysis.
In the case of the problem-oriented analysis of biodiversity loss and conservation, there is another reason why an interdisciplinary division of labor will
not do, but an interdisciplinary integrated analysis is required. As the problem
is at the intersection of two mutually interacting systems, the ecological system
and the economic system, there are many potential feedbacks from one system
onto the other. Due to these feedbacks, a clear division of labor between the
natural sciences and economics is not possible, because in a division of labor one
would loose the feedbacks from the analysis. Instead, an encompassing analysis of complex problems, such as biodiversity loss and conservation, requires
an interdisciplinary integrated analysis (Wätzold et al., forthcoming).
The natural science constraints, which in the case of biodiversity mainly
come from ecology, can no longer be taken to be fixed and exogenous to ecological, environmental and resource economics. Instead, while ecology still
imposes constraints on economic action, the choice of a particular allocation
by an economic agent has repercussions on the ecological system, which, in
turn, influences the ecological constraints. These feedbacks have to be taken
into account in a problem-oriented ecological-economic analysis of biodiversity
loss and conservation. As a consequence, the analysis of biodiversity loss and
Introduction
13
conservation in Part II of this study is an interdisciplinary integrated analysis,
in which the economic system is modelled based on economic concepts and relationships, and the ecological system is modelled based on ecological concepts
and relationships, with both sets of concepts and relationships highly adapted
to each other and to the joint research aims, so that the impact of economic
action onto the ecological system, and the resulting feedback for the set of
feasible economic actions, can be studied explicitly.
1.3.4
Conceptual Analysis
This study is a conceptual (as opposed to: empirical or applied) analysis. It
focuses on the definition, clarification and interdisciplinary application of concepts that are of central relevance for structuring the analysis of economyenvironment interactions from the interdisciplinary perspective of economics
and the natural sciences. This includes the exploration of relationships among
these concepts with the help of conceptual models. It leads to the development of policy recommendations on a conceptual level, e.g. for the control of
emissions (Chapter 6) or for ecosystem management (Chapter 11). Examples
of concepts, which are central to this study, include conservation of mass, irreversibility and joint production in Part I; and biodiversity, ecosystem services
and insurance in Part II.
Conceptual analysis is an accepted approach in modern economics, and
has been advocated before on several occasions, also for the field of ecological,
environmental and resource economics. For example, in their pioneering book
on Economic Theory and Exhaustible Resources, Dasgupta and Heal (1979:
9–10) justify conceptual analysis as follows:
What one aims at in constructing an economic model, whose purpose is the development of understanding at a basic conceptual level
(as opposed for example to the prediction of the values to be assumed by a particular set of variables at a future date), is to strip
away detail and in the process sacrifice precision, in order to grasp
at general principles which would be obscured but by no means
invalidated by the inclusion of detail. What one aims at in other
words is the construction of a framework which is simple enough to
reveal the principles at work but whose basic structure is robust to
the kinds of additions and extensions generally needed to implement
the analysis in any particular situation.
Beyond its use in individual scientific disciplines, conceptual analysis is necessary and essential in order to lay a solid basis for the interdisciplinary integration of established scientific disciplines. It helps to identify the potential
as well as the pitfalls of interdisciplinary integration and is a prerequisite for
asking relevant and meaningful research questions (cf. Section 1.3.2).
14
Natural Science Constraints in Environmental and Resource Economics
After the methodological basis is now prepared, the remaining sections of this
chapter introduce the contents of this study. Section 1.4 previews Part I (Thermodynamics), while Section 1.5 previews Part II (Biodiversity).
1.4
The Method of Thermodynamics
This section gives a brief outline of the analysis and results in Part I of this
study (Chapters 2–6), which deals with thermodynamic analysis of economyenvironment interactions and is characterized by method-orientation. Thermodynamics is the branch of physics that deals with macroscopic transformations of energy and matter. The origins of thermodynamics are to be
found in the nineteenth century when practitioners, engineers and scientists
like James Watt (1736–1819), Sadi Carnot (1796–1832), James Prescott Joule
(1818–1889), Rudolph Clausius (1822–1888) and William Thomson (the later
Lord Kelvin, 1824–1907) wanted to understand and increase the efficiency with
which steam engines perform useful mechanical work. From the beginning, this
endeavor has combined the study of natural systems and the study of engineered systems – created and managed by purposeful human action – in a very
peculiar way, which is rather unusual for a traditional natural science such as
physics.
Not surprisingly then, the laws of thermodynamics were found by economists
to be concepts with considerable implications for economics. In the late 1960s
and early 1970s economists discovered the relevance of thermodynamics for
environmental and resource economics (Pethig 2003, Spash 1999: 418, Turner
1999a: Section 2). For instance, economists like Kenneth Boulding (1966),
Robert Ayres and Allen Kneese (1969), and Nicolas Georgescu-Roegen (1971)
turned to thermodynamics when they wanted to analyze economy-environment
interactions in an encompassing way, and root the economy in its biogeophysical basis analytically.
1.4.1
Thermodynamic Analysis: Rationale, Concepts, and Caveats
Chapter 2 opens the discussion of how to integrate thermodynamic analysis
into ecological, environmental and resource economics. It lays out the fundamental rationale of this endeavor, which is crucially based on the duality
between real and monetary descriptions of economic action, and sketches its
historical origins. It also addresses the question ‘How can thermodynamic
concepts, laws and results be incorporated in a fruitful manner into economic
analysis?’ This has been attempted in four basic ways, which are very different
in the intellectual approach taken:
1. Isomorphism of formal structure,
Introduction
15
2. Analogies and metaphors,
3. Energy, entropy and exergy theories of value,
4. Thermodynamic constraints on economic action.
It is the last one of these approaches, which is taken in this study. It builds on
a clear division of labor between the disciplines of thermodynamics and economics. The laws of thermodynamics are used to capture the constraints on
transformations of energy and matter. Their role is limited to this particular
task. Based on this conceptualization of constraints, methods and concepts
from economics are then used to study allocations in an economy which result
from the optimizing behavior of firms and households, e.g. profit-maximizing
resource-extraction and production firms as well as utility-maximizing households purchasing the consumer goods produced. This approach can be operationalized directly, and is empirically meaningful for ecological, environmental
and resource economics. It lends itself quite naturally to modeling.
Chapter 2 also surveys the literature on implications and insights of thermodynamic analysis in ecological, environmental and resource economics. The
chapter concludes by assessing the role of thermodynamics for ecological, environmental and resource economics, and for the discussion of sustainability.
There is a brief and basic introduction into the elementary concepts and laws
of thermodynamics in the Appendix.
1.4.2
The Inada Conditions for Material Resource Inputs
Reconsidered
Chapter 3 formally explores one particular implication that the thermodynamic law of conservation of mass, the so-called Materials-Balance-Principle,
has for modeling production. It is shown that the marginal product as well as
the average product of a material resource input are bounded from above. This
means that the usual Inada conditions (Inada 1963), when applied to material
resource inputs, are inconsistent with a basic law of nature. This is important
since the Inada conditions are usually held to be crucial for establishing steady
state growth under scarce exhaustible resources.
While the advocates of a thermodynamic-limits-to-economic-growth perspective (e.g. Boulding 1966, Daly 1991[1977], Georgescu-Roegen 1971) usually
stress the universal and inescapable nature of limits imposed by laws of nature, pro-economic-growth advocates usually claim that there is plenty of scope
for getting around particular thermodynamic limits by substitution, technical
progress and ‘dematerialization’ (e.g. Beckerman 1999, Smulders 1999, Stiglitz
1997). The latter therefore often conclude that, on the whole, thermodynamic
constraints are simply irrelevant for economics. This chapter takes a more
differentiated stand, by analyzing in detail
16
Natural Science Constraints in Environmental and Resource Economics
(i) what exactly are the implications of thermodynamics for modeling production at the level of a single production process, and
(ii) how these constraints carry over to the level of aggregate production,
considering that there is scope for substitution in an economy between
different resources and different production technologies.
1.4.3
Temporal and Thermodynamic Irreversibility in Production
Theory
From a physical point of view, irreversibility is an essential dynamic feature of
real production. Therefore, it should be properly taken into account in dynamic
analyses of production systems. The idea of irreversibility can be rigorously
rooted in the laws of thermodynamics (Kondepudi and Prigogine 1998: 84ff, Zeh
2001). The importance of thermodynamic irreversibility, and the physicists’
preoccupation with this concept, lies in the fact that it precludes the existence
of perpetual motion machines, that is, devices which use a limited reservoir of
available energy to perform work forever (Second Law of Thermodynamics). It
is an everyday experience that no such thing as a perpetual motion machine
exists. In order to make this insight accessible to economic analysis, and to
the study of long term economy-environment interactions, it is necessary to
adequately represent thermodynamic irreversibility as a constraint for economic
action (Georgescu-Roegen 1971).
Economists have devoted some effort to incorporating irreversibility into
production theory. However, irreversibility has often been introduced into the
theory as an ad-hoc-assumption. As a result, the assumption did not always
achieve what it actually should achieve from a thermodynamic point of view,
namely to imply irreversibility of the system’s evolution as stated by the Second
Law of Thermodynamics.
Chapter 4 introduces a formal and rigorous definition of thermodynamic
irreversibility, which is (i) sound from a physical point of view and (ii) formulated such that it is compatible with formal modelling in economic production
theory. In order to assess, whether – and to what extent – different notions
of irreversibility from production theory capture thermodynamic irreversibility, two prominent irreversibility concepts – the one due to Koopmans (1951b)
and the one due to Arrow-Debreu (Arrow and Debreu 1954, Debreu 1959) –
are reexamined against the definition of thermodynamic irreversibility. It is
shown that Koopmans’ notion of irreversibility fully captures thermodynamic
irreversibility, and that the notion of Arrow-Debreu, which has become the
standard one in economic theory, does not capture thermodynamic irreversibility but only the weaker aspect of temporal irreversibility. This means, the
standard irreversibility concept of production theory is too weak to be in full
accordance with the laws of nature.
Introduction
1.4.4
17
Necessity and Inefficiency in the Generation of Waste
It has been argued, based on the thermodynamic laws of mass conservation and
entropy generation, that in industrial production processes the occurrence of
waste is as necessary as the use of material resources (Ayres and Kneese 1969,
Faber et al. 1998, Georgescu-Roegen 1971).12 On the other hand, it seems to
be quite obvious that the sheer amount of waste currently generated in modern
industrial economies is to some extent due to various inefficiencies and might,
in principle, be reduced.
Chapter 5 discusses to which extent the occurrence of waste is actually
an unavoidable necessity of industrial production, and to which extent it is
an inefficiency that may, in principle, be reduced. For that sake, the laws of
thermodynamics are employed as an analytical framework within which results about current ‘industrial metabolism’ (Ayres and Simonis 1994) may be
rigorously deduced in energetic and material terms. It is demonstrated that
the occurrence of waste by-products is an unavoidable necessity in the industrial production of desired goods. While waste is thus an essential qualitative
element of industrial production, the quantitative extent to which waste occurs may vary within certain limits according to the degree of thermodynamic
(in)efficiency with which these processes are operated. The chapter discusses
the question of which proportion of the amount of waste currently generated
is due to thermodynamic necessity, and which proportion is due to thermodynamic inefficiency.
1.4.5
Optimal Dynamic Scale and Structure of a Multi-Pollution
Economy
While Chapters 3–5 have dealt with the implications from thermodynamics for
modelling the production process at both the micro- and macro-level, the question of optimal allocation of resources has not been addressed so far. Chapter 6
closes this gap. It takes as its starting point a thermodynamic representation
of production as joint production of consumption and environmental pollution,
and explores the implications for optimal macroeconomic dynamics.
Chapter 6 looks into the coupled environmental-economic dynamics of a
multi-sector-multi-pollution-economy. It addresses the following questions: How
should the macroeconomic scale and structure change over time in response to
the dynamics of environmental pollution?13 Is this dynamic process monotonic
over time, or can a trade-off between long-run and short-run considerations
12
For example, Georgescu-Roegen (1975: 357) has argued that ‘waste is an output just as
unavoidable as the use of natural resources’.
13
Scale means the overall level of economic activity, measured by total factor input; and
structure means the composition of economic activity, measured by relative factor inputs to
different sectors.
18
Natural Science Constraints in Environmental and Resource Economics
(e.g. lifetime versus harmfulness of pollutants) induce a non-monotonic economic dynamics? What is the time scale of economic dynamics (i.e. change of
scale and structure), and how is it influenced by the different time scales and
constraints of the economic and environmental systems? These questions are
relevant for the current policy discussion on the sustainable biophysical scale
of the aggregate economy relative to the surrounding natural environment (e.g.
Arrow et al. 1995, Daly 1992a, 1996, 1999), and how economic policy should
promote structural economic change as a response to changing environmental
pressures (e.g. de Bruyn 1997, Winkler 2005).
The analysis shows that along the optimal time-path (i) the overall scale
of economic activity may be less than maximal, (ii) the time scale of economic
dynamics (change of scale and structure) is mainly determined by the longestlived pollutant, (iii) the optimal control of emissions may be non-monotonic. In
particular the last result raises important questions about the design of optimal
environmental policies.
1.5
The Problem of Biodiversity Loss and
Conservation
This section gives a brief outline of the analysis and results in Part II of this
study (Chapters 7–11), which deals with biodiversity loss and conservation and
is characterized by problem-orientation. Biological diversity (or ‘biodiversity’,
for short), which has been defined as ‘the variability among living organisms
from all sources ... and the ecological complexes of which they are part’ (CBD
1992), is valuable for humans for a number of reasons. Many species have direct
use value as food, fuel, construction material, industrial resource or pharmaceutical substance. Biodiversity also has an important indirect use value in so
far as entire ecosystems perform valuable services such as nutrient cycling, control of water runoff, purification of air and water, soil regeneration, pollination
of crops and natural vegetation, control of pests and diseases, or local climate
stabilization (Daily 1997b, Millennium Ecosystem Assessment 2005). These
ecosystem services can only be provided by more or less intact ecosystems and
result from the complex – and up to now not well understood – interplay of
many different species in these ecosystems (Holling et al. 1995, Hooper et al.
2005, Kinzig et al. 2002, Loreau et al. 2001, 2002b, Schulze and Mooney 1993,
Tilman 1997a).
Biodiversity is currently being lost at rates that exceed the natural extinction rates of the past by a factor of somewhere between 100 and 1,000 (Watson
et al. 1995b). This is one of the most eminent environmental problems of our
time (Wilson 1988). By now, the international community has acknowledged
Introduction
19
the problem of biodiversity loss, and the need to enact policies to halt or even
reverse this problem. For example, in June 1992, the Convention on Biological Diversity was signed by 156 states at the United Nations Conference on
Environment and Development in Rio de Janeiro, Brazil, with the aim of safeguarding the sustainable conservation and use of biodiversity at the global level
(CBD 1992).
1.5.1
Biodiversity as an Economic Good
Chapter 7 opens the discussion of biodiversity loss and conservation by addressing, on a fundamental level, the question of what economics can contribute to an encompassing discussion of biodiversity loss and conservation.
More specifically, it addresses the following questions:
(i) In what sense can one think of biodiversity as an economic good?
(ii) In what sense does biodiversity have economic value?
(iii) What can economic analysis contribute to the explanation of biodiversity
loss?
(iv) What is the relevance of economic valuation for biodiversity conservation?
Discussing these questions helps to clarify the conceptual foundations upon
which an ecological-economic analysis of biodiversity loss and conservation is
possible. At the same time, it sheds light on the question of how exactly the
two disciplines of economics and ecology need to interact in order to generate
fruitful and relevant contributions to this analysis.
The chapter is written from an economic perspective and serves as a survey
of the relevant literature. As its starting point, it takes the hypothesis that
biological diversity can be thought of as an economic good which has economic
value. This hypothesis is vindicated in detail. Its fruitfulness is then tested
by applying it to explain the large-scale loss of biodiversity currently observed
and to develop recommendations for biodiversity conservation. Viewing biodiversity as an economic good, which has economic value, makes obvious the
potential and limits of economics as an academic discipline for the discussion
of biodiversity loss and conservation. These insights form the working basis of
the remaining chapters of Part II, which display different degrees of interdisciplinary integration of economics with ecology.
1.5.2
Ecological and Economic Measures of Biodiversity
For analyses of how biodiversity contributes to ecosystem functioning, how it
enhances human well-being, and how these services are currently being lost,
a quantitative measurement of biodiversity is crucial. Ecologists, for that
20
Natural Science Constraints in Environmental and Resource Economics
sake, have traditionally employed different concepts such as species richness,
Shannon-Wiener-entropy, or Simpson’s index (see e.g. Magurran 2004). Recently, economists have added to that list measures of (bio)diversity that are
based on pairwise dissimilarity between species (e.g. Weitzman 1992, 1998) or,
more generally, weighted features of species (Nehring and Puppe 2002, 2004).
In Chapter 8, I give a review and conceptual comparison of the two broad
classes of biodiversity measures currently used, the ecological ones and the economic ones. It turns out that the two classes are distinct by the information
they use for constructing a diversity index. While the ecological measures use
the number of different species in a system as well as their relative abundances,
the economic ones use the number of different species as well as their characteristic features. In doing so, the two types of measures aim at characterizing
two very different aspects of the ecological-economic system. The economic
measures characterize the abstract list of species existent in the system, while
the ecological measures target the actual, and potentially unevenly distributed
allocation of species.
I argue that the underlying reason for this difference is in the philosophically
distinct perspective on diversity between ecologists and economists. Ecologists
traditionally view diversity more or less in what may be called a ‘conservative’
perspective, while economists predominantly adopt what may be called a ‘liberal’ perspective on diversity (Kirchhoff and Trepl 2001). In the conservative
view, which goes back to Leibniz and Kant, diversity is an expression of unity.
By viewing a system as diverse, one stresses the integrity and functioning of the
entire system. The ultimate concern is with the system at large. In this view,
diversity may have an indirect value in that it contributes to certain overall
system properties, such as stability, productivity or resilience at the system
level. In contrast, in the liberal view, which goes back to Descartes, Hume and
Locke, diversity guarantees the freedom of choice for autonomous individuals
who choose from a set of diverse alternatives. The ultimate concern is with
the well-being of individuals. In this view, diversity of a choice set has a direct
value in that it allows individuals to make a choice that better satisfies their
individual and subjective preferences.
The question of how to measure biodiversity, thus, is ultimately linked to
the question of what is biodiversity good for. Do we see biodiversity as valuable
for individuals who want to make a choice from a diverse resource base, e.g.
when choosing certain desired genetic properties of crops or pharmaceutical
substances? Or do we see it as valuable for overall ecosystem functioning, e.g.
out of a concern for conserving certain desired ecosystem services such as water
purification or soil regeneration? Of course, there is a continuous spectrum in
between these two extremes. But in any case, so the conclusion of this chapter,
the measurement of biodiversity requires a prior normative judgment as to what
purpose biodiversity serves in ecosystems and economic systems.
Introduction
1.5.3
21
The Insurance Value of Biodiversity in the Provision of
Ecosystem Services
Biodiversity is useful and valuable to humans for a number of reasons (see the
discussion in Chapter 7). One particular reason is that biodiversity provides
insurance by stabilizing the provision of ecosystem services which are being
used by risk-averse economic agents. In Chapter 9, I present a conceptual
ecological-economic model that combines (i) ecological results about the relationships between biodiversity, ecosystem functioning, and the provision of
ecosystem services with (ii) economic methods to study decision-making under
uncertainty. In this framework I (1) determine the insurance value of biodiversity, (2) study the optimal allocation of funds in the trade-off between
investing into biodiversity protection and the purchase of financial insurance,
and (3) analyze the effect of different institutional settings in the market for
financial insurance on biodiversity protection.
The conclusion from this analysis is that biodiversity can be interpreted as
a form of natural insurance for risk averse ecosystem managers against the overor under-provision with ecosystem services, such as biomass production, control
of water run-off, pollination, control of pests and diseases, nitrogen fixation, soil
regeneration etc. Thus, biodiversity has an insurance value, which is a value
component in addition to the usual value arguments (such as direct or indirect
use or non-use values, or existence values) holding in a world of certainty. This
insurance value should be taken into account when deciding upon how much
to invest into biodiversity protection. It leads to choosing a higher level of
biodiversity than without taking the insurance value into account, with a higher
degree of risk aversion leading to a higher optimal level of biodiversity. As far
as the insurance function is concerned, biodiversity and financial insurance
against income risk, e.g. crop yield insurance, may be seen as substitutes. If
financial insurance is available, a risk averse ecosystem manager, say, a farmer,
will partially or fully substitute biodiversity’s insurance function by financial
insurance, with the extent of substitution depending on the costs of financial
insurance. Hence, the availability, and exact institutional design, of financial
insurance influence the level of biodiversity protection.
1.5.4
Insurance and Sustainability through Ecosystem
Management
As shown in Chapter 9, biodiversity has an insurance value which is relevant
for decisions about how to manage ecosystems. While the analysis in Chapter 9
was based on a very simple and stylized ecological-economic model, in order to
focus on the conceptual structure of the argument, Chapter 10 develops this
argument further by looking in detail at a realistic case: grazing management
in semi-arid rangelands. Livestock farmers in semi-arid regions make use of
22
Natural Science Constraints in Environmental and Resource Economics
the ecosystems’ insurance function by choosing grazing management strategies
so as to hedge against their income risk which stems from the stochasticity of
precipitation.
The analysis in Chapter 10 is based on a dynamic and stochastic ecologicaleconomic model of grazing management in semi-arid rangelands. The non-equilibrium ecosystem is driven by stochastic precipitation. A risk averse farmer
chooses a grazing management strategy under uncertainty so as to maximize
expected utility from farming income. Grazing management strategies are rules
about which share of the rangeland is given rest depending on the actual rainfall
in that year. In a first step, the farmer’s short-term optimal grazing management strategy is determined. It is shown that a risk-averse farmer chooses a
strategy so as to obtain insurance from the ecosystem: the optimal strategy
reduces income variability, but yields less mean income than possible. In a
second step, the long-run ecological and economic impact of different strategies
is analyzed. The conclusion is that the more risk-averse a farmer is, the more
conservative and sustainable is his short-term optimal grazing management
strategy, even if he has no specific preference for the distant future.
1.5.5
Optimal Investment in Multi-Species Protection
From the discussion in Chapter 7 it has become apparent that biodiversity is
useful and valuable to humans for many reasons, with one particular reason
– its insurance function – discussed in detail in Chapters 9 and 10. It has
also become apparent that biodiversity is currently being lost at, on average,
suboptimally high rates. This raises the question of how to protect biodiversity
in a manner that is ecologically effective and economically efficient.
In Chapter 11, I contribute to this discussion by studying optimal investment in multi-species protection when species interact in an ecosystem. The
analysis is based on a model of stochastic species extinction in which survival
probabilities are interdependent. Individual species protection plans can increase a species’ survival probability within certain limits and contingent upon
the existence or absence of other species. Protection plans are costly and the
conservation budget is fixed. It is assumed that human well-being depends
solely on the services provided by one particular species, but other species contribute to overall ecosystem functioning and thus influence the first species’
survival probability.
The analysis shows that taking into account species interactions in an
ecosystem is crucial for the optimal allocation of a conservation budget. Compared with policy recommendations obtained under the assumption of independent species, interactions in an ecosystem can reverse the rank ordering of
spending priorities among species conservation projects. Hence, an approach
to species protection that is efficient in terms of both species conservation and
budget resources should be based on a multi-species framework and should
Introduction
23
take into account the basic underlying ecological relations. Another interesting
result is that even if biological conservation decisions are exclusively derived
from a utilitarian framework, with species interaction it may be optimal to
invest in the protection of species that do not directly contribute to human
well-being. This is due to their role for overall ecosystem functioning and for
safeguarding the existence of those species that are the ultimate target of environmental policy. The conclusion is that effective species protection should
go beyond targeting individual species, and consider species relations within
whole ecosystems as well as overall ecosystem functioning.
24
Natural Science Constraints in Environmental and Resource Economics
PART I
Thermodynamics
2. Thermodynamic Analysis in Ecological,
Environmental and Resource Economics:
Rationale, Concepts, and Caveats∗
2.1
Introduction
Integrating methods and models from thermodynamics and from economics
promises to yield encompassing insights into the nature of economy-environment
interactions. At first sight, the division of labor between thermodynamics and
economics seems obvious. Thermodynamics should provide a description of
societies’ physical environment, while economics should provide an analysis
of optimal individual and social choice under the restriction of environmental
scarcities.
But the task is more difficult. Being a branch of physics, thermodynamics
is a natural science. It explains the world in a descriptive and causal, allegedly value-free manner. On the other hand, economics is a social science.
While it pursues descriptive and causal (so-called ‘positive’) explanations of
social systems to a large extent, it also has a considerable normative dimension. Valuation is one of its basic premises and purposes. Bringing together
thermodynamics and economics in a common analytical framework therefore
raises all kinds of questions, difficulties and pitfalls.
This chapter discusses the rationale, concepts, and caveats for integrating
thermodynamic analysis into ecological, environmental and resource economics.
Section 2.2 lays out the fundamental rationale of this endeavor and sketches its
historical origins. Section 2.3 identifies different approaches to incorporating
thermodynamic concepts into economic analysis and assesses their respective
potential for ecological, environmental and resource economics. Section 2.4
surveys various implications and insights that thermodynamic analysis has already yielded for ecological, environmental and resource economics. Section 2.5
∗
Revised version of ‘Thermodynamic Models’, previously published in J. Proops and P.
Safonov (eds), Modelling in Ecological Economics, Cheltenham: Edward Elgar, 2004, pp.
102–129.
27
28
Natural Science Constraints in Environmental and Resource Economics
concludes by assessing the role of thermodynamics for ecological, environmental and resource economics, and for the discussion of sustainability. There is a
brief and basic introduction into the elementary concepts and laws of thermodynamics in the Appendix.
2.2
2.2.1
Fundamental Rationale and Historical Origins
Different Perspectives on Economy-Environment
Interactions
When economists started to analyze the flow of resources, goods, services and
money in an economy, the picture was pretty simple: there are two groups
of economic agents, consumers and producers; producers deliver goods and
services to consumers, and consumers provide the resources with which they
are endowed, labor in particular, to producers. Thus, there is a circular flow
of commodities in an economy. There is an equivalent circular flow of money
counter to that primary flow, as consumers pay money to producers for the
goods they consume, and producers remunerate the labor force they receive
from the consumers/laborers.1
Since the two corresponding flows, the primal flow of real commodities
and the dual flow of monetary compensation, are exactly equivalent, it seems
superfluous to always study both of them when analyzing economic transactions
and allocations. Hence, the convention was established in economics to focus on
the monetary flow. The current system of national economic accounts, which
is meant to be a full representation of economic activity in an economy over
one time period, therefore captures all transactions in monetary units, e.g. the
provision of labor and capital, the trading of intermediate goods and services
between different sectors of the economy, and final demand for consumer goods.
Of course, this picture is too simple. It neglects the use of natural resources
and the emission of pollutants and wastes. Both activities are unavoidable aspects of economic action (see Chapter 5 below). In the early twentieth century,
the subdiscipline of environmental and resource economics emerged to deal with
the question of how to account, in an economic sense, for the use of natural
resources on the one hand and the emission of pollutants and wastes on the
other (Gray 1913, 1914, Hotelling 1931, Pigou 1912, 1920). The picture now
appeared as follows: there is a circular flow – actually: two equivalent circular flows – between consumers and producers which form the core of economic
activity. In addition, there is an inflow of natural resources and an outflow of
1
Later, this system was extended to include savings and investment, as well as imports
and exports.
Thermodynamic Analysis: Rationale, Concepts, and Caveats
29
emissions and wastes. Thus, a linear throughflow of energy and matter drives
the circular flow of economic exchange.
A further step in the development of thinking about economy-environment
interactions, was the insight that the inflow of natural resources (resource economics) and the outflow of emissions and wastes (environmental economics) are
not independent. Obviously, these two flows are linked by economic activity,
i.e. economic activity transforms natural resources into emissions and wastes.
But these two flows are also linked because they originate and terminate in the
natural geobiophysical environment. For example, environmental pollutants
released into natural ecosystems may impair the ecosystems’ ability to produce
the ecosystem goods and services, which are then used as a natural resource by
the economy. This means, the extraction of natural resources, the production
of goods and services within the economy, as well as the emission of pollutants and wastes all happen within the system of the natural geobiophysical
environment.
This is the ‘vision’ (in the sense of Schumpeter)2 of ecological economics:
ecological economics views the human economy as an open subsystem of the
larger, but finite, closed, and non-growing system of non-human nature (Ayres
1978, Boulding 1966, Daly 1991[1977], Faber and Proops 1998, GeorgescuRoegen 1971, and many more). In this view, the human economy is a part
of nature. In contrast, in the view of traditional environmental and resource
economics Nature is treated as a part of the human economy. Both ‘resources’
and ‘environment’ are treated as additional economic sectors in the system of
national economic accounts, and flows to and from these sectors are accounted
for in monetary units.3
2.2.2
Duality Between the Real and Monetary Descriptions and
the Role of Thermodynamics
Environmental and resource economics faced one conceptual problem from the
very beginning. Economic analysis, including environmental and resource economics, is based on the idea of duality (i.e. equivalence) between the flow of
real commodities and services (measured in physical units) and an equivalent
value flow (measured in monetary units), and consequently focuses on the value
dimension. But the inflow of natural resources, as well as the outflow of emissions and waste, do not have an apparent value dimension. Markets do not
indicate these values, as markets often do not exist in this domain. And where
2
Schumpeter (1954: 42) defines a vision as the ‘preanalytic cognitive act that necessarily
precedes any scientific analysis’.
3
Brown (2001: 5) argues that the change of perspective from nature-as-part-of-theeconomy to the-economy-as-part-of-nature amounts to a scientific revolution not unlike the
transition from the geocentric to the heliocentric world view in the Copernicanean revolution.
30
Natural Science Constraints in Environmental and Resource Economics
they exist, the resulting values are distorted due to ubiquitous externalities and
public goods.
As a result, the valuation of natural goods and services has to be set up
explicitly as a non-market process, and elaborate theories and techniques have
been proposed for this purpose.4 All these techniques require, to a greater or
lesser extent, an adequate, prior description – in real terms – of the particular commodity or service to be valued. In other words, before individuals or
society can value something, they have to have an adequate idea about what
exactly that something is. This holds, in particular, for the energy and material
resources used in production as well as for the emissions and wastes generated
as by-products of desired goods.
And here lies the relevance of thermodynamics. Being the branch of physics
that deals with transformations of energy and matter, thermodynamics is an
appropriate foundation in the natural sciences to provide a description in real
terms of what goes on when humans interact with the non-human environment. In particular, thermodynamics captures the energy/matter dimension
of economy-environment interactions. Thus, it is a necessary complement and
prerequisite for economic valuation.
2.2.3
Historical Origins of Thermodynamic Analysis in Ecological,
Environmental and Resource Economics
The origins of thermodynamics are to be found in the nineteenth century
when practitioners, engineers and scientists like James Watt (1736–1819), Sadi
Carnot (1796–1832), James Prescott Joule (1818–1889), Rudolph Clausius (1822–
1888) and William Thomson (the later Lord Kelvin, 1824–1907) wanted to understand and increase the efficiency with which steam engines perform useful
mechanical work. From the beginning, this endeavor has combined the study
of natural systems and the study of engineered systems – created and managed
by purposeful human action – in a very peculiar way, which is rather unusual
for a traditional natural science such as physics.
Not surprisingly then, the laws of thermodynamics were found by economists
to be concepts with considerable implications for economics. In the late 1960s
and early 1970s economists discovered the relevance of thermodynamics for
environmental and resource economics (Pethig 2003, Spash 1999: 418, Turner
1999a: Section 2). For instance, economists like Kenneth Boulding (1966),
Robert Ayres and Allen Kneese (1969), and Nicolas Georgescu-Roegen (1971)
turned to thermodynamics when they wanted to analyze economy-environment
interactions in an encompassing way, and root the economy in its biogeophysical basis analytically.
4
For an overview see e.g. Freeman (2003) or Hanley and Spash (1993).
Thermodynamic Analysis: Rationale, Concepts, and Caveats
31
In a first step, the Materials Balance Principle was formulated based on
the thermodynamic Law of Conservation of Mass (Ayres and Kneese 1969,
Boulding 1966, Kneese et al. 1972). In view of this principle, all resource
inputs that enter a production process eventually become waste. This is now
an accepted and undisputed piece of ecological, environmental and resource
economics.
At the same time, Georgescu-Roegen (1971) developed an elaborate and
extensive critique of neoclassical economics based on the laws of thermodynamics, and, in particular, the Entropy Law, which he considered to be ’the
most economic of all physical laws’ (Georgescu-Roegen 1971: 280).5 His contribution initiated a heated debate over the question of whether the Entropy
Law – and thermodynamics in general – is relevant to economics (Burness et
al. 1980, Daly 1992b, Kåberger and Månsson 2001, Khalil 1990, Lozada 1991,
1995, Norgaard 1986, Townsend 1992, Williamson 1993, Young 1991, 1994).6
While Georgescu-Roegen had, among many other points, formulated an essentially correct insight into the irreversible nature of transformations of energy
and matter in economies, his analysis is flawed to some extent by positing of
what he calls a ‘Fourth Law of Thermodynamics’ (Ayres 1999b).7 This may be
the reason that the Second Law and the entropy concept have not yet acquired
the same undisputed and foundational status for ecological, environmental and
resource economics as have the First Law and the Materials Balance Principle.
But as Georgescu-Roegen’s work and the many studies following his lead
have shown, the Entropy Law, properly applied, yields insights into the irreversible nature of economy-environment interactions that are not available
otherwise (Baumgärtner et al. 1996). Both the First and the Second Laws of
Thermodynamics therefore need to be combined in the study of how natural
resources are extracted, used in production, and give rise to emissions and
waste, thus leading to integrated models of economy-environment interactions
(e.g. Baumgärtner 2000, Faber et al. 1995, Perrings 1987, Ruth 1993, 1999).
2.3
Different Approaches
How can thermodynamic concepts, laws and results be incorporated in a fruitful manner into economic analysis? This has been attempted in four basic
5
The works of Georgescu-Roegen are surveyed in a number of recent volumes (e.g. Beard
and Lozada 1999, Mayumi 2001, Mayumi and Goody 1999) and a special edition of the
journal Ecological Economics (Vol. 22, No. 3, 1997).
6
See Baumgärtner et al. (1996) for a summary of this discussion.
7
Georgescu-Roegen posited that in a closed system, matter is distributed in a more and
more disordered way. He called this the ‘Fourth Law’, in extension of the three, well established laws of classical thermodynamics, described in the Appendix.
32
Natural Science Constraints in Environmental and Resource Economics
ways,8 which are very different in the intellectual approach taken. In the following, I describe each of them in detail and assess their potential for ecological,
environmental and resource economics.
2.3.1
Isomorphism of Formal Structure
Both thermodynamics and economics can be set up formally as problems of
optimization under constraints. For example, equilibrium allocations in an
economy can be viewed as a result of the simultaneous utility maximization
under budget constraints of many households and profit maximization under technological constraints of many firms. Likewise, equilibrium micro- or
macrostates of a thermodynamic system can be derived from the minimization
of a thermodynamic potential, such as e.g. Helmholtz or Gibbs free energy,
under the constraints of constant pressure, volume, chemical potential etc.9
The mathematical structure of both economic and thermodynamic problems
is, thus, formally equivalent. There is an isomorphism between the two types
of problems and their respective solutions.10
As a result, one may exploit this formal isomorphism to obtain insights into
the structure of economic equilibrium allocations from studying the structural
properties of thermodynamic equilibria. Of course, these insights pertain to the
formal structure of equilibrium solutions only, and do not contain any substantive content about thermodynamics or economics in themselves. For instance,
based on what is known as the Le Chatelier Principle in thermodynamics (Kondepudi and Prigogine 1998: 239–240), Samuelson (1947, 1960a, 1960b) established the method of comparative statics in economics. This method explains
the changes in the equilibrium solution of a constrained maximization problem (economic or thermodynamic) when one of the constraints is marginally
tightened or relaxed. It has proven to be a very powerful tool and has found
widespread use in modern economics.
Another example is the formal isomorphism between entropy and utility
(Candeal et al. 2001a, 2001b), which becomes apparent in a particular entropy
representation whereby entropy is constructed as an order preserving function
that satisfies a continuity property (Candeal et al. 2001a, Cooper 1967).
8
Söllner (1997) makes a similar distinction.
This becomes most apparent in the Tisza (1966)/Callen (1985)-axiomatization of thermodynamics (Smith and Foley 2004, Sousa and Domingo, forthcoming).
10
Mirowski (1989) has argued that modern economics is essentially built after the logic and
formal structure of classical mechanics, i.e. Lagrangian and Hamiltonian formalism. Some
authors in the ecological-economics community have taken this observation as a starting point
for a methodological critique of the conventional economic approach to studying economyenvironment interactions and proposed that ecological economics should be inspired more by
thermodynamics instead, in order to get hands on the fundamental irreversibility of economyenvironment interactions (Amir 1995, 1998, Costanza et al. 1997a, Georgescu-Roegen 1971,
Lozada 1995, Martinez-Alier 1997).
9
Thermodynamic Analysis: Rationale, Concepts, and Caveats
33
But overall, it seems as if the potential of exploiting the isomorphism of
formal structures in thermodynamic and economic equilibria was fairly limited
and is, by now, largely exhausted.
2.3.2
Analogies and Metaphors
A second approach takes thermodynamic concepts and transfers them into economic thinking as analogies and metaphors (Faber and Proops 1985, Proops
1985, 1987). For example, under this approach, ‘order’ and ‘disorder’ in an
economy are interpreted as expressions of ‘social entropy’, or the economy is
seen as a ‘self-organizing dissipative system far from thermodynamic equilibrium’. Typically, no attempt is made under this approach to clearly define
the various terms, such as ‘order’, ‘entropy’ or ‘equilibrium’, in either thermodynamic or economic terms. Instead, these terms are used to evoke certain
associations with the reader.
To a reader who is well trained in both thermodynamics and economics,
it remains unclear whether a term like e.g. ‘equilibrium’ refers to thermodynamic equilibrium (in the sense of a thermodynamic system being in a state of
minimal thermodynamic potential, e.g. Helmholtz free energy) or to economic
equilibrium (in the sense of an economy of households and firms being in a
state of market equilibrium where demand equals supply). Certainly, using
these terms in such a loose manner cannot have the status of making exact and
deductive scientific statements about economic systems.
Despite these large unclarities, the analogies-and-metaphors-approach has
merit as a heuristic, since it allows one to see economic phenomena in a new
light. Thus, it generates new and potentially fruitful questions, rather than
answering existing ones. In that sense, it is more a ‘vision’ in the sense of
Schumpeter (1954: 42),2 than a rigorous analytical approach.
2.3.3
Energy, Entropy and Exergy Theories of Value
It has been argued that economic values based on subjective individual preferences are to some extent arbitrary and might be misleading in achieving
sustainable solutions for environmental problems. In contrast, the argument
goes, sustainability requires the identification of the ‘true’ and ‘objective’ value
of nature’s goods and services, and of damages to these. Often, thermodynamic quantities are proposed to give such an ‘objective’ value rod, e.g. energy
(Costanza 1981, Hannon 1973, 1979, Hannon et al. 1986, Odum 1971), (low)
34
Natural Science Constraints in Environmental and Resource Economics
entropy11 or exergy (Bejan et al. 1996. 407).12 In all these cases, the argument
is essentially as follows: Energy (or, alternatively: exergy, low entropy) is the
only really scarce factor here on Planet Earth. It therefore measures the ultimate scarcity that we face in dealing with nature. As a result, the amount of
energy (exergy, low entropy) contained in every good or service measures its
‘true’ scarcity, and should therefore be taken as its value. Decisions concerning
sustainability, so the argument, must be based on such energy/entropy/exergyvalues, as they represent the ultimate scarcities.
From an economic point of view, this argument is untenable. It is untenable for the very same reasons that, for instance, a labor theory of value as
advocated by David Ricardo or Karl Marx is untenable, and any other singlefactor-theory of value would be untenable, be that factor energy, labor, oxygen,
or anything else. ‘Value’, as it is understood in economics, results from the interplay of human goals and ends on the one hand (e.g. profit maximization,
utility maximization, or sustainability), and scarcity of means to achieve these
ends on the other hand (e.g. natural resources, capital, labor, or time). The
higher the goals and the scarcer the resources necessary to achieve them, the
more valuable are these resources. There is an economic theorem which states
that only under very limiting assumptions the value of a good or service is given
by the total amount of a factor of production (e.g. energy or labor) which has
been used, directly or indirectly, in producing it. This is the so-called nonsubstitution theorem, proven in 1951 independently by four masterminds of
economics: Arrow (1951), Koopmans (1951a), Georgescu-Roegen (1951) and
Samuelson (1951).13 This theorem identifies the conditions, under which a
single-factor-theory of value holds:
(A1) There is only one primary, i.e. non-producible, factor of production.
(A2) This factor is directly used in the production of every intermediate or
final good or service.
(A3) All production processes are characterized by constant returns to scale,
i.e. scaling the amounts of all inputs by a factor of λ > 0 also scales the
amount of output produced by the same factor λ.
11
Burness et al. (1980: 7) and Patterson (1998) wrongly claim that Georgescu-Roegen
(1971: Chapter 5) proposes a (low) entropy theory of value. On the contrary, GeorgescuRoegen (1971: 282) explicitly warns against such an interpretation. Georgescu-Roegen (1979)
also gives an explicit rebuttal of energy theories of value. See Baumgärtner et al. (1996: 123–
125) for details.
12
Patterson (1998) surveys different theories of value in ecological economics.
13
Three of these – Arrow, Koopmans, and Samuelson – were awarded the Nobel Prize in
Economics. Some claim that the fourth one – Georgescu-Roegen – would have deserved it
as well.
Thermodynamic Analysis: Rationale, Concepts, and Caveats
35
(A4) There is no joint production, i.e. every process of production yields exactly one output.
These are very restrictive assumptions. Only if (A1)–(A4) are fulfilled does a
single-factor-theory of value fully explain the value of goods and services. If
one of them does not hold, a single-factor-theory of value cannot provide a
satisfactory explanation of value.
As for energy/entropy/exergy as a factor of production, one may safely assume that (A2) is fulfilled, and one may concede that (A1) can be taken to be
fulfilled as well.14 But in general, (A3) is not fulfilled, as many technologies
are characterized by either increasing or decreasing returns to scale. And thermodynamic considerations, to which we will turn in detail later, imply that
every process of production is joint production, so that (A4) is violated. This
means, while energy, entropy or exergy theories of value are conceivable in very
restricted models (characterized by Conditions A1–A4) they must be refuted
for real economy-environment systems. To be sure, while energy, entropy or
exergy are important factors in explaining value, value is a complex and encompassing phenomenon, and thermodynamic quantities alone cannot provide
a satisfactory explanation.
2.3.4
Thermodynamic Constraints on Economic Action
Another approach to integrate insights from thermodynamics into economics
starts from the observation that the laws of thermodynamics constrain economic action. Thermodynamic laws specify what is possible and what is not
possible in the transformation of energy and matter. Such transformations play
an important role in any economy, for example in
• the extraction of natural resources from the geo-bio-chemical-physical
environment,
• the use of these resources in the production of goods and services,
• the generation and emission of wastes and environmental pollutants as
by-products of desired goods, and
• the recycling of wastes into secondary resources.
All these transformations of energy and matter are at the center of interest in
the field of ecological, environmental and resource economics. Hence, the laws
of thermodynamics play an important role in describing relevant constraints
14
One may also consider space and time as primary production factors, as they surely enter
every process of production in some sense. But then, energy is not the only primary factor
any more.
36
Natural Science Constraints in Environmental and Resource Economics
and scarcities for the economic analysis of economy-environment interactions
(Cleveland and Ruth 1997).
This approach builds on a clear division of labor between the disciplines
of thermodynamics and economics. The laws of thermodynamics are used to
capture the constraints on transformations of energy and matter. Their role is
limited to this particular task. Based on this conceptualization of constraints,
methods and concepts from economics are then used to study allocations in an
economy which result from the optimizing behavior of firms and households, e.g.
profit-maximizing resource-extraction and production firms as well as utilitymaximizing households purchasing the consumer goods produced.
This approach can be operationalized directly, and is empirically meaningful for ecological, environmental and resource economics. It lends itself quite
naturally to modeling. One can distinguish between different model types for
integrated thermodynamic-economic analysis, according to the thermodynamic
concepts and laws they incorporate:
• models incorporating mass and conservation of mass (First Law), either
for one particular material (say, copper) or for a number of materials,
• models incorporating energy and conservation of energy (First Law),
sometimes in variants such as emergy (‘embodied energy’),
• models incorporating entropy and entropy generation (Second Law),
• models incorporating energy and entropy, sometimes in the form of exergy
(First and Second Law),
• models incorporating mass, energy and entropy (First and Second Law).
Models based on the First Law are useful for studying the economic implications from the scarcities due to physical conservation of mass and energy in the
throughflow of materials and energy through the economy. Models based on the
Second Law are useful for studying the economic implications from the scarcities due to the temporal directedness of this throughflow and its qualitative
degradation by dissipation of energy and dispersal of matter.
2.4
Implications of and Insights from
Thermodynamic Models
The use of thermodynamic concepts, laws and models in ecological, environmental and resource economics is an ongoing endeavor. So far, it has revealed
Thermodynamic Analysis: Rationale, Concepts, and Caveats
37
a number of significant implications and insights about different aspects of
economy-environment interactions.15
2.4.1
Materials Balance: The ‘Planet Earth’-Perspective
The Materials Balance Principle is based on the Law of Conservation of Mass as
implied by the First Law of Thermodynamics (Ayres 1978, Ayres and Kneese
1969, Boulding 1966, Kneese et al. 1972).16 Since mass cannot be created, but
is conserved in all transformations, all material resource inputs that enter a
production process (i) diminish the corresponding resource reservoir, and (ii)
eventually become waste.
This principle has lead to a view of the Earth, including the human society,
as a ‘spaceship’ (Boulding 1966), which is completely closed to the surrounding
space in material terms. Thus, all material transformations on Earth should
be managed in a self-reliant and sustainable way.
2.4.2
Irreversibility of (Micro- and Macro-)Economic Processes
All processes of macroscopic change are irreversible. Examples include natural
processes, such as the growing and blooming of a flower, as well as technical
processes, such as the burning of fossil fuels in combustion engines. The entropy
concept and the Second Law of Thermodynamics have been coined such as to
capture this fact of nature (Kondepudi and Prigogine 1998, Zeh 2001).
The relevance of thermodynamic irreversibility for economics lies in the fact
that it precludes the existence of perpetual motion machines, i.e. devices which
use a limited reservoir of available energy to perform work forever. It is an
everyday experience that no such thing as a perpetual motion machine exists.
This holds for the micro-level, i.e. individual production processes, as well as
for the macro-level, i.e. the economy at large (Georgescu-Roegen 1971).
In order to make this insight accessible to economic analysis it is necessary
to adequately represent thermodynamic irreversibility as a constraint for economic action. Modern economic theory has devoted some effort to incorporating irreversibility into production theory. However, the standard irreversibility
concept of economics, which is due to Arrow and Debreu (1954) and Debreu
(1959), does not encompass thermodynamic irreversibility; it only establishes
temporal irreversibility – a weaker form of irreversibility (Baumgärtner 2005).
15
Surveys of this area of research include Baumgärtner et al. (1996), Beard and Lozada
(1999), Burley and Foster (1994), Daly (1997a), Mayumi and Gowdy (1999), Pethig (2003)
and Ruth (1999).
16
Pethig (2003) surveys the Materials-Balance-Principle’s origin and impact for environmental and resource economics.
38
Natural Science Constraints in Environmental and Resource Economics
2.4.3
Resource Extraction and Waste Generation
The insights described in Sections 2.4.1 and 2.4.2 have been applied, in particular, to the analysis of mineral resource extraction (e.g. Ruth 1995a, 1995b,
1995c), the generation of wastes and pollution (Kümmel 1989, Kümmel and
Schüssler 1991), and the relation between the two (Faber 1985, Faber et al.
1995[1983]). At a very abstract level, high entropy (or: exergy lost) may be
seen as the ultimate form of waste (Ayres et al. 1998, Ayres and Martinás 1995,
Faber 1985, Kümmel 1989, Kümmel and Schüssler 1991).17
2.4.4
Representation of the Production Process
Every process of production is, at core, a transformation of energy and matter
(Ayres and Kneese 1969). Hence, the laws of thermodynamics provide a suitable analytical framework for a rigorous deduction of insights into the physical
aspects of production (Baumgärtner 2000). In particular, any representation of
production in economic models should be in accordance with the laws of thermodynamics.18 For this reason, the neoclassical production function, which is
the standard way of representing the production process in economic models,
has been critically discussed against the background of thermodynamics. It
has become apparent that this concept is incompatible with the laws of thermodynamics for a number of reasons:
(i) Georgescu-Roegen (1971) claims that the neoclassical production function is incompatible with the laws of thermodynamics, basically, because
it does not properly reflect the irreversible nature of transformations of
energy and matter, and because it confounds flow and fund quantities
(Daly 1997b, Kurz and Salvadori 2003).
(ii) One essential factor of production, which is very often omitted from the
explicit representation, is energy (actually: exergy) (Ayres 1998, Kümmel
1989).19 Its exact role for the production process, and its interplay with
17
Waste materials deposited in the natural environment, however, might cause environmental problems not because of their high entropy, but precisely because their entropy is not
yet maximal. In other words, it is the exergy still contained in waste materials, i.e. the potential to initiate chemical reactions and perform work, which makes these wastes potentially
harmful to the natural environment (Ayres et al. 1998, Ayres and Martinás 1995, Perrings
1987). However, the view that the ‘waste exergy’ of by-products can be seen as a measure
for potential harm done to natural ecosystems is limited. It does not take into account the
(eco-)toxicity of some inert materials, nor does it take into account purely physical effects of
inert materials, e.g. global warming due to the carbon dioxide emitted into the atmosphere.
18
Krysiak and Krysiak (2003) discuss whether the neglect of the laws of thermodynamics
in economics leads to any substantial problems.
19
Econometric studies show that the production factor energy (exergy) explains a surprisingly large share of economic growth observed over the 20th century in the US, German or
Japanese economies (Ayres et al. 2003, Kümmel et al. 1985, 2000).
Thermodynamic Analysis: Rationale, Concepts, and Caveats
39
other production factors, such as capital or material resources, is studied
in engineering thermodynamics (e.g. Bejan 1996, 1997, Bejan et al. 1996;
see also Section 2.4.5 below). This has lead to a new understanding of
the role of energy for economic production processes, which goes beyond
simply treating it as a factor of production (Buenstorf 2004).
(iii) The conservation laws for energy and matter imply that there are limits
to substitution between energy-matter inputs, which are subject to the
laws of thermodynamics, and other inputs such as labor or capital, which
lie outside the domain of thermodynamics (Berry and Andresen 1982,
Berry et al. 1978, Dasgupta and Heal 1979: Chapter 7).20
(iv) From the First and Second Laws of Thermodynamics it becomes obvious
that ‘[g]iven the entropic nature of the economic process, waste is an
output just as unavoidable as the input of natural resources’ (GeorgescuRoegen 1975: 357). This holds not only for the economy at large, but for
every individual process of production at the micro-level (Baumgärtner
2000: Chapter 5, 2002; Baumgärtner and de Swaan Arons 2003, Faber et
al. 1998). As a result, there is no such thing as ‘single production’, i.e.
the production of just one single output as modelled by the neoclassical
production function. Rather, all production is joint production, i.e. there
is necessarily more than one output (Baumgärtner et al. 2001, 2006, Faber
et al. 1998).
All these apparent inconsistencies between the laws of thermodynamics and the
standard assumptions about the neoclassical production function have led to
more general descriptions of the production process, which blend the traditional
theory of production with thermodynamic principles. Some of them use partialanalysis models (Anderson 1987, Ayres 1995, Ethridge 1973, Baumgärtner
2000: Chapter 4); others use general equilibrium models of the whole economy (Ayres and Kneese 1969, Perrings 1986, Krysiak and Krysiak 2003, Pethig
2006, Noll and Trijonis 1971).
2.4.5
Finite-Time/Finite-Size Thermodynamics:
Exergy-Engineering
Recent research in the applied field of engineering thermodynamics has addressed the circumstance that chemical and physical processes in industry never
20
Krysiak and Krysiak (2003) show that most abstract economic models, and general
equilibrium theory in particular, are consistent with physical conservation laws. On the
other hand, most applied models, also from environmental and resource economics, which
use specific specifications of production functions, are not. As for the latter, Pethig (2006)
shows that production functions with emissions treated as inputs can be reconstructed as a
subsystem of a comprehensive production-cum-abatement technology that is in line with the
materials balance principle.
40
Natural Science Constraints in Environmental and Resource Economics
happen in a completely reversible way between one equilibrium state and another equilibrium state. Rather, these processes are enforced by the operator
of the process and they are constrained in space and time. This has led to
an extension of ideal equilibrium thermodynamics, known as finite-time/finitesize thermodynamics (e.g. Andresen et al. 1984, Bejan 1996, 1997, Bejan et al.
1996).
From the point of view of finite-time/finite-size thermodynamics it becomes
obvious that the minimum exergy requirement and minimum waste production
in chemical or physical processes is considerably higher than that suggested
by ideal equilibrium thermodynamics. The reason for the increased exergy
requirement (which entails an increased amount of waste at the end of the
process) lies in the fact that chemical and physical transformations are forced
to happen over a finite time by the operator of the production plant, which
necessarily causes some dissipation of energy.
The finite-time/finite-size consideration is a very relevant consideration for
many production processes, in particular in the chemical industry. Finitetime/finite-size thermodynamics allows one to exactly identify, trace down and
quantify exergetic inefficiencies at the individual steps of a production processes
(Bejan 1996, 1997, Bejan et al. 1996, Brodyansky et al. 1994, Creyts 2000,
Szargut et al. 1988), along the entire chain of a production process (Ayres et al.
1998, Cornelissen and Hirs 1999, Cornelissen et al. 2000), for whole industries
(Dewulf et al. 2000, Hinderink et al. 1999, Ozdogan and Arikol 1981), and for
entire national economies (Nakićenović et al. 1996, Schaeffer and Wirtshafter
1992, Wall 1987, 1990, Wall et al. 1994). Thus, it yields valuable insights into
the origins of exergy losses and forms a tool for designing industrial production
systems in an efficient and sustainable manner (Connelly and Koshland 2001,
de Swaan Arons and van der Kooi 2001, de Swaan Arons et al. 2003).
Furthermore, it becomes apparent that energy/exergy and time are substitutes as factors of production in many production processes (Andresen et
al. 1984; Berry and Andresen 1982, Spreng 1993). A production process may
be speeded up at the expense of employing more energy/exergy, and the use
of energy/exergy may be reduced by allowing the production process to just
take longer. Prominent examples for such a trade-off-relationship are transport
services or chemical reaction processes.
2.4.6
Thermodynamic and Economic Efficiency
Both thermodynamics and economics analyze systems in terms of their ‘efficiency’. Both concepts may be applied to the very same system, e.g. a production plant or a whole national economy. Yet, the thermodynamic and the
economic notions of efficiency differ fundamentally, as they refer to very different variables of the system. In fact, the two notions are completely independent
(Berry et al. 1978, Dasgupta and Heal 1979: Chapter 7, Baumgärtner 2001).
Thermodynamic Analysis: Rationale, Concepts, and Caveats
41
As a result, thermodynamic efficiency is neither necessary nor sufficient for economic efficiency, even when economic efficiency includes concerns over energy,
resources and environmental quality.
2.4.7
Sustainability: Limits to Economic Growth
From the very beginning, the recourse to thermodynamic arguments in ecological, environmental and resource economics was motivated by a long-term and
global concern for the sustainable existence of humankind on ‘Planet Earth’
(Boulding 1966, Daly 1973, 1991[1977], Georgescu-Roegen 1971). The preanalytic ‘vision’ (Schumpeter 1954: 42)2 behind this concern was that of the
human economy as an open subsystem of the larger, but finite, closed, and
non-growing system of the biogeophysical environment.
In this view, thermodynamic analysis has helped to sketch the potential and
limits of economic growth. It has turned out that limits to the growth of energymatter throughput through the economy exist, which may ultimately set limits
to economic growth. This claim is vindicated by the following arguments:21
(i) Conservation of mass implies that the marginal product as well as the
average product of a material resource input may be bounded from above
(Baumgärtner 2004a). This means that the usual Inada conditions (Inada
1963) do not hold for material resource inputs. This is important since
the Inada conditions are usually held to be crucial for establishing steady
state growth under scarce exhaustible resources (e.g. Dasgupta and Heal
1974, Solow 1974, Stiglitz 1974).
(ii) As described in Section 2.4.4 above, the conservation laws for energy and
matter imply that there are limits to substitution between energy-matter
inputs, which are subject to the laws of thermodynamics, and other inputs
such as labor or capital, which lie outside the domain of thermodynamics
(Berry and Andresen 1982, Berry et al. 1978, Dasgupta and Heal 1979:
Chapter 7). This is important since substitutability among essential and
scarce production factors (with an elasticity of substitution not smaller
than one) is usually held to be crucial for establishing steady state growth
(e.g. Dasgupta and Heal 1974, Solow 1974, Stiglitz 1974).
(iii) Some have posited that resource scarcity can be overcome by recycling.
However, thermodynamic analysis clearly shows that there are limits to
recycling as well (Ayres 1999b, Craig 2001).
(iv) Others still have posited that technical progress is an important driver
of economic growth, and that technical progress will continue. However,
21
Cleveland and Ruth (1997) present these arguments in more detail and review the relevant literature.
42
Natural Science Constraints in Environmental and Resource Economics
thermodynamic analysis clearly shows that there are limits to technical
progress (Ruth 1995a, 1995b, 1995c).
2.5
Conclusion and Caveat: Thermodynamics and
Sustainability
In conclusion, thermodynamic concepts, laws and models are relevant for ecological, environmental and resource economics in various ways and on different
levels of abstraction.
(i) As all processes of change are essentially processes of energy and material transformation, the concepts and laws of thermodynamics apply
to all of them. The framework of thermodynamics thus creates a unifying perspective on ecology, the physical environment, and the economy.
This unifying framework, combined with economic and ecological analysis, allows asking questions which would not have been asked from the
perspective of one scientific discipline alone.
(ii) On a more specific level thermodynamic concepts allow the incorporation of physical driving forces and constraints in models of economyenvironment interactions, both microeconomic and macroeconomic. They
are essential for understanding the extent to which resource and energy
scarcity, nature’s capacity to assimilate human wastes and pollutants,
as well as the irreversibility of transformation processes, constrain economic action. Thermodynamic concepts thus allow economics to relate
to its biogeophysical basis, and yield insights about that relation which
are otherwise not available.
(iii) On an even more applied level, thermodynamic concepts provide tools
for quantitative analysis of energetic and material transformations for
engineers and managers. They may be used to design industrial production plants or individual components, such as to maximize their energetic
efficiency and minimize their environmental impact.
With its rigorous but multifarious character as a method of analysis, its rich set
of fruitful applications, and its obvious potential to establish relations between
the natural world and purposeful human action, thermodynamics is therefore
one of the cornerstones in the conceptual foundation of ecological, environmental and resource economics.
However, one important caveat seems to be in place. Thermodynamics is a
purely descriptive science. That means, it only allows one to make statements
of the kind ‘If A, then B’. In particular, it is not a normative science. By itself,
Thermodynamic Analysis: Rationale, Concepts, and Caveats
43
it neither includes nor allows value statements (Baumgärtner 2000: 65–66) or
statements of the kind ‘C is a good, and therefore desirable, state of the world,
but D is not’.22 In contrast, sustainability is essentially a normative issue
(Faber et al. 1995, Faber et al. 1996: Chapter 5). Sustainability is about the
question ‘In what kind of world do we want to live today and in the future?’,
thus, inherently including a dimension of desirability. A purely descriptive
science alone, like thermodynamics, cannot give an answer to that question.
Thermodynamics is necessary, however, to identify clearly the feasible options of development and their various properties, before a choice is then made
about which option to choose based on some normative criteria. That choice
requires a valuation or, more generally, a normative judgment of the different
options at hand. It is therefore necessary not only to know the energetic and
material basis of society’s metabolism – both current and feasible alternatives
– but also to link these thermodynamic aspects to the human perception and
valuation of natural resources, commodity products and waste joint products,
and the state of the natural environment.
The role of thermodynamics for conceiving sustainable modes of societal
metabolism, therefore, is relative but essential. Thermodynamics is necessary
to identify which options and scenarios of resource use, economic production,
and waste generation are feasible and which are not. It thereby contributes to
making informed choices about the future.
Appendix: Concepts and Laws of Thermodynamics
Thermodynamics is the branch of physics that deals with macroscopic transformations of energy and matter. Briefly summarized, the fundamental concepts
and laws of phenomenological thermodynamics can be stated as follows.23
A2.1
Systems and Transformations
With respect to the potential exchange of energy and matter between the inside and the outside of the system under study, one distinguishes between the
following types of thermodynamic system:
• Isolated systems exchange neither energy nor matter with their surrounding environment.
22
This holds even for the notion of thermodynamic efficiency, which is a purely technical
notion (see the discussion in Section 2.4.6 above).
23
This appendix is taken from Baumgärtner (2002: Section 2.3). For a comprehensive introduction to (phenomenological) thermodynamics see Callen (1985), Kondepudi and Prigogine
(1998) or Zemansky and Dittman (1997).
44
Natural Science Constraints in Environmental and Resource Economics
• Closed systems exchange energy, but not matter, with their surrounding
environment.
• Open systems exchange both energy and matter with their surrounding
environment.
A system is said to be in thermodynamic equilibrium when there is complete
absence of driving forces for change in the system. Technically, the various
potentials of the system are at their minimum, such that there are no spatial variations of any of the intensive variables within the system. Intensive
variables are quantities which do not change when two separate but identical
systems are coupled. In contrast, extensive variables are quantities whose value
for the total system is simply the sum of the values of this quantity in both
systems. For example, temperature and pressure are intensive variables while
mass and volume are extensive ones. As long as there are spatial variations in,
say, temperature within a system, it is not yet in thermodynamic equilibrium,
because a potential for change exists. The equilibrium state is characterized
by a uniform temperature throughout the system.
Consider an isolated system which undergoes a transformation over time
between some initial equilibrium state and some final equilibrium state, either
by interaction with its environment or by interaction between different constituents within the system. If the final state is such that no imposition or
relaxation of constraints upon the isolated system can restore the initial state,
then this process is called irreversible. Otherwise the process is called reversible.
For example, at some initial time a gas is enclosed in the left part of an isolated box; the right part is separated from the left part by a wall and is empty.
Now, the separating wall is removed. The molecules of the gas will then evenly
distribute themselves over the entire volume of the box. The thermodynamic
equilibrium of the final state is characterized by a uniform density of molecules
throughout the entire volume. Reintroducing the wall into the isolated system
separating the left part from the right half would not restore the initial state
of the system. Nor would any other imposition or relaxation of constraints on
the isolated system be able to restore the initial state. Therefore, the transformation given by the removal of the wall is an irreversible transformation of the
isolated system.24 Generally, a process of transformation can only be reversible
if it does not involve any dissipation of energy, such as through e.g. friction,
viscosity, inelasticity, electrical resistance or magnetic hysteresis.
24
Note that this does not mean that the initial state of the system can never be restored.
However, in order to restore the system’s initial state, the initially isolated system has to be
opened to the influx of energy. For instance, the initial state could be restored by removing
the system’s insulation and performing work on the system from the outside, e.g. by pressing
all the molecules into the left part with a mobile wall that is initially at the right hand end
of the system and from there on moves left.
Thermodynamic Analysis: Rationale, Concepts, and Caveats
A2.2
45
The Fundamental Laws of Thermodynamics
The First Law of Thermodynamics states that in an isolated system (which may
or may not be in equilibrium) the total internal energy is conserved. This means
that energy can be neither created nor destroyed. It can, however, appear
in different forms, such as heat, chemical energy, electrical energy, potential
energy, kinetic energy, work, etc. For example, when burning a piece of wood or
coal the chemical energy stored in the fuel is converted into heat. In an isolated
system the total internal energy, i.e. the sum of energies in their particular
forms, does not change over time. In any process of transformation only the
forms in which energy appears change, while its total amount is conserved.
Similarly, in an isolated system the total mass is conserved (Law of Conservation of Mass). Obviously, if matter cannot enter or leave an isolated system,
the number of atoms of any chemical element within the system must remain
constant. In an open system which may exchange matter with its surrounding,
a simple Materials Balance Principle holds: the mass content of a system at
some time is given by its initial mass content plus inflows of mass minus outflows of mass up to that point in time. The law of mass conservation, while
often regarded as an independent conservation law besides the law of energy
conservation, is actually an implication of the First Law of Thermodynamics.
According to Einstein’s famous relation E = mc2 mass is a form of energy,
but mass can only be transformed into non-material energy, and vice versa, in
nuclear reactions. Therefore, neglecting nuclear reactions, it follows from the
First Law of Thermodynamics that mass and non-material energy are conserved
separately.
In any process transforming energy or matter, a certain amount of energy
is irrevocably transformed into heat. The variable entropy has been defined
by Rudolph Clausius (1854, 1865) such as to capture this irrevocable transformation of energy: if a certain amount of heat dQ is reversibly transferred
to or from a system at temperature T , then dS = dQ/T defines the entropy
S. Clausius showed that S is a state variable of the system, i.e. it remains
constant in any reversible cyclic process, and increases otherwise. The Second
Law of Thermodynamics, the so-called Entropy Law, states the unidirectional
character of transformations of energy and matter: With any transformation
between an initial equilibrium state and a final equilibrium state of an isolated
system, the entropy of this system increases over time or remains constant.
It strictly increases in irreversible transformations, and it remains constant in
reversible transformations, but it cannot decrease.
Entropy, in this view, can be interpreted as an indicator for the system’s
capacity to perform useful work. The higher the value of entropy, the higher
the amount of energy already irreversibly transformed into heat, the lower the
amount of free energy of the system and the lower the system’s capacity to
perform work. Expressed the other way round, the lower the value of entropy,
46
Natural Science Constraints in Environmental and Resource Economics
the higher the amount of free energy in the system and the higher the system’s
capacity to perform work. Hence, the statement of the Second Law of Thermodynamics amounts to saying that, for any process of transformation, the
proportion of energy in the form of heat to total energy irreversibly increases
or remains constant, but certainly never decreases. In other words, with any
transformation of energy or matter, an isolated system loses part of its ability to perform useful mechanical work and some of its available free energy is
irreversibly transformed into heat. For that reason, the Second Law is said
to express an irreversible degradation of energy in isolated systems over time.
Through this, the economic relevance of the Second Law becomes obvious.
While the notion of entropy introduced to phenomenological thermodynamics by Clausius is based on heat, Ludwig Boltzmann (1877) introduced a
formally equivalent notion of entropy that is based on statistical mechanics and
likelihood. His notion reveals a different interpretation of entropy and helps to
show why it irreversibly increases over time. Statistical mechanics views gases
as assemblies of molecules, described by distribution functions depending on position and velocity. This view allows the establishment of connections between
the thermodynamic variables, i.e. the macroscopic properties such as temperature or pressure, and the microscopic behaviour of the individual molecules
of the system, which is described by statistical means.25 The crucial step is to
distinguish between microstates and macrostates of a system. The microstate is
an exact specification of the positions and velocities of all individual particles;
the macrostate is a specification of the thermodynamic variables of the whole
system.
Boltzmann assumed that all microstates have equal a priori probability,
provided that there is no physical condition which would favour one configuration over the other. He posited that every macrostate would always pass to one
of higher probability, where the probability of a macrostate is determined by
the number of different microstates realising this macrostate. The macroscopic
thermal equilibrium state is then the most probable state, in the sense that it
is the macrostate which can be realized by the largest number of different microstates. Boltzmann defined the quantity Ω, counting the number of possible
microstates realising one macrostate, and related this to the thermodynamic
entropy S of that macrostate. He used S = k log Ω, with k as a factor of
proportionality called Boltzmann’s constant. Entropy can thus be taken as a
measure of likelihood: highly probable macrostates, that is macrostates which
can be realized by a large number of microstates, also have high entropy. At
the same time, entropy may be interpreted as a measure of how orderly or
mixed-up a system is. High entropy, according to the Boltzmann interpretation, characterizes a system in which the individual constituents are arranged
25
Balian (1991), Huang (1987) and Landau and Lifshitz (1980) give an introduction to
statistical mechanics.
Thermodynamic Analysis: Rationale, Concepts, and Caveats
47
in a spatially even and homogeneous way (‘mixed-up systems’), whereas low entropy characterizes a system in which the individual constituents are arranged
in an uneven and heterogeneous way (‘orderly systems’). The irreversibility
stated by the Second Law in its phenomenological formulation (in any isolated
system entropy always increases or remains constant) now appears as the statement that any isolated macroscopic system always evolves from a less probable
(more orderly) to a more probable (more mixed-up) state, where Ω and S are
larger.
Whereas the Second Law in its Clausius or Boltzmann formulation makes
a statement about isolated systems in thermodynamic equilibrium only, the
study of closed and open systems far from equilibrium has shown (Prigogine
1962, 1967) that entropy is also a meaningful and useful variable in closed and
open systems. Any open system is a subsystem of a larger and isolated system.
According to the conventional formulation of the Second Law, the entropy of
the larger and isolated system has to increase over time, but the entropy of any
open subsystem can, of course, decrease. Viewing open systems as subsystems
of larger and isolated systems reveals, however, that an entropy decrease in an
open subsystem necessarily has to be accompanied by an entropy increase in
the system’s environment, that is the rest of the larger, isolated system, such
that the entropy of the total system increases.
A generalization of the Second Law is possible so that it does not only refer
to isolated systems. Irrespective of the type of thermodynamic system under
study, and irrespective of whether the system is in thermodynamic equilibrium
or not, it is true that entropy cannot be annihilated; it can only be created (Falk
and Ruppel 1976: 353). This more general, system-independent formulation
of the Second Law implies the usual formulations for isolated systems. The
relevance of the system-independent formulation of the Second Law lies in the
fact that most real systems of interest are not isolated but closed or open.
Hence, the latter formulation is the form in which the Second Law is apparent
in everyday life.
A2.3
Quantification and Application
The entropy concept is essential for understanding how resource and energy
scarcity, as well as the irreversibility of transformation processes, constrain
economic action (Baumgärtner 2003, Georgescu-Roegen 1971). However, it
is a very abstract concept and it is notoriously difficult to apply in specific
contexts. One of the complications is due to the fact that a system’s capacity to
perform work depends not only on the state of the system, but also on the state
of the system’s environment. Therefore, for applications of the fundamental
thermodynamic insights in the areas of mechanical and chemical engineering,
as well as in economics, it is useful to relate the system’s ability to perform
work to a certain standardized reference state of its environment. Exergy is
48
Natural Science Constraints in Environmental and Resource Economics
defined to be the maximum amount of work obtainable from a system as it
approaches thermodynamic equilibrium with its environment in a reversible
way (Szargut et al. 1988: 7). Exergy is also commonly called available energy,
or available work, and corresponds to the ‘useful’ part of energy, thus combining
the insights from both the First and Second Laws of Thermodynamics. Hence,
exergy is what most people mean when they use the term ‘energy’ carelessly,
e.g. when saying that ‘energy is used’ to carry out a certain process.
The relationship between the concepts of entropy and exergy is simple, as
Blost = T0 Sgen (Law of Gouy and Stodola), where Blost denotes the potential
work or exergy lost by the system in a transformation process, T0 denotes the
temperature of the system’s environment, and Sgen denotes the entropy generated in the transformation. This means, as the system’s entropy increases as a
consequence of irreversible transformations according to the Second Law, the
system loses exergy or some of its potential to perform work. Exergy, unlike
energy, is thus not a conserved quantity. While the entropy concept stresses
that with every transformation of the system something useless is created, the
exergy concept stresses that something useful is diminished. These developments are two aspects of the same irreversible character of transformations of
energy and matter.
As the system might consist simply of a bulk of matter, exergy is also a
measure for the potential work embodied in a material, whether it is a fuel,
food or other substance (Ayres 1998, Ayres et al. 1998). The exergy content of
different materials can be calculated for standard values specifying the natural
environment, by considering how that material eventually reaches thermodynamic equilibrium with its environment with respect to temperature, pressure,
chemical potential and all other intensive variables.26 While taking a particular
state of the system’s environment as a reference point for the definition and
calculation of exergy may be considered a loss of generality as compared to
the entropy concept, this referencing seems permissible since all processes of
transformation – be it in nature or in the economy – are such that:
(i) all the materials involved eventually do reach thermodynamic equilibrium
with the natural environment;
(ii) the environment is so large that its equilibrium will not be affected by
the particular transformation processes under study.
While both the entropy and the exergy concept yield the same qualitative insights into the fundamentally irreversible character of transformations of energy
and matter, the exergy concept is more tangible, as it is directly related to the
26
Exergy values for many materials are typically calculated for an environmental temperature of 298.15 K and pressure of 101.325 kPa and can be found in tables, such as e.g. in
Szargut et al. (1988: Appendix).
Thermodynamic Analysis: Rationale, Concepts, and Caveats
49
very compelling idea of ‘available work’ and can be more easily quantified than
entropy.
50
Natural Science Constraints in Environmental and Resource Economics
3. The Inada Conditions for Material Resource
Inputs Reconsidered∗
3.1
Introduction
It is characteristic for many of the pioneering theoretical contributions to the
analysis of economic growth under scarcity of exhaustible natural resources
(Dasgupta and Heal 1974, Hoel 1978, Mäler 1974, Schulze 1974, Smith 1977,
Solow 1974, Stiglitz 1974, Weinstein and Zeckhauser 1974) as well as large
parts of the vast strand of literature which they initiated that they take the
well established neoclassical growth theory as their starting point and extend
it in the most simple way to also include natural resources, namely by adding
one additional variable representing material resource input into a neoclassical
aggregate production function. This production function is usually assumed to
display the standard properties concerning the resource factor (substitutability
between factors of production, positive decreasing marginal product approaching zero and infinity in the two limits of infinite and vanishing resource input).1
However, this procedure does not appropriately account for the fact that
the extraction of material resource inputs, their transformation within the production process, and their emission or disposal after use are, at root, transformations of energy and matter. As such they are subject to the laws of
thermodynamics, which is the branch of physics dealing with transformations
of energy and matter. Thermodynamic relations, thus, may impose additional
constraints on economic action (Daly 1997b, Solow 1997).
This chapter formally explores one particular implication that the thermodynamic law of conservation of mass, the so-called Materials-Balance-Principle,
has for modeling production. It is shown that the marginal product as well as
the average product of a material resource input are bounded from above. This
∗
Previously published in Environmental and Resource Economics, 29(3), 307–322 (2004).
Historically, the concept of a production function has been introduced by Wicksell (1893)
and Wicksteed (1992[1894]) to analyze the distribution of income among the factor owners,
and not its physical production (Schumpeter 1954: 1028, Sandelin 1976). But later on, the
concept has come to dominate economists’ thinking about the physically feasible production
possibilities, as in the discussion of economic growth under scarcity of natural resources.
1
51
52
Natural Science Constraints in Environmental and Resource Economics
means that the usual Inada conditions (Inada 1963), when applied to material
resource inputs, are inconsistent with a basic law of nature. This is important since the Inada conditions are usually held to be crucial for establishing
steady state growth under scarce exhaustible resources. While the advocates of
a thermodynamic-limits-to-economic-growth perspective (e.g. Boulding 1966,
Daly 1991[1977], Georgescu-Roegen 1971) usually stress the universal and inescapable nature of limits imposed by laws of nature, pro-economic-growth advocates usually claim that there is plenty of scope for getting around particular
thermodynamic limits by substitution, technical progress and ‘dematerialization’ (e.g. Beckerman 1999, Smulders 1999, Stiglitz 1997). The latter therefore
often conclude that, on the whole, thermodynamic constraints are simply irrelevant for economics. This chapter takes a more differentiated stand, by
analyzing in detail
(i) what exactly are the implications of thermodynamics for modeling production at the level of a single production process, and
(ii) how these constraints carry over to the level of aggregate production,
considering that there is scope for substitution in an economy between
different resources and different production technologies.
This chapter continues and merges two strands in the literature on the production function. In a first strand, the neoclassical production function has been
critically discussed against the background of thermodynamics. GeorgescuRoegen (1971) claims that the neoclassical production function is incompatible
with the laws of thermodynamics, basically because it does not properly reflect
the irreversible nature of transformations of energy and matter, and because
it confounds flow and fund quantities (Daly 1997a, Kurz and Salvadori 2003).
Berry et al. (1978) and Dasgupta and Heal (1979: Chapter 7) demonstrate
that the conservation laws for energy and matter imply that substitutability
between energy-matter inputs, which are subject to the laws of thermodynamics, and other inputs such as labor or capital, which lie outside the domain of
thermodynamics, is restricted. All these apparent inconsistencies between the
laws of thermodynamics and the standard assumptions about the neoclassical
production function have led to more general descriptions of the production
process, which blend the traditional theory of production with the thermodynamic principle of conservation of mass (Anderson 1987, Baumgärtner 2000:
Chapter 4, Pethig 2003: Section 3.3).
Another strand in the literature, more narrowly concerned with production
theory (Shephard 1970), has focused specifically on the Inada conditions. It has
been demonstrated that the Inada conditions follow from other basic properties
of the neoclassical production function (Dyckhoff 1983), and that they impose
strong restrictions on the asymptotic behavior of the elasticity of substitution
The Inada Conditions for Material Resource Inputs Reconsidered
53
between capital and labor (Barelli and de Abreu Pessôa 2003). Furthermore,
the Inada conditions have been shown to be incompatible with another basic
principle within economics, the Law of Diminishing Returns (Färe and Primont
2002).
The chapter is organized as follows. In Section 3.2, the Inada conditions
are briefly reviewed in the context of neoclassical growth theory with and without natural resources. Section 3.3 provides a thermodynamic analysis of the
production process at the micro level, i.e. for a micro production function for
a single commodity. Section 3.4 explores the implications for the Inada conditions at the macro-level, i.e. for an aggregate production function for an all
purpose commodity. Section 3.5 concludes.
3.2
The Inada Conditions on Resource Inputs
With just two inputs, capital K and labor L, the aggregate neoclassical production function for output Y takes the form Y = F (K, L). It is usually assumed
to exhibit constant returns to scale and positive and diminishing marginal
products with respect to each input for all K, L > 0 (Solow 1956, Swan 1956):
∂F
> 0,
∂K
∂2F
< 0,
∂K 2
∂F
> 0,
∂L
∂2F
< 0.
∂L2
(3.1)
Furthermore, following Inada (1963) the marginal product of an input is assumed to approach infinity as this input goes to zero and to approach zero as
the input goes to infinity:
∂F
∂F
= lim
= +∞,
K→0 ∂K
L→0 ∂L
lim
∂F
∂F
= lim
= 0.
K→+∞ ∂K
L→+∞ ∂L
lim
(3.2)
In growth models these so-called Inada conditions are crucial for the existence
of interior steady state growth paths: they are sufficient (yet not necessary) for
the existence of an interior solution in which the economy grows at a constant
and strictly positive rate. Assumptions (3.1) and (3.2) imply that each input is
essential for production, that is, F (0, L) = F (K, 0) = 0, and that output goes
to infinity as either input goes to infinity.
When extending the framework of neoclassical growth theory to also include
natural resources this is usually done by including one additional variable,
R, into the production function, representing material resource input: Y =
F (K, L, R) (Dasgupta and Heal 1974: 9, Solow 1974: 34, Stiglitz 1974: 124).
The same standard assumptions are then made about this resource dependent
production function F as made before about the capital-labor-only-production
function. For instance, F is assumed to be increasing, strictly concave, twice
54
Natural Science Constraints in Environmental and Resource Economics
differentiable, and linear homogeneous (Dasgupta and Heal 1974: 9, Solow 1974:
34, Stiglitz 1974: 124). Furthermore, some more or less direct analogue to
the Inada conditions is assumed in order to guarantee existence of non-trivial
(interior) solutions. For example, Solow (1974: 34) assumes that resources are
essential for production, i.e. F (K, L, 0) = 0, and, at the same time the average
product of R has no upper bound, i.e. there is no α < +∞ with F/R < α.
While this is a particular form of the Inada condition, since it necessarily follows
from limR→0 ∂F/∂R = +∞, Dasgupta and Heal (1974: 11) directly assume that
limR→0 ∂F/∂R = +∞.
Based on one or the other form of Inada conditions, the result is that
even with a limited reservoir of an exhaustible natural resource and with that
resource being essential for production it is possible to maintain a positive
and constant level of consumption forever (Solow 1974). If there is technical progress there might even be exponentially growing consumption (Stiglitz
1974). And while the remaining stock of the resource will approach zero along
the optimal path, the resource will never completely be exhausted (Dasgupta
and Heal 1974).
So, in a sense, these analyses seem to have produced a rather optimistic
answer to the ‘Limits to growth’-concern (Meadows et al. 1972). However, the
Inada conditions as applied (in whatever form) to material resource inputs may
be inconsistent with the thermodynamic law of conservation of mass. This is
demonstrated in the following.
3.3
Thermodynamic Limits to Resource
Productivity at the Micro Level
The First Law of Thermodynamics implies that matter can neither be created
nor annihilated, i.e. in a closed system it is conserved.2 This law establishes
what is known as ‘Materials-Balance-Principle’ in environmental and resource
economics (Ayres 1999a, Pethig 2003).
In order to infer this Law’s implications for the production process, consider
the following simple model of production at the micro level, i.e. production of a
particular good by a particular elementary technology. For the moment assume
that there is only one single natural resource.3 Production of output Y depends
– besides capital K and labor L – on the resource material R:
Y = F (K, L, R).
2
(3.3)
A closed thermodynamic system is one that does not exchange matter with its surrounding. It may, however, exchange energy with its surrounding. For an elementary introduction
into thermodynamics, see e.g. Kondepudi and Prigogine (1998).
3
The generalization to many different natural resources will be done in Section 3.4 below.
The Inada Conditions for Material Resource Inputs Reconsidered
55
As a by-product the production process yields the non-negative amount W of
waste. All three, R, Y and W , are measured in physical (mass) units. Let ρ
with 0 < ρ ≤ 1 denote the (mass) fraction of resource material contained in the
output, and μ with 0 ≤ μ ≤ 1 the (mass) fraction of resource material contained
in the waste.4 While ρ is, in general, a function of K and L, i.e. the resource
content of the final product may be decreased by using more capital and labor
(‘dematerialization’), there obviously are physical limits to dematerialization.
For instance, in order to produce one kilogram of iron screws one needs to
employ at least one kilogram of pure iron. This means that ρ is physically
bounded from below, in particular ρ > 0. Therefore, one may take ρ as a
constant denoting the lower bound to dematerialization, i.e. ρ = const. with
0 < ρ ≤ 1 denotes the minimum (mass) fraction of resource material contained
in the output.
Applying the materials-balance-principle to the production process results
in the following formal balance equation:
R = ρF (K, L, R) + μW,
(3.4)
which states that the resource material which enters the process also eventually
has to come out of the process, be it in the desired product or in the waste.
Rearranging Equation (3.4) into
F (K, L, R)
1
μW
=
1−
R
ρ
R
and noting that μW/R ≥ 0 yields the following upper bound for the average
resource product F/R:
1
F (K, L, R)
≤ .
(3.5)
R
ρ
This establishes the following result.
Proposition 3.1
The average product of resource input, F (K, L, R)/R, is bounded from above
by the inverse of the resource fraction in the good produced, 1/ρ.
Proposition 3.1 has the following implication for the shape of the production
function. Equation (3.5) can be rearranged into
1
F (K, L, R) < R,
ρ
4
That ρ and μ are allowed to be less than 1 is due to materials other than the natural
resource R considered here. These other materials might enter the production process and
be part of the product as well as of the waste. Yet, they are not explicitly represented here.
56
Natural Science Constraints in Environmental and Resource Economics
which states that for fixed values of K and L the graph of F plotted as a
function of R always stays below a line of slope 1/ρ starting at the origin
(Figure 3.1). As ρ becomes smaller, the upper limit on the average resource
product will grow. However, the upper limit will always remain finite, since ρ
is strictly positive.
Y 6
1
ρR
F (K̄, L̄, R)
-
R
Figure 3.1 The materials-balance-principle implies that the graph of
F (K̄, L̄, R) is bounded from above by a line of slope 1/ρ starting at the origin.
With the average resource product F/R bounded from above by the inverse
of the resource fraction in the good produced, 1/ρ, a similar argument applies
to the marginal resource product. Taking the total differential of the material
balance Equation (3.4) and considering only variations in R, i.e. dK = dL = 0,
yields
∂F (K, L, R)
∂W (K, L, R)
dR = ρ
dR + μ
dR .
∂R
∂R
This holds for all dR ≥ 0 and, thus, implies
1=ρ
∂W (K, L, R)
∂F (K, L, R)
+μ
.
∂R
∂R
(3.6)
Equation (3.6) simply is the materials-balance-equation for one additional unit
of resource input employed in the production process. It leaves the process
either as part of the desired product or as waste. The amount of the latter,
∂W/∂R, obviously, cannot be negative. Therefore, rearranging Equation (3.6)
into
∂F (K, L, R)
1
∂W (K, L, R)
=
1−μ
∂R
ρ
∂R
The Inada Conditions for Material Resource Inputs Reconsidered
57
and noting that μ∂W/∂R ≥ 0 yields the following upper bound for the marginal
resource product ∂F/∂R:
∂F (K, L, R)
1
≤ .
(3.7)
∂R
ρ
This establishes the following result.
Proposition 3.2
The marginal product of resource input, ∂F/∂R, is bounded from above by the
inverse of the resource fraction in the good produced, 1/ρ.
It is immediately obvious that if the marginal resource product FR is
bounded from above by 1/ρ, then the marginal resource product as resource
input approaches zero is also bounded from above be the same value:
1
∂F (K, L, R)
≤ .
R→0
∂R
ρ
lim
(3.8)
This is the content of the following corollary to Proposition 3.2.
Corollary
The marginal product of resource input as resource input approaches zero is
bounded from above by the inverse of the resource fraction in the good produced,
1/ρ.
By this corollary it becomes apparent that the Inada conditions (in whatever form), when applied to a micro level production function with material
resource inputs, are inconsistent with the Materials-Balance-Principle.
The intuition behind the simple formal exercise carried out in this section is
that matter cannot be created and, consequently, the produced output cannot
contain more of some material than has been supplied as input. If, for instance,
one needs 100 gram of some resource material in order to produce 1 kilogram of
a good (ρ = 1/10), then, out of 1 kilogram of the resource one can produce at
maximum (i.e. with no waste) 10 kilogram of output. This means, the average
as well as the marginal resource product is bounded from above by 10 (= 1/ρ).
3.4
Thermodynamic Limits to Resource
Productivity at the Macro Level
The simple model of production specified by Equation (3.3) was confined to the
description of one particular micro level production process and one particular
natural resource. In order to analyse how the thermodynamic law of conservation of mass may restrict aggregate production, we should think of production
in a more general way:
58
Natural Science Constraints in Environmental and Resource Economics
• There are many different natural resources, such that one can substitute from one resource to another, in order to avoid thermodynamic constraints on micro level production, such as Conditions (3.5) or (3.7), to
become binding.
• Production of an aggregate output, such as an all purpose commodity
or GDP, is a multi-level process. On a first level, a number of different
intermediate goods are produced from elementary resources (micro level
production). On a second level, the final output is produced from the
intermediate goods (macro level production).5
• In the production of final output there is scope for substitution between
the input of different intermediate goods.
In such a setting, there is plenty of scope for substitution both between different elementary resource materials and between production processes. The
question then is: To what extent do thermodynamic constraints on micro level
production processes, such as Conditions (3.5) or (3.7), carry over to the macro
level? And how do the laws of thermodynamics restrict aggregate production
in such a general setting?
To answer these questions, consider the following model of production of
an all purpose commodity from intermediate goods, which are themselves produced from a variety of elementary natural resources. There are n different
elementary natural resources, numbered by i = 1, . . . , n. Assume that this is a
complete and exhaustive list of material resources actually or potentially used
in production. For example, one may think of this list of natural resources as
the complete list of known chemical elements, in which case n = 112.6 These
are used as inputs in the production of m different intermediate goods, each of
which is produced by a single process of production, numbered by j = 1, . . . , m.
Production of theses intermediates is described by production functions
Yj = F j (Kj , Lj , R1j , . . . , Rnj )
for all j = 1, . . . , m,
(3.9)
where Kj and Lj denote input of capital and labor into production of intermediate good j. Similarly, Rij (with i = 1, . . . , n and j = 1, . . . , m) denotes input
of resource material i into production of intermediate good j. Then,
Ri =
m
Rij
(3.10)
j=1
5
In general, aggregate production may be over more than two levels. But the essential
insights can already be grasped from considering a two-level-system.
6
As of 2003, there are 112 known chemical elements, 83 of which are naturally occurring.
Examples include hydrogen, carbon, oxygen, iron, copper, aluminum, gold and uranium.
Elements 113 through 118 are known to exist, but are not yet discovered (IUPAC 2003).
The Inada Conditions for Material Resource Inputs Reconsidered
59
is the total amount of resource material i utilized in production. Each production process F j also yields a certain amount of waste, Wj .
Let ρij with 0 ≤ ρij ≤ 1 denote the (mass) fraction of resource material i
contained in the intermediate good j, and μij with 0 ≤ μij ≤ 1 the (mass)
fraction of resource material i contained in the waste from producing intermediate good j. Note that ρij will be zero if the intermediate good j does not
contain any material of type i. This may include cases in which some resource
material has been used in, or is even essential for, the production of the intermediate, say as a catalyst, yet the material is not contained in the good
produced. Nonetheless, every intermediate good j – as long as it is a material
good and not an immaterial service – contains some amount of some of the
materials, while not containing anything of other materials. In order to make
this distinction explicit, let
Ij = {i | ρij > 0} ⊆ {1, . . . , n}
(3.11)
be the set of all resources which make up – as far as material content goes –
the intermediate good j. The complement set I¯j = {1, . . . , n} \ I j then denotes
the set of all resources which are not materially contained in the intermediate
good j.
The final good, an all purpose commodity, is produced from capital K,
labor L and the intermediate goods Yj (with j = 1, . . . , m):
Y = F (K, L, Y1, . . . , Ym ),
(3.12)
where Yj (with j = 1, . . . , m) denotes input of intermediate good j as produced
on the first level of production (Equation 3.9). On this level, elementary resources do not enter directly into production, but only indirectly insofar as they
are embedded in the intermediates.7 The final good production function (3.12)
can be interpreted as an aggregate production function of the economy, specifying how the final good is produced from elementary resources, when the Yj ’s
are replaced by the respective micro level production functions (Equation 3.9).
The production of the final good also yields a certain amount of waste,
W . Let ρi with 0 ≤ ρi ≤ 1 denote the (mass) fraction of resource material i
contained in the final output, and μi with 0 ≤ μi ≤ 1 the (mass) fraction of
the resource material i contained in the waste. Note that ρi will be zero if the
final good does not contain any material of type i. Nevertheless, the final good
– as long as it is a material good and not an immaterial service – contains
some amount of some of the materials, while not containing anything of other
materials. For example, a passenger car may contain aluminum, carbon and
7
This assumption, which is also quite plausible, only serves to simplify the notation and
does not restrict the validity of results. It could easily be relaxed.
60
Natural Science Constraints in Environmental and Resource Economics
thallium, but no gold or plutonium. In order to make this distinction explicit,
let
(3.13)
I = {i | ρi > 0} ⊆ {1, . . . , n}
be the set of all resources which make up – as far as material content goes –
the final good. The complement set I¯ = {1, . . . , n} \ I then denotes the set of
all resources which are not materially contained in the final good. Assume that
the final good is material, that is, it contains at least one type of material.
Assumption 3.1
I is non-empty.
With this setting and notation, Propositions 3.1 and 3.2 derived in Section 3.3 above can obviously be translated and generalized into the following
statement:
Lemma 3.1
The thermodynamic law of conservation of mass implies that the micro level
production functions F j (Kj , Lj , R1j , . . . , Rnj ) for all j = 1, . . . , m have the following properties:
F j (Kj , Lj , R1j , . . . , Rnj )
1
≤
Rij
ρij
j
1
∂F (Kj , Lj , R1j , . . . , Rnj )
≤
∂Rij
ρij
and
for all i ∈ Ij .
In words, the average and marginal resource product of resource material i in
producing the intermediate good j is bounded from above by 1/ρij in all cases
where material i is contained in intermediate good j. If, in contrast, material i
is not contained in intermediate good j, the average and marginal resource
product of resource material i do not need to be bounded from above.8
Considering the overall two-level production system, the formal balance
conditions for resource material i (for all i = 1, . . . , n) then read as follows:
Ri =
m
Rij ,
(3.14)
j=1
m
Rij = ρij F j (Kj , Lj , R1j , . . . , Rnj ) + μij Wj
for all j = 1, . . . , m ,
(3.15)
ρij F j (Kj , Lj , R1j , . . . , Rnj ) = ρi F (K, L, Y1, . . . , Ym ) + μi W .
(3.16)
j=1
8
Of course, in the latter case they may still be bounded from above for reasons other than
thermodynamic necessity.
The Inada Conditions for Material Resource Inputs Reconsidered
61
Equation (3.14) states that the total amount of resource material i employed
in production, Ri , may be used in any of the m production processes for intermediate goods. Equation (3.15) expresses conservation of mass of resource
material i on the first level of production in all of the m intermediate good
production processes: the total amount of material utilized in one of these processes, Rij , leaves the process either as part of the intermediate good j or as
part of the waste generated by that process. Equation (3.16) expresses conservation of mass on the second level of production: resource material i enters
production of the final product indirectly, namely embedded in the m intermediate goods, each of which has a material content of that material of ρij Yj .
It leaves the production process either as part of the final good or as part of
the waste generated by that process. Summing balance Equation (3.15) over
all micro level processes j, adding balance Equation (3.16) for the macro level,
and using (3.14) yields an overall balance condition for material i:
Ri = ρi F (K, L, Y1, . . . , Ym ) + μiW +
m
μij Wj .
j=1
This condition states that the material utilized in production, Ri , leaves the
two-level production system either as part of the final good, or as part of the
waste generated by the final good production process, or as part of the waste
generated in any of the m intermediate good production processes.
Rearranging Equation (3.17) into
⎡
⎤
m
μij Wj
⎥
F (K, L, Y1 , . . . , Ym )
μi W
1 ⎢
j=1
⎥
1
−
(3.17)
= ⎢
−
⎦
Ri
ρi ⎣
Ri
Ri
m
and noting that μi W/Ri ≥ 0 as well as
j=1 μij Wj /Rj ≥ 0, the following
inequality holds:
1
F (K, L, Y1, . . . , Ym)
≤ .
(3.18)
Ri
ρi
For all materials i ∈ I which make up the final good, ρi is strictly positive so
that 1/ρi < +∞ is a a finite upper bound for the average resource product of
material i in aggregate production, F/Ri . From Assumption 3.1 it follows that
there is at least one such material. For all other materials with i ∈
/ I, ρi is zero
so that 1/ρi is not a finite upper bound. This establishes the following result.
Proposition 3.3
(i) For all materials i ∈ I, which make up the final good, the average product
of resource material i in aggregate production, F/Ri , is bounded from
above by the inverse of this material’s mass fraction in the final good,
1/ρi .
62
Natural Science Constraints in Environmental and Resource Economics
(ii) There exists at least one such material for which the average product is
bounded from above.
(iii) For all materials i ∈
/ I, which are not contained in the final good, the
average product of resource material i in aggregate production, F/Ri , does
not need to be bounded from above.
In order to derive an analogue statement about the marginal resource products,
take the total differential of the material balance Equation (3.17) for material i,
with the Yj in production function F replaced my the intermediate good production functions F j (Equation 3.9), and consider only variations in resource
material i (i.e. dK = dL = 0, dKj = dLj = 0 for all j and dRi j = 0 for all
i = i):
m
m
m
∂F ∂F j
∂W ∂F j
∂Wj
dRij + μi
dRij +
μij
dRij . (3.19)
dRi = ρi
∂Yj ∂Rij
∂Yj ∂Rij
∂Rij
j=1
j=1
j=1
From balance Equation (3.14) it follows that
dRi =
m
dRij .
(3.20)
j=1
Replacing dRi in Equation (3.19) by expression (3.20) and rearranging terms
yields
m j=1
∂F ∂F j
∂W ∂F j
∂Wj
− μi
− μij
1 − ρi
dRij = 0 .
∂Yj ∂Rij
∂Yj ∂Rij
∂Rij
(3.21)
This holds for all dRij ≥ 0 and, thus, implies that the term in brackets equals
zero. This can be rearranged into
∂W ∂F j
∂Wj
1
∂F ∂F j
=
− μij
1 − μi
.
(3.22)
∂Yj ∂Rij
ρi
∂Yj ∂Rij
∂Rij
Noting that the second and third term in brackets are non-negative yields the
following inequality, which holds for all i and j:
1
∂F ∂F j
≤
.
∂Yj ∂Rij
ρi
(3.23)
On the other hand, taking the total differential of the defining equation for
production function F (Equation 3.12), with the Yj in production function F
replaced my the intermediate good production functions F j (Equation 3.9),
The Inada Conditions for Material Resource Inputs Reconsidered
63
and considering only variations in resource material i (i.e. dK = dL = 0,
dKj = dLj = 0 for all j and dRi j = 0 for all i = i), yields:
m
∂F ∂F j
dRij .
dF =
∂Yj ∂Rij
j=1
(3.24)
From (3.24) one obtains, using Inequality (3.23) and Equation (3.20)
dF =
m
m
m
∂F ∂F j
1
1 1
dRij ≤
dRij =
dRij = dRi ,
∂Yj ∂Rij
ρ
ρi j=1
ρi
j=1
j=1 i
(3.25)
so that we have the following inequality:
dF ≤
1
dRi .
ρi
(3.26)
Interpreting this inequality for differentials as an algebraic expression and rearranging finally yields:
dF
1
≤ .
(3.27)
dRi
ρi
Since the production function F (Equation 3.12) does neither directly nor indirectly depend on Ri , the expression dF/dRi should not be interpreted as a
(total) derivative in the strict sense. However, in a rather loose sense, it may
be interpreted as something like a total derivative. It tells us by how much
the aggregate output Y changes when an additional marginal unit of resource
material Ri is used in production, by dividing it up among
the m intermediate
9
good production processes in such a manner that dRi = m
j=1 dRij .
Again, for all materials i ∈ I which make up the final good, ρi is strictly
positive so that, according to Inequality (3.27), 1/ρi < +∞ is a a finite upper
bound on dF/dRi. From Assumption 3.1 it follows, that there is at least one
such material. For all other materials with i ∈
/ I, ρi is zero so that 1/ρi is not
a finite upper bound. This establishes the following result.
Proposition 3.4
(i) For all materials i ∈ I, which make up the final good, the marginal product
of resource material i in aggregate production, dF/dRi , is bounded from
9
In that sense, one could define
m
∂F ∂F j
dF
≡
dRi
∂Yj ∂Rij
j=1
subject to
dRi =
m
dRij .
j=1
However, this definition is not unique. While there is a multitude of ways in which dRi may be
divided up among the m different intermediate good production processes, Inequality (3.26)
holds in any case. Hence, result (3.27) holds irrespective of the exact way in which dF/dRi
is defined.
64
Natural Science Constraints in Environmental and Resource Economics
above by the inverse of this material’s mass fraction in the final good,
1/ρi .
(ii) There exists at least one such material for which the marginal product is
bounded from above.
(iii) For all materials i ∈
/ I, which are not contained in the final good, the
marginal product of resource material i in aggregate production, dF/dRi ,
does not need to be bounded from above.
Comparing Propositions 3.3 and 3.4 for macro level production with Propositions 3.1 and 3.2 for micro level production, we see that all results that were
obtained in the simple micro-level setting essentially carry over to the general
two-level-multi-resources-multi-processes setting. The only qualification is that
the boundedness-results only hold for materials in the set I which make up the
final good.
3.5
Discussion
It has been shown that the Inada conditions, when applied to material resource
inputs, may be inconsistent with the thermodynamic law of conservation of
mass, the so-called Materials-Balance-Principle. In particular, the analysis has
revealed that the average and marginal product of a natural resource material
in aggregate production are bounded from above due to the thermodynamic
law of mass conservation if the final good, an all-purpose commodity, contains
this material. An upper bound is given by the inverse of this material’s mass
fraction in the final good.
The analysis was based on a model of multi-level production where different
intermediate goods are produced from different elementary resources, and an
all-purpose final commodity is produced from these intermediates. Note that
no limits on substitution between resource materials or between intermediate
products have been assumed. Another thing to note is that the upper bound
specified by inequalities (3.18) and (3.27) is certainly not the lowest upper
bound, but comes out of a more or less crude estimation (from Equations (3.17),
(3.25) to Inequalities (3.18), (3.26)). For this reason, the upper bound given
here does not depend on any model parameters other than ρi .
When discussing the relevance of these results for the natural-resourcesand-economic-growth-debate, the crucial questions are:
(i) How many, and which natural resource materials are elements of the
set I? That is, what are the natural resource materials that make up,
materially, the final good?
The Inada Conditions for Material Resource Inputs Reconsidered
65
(ii) What is these materials’ mass fraction ρi in the final good?
(iii) How do the set I and the relevant parameter values ρi change over time?
It is probably the difference in opinion on these questions which make a difference between the ‘optimists’ and the ‘pessimists’ in the discussion about the
thermodynamic limits to economic growth.
This analysis has revealed that there are stringent thermodynamic limits to
resource productivity in aggregate production for a number of natural resource
materials. The analysis has also revealed that not all resource materials need
to be limiting. Hence, the overall conclusion is that the question of thermodynamic limits to economic growth requires a detailed investigation, with separate
analyses and results for each material. This shifts the focus of the debate from
overall growth to the more detailed level of factor substitution and structural
change.
66
Natural Science Constraints in Environmental and Resource Economics
4. Temporal and Thermodynamic
Irreversibility in Production Theory∗
4.1
Introduction
From a physical point of view, irreversibility is an essential dynamic feature
of real production. Therefore, it should be properly taken into account in
dynamic analyses of production systems.1 For example, engineers account
for irreversibility when designing and optimizing production processes (Bejan 1997, Bejan et al. 1996, Brodyansky et al. 1994, Szargut et al. 1988), and
economists consider irreversibility when studying economy-environment interactions (Ayres 1998, 1999b, Baumgärtner et al. 2006, Faber et al. 1995[1983],
Georgescu-Roegen 1971, Mäler 1974, Perrings 1987, Pethig 1979).
The idea of irreversibility can be rigorously rooted in the laws of nature
(Zeh 2001), in particular in thermodynamics (Kondepudi and Prigogine 1998:
84ff). The importance of thermodynamic irreversibility, and the physicists’
preoccupation with this concept, lies in the fact that it precludes the existence
of perpetual motion machines, that is, devices which use a limited reservoir of
available energy to perform work forever (Second Law of Thermodynamics). It
is an everyday experience that no such thing as a perpetual motion machine
exists. In order to make this insight accessible to economic analysis, and to
the study of long term economy-environment interactions, it is necessary to
adequately represent thermodynamic irreversibility as a constraint for economic
action (Georgescu-Roegen 1971).2
∗
Sections 4.2 and 4.3.2 of this Chapter have previously been published in Economic Theory, 26(3), 725–728 (2005). They formally elaborate an idea which has originally been
proposed by Baumgärtner (2000: Section 11.1).
1
In this chapter, the focus is on irreversibility in processes of production. Other origins
of irreversibility, for instance investment and learning under uncertainty (e.g. Dixit 1992,
Dixit and Pindyck 1994, Pindyck 1991) or ‘lock-in’ due to increasing returns (Arthur 1989),
are not considered here. For a comprehensive survey of various notions of irreversibility in
economics see Dosi and Metcalfe (1991).
2
The contribution of Georgescu-Roegen (1971), who considered the Second Law of Thermodynamics ‘the most economic of all physical laws’ (p. 280), initiated a heated debate over
67
68
Natural Science Constraints in Environmental and Resource Economics
Economists have devoted some effort to incorporating irreversibility into
production theory. The reason is primarily a concern for physical realism in
the description of the set of ‘feasible’ production processes.3 However, irreversibility has often been introduced into the theory as an ad-hoc-assumption.
As a result, the assumption did not always achieve what it actually should
achieve from a thermodynamic point of view, namely to imply irreversibility of
the system’s evolution as stated by the Second Law of Thermodynamics.
In this chapter, I will introduce a formal and rigorous definition of thermodynamic irreversibility, which is (i) sound from a physical point of view and
(ii) formulated such that it is compatible with formal modelling in economic
production theory. In order to assess, whether – and to what extent – different notions of irreversibility from production theory capture thermodynamic
irreversibility, I will then reexamine two prominent irreversibility concepts –
the one due to Koopmans (1951b) and the one due to Arrow-Debreu (Arrow
and Debreu 1954, Debreu 1959) – against the definition of thermodynamic
irreversibility.
The chapter is organized as follows. In Section 4.2, I briefly review the concept of thermodynamic irreversibility and distinguish it from the weaker concept of temporal irreversibility. I propose formal definitions of both concepts.
In Section 4.3, the irreversibility concepts of Koopmans and of Arrow-Debreu
are reexamined against these definitions. I show that Koopmans’ notion of
irreversibility fully captures thermodynamic irreversibility, and that the notion of Arrow-Debreu does not capture thermodynamic irreversibility but only
the weaker aspect of temporal irreversibility. I conclude with Section 4.4, by
putting the results into perspective.
4.2
The Thermodynamic Notion of Irreversibility
The textbook definition of thermodynamic irreversibility builds on the consideration of an isolated system, defined by its boundaries.4 A transformation
the question of whether the Entropy Law is relevant for economics (e.g. Burness et al. 1980,
Daly 1992b, Kåberger and Månsson 2001, Khalil 1990, Lozada 1991, 1995, Norgaard 1986,
Townsend 1992, Williamson 1993, Young 1991, 1994). See Baumgärtner et al. (1996) for a
summary of this discussion.
3
In economic theory, irreversibility of the total production set has originally been considered as fundamental (beside closedness, convexity and the no-free-lunch condition) for the
existence proof of general competitive equilibrium in economies with production (Arrow and
Debreu 1954, Debreu 1959). Yet, later it has become obvious that, while this assumption
simplifies the proof, it is not necessary (Debreu 1962: 258; Koopmans 1957: 78, Footnote 4;
McKenzie 1959: 55, 1961).
4
Recall (from Section A2.1) that a thermodynamic system is called isolated if it exchanges
neither energy nor matter with its surrounding environment; it is called closed if it exchanges
Temporal and Thermodynamic Irreversibility in Production Theory
69
over time of an isolated system between some initial state and some final state
is called irreversible if there is no means by which the system can be exactly
restored to its initial state (Kondepudi and Prigogine 1998, Zeh 2001). Otherwise, the transformation is called reversible.
As for closed or open systems, one can consider the system and its environment, such that the overall system is once again isolated. Thus, a transformation in an open system is called irreversible if there is no means by which the
system and its environment can be exactly restored to their respective initial
states. Since the economy is an open system in the thermodynamic sense, any
physically meaningful analysis of economic irreversibility should consider the
economy plus its natural environment (‘Planet Earth’).
Formalizing this notion of thermodynamic irreversibility in a way customary
to economic theory requires one
(i) to distinguish between states of the system (stock variables) and transformations of the system (flow variables), and
(ii) to consider time as an explicit variable.
Consider an economy with n physically different goods, including natural resources and wastes, and T discrete points in time. Let si (t) ∈ IR+ denote the
stock of good i (i = 1, . . . , n) at time t ∈ [1, . . . , T ] and s(t) = (s1 (t), . . . , sn (t)) ∈
IRn+ . At every point in time, s completely characterizes the state of the economy
in terms of the different state variables si . This is an explicit time representation (ETR) of the commodity space, since time shows up explicitly. One could
also adopt an implicit time representation (ITR), by making the following assumption (Arrow and Debreu 1954: 266, Debreu 1959: 29).
Assumption 4.1
The same physical commodity at two different points in time is regarded as
two different economic commodities. (Dated goods)
In order to compare different notions of irreversibility it is helpful to have
a mapping from ITR to ETR-representations. Let Y ⊆ IRnT be the set of
all feasible aggregate ITR-production vectors y, that is, the set of all feasible
transformations of the system. A production vector y has as components the
net output of all dated commodities. For the sake of notational convenience
assume that all inputs enter a production process y simultaneously at time t(y)
and that all outputs are simultaneously obtained at t(y).5 One may then define
the mapping
Π : IRnT → IRn+2
with
Π(y) = (ŷ, t(y), t(y)) ,
(4.1)
energy, but not matter; and it is called open if it exchanges both energy and matter with its
surrounding environment.
5
In other words, consider elementary production processes (Takayama 1985: 487–488).
70
Natural Science Constraints in Environmental and Resource Economics
where ŷ ∈ IRn is the vector of physical net output and t (t) denotes the point in
time at which inputs (outputs) are supplied (obtained) under transformation
y. For every ITR-production vector y ∈ Y the image Π(y) is the corresponding
ETR-production vector. With this notation, the effect of a transformation y
on the state s of the system is given in ETR-terminology by
s(t(y)) = s(t(y)) + ŷ .
(4.2)
In order to assess the dynamic effect of several transformations on the state
of the system, one needs to make an assumption about what combinations
of transformations are feasible. For that sake, I will assume throughout this
chapter that the ITR-production set Y has the following property:
Assumption 4.2
If y 1 , y 2 ∈ Y and t(y j ) ≥ t(y i) for i, j = 1, 2 and i = j, then y 1 + y 2 ∈ Y .
(Temporal additivity)
In words, if both y 1 and y 2 are feasible ITR-production vectors and one
of them (y j ) begins after the other one (y i) has ended, then it is feasible to
carry out first the physical transformation described by y j and then, later
in time, the physical transformation described by y i. The property is called
temporal additivity since it refers solely to adding two production processes
in the time dimension. This assumption is considerably weaker than the usual
assumption of additivity (y 1, y 2 ∈ Y implies y 1 +y 2 ∈ Y ) since only the addition
of production vectors which are carried out one after the other is assumed to
be feasible. In contrast, ordinary additivity would also allow for the addition
of simultaneous physical transformations, thus ruling out decreasing returns to
scale.6 One can now define thermodynamic irreversibility as follows.
Definition 4.1
An ITR-production set Y has the property of thermodynamic irreversibility if
and only if for every y ∈ Y with ŷ = 0 there exists no y ∈ Y with ŷ = −ŷ.
In words, if y is a feasible non-trivial production vector there exists no
feasible production vector which reverses the physical net effect of y. Hence,
6
While temporal additivity (Assumption 4.2) is considerably weaker than full additivity
(y , y 2 ∈ Y implies y 1 + y 2 ∈ Y ) one may, however, still imagine examples where this
assumption is not fulfilled. For instance, when the change in the state of the system induced
by the first transformation makes it impossible to carry out the second transformation later
on. The following system would be a (disgusting!) example: Transformation y 1 uses one cup
of black coffee and a drop of vanilla syrup at time t0 to produce vanilla flavored coffee at
time t1 > t0 ; transformation y 2 uses one cup of black coffee and one drop of banana syrup
at time t2 to produce banana flavored coffee at time t3 > t2 . If the system has an initial
stock of only one cup of black coffee at time t0 , both transformations y 1 and y 2 are feasible
separately, but it is not feasible to carry out both of them. What Assumption 4.2 essentially
states is that stocks are not constraining the transformations of the system.
1
Temporal and Thermodynamic Irreversibility in Production Theory
71
under thermodynamic irreversibility the initial state of the system cannot be
restored: if the system has initially been in some state s0 , and as a consequence
of the transformation y has evolved into some final state sf = s0 + ŷ, then there
is no transformation y which brings the system into a state which is identical
to the initial state s0 .7
Note that the crucial condition (ŷ = −ŷ) in Definition 4.1 is in terms of the
physical net effect of transformations y and y . It does not constrain in any way
the time structure of these transformations, that is, when inputs are supplied
and outputs are obtained. A weaker restriction on the set of feasible production
vectors than thermodynamic irreversibility is temporal irreversibility.
Definition 4.2
An ITR-production set Y has the property of temporal irreversibility if and
only if for every y ∈ Y with ŷ = 0 there exists no y ∈ Y with ŷ = −ŷ and
t(y ) = t(y), t(y ) = t(y).
In words, if y is a feasible non-trivial production vector, there exists no y which reverses both the physical net effect of transformation y and its temporal
input-output-structure.
While thermodynamic irreversibility excludes the possibility that the system returns into its initial state s0 at any, possibly later, point in time t ≥ t0 ,
temporal irreversibility only excludes the possibility that the system returns
into its initial state s0 at initial time t0 . Temporal irreversibility, thus, states
that one cannot reverse physical transformations by going back in time. Obviously, temporal irreversibility is a weaker concept than thermodynamic irreversibility, in the sense that thermodynamic irreversibility implies temporal
irreversibility, but not vice versa.8
Proposition 4.1
If an ITR-production set Y has the property of thermodynamic irreversibility,
it also has the property of temporal irreversibility.
4.3
Notions of Irreversibility in Production Theory
Having now a formal and rigorous definition of thermodynamic irreversibility (Definition 4.1), and also of the weaker concept of temporal irreversibility
The condition sf + ŷ = s0 + ŷ + ŷ = s0 can only be fulfilled for ŷ = −ŷ, which is
precluded by thermodynamic irreversibility according to Definition 4.1.
8
The condition on Y in Definition 4.2 of temporal irreversibility includes the condition in
Definition 4.1 of thermodynamic irreversibility (ŷ = −ŷ), and puts additional conditions on
Y (t(y ) = t(y), t(y ) = t(y)).
7
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Natural Science Constraints in Environmental and Resource Economics
(Definition 4.2), one can now reexamine the irreversibility concept of Koopmans (1951b) and the one of Arrow-Debreu (Arrow and Debreu 1954, Debreu
1959) against these definitions.9
4.3.1
Koopmans’ Notion of Irreversibility
Koopmans’ (1951b) theory of production is based on the analysis of activities,
that is, vectors representing feasible combinations of inputs and outputs. His
analysis is static. In the language introduced in Section 4.2 above, all of Koopmans’ statements refer to the physical net effect ŷ of transformations y. There
are two basic assumptions associated with the notion of an activity: divisibility
and additivity (Koopmans 1951b: 36).
Assumption 4.3
Each activity is capable of continuous proportional expansion or reduction.
(Divisibility)
Assumption 4.4
Any number of activities can be carried out simultaneously without modification in the technical rations by which they are defined, provided only that the
total resulting net output of any commodity, whenever negative, is within the
limitations on primary resources. The joint net output of any commodity from
all activities then equals the sum of the net outputs of that commodity from
the individual activities. (Additivity)
Assumptions 4.3 and 4.4 imply that all statements about feasible activities also hold for linear combinations of feasible activities. Thus, they exclude
economies or diseconomies of scale. Note that Koopmans’ additivity assumption (Assumption 4.4) is considerably stronger then the assumption of temporal additivity (Assumption 4.2), since it refers to the simultaneous addition of
physical transformations.
Koopmans (1951b: 48) then introduces the idea of irreversibility in the form
of a fundamental postulate:
Postulate A
It is impossible to find a set of positive amounts of some or all activities, of
which the joint effect is a zero net output for all commodities.
This is to say (Koopmans 1951b: 48–49) that it is not possible to find activity vectors, such that the net output resulting from one of them is being exactly
9
The treatment of Arrow-Debreu has become the state-of-the-art way of incorporating
irreversibility into production theory. It is essentially what is taught in many economic
textbooks (e.g. Mas-Colell et al. 1995: 132).
Temporal and Thermodynamic Irreversibility in Production Theory
73
offset by the net output brought about by a linear combination of the other activities. Koopmans’ Postulate A thus excludes the possibility that by a suitably
chosen combination of activity vectors the system undergoes some activites as a
result of which it returns back to its initial state. Obviously, Koopmans’ notion
of irreversibility is exactly one of thermodynamic irreversibility as specified by
Definition 4.1 above.10
4.3.2
The Arrow-Debreu Notion of Irreversibility
A formalized and slightly, but significantly, altered version of Koopmans’ irreversibility concept is introduced by Arrow and Debreu (1954) as well as Debreu
(1959). Their irreversibility concept consists of two elements:
(i) a formal statement about the set of feasible production vectors and
(ii) the ITR-convention of dated goods (Assumption 4.1)
The formal statement (i) is the following (Arrow and Debreu 1954: 267, Debreu
1959: 40):
Assumption 4.5
Y ∩ −Y ⊆ {0}. (Arrow-Debreu-Irreversibility)
This can be reformulated as saying that if a production vector y is feasible,
the reverse production vector −y is not feasible unless y describes null-activity
(y = 0). While this is, as it stands, just a rephrasement of Koopmans’ Postulate A in more technical language, the ITR-convention (ii) constitutes a new
element for the notion of irreversibility.
To illustrate how the dated-goods-interpretation (Assumption 4.1) affects
the working of Assumption 4.5, consider an economy with two physically distinct commodities, say metal and screws, and two points in time, t0 and t1 > t0 .
Initially, the state of the system is
s(t0 ) = (M, S) ,
(4.3)
with M denoting the total initial stock of metal in the economy and S the total
initial stock of screws. Assume that there exists a feasible production process
y which turns m > 0 units of metal at time t0 into one unit of screws at time
t1 . The corresponding ETR-production vector reads Π(y) = ((−m, +1), t0 , t1 ).
This transformation changes the state of the system to
s(t1 ) = s(t0 ) + ŷ
= (M, S) + (−m, +1)
= (M − m, S + 1) .
10
(4.4)
Note that Tjalling C. Koopmans by training was a physicist and his first two publications
were in physics (Niehans 1990: 408).
74
Natural Science Constraints in Environmental and Resource Economics
By Assumption 4.1, there are four ITR-goods: metal at time t0 , metal at time
t1 , screws at time t0 and screws at time t1 . With the convention that production
vectors have as components net output of metal at time t0 , net output of
metal at time t1 , net output of screws at time t0 , and net output of screws at
time t1 , the production process y is represented by the ITR-production vector
y = (−m, 0, 0, +1). By Assumption 4.5, one has −y = (+m, 0, 0, −1) ∈
/ Y.
This means, it is not possible to turn one unit of screws at time t1 back into
m units of metal at time t0 < t1 , thus bringing back the system to its original
state s(t0 ) = (M, S) at the initial point in time t0 (Figure 4.1).
state
s
1
6
y
s0
s
y∈Y
−y ∈
/Y
−y
s
-
t0
t1
time
Figure 4.1 The Arrow-Debreu notion of irreversibility establishes temporal irreversibility. (Figure from Baumgärtner 2000: 236).
Obviously, Assumptions 4.1 and 4.5 establish temporal irreversibility. But
they do not suffice to establish thermodynamic irreversibility.
Proposition 4.2
(i) Assumptions 4.1 and 4.5 imply that Y has the property of temporal irreversibility.
(ii) Assumptions 4.1 and 4.5 do not imply that Y has the property of thermodynamic irreversibility.
Proof: (i) is obvious. (ii) is proven by giving an example for an ITR-production
set Y which satisfies Assumptions 4.1 and 4.5, but does not have the property
of thermodynamic irreversibility.
Consider the metal-and-screws-economy introduced above with four points
in time, t0 < t1 < t2 < t3 . Assume that there are two feasible production
processes: the first one (y 1 ) turns m > 0 units of metal at time t0 into one unit
of screws at time t1 ; the other one (y 2) turns one unit of screws at time t2 into
Temporal and Thermodynamic Irreversibility in Production Theory
75
m units of metal at time t2 . The corresponding ETR-production vectors read
Π(y 1) = ((−m, +1), t0 , t1 ) and Π(y 2 ) = ((+m, −1), t2 , t3 ).
Under Assumption 4.1, there are eight distinct ITR-goods: metal at time
t0 , metal at time t1 , metal at time t2 , metal at time t3 , screws at time t0 ,
screws at time t1 , screws at time t2 and screws at time t3 . With the convention
that the components of the production vectors represent net output of metal at
times t0 , t1 , t2 , t3 , and net output of screws at time t0 , t1 , t2 , t3 , the production
possibilities can be represented by the ITR-production set Y = {y 1, y 2 } with
y 1 = (−m, 0, 0, 0, 0, +1, 0, 0) and y 2 = (0, 0, 0, +m, 0, 0, −1, 0).
Assumption 4.5 is satisfied, as y 2 = −y 1 . But with ŷ 1 = (−m, +1) and ŷ 2 =
(+m, −1) one has ŷ 2 = −ŷ 1 , in contradiction of thermodynamic irreversibility.
2
state
s1
y 1, y 2 ∈ Y
6
s
y1
s0
s
s
@
@
2
1
@ y = −y
@
@
Rs
@
-
t0
t1
t2
t3 time
Figure 4.2 The Arrow-Debreu notion of irreversibility does not establish thermodynamic irreversibility. (Figure modified from Baumgärtner 2000: 237).
The example used in the proof can be interpreted as follows (Figure 4.2).
Initially, the state of the system is again
s(t0 ) = (M, S) ,
(4.5)
with M denoting the total initial stock of metal in the economy and S the total
initial stock of screws. As a consequence of carrying out production process y 1,
which turns m > 0 units of metal at time t0 into one unit of screws at time t1 ,
the state of the economy changes from its initial state s(t0 ) to
s(t1 ) = s(t0 ) + ŷ 1
= (M, S) + (−m, +1)
= (M − m, S + 1) .
(4.6)
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Natural Science Constraints in Environmental and Resource Economics
That is, the stock of metal has decreased by m units, and the stock of screws
has increased by one unit. If then production process y 2 is carried out, which
is also feasible and turns one unit of screws at time t2 into m units of metal at
time t2 , the state of the economy changes from s(t2 ) = s(t1 ) to
s(t3 ) = s(t2 ) + ŷ 2
= (M − m, S + 1) + (+m, −1)
= (M, S) ,
(4.7)
thus restoring the initial state of the system: s(t3 ) = s(t0 ). Hence, by carrying
out first y 1 and subsequently y 2 the economy would undergo a cyclical transformation process: some amount of metal is turned into screws, and later all
screws are transformed back into the original amount of metal. This simple
model would be the blueprint for a perpetual motion machine, in contradiction
of thermodynamic irreversibility
4.4
Conclusion
Summing up, while Koopmans’ notion of irreversibility fully captures thermodynamic irreversibility, the Arrow-Debreu notion does not capture thermodynamic irreversibility but only the weaker aspect of temporal irreversibility. The
crucial difference between the two concepts is Arrow-Debreu’s interpretation of
goods as being dated (Assumption 4.1). While the dated-goods-interpretation
allows one to obtain a theory of time and uncertainty with seemingly no formal
effort (Debreu 1959: 98), it considerably reduces the physical content of the
irreversibility assumption proper (that is, Assumption 4.5).11, 12 Had Arrow11
It is interesting to note that Debreu uses the dated-goods-interpretation only when dealing with uncertainty (Debreu 1959: Chapter 7). Yet, for this sake he considers an exchange
economy without production (or irreversibility). On the other hand, when studying production (and irreversibility) he does not make any use of the dated-goods-interpretation.
12
The usefulness of introducing time into production theory by considering goods as being
dated may be questioned anyway (Baumgärtner 2000: 239). If one commodity (as described
by its physical properties) at one given point in time and the very same commodity (as
described by its physical properties) at another point in time are taken as two different
economic goods, then such an economic description creates artificial processes of production
where actually no physical transformation of energy and matter takes place. For example,
simply letting an object rest in some place for a while would constitute a process of production
under such an interpretation, since one economic good – the object at an earlier time – is
transformed into a different economic good – the very same object at some later time. In
this view, economic production may happen where actually no physical transformation of
energy and matter takes place and no positive amount of entropy is created. So, the datedgoods-interpretation creates a notion of production which is at odds with the physical view
of production as a transformation of energy and matter.
Temporal and Thermodynamic Irreversibility in Production Theory
77
Debreu not made the dated-goods-interpretation, their notion of irreversibility
would be fully equivalent to Koopmans’ one.
In order to put these results into perspective, one should add two comments. First, Arrow and Debreu were not primarily concerned with making
realistic assumptions, but with identifying the weakest, and thus most general,
assumptions under which a general competitive equilibrium could be shown
to exist.13 And indeed, temporal irreversibility is a weaker assumption than
thermodynamic irreversibility (Proposition 4.1). As a consequence, all results
obtained in an Arrow-Debreu-framework, including their assumption of irreversibility, apply as well to systems for which the more restrictive assumption
of thermodynamic irreversibility is made.
Second, while thermodynamic irreversibility is a fact of nature for completely specified thermodynamic systems – that is, isolated, closed or open
systems which are described in terms of all state variables – it does not need
to hold in an incompletely specified system (Dyckhoff 1994: 78). For example, turning metal first into screws and then completely back into the original
amount of metal, may appear feasible as long as one neglects energy. But transforming metal into screws requires energy, thus reducing the stock of available
energy in the system; and so does the recycling of metal from screws, which
further decreases the stock of available energy in the system. Therefore, the
transformation of metal into screws, which seems to be reversible when neglecting the state variable ‘available energy’, turns out to be actually irreversible
when properly taking all physical state variables into account.14
In the end, the relevance of thermodynamic irreversibility for economic
analysis comes down to the question of which system is under study. Arrow
and Debreu’s description of production vectors includes ‘only the components
which correspond to marketable commodities’ (Arrow and Debreu 1954: 267).
This system is incompletely specified from the physical point of view, because
the commodity space does not include essential physical state variables such
as available energy or entropy. Therefore, there is no reason to take thermodynamic irreversibility to be a relevant property of this system. But if one
aims at an encompassing analysis of economy-environment interactions, essential physical state variables have to be included in the description of the system.
Thermodynamic irreversibility then is a relevant property of the system, and
the Arrow-Debreu notion of irreversibility is too weak to be in full accordance
with the laws of nature.
13
But see Footnote 3 above.
Baumgärtner (2000: Chapter 11) further elaborates on the ‘complete-representation’approach to modeling irreversibility, based on the concept of joint production.
14
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Natural Science Constraints in Environmental and Resource Economics
5. Necessity and Inefficiency in the Generation
of Waste∗
with Jakob de Swaan Arons
5.1
Introduction
The sheer amount of waste generated in modern industrial economies is enormous. For example, in 1990 the amount of waste in West Germany (measured
in physical units, such as tons) exceeded the amount of useful economic output (also measured in physical units) by more than a factor of four: out of a
total material output of 59,474.6 million metric tons generated by all sectors
of the economy, only 3,602.6 million metric tons (6.1 %) were contained in the
different components of GDP, while 7,577.2 million metric tons (12.7 %) were
intermediate outputs for reuse within the economy (including recovered and
recycled materials) and 48,294.8 million metric tons (81.2 %) were final wastes
(Statistisches Bundesamt 1997). This huge dimension of material waste generation is also confirmed for other industrialized countries, e.g. Denmark, Italy
and the USA (Acosta 2001).
The notion of ‘waste’ is a difficult one, as a proper definition should build
on descriptive materials-balance on the one hand, and normative human attitudes and valuation on the other hand (Bisson and Proops 2002). ‘Waste’
essentially denotes an ultimately unwanted by-product in the production of
some desired good or service. We will use the term here in a slightly more
general sense, to refer to a by-product of a desired good on the level of a single
production process. For example, the process of enriching uranium generates
depleted uranium as a by-product together with the desired enriched uranium.
Although this by-product may be used to produce other products (e.g. special
ammunition), it is a ‘waste’ in the process of enriching uranium. Thus, we
focus on the material origin of waste. At the same time, we disregard two
related issues which are most relevant in the discussion of waste, and which
∗
Previously published, without the Appendix, in Journal of Industrial Ecology, 7(2), 113–
123 (2003). The Appendix is a revised version of Baumgärtner (2000: 72–77).
79
80
Natural Science Constraints in Environmental and Resource Economics
are treated in detail elsewhere in the literature. First, we do not explicitly
analyze whether the by-product (‘waste’) considered here is actually positively
or negatively valued, as this would require an economic analysis (Baumgärtner
2000, 2004d, Powell et al. 2002). Instead, we argue that in many cases one can
safely assume that it is unwanted. Second, by looking only at by-products on
the level of a production process, we do not follow the by-products’ broader
impact in an economy. In particular, we do not consider the possibility that
what is an unwanted by-product for one producer may be a valuable resource
for another producer, giving rise to the idea of an ‘industrial ecology’ (Ayres
and Ayres 2002, Hardy and Graedel 2002). With such a notion of ‘waste’,
our analysis is relevant for the field of industrial ecology, because it deals with
the qualitative and quantitative conditions under which all those ‘waste’ byproducts come into existence that give rise to the problems commonly studied
in the industrial ecology literature.
It has been argued, based on the thermodynamic laws of mass conservation
and entropy generation, that in industrial production processes the occurrence
of waste is as necessary as the use of material resources (Ayres and Kneese
1969, Faber et al. 1998, Georgescu-Roegen 1971).1 On the other hand, it seems
to be quite obvious that the sheer amount of waste currently generated is to
some extent due to various inefficiencies and might, in principle, be reduced.
In this chapter we analyze the question to what extent the occurrence of
waste is actually an unavoidable necessity of industrial production, and to what
extent it is an inefficiency that may, in principle, be reduced. For that sake, we
employ the laws of thermodynamics as an analytical framework within which
results about current ‘industrial metabolism’ (Ayres and Simonis 1994) may
be rigorously deduced in energetic and material terms.
In Section 5.2, we demonstrate that industrial production is necessarily and
unavoidably joint production. This means waste outputs are an unavoidable byproduct in the industrial production of desired goods. In Section 5.3, we analyze
the degree of thermodynamic (in)efficiency of industrial production processes,
and the associated amounts of waste due to these inefficiencies. Section 5.4
concludes.
5.2
5.2.1
Joint Production of Desired Goods and Waste
The Thermodynamic View of Production
Production can in the most general way be conceived of as the transformation
of a number of inputs into a number of outputs. In thermodynamic terms, en1
For example, Georgescu-Roegen (1975: 357) has argued that ‘waste is an output just as
unavoidable as the use of natural resources’.
Necessity and Inefficiency in the Generation of Waste
81
ergy (actually: exergy) and matter are the fundamental factors of production
(Ayres 1998, Baumgärtner 2000, Faber et al. 1998, Ruth 1993). From a thermodynamic view,2 two quantifiable characteristics of an input or an output are
its mass, m, and its entropy, S. Alternatively, one could use its exergy instead
of its entropy; this will be done in the next section. Because both mass and
entropy are extensive quantities, it is useful to introduce the ratio of the two,
σ = S/m, for m > 0 as an intensive quantity. σ is called specific entropy and
measures the entropy per unit mass of a bulk of matter irrespective of that
bulk’s size.3
5.2.2
Joint Products are Unavoidable in Industrial Production
Let us now narrow down the analysis to the particular type of production
which is found in most developed countries and which is most relevant as far
as economy-environment interactions are concerned. This is what one may call
regular industrial production. For that sake, consider the following reference
model of regular industrial production (Baumgärtner 2000: Chapter 4). A raw
material is transformed into a final product. The exergy necessary to carry out
that transformation is typically provided by a material fuel. As the analysis
of the reference model will reveal, it is then unavoidable that a by-product is
jointly produced with the desired product. The analysis will also suggest that
this by-product may often be considered an unwanted waste. The industrial
production process can, thus, be depicted as in Figure 5.1. An example of such
an industrial production process is the production of pure iron as a desired
product from iron ore as raw material (see e.g. Ruth 1995a). The fuel in
that example is coke, and there are slag, carbon dioxide and heat as waste
by-products.
The focus on regular industrial production processes justifies building the
reference model on the assumption of two kinds of inputs, raw material and
fuel, and not more than two kinds of outputs, desired product and by-product.4
2
For those readers not familiar with classical thermodynamics we recommend the work
of Callen (1985), Kondepudi and Prigogine (1998) or Zemansky and Dittman (1997) as
comprehensive, yet accessible introductions. The appendix to Chapter 2 provides a short
and basic introduction to classical thermodynamics in non-technical terms. Bejan (1997)
gives a good introduction to engineering thermodynamics.
3
Thermodynamic variables, such as volume and particle number, which are proportional
to the size of the system, are called extensive variables. Variables, such as temperature or
pressure, that specify a local property and are independent of the size of the system, are
called intensive variables. If one doubled a bulk of matter, then the two extensive quantities
m and S would double as well while the ratio of the two, σ = S/m, would remain constant.
Specific entropy, thus, is an intensive variable.
4
This assumption may be relaxed. It may be assumed that there are a number of additional inputs and outputs besides the ones mentioned in the text. The joint production
result is not altered by the assumption of additional inputs or outputs.
82
Natural Science Constraints in Environmental and Resource Economics
'
'
$
low entropy fuel:
@
mf , Sf
&
@
%
@ production
R
@
process
'
$
high entropy
raw material:
mrm , Srm
&
%
Sgen ≥ 0
$
low entropy product:
mp , Sp
&
@
'
@
high entropy
@
R
@
by-product
&
mbp , Sbp
%
$
%
Figure 5.1 The thermodynamic structure of regular industrial production in
terms of mass and (specific) entropy. (Figure modified from Baumgärtner 2000:
74).
In the notation introduced above, mj and Sj are the mass and the entropy
of the inputs and outputs involved and σj is their respective specific entropy
(j = rm, f, p, bp which stands for raw material, fuel, product, by-product). One
may then formally define the notion of industrial production in thermodynamic
terms.
Definition 5.1
Within the formal framework of the reference model, a process of production is
called industrial production if and only if it exhibits the following three properties:
mrm , mp > 0,
(5.1)
σrm > σp ,
(5.2)
mf > 0.
(5.3)
Property (5.1) means that the production process essentially consists of a material transformation, that is, a raw material is transformed into a material
desired product. Property (5.2) states that the direction of this material transformation is such as to transform a raw material of relatively high specific
entropy into a desired product of lower specific entropy. In our example, iron
oxide (Fe2 O3 ) and pure iron (Fe) have a specific entropy of 87.4 J/mole K
and 27.3 J/mole K respectively (see Table 5.1; Kondepudi and Prigogine 1998.
Appendix). The underlying idea is that most raw materials are still impure
and, therefore, can be thought of as mixtures from which the desired product
is to be obtained by de-mixing of the different components of the raw material.
More generally, desired products are thought of as matter in a more orderly
state than the raw material. From basic thermodynamics we know that such
Necessity and Inefficiency in the Generation of Waste
83
a transformation process requires the use of exergy. Property (5.3) states that
the exergy input also has mass, that is, the exergy necessary to carry out the
desired transformation is provided by a material fuel, such as for example, oil,
coal or gas.
The constraints imposed on production processes by the laws of thermodynamics can be formalized as follows:
mrm + mf = mp + mbp ,
Srm + Sf + Sgen = Sp + Sbp with Sgen ≥ 0.
(5.4)
(5.5)
The Law of Mass Conservation (Equation 5.4) states that the total ingoing
mass has to equal the total outgoing mass because mass is conserved in the
production process. The Second Law of Thermodynamics (Equation 5.5), the
so-called Entropy Law, states that in the production process a non-negative
amount of entropy is generated, Sgen , which is added to the total entropy of all
inputs to yield the total entropy of all outputs.
Within the framework of that reference model, the two laws of thermodynamics, Equations (5.4) and (5.5), together with the assumption of industrial
production, Properties (5.1)–(5.3), imply that the second output necessarily
exists (Baumgärtner 2000: 77).
Proposition 5.1
For any process of industrial production of a desired product (Properties 5.1–
5.3), the laws of thermodynamics (Equations 5.4 and 5.5) imply the existence
of at least one by-product.
Proof: see Appendix A5.1.
This means, the occurrence of a by-product is necessary and unavoidable
in every process of regular industrial production. In economic terms, one may
speak of ‘joint production’, as the desired product and the by-product are
necessarily produced together (Baumgärtner et al. 2001, 2006).
The intuition behind this result is the following. One obvious reason for the
existence of joint outputs besides the desired product is simply conservation of
mass. If, for instance, pure iron is produced from iron ore with a carbon fuel,
the desired product, which is pure iron, does not contain any carbon. Yet, the
carbon material from the fuel has to go somewhere. Hence, there has to be
a joint product containing the carbon. But there is a second reason for the
existence of joint products besides and beyond conservation of mass, and that
is the generation of entropy according to the Second Law of Thermodynamics.
Think of a production process where all of the raw material and the material
fuel end up as part of the desired product, for example, the production of
cement. In that case, mass conservation alone would not require any joint
84
Natural Science Constraints in Environmental and Resource Economics
product. But because the desired product has lower specific entropy than the
raw material, and there is some non-negative amount of entropy generated by
the process, there is a need for a joint output taking up the excess entropy. In
many cases, as in the example of cement production, this happens in the form
of low-temperature heat, which may be contained in the product, a by-product
or transferred to the environment.
In most cases of industrial production, both of these reasons – the one
based on mass conservation and the one based on entropy generation – hold
at the same time. Therefore, the joint product is typically a high entropy
material. Due to its high entropy it will most often be considered useless and,
therefore, an undesired waste; however, one should be careful to note that the
classification of an output as ‘waste’ carries a certain value judgment, which
cannot be inferred from thermodynamics alone.5
5.3
Thermodynamic (In)Efficiency of Industrial
Production
The thermodynamic analysis in the previous section has demonstrated that
the existence of a high entropy joint product is necessary and unavoidable in
every process of regular industrial production. In reality, however, much of the
waste currently generated is obviously avoidable. Yet this observation is not
in contradiction to the result derived above. While the reference model was
based on the assumption of thermodynamic efficiency, current technology and
production practices are to a large extent thermodynamically inefficient. As
a consequence, while a certain amount of waste is necessary and unavoidable
for thermodynamic reasons, the actual amount of waste produced with current
technologies is an expression of inefficiency. Thermodynamic considerations
which originated in the applied field of engineering thermodynamics, in particular the exergy concept, can tell us exactly what amount of waste is due
to inefficiency and may, in principle, be reduced (e.g. Ayres 1999, Bejan et al.
1996, Brodyansky et al. 1994, Cleveland and Ruth 1997, Creyts 2000, de Swaan
Arons and van der Kooi 2001, de Swaan Arons et al. 2003, Dewulf et al. 2000,
Ruth 1995b, 1995c).6
5
For a review of various attempts to construct a so-called ‘entropy theory of value’, and
a refutation of these endeavours see Baumgärtner et al. (1996).
6
For a more general discussion of the relevance of the exergy concept for the field of
Industrial Ecology see Connelly and Koshland (2001).
Necessity and Inefficiency in the Generation of Waste
5.3.1
85
Engineering Thermodynamics: The Exergy Concept
Exergy is defined to be the maximum amount of work obtainable from a system as it approaches thermodynamic equilibrium with its environment in a
reversible way (Ayres 1998: 192, Szargut et al. 1988: 7). Exergy is also commonly called ‘available energy’, or ‘available work’, and corresponds to the
useful part of energy, thus combining the insights from both the First and Second Laws of Thermodynamics. Hence, exergy is what most people mean when
they use the term ‘energy’, for example, when saying that ‘energy is used’ to
carry out a certain process. As the system might consist simply of a bulk of
matter, exergy is also a measure of the potential work embodied in a material,
whether it is a fuel, food or other substance (Ayres et al. 1998). The exergy
content of different materials can be calculated for standard values specifying
the natural environment, by considering how that material eventually reaches
thermodynamic equilibrium with its environment with respect to temperature,
pressure, chemical potential and all other intensive variables.
The relationship between the concepts of entropy and exergy is simple, as
Blost = T0 Sgen (Law of Gouy and Stodola), where Blost denotes the potential
work or exergy lost by the system in a transformation process, T0 denotes the
temperature of the system’s environment, and Sgen denotes the entropy generatedin the transformation. This means, as the system’s entropy increases as a
consequence of irreversible transformations according to the Second Law, the
system loses exergy or some of its potential to perform work. Exergy, unlike
energy, is thus not a conserved quantity. While the entropy concept stresses
that with every transformation of the system something useless is created, the
exergy concept stresses that something useful is diminished. These developments are two aspects of the same irreversible character of transformations of
energy and matter. The character of regular industrial production, as sketched
in Figure 5.1 above, therefore has a corresponding description in terms of exergy
(Figure 5.2).
With the strict correspondence, established by the Law of Gouy and Stodola,
between the entropy generated in an irreversible transformation and exergy lost
in this process, our entire analysis could, in principle, be based either on the
entropy concept or on the exergy concept. Physicists usually prefer the entropy
route, as entropy is the concept traditionally established in physics. On the
other hand, exergy seems to be more popular with engineers and people interested in applied work. Instead of preferring one route to the other, or going
all the way along both routes in parallel, we illustrate the fruitfulness of both
approaches by employing them at different stages of the argument. While we
have demonstrated above the result (Proposition 5.1) that regular industrial
production necessarily yields waste by-products based on the entropy concept,
we now switch to the exergy concept to analyze the efficiency of regular industrial production.
86
Natural Science Constraints in Environmental and Resource Economics
'
$
'
high exergy fuel:
@
mf , Bf
&
@
%
@ production
R
@
process
'
$
low exergy
raw material:
mrm , Brm
&
%
Blost ≥ 0
$
high exergy product:
mp , Bp
&
@
'
@
low exergy
@
R
@
by-product
&
mbp , Bbp
%
$
%
Figure 5.2 The thermodynamic structure of regular industrial production in
terms of mass and exergy.
5.3.2
(In)Efficiency in Thermodynamic Equilibrium
In this section, regular industrial production is quantitatively analyzed in exergy terms with regard to thermodynamic (in)efficiency. For that sake, we turn
in detail to one particular step in the production process introduced above as
an illustrative example. In the production of pure iron from iron ore, the first
step is to extract the ore from the deposit. In the next step, the ore is separated
by physical means into iron oxide and silicates. The third step, which we shall
analyze in detail in this section, then consists of chemically reducing the iron
oxide to pure iron. This reduction requires exergy. It is typically provided by
burning coke, which, for the purpose of this analysis, can be taken to be pure
carbon. So, in the terminology outlined earlier, the desired product of this
transformation is pure iron (Fe), the raw material is iron oxide (Fe2 O3 ) and the
fuel is carbon (C). As a waste joint product in this reaction, carbon dioxide
(CO2 ) is generated. The chemical reaction in this production process may be
written down as follows:
2 Fe2 O3 + 3 C → 4 Fe + 3 CO2 .
(5.6)
The molecular weight, specific entropy and exergy of the chemicals involved in
the reaction are given in Table 5.1.
As one sees, the desired product Fe has a much higher exergy (i.e. lower
specific entropy) than the raw material Fe2 O3 . It is the relatively high exergy
content (i.e. low specific entropy) of the fuel C that provides the exergy for this
transformation to happen. The waste CO2 is then characterized by low exergy
content (i.e. high specific entropy).
Necessity and Inefficiency in the Generation of Waste
chemical
Fe
Fe2 O3
C
CO2
O2
molecular weight
[g/mole]
56
160
12
44
32
specific entropy
[J/mole K]
27.3
87.4
5.7
213.8
161.1
87
exergy
[kJ/mole]
376.4
16.5
410.3
19.9
4.0
Table 5.1 Molecular weight, specific entropy of the different chemicals involved
in the reduction of iron oxide to pure iron. One mole is, by definition, the
amount of some material that contains as many atoms as 12 g of carbon isotope 12 C. For every material, one mole contains 6.022 × 1023 particles. The
molecular weight of a material is its mass per mole. Source: Kondepudi and
Prigogine (1998: Appendix), Szargut et al. (1988: Appendix, Table I).
Mass balance
The chemical reaction equation (5.6) is correct in terms of the mass balance:
all atoms of an element that go into the reaction come out of the reaction as
well. Conservation of mass is the reason for the existence of the joint product CO2 . Producing four moles of Fe, thus, entails three moles of CO2 as
waste. That makes 0.75 moles of waste CO2 emissions per mole of Fe produced
(corresponding to 0.59 kg CO2 per kg of Fe) for mass balance reasons alone.
Thermodynamically efficient energy balance
Checking the reaction equation (5.6) with the exergy values given in Table 5.1
reveals that while the reaction equation is written down correctly in terms of
the mass balance, it is not yet correct in energetic terms. For, in order to
produce four moles of Fe with an exergy content of 1,505.6 kJ, one needs the
input of at least 1505.6 kJ as well. (Recall that exergy cannot be created,
but always diminishes in the course of a transformation due to irreversibility.)
But three moles of C only contain 1,230.9 kJ. Therefore, one actually needs
more than three moles of C to deliver enough exergy for four moles of Fe to
be produced from Fe2 O3 . We compensate for this shortage of exergy on the
input side by introducing 0.76 additional units of the exergy source C. The
reaction equation should, thus, be written down as follows to obey both laws
of thermodynamics:
2 Fe2 O3 +
3.76 C + 0.76 O2 →
4 Fe
+ 3.76 CO2 .
(33.0)
(1, 542.7)
(3.0)
(1, 505.6)
(74.8)
(5.7)
The numbers in brackets below each input and output give the exergy content
in kJ of the respective amounts of inputs and outputs. On the input side 0.76
88
Natural Science Constraints in Environmental and Resource Economics
moles of oxygen (O2 ) have been added to fulfill the mass balance with the
additional 0.76 moles of C involved. This oxygen comes from the air and enters
the transformation process when carbon is burned.
From reaction equation (5.7) we see that the exergy supplied to the reaction by its inputs (1,580 kJ) now suffices to yield the exergy of the outputs
(1,580 kJ). As no exergy is lost in the reaction, that is, the exergy of the
inputs exactly equals the exergy of the outputs, this corresponds to a thermodynamically 100 %-efficient and reversible transformation, in which no entropy
is generated and no exergy is lost. In mass terms, reaction equation (5.7) tells
us that even in thermodynamically ideal transformations of Fe2 O3 into Fe, 3.76
moles of CO2 are generated as material waste when producing four moles of
Fe. That makes 0.94 moles of waste CO2 emissions per mole of Fe produced
(corresponding to 0.74 kg CO2 per kg of Fe). This amount is the minimum
waste generation required by the two laws of thermodynamics, as shown in
Section 5.2 above to necessarily exist (Proposition 5.1).
Thermodynamic inefficiency
In real production processes the exergy content of carbon of 410.3 kJ/mole
is never put to work with an efficiency of 100 %. Detailed data on pig iron
production in real blast furnaces in Poland (Szargut et al. 1988: Table 7.3),
where coke is burned together with atmospheric oxygen, imply that the efficiency of exergy conversion is only about 33 %.7 This means that out of one
mole of C one obtains only 135.4 kJ instead of the ideal value of 410.3 kJ. As a
consequence, in order to deliver the exergy necessary to carry out the chemical
reaction one needs to employ at least 12.42 moles of C. The reaction equation
for a transformation that is only 33 %-efficient in energy conversion would thus
read:
2 Fe2 O3 + 12.42 C + 9.42 O2 →
4 Fe
+ 12.42 CO2 + heat .
(33.0)
(5, 095.9)
(37.7)
(1, 505.6)
(247.2)
(5.8)
Out of the 5,095.9 kJ of exergy supplied by 12.42 moles of C only 33 %, or
1,681.6 kJ, are put to work in the reaction due to the inefficiency in energy
conversion. The amount of exergy supplied by the inputs but not contained in
the outputs of the reaction corresponds to exergy lost in the process, Blost =
3,413.8 kJ, which is mainly emitted from the reaction as waste heat.
From Equation (5.8) we see that due to the inefficiency in energy conversion
the amount of material fuel that is necessary to drive the transformation has
7
Typical exergy conversion efficiencies in the process industry range from values as low
as 4 %, 6 % and 9 % in the production of nitric acid, oxygen and copper respectively up to
values of 58 % and 63 % in the production of hydrogen and methanol (Hinderink et al. 1999:
Table 1).
Necessity and Inefficiency in the Generation of Waste
89
more than tripled. In order to produce four moles of Fe with a 33 %-efficiency
one needs 12.42 moles of C (Equation 5.8) instead of just 3.76 moles in the
efficient case (Equation 5.7). As a consequence, the reaction generates 12.42
moles of CO2 (Equation 5.8) instead of just 3.76 moles in the efficient case
(Equation 5.8). That makes 3.11 moles of waste CO2 emissions per mole of Fe
produced (corresponding to 2.45 kg CO2 per kg of Fe), with only 30 % (0.94
moles) due to thermodynamic necessity (cf. the discussion of Equation 5.7
above) and 70 % (2.17 moles) due to thermodynamic inefficiency.
From this analysis one might conclude that roughly two thirds of the waste
currently generated in iron production is due to thermodynamic inefficiency,
while one third is actually necessary for thermodynamic reasons. Therefore,
even increasing thermodynamic process efficiency to the ideal value of 100 %
will not reduce the amount of waste to zero, but only to one third of the amount
currently generated.
5.3.3
Finite-Time/Finite-Size Thermodynamics
Pointing to the thermodynamic inefficiency of a real production process, and
how it implies the occurrence of large amounts of waste, seems to suggest that
the amount of waste can easily be reduced by increasing the thermodynamic
efficiency at which the process is carried out. However, there are good economic
reasons why this form of thermodynamic inefficiency may actually be desired.
The analysis so far was entirely based on concepts and methods from ideal
equilibrium thermodynamics, which means that a level of 100 %-efficiency in
this framework is reached by operating processes in a completely reversible way
between one equilibrium state and another equilibrium state, resulting in zero
entropy generation (or: exergy loss) during the process. Recent research in the
applied field of engineering thermodynamics has addressed the circumstance
that chemical and physical processes in industry never happen in a completely
reversible way between one equilibrium state and another equilibrium state.
Rather, these processes are enforced by the operator of the process and they
are constrained in space and time. This has led to an extension of ideal equilibrium thermodynamics, known as finite-time/finite-size thermodynamics (e.g.
Andresen et al. 1984; Bejan 1996, 1997, Bejan et al. 1996).
From the point of view of finite-time/finite-size thermodynamics it becomes
obvious that the minimum exergy requirement and minimum waste production
in chemical or physical processes is considerably higher than that suggested by
the ideal equilibrium thermodynamics analysis carried out so far. The reason
for the increased exergy requirement (which entails an increased amount of
waste at the end of the process) lies in the fact that chemical and physical
transformations are forced to happen over a finite time by the operator of
the production plant, which necessarily causes some dissipation of energy. In
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Natural Science Constraints in Environmental and Resource Economics
the language of the reference model described earlier, this shows in a strictly
positive amount Sgen of entropy generated in the process.
The finite-time/finite-size consideration is a very relevant consideration for
many production processes, in particular in the chemical industry. Finitetime/finite-size thermodynamics allows one to exactly identify, track down and
quantify exergetic inefficiencies at the individual steps of a production processes
(Bejan 1996, 1997, Bejan et al. 1996, Brodyansky et al. 1994, Creyts 2000,
Szargut et al. 1988), along the entire chain of a production process (Ayres et
al. 1998, Cornelissen and Hirs 1999, Cornelissen et al. 2000), and for whole
industries (Dewulf et al. 2000, Hinderink et al. 1999). Thus, it yields valuable
insights into the origins of exergy losses and forms a tool for designing industrial production systems in an efficient and sustainable manner (Connelly and
Koshland 2001, de Swaan Arons and van der Kooi 2001, de Swaan Arons et al.
2003).
An example which demonstrates how large Sgen can actually be is the enrichment of uranium (Balian 1991: 347–348, 383–385). In the production of enriched uranium the actual exergy input is larger than the theoretical minimum
calculated from ideal equilibrium thermodynamics by a factor of 70 million!
At the Eurodif factory, the French enriching plant from which the data are
taken, the process of enriching by isotope separation is realized by gas diffusion through a semipermeable membrane. An ideal process realization would
require letting the gas diffuse in thermodynamic equilibrium, which would take
an infinite time span. In order to carry out the process in finite time, diffusion
is enhanced by building up an enormous pressure difference between the two
sides of the membrane, which requires an equally enormous amount of energy.
Then, the process of diffusion is no longer an equilibrium process. Instead,
it is irreversible and Sgen > 0. A comparison of the ideal separation process
and the real process realization shows that the huge irreversible loss of energy
in the actual separation process is entirely due to the dissipation of energy
in the many compressions and decompressions which are necessary to run the
separation process under a pressure difference and, thus, in finite time. This
dissipated energy leaves the process as waste heat.8
5.4
Conclusion
Our basic result is twofold. First, based on a thermodynamic analysis we have
confirmed previous assertions that waste is an unavoidable and necessary joint
8
Note that the efficiency of uranium enrichment in a centrifuge plant is about two orders of
magnitude higher than in a membrane plant. Despite the tremendous exergetic inefficiency,
enrichment of uranium makes sense, as the exergy loss in enrichment is small with respect
to the exergy content of enriched uranium as a fuel for nuclear fission.
Necessity and Inefficiency in the Generation of Waste
91
output in the regular industrial production of desired goods. This is the type
of production technology that is currently in use and dominates production
in industrial economies. Second, thermodynamic analysis has also allowed us
to quantify the amount of waste that – beyond the thermodynamic minimum
required – is due to inefficiencies. We have identified three major reasons for the
occurrence of large amounts of excessive material waste from regular industrial
production:
1. The first reason is simply conservation of mass. Starting with a raw material, which is a mixture of different chemical elements, to produce a
desired product, which is made up of only one particular chemical element, necessarily leaves a material waste.
2. The second reason is the use of a material fuel, which is a characteristic
property of many industrial production technologies currently in use. The
fuel – carbon in our example – only serves to provide the exergy for the
chemical reaction. The carbon material itself is actually neither wanted
nor needed in the reaction. Because mass is conserved, the fuel material
has to go somewhere after its exergy content has been stripped off. And
that makes the waste. An alternative, immaterial way of providing exergy
to production processes would be the use of renewable energy sources,
such as solar, wind, tidal or hydro-energy.9
3. The third reason is the thermodynamically inefficient performance of current technologies when it comes to the conversion of exergy, which is a
necessary factor of production in all production processes. In particular,
this is due to the operation of production processes under non-equilibrium
conditions, in order to have them completed in finite time. The shorter
the time span within which one wants the process to be completed, the
more energy will irreversibly be dissipated. This inefficiency not only
considerably increases the need for fuel beyond the minimum exergy requirement; it also increases the amount of material waste generated far
beyond the thermodynamic necessity. This holds, in particular, for carbon dioxide emissions when carbon (e.g. coal or coke) or hydrocarbons
(e.g. oil or natural gas) are used as a fuel.
9
Note that, because the primary goal of carrying out the transformation studied using
reaction equations (5.6)–(5.8) is to split Fe2 O3 into Fe and O2 , the minimal way of doing
that would be: 2 Fe2 O3 + direct exergy → 4 Fe + 3 O2 . The exergy necessary to achieve the
splitting of Fe2 O3 into Fe and O2 could, for instance, be delivered by solar energy directly.
Without any material fuel the amount of material waste would be considerably reduced.
With four moles of Fe there would be three moles of O2 jointly produced. That makes 0.75
moles of waste O2 emissions per mole of Fe produced (corresponding to 0.43 kg O2 per kg
of Fe). However, running the chemical process in this direct way, that is, powered by solar
energy instead of material fuel input, would require technologies very different from the ones
we are currently using.
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Natural Science Constraints in Environmental and Resource Economics
Summing up, notwithstanding the fundamental insight that waste is a necessary joint output in regular industrial production technologies as they are
currently in use, a large potential exists for the reduction of waste. Thermodynamics has proven to be a very useful analytical tool for studying and exploiting
this potential. In order to translate this potential into real solutions, new production technologies are needed. In particular, new technologies should have
higher exergetic fuel efficiency than existing ones. Or, even better, they should
not use a material fuel at all for their exergy input, but use renewable energy
sources.
Appendix
A5.1
Proof of Proposition 5.1
In order to prove the proposition (see Baumgärtner 2000: 72–77), one should
distinguish between different chemical elements, such as e.g. oxygen (O), carbon
(C) or iron (Fe). Each input and each output of a production process may, in
general, be composed of various such elements. For example, a raw material
input into production may be iron oxide (Fe2 O3 ), which consists of the two
chemical elements of iron (Fe) and oxygen (O). In any chemical reaction, the
mass of each element is conserved separately,10 so that the thermodynamic Law
of Mass Conservation (Equation 5.4) should be formulated more precisely as
∀e
mrm (e) + mf (e) = mp (e) + mbp (e)
with e = . . . , O, . . . , C, . . . , Fe, . . . ,
(A5.1)
where m(e) denotes the mass of chemical element e. For instance, in the reduction of iron oxide (Fe2 O3 ) into pure iron (Fe) by means of coke (C), the
mass of all chemical elements – oxygen, carbon, iron and others – is conserved
separately; that is, the mass of iron in the inputs must equal the mass of iron
in the outputs, and similarly for the mass of oxygen, carbon, etc.
In the proof, a mass balance will explicitly be considered for only one chemical element, the so-called ‘element under consideration’. Beyond that, it is only
important that there exists more than one chemical element. But these other
elements’ mass balance is of no explicit interest. In order to simplify the presentation, we therefore omit the argument e where it is clear that we refer to the
first element. The detailed mass balance (A5.1) then reduces to Equation (5.4),
but nevertheless is meant to refer to the first element.
10
In principle, chemical elements may be transformed into each other by nuclear reactions.
However, this possibility is neglected here.
Necessity and Inefficiency in the Generation of Waste
93
In this interpretation, Properties (5.1) and (5.3) may be relaxed to
mrm , mp ≥ 0,
mf ≥ 0,
(5.1 )
(5.3 )
with each of these quantities, mrm , mp and mf , being strictly positive for at least
one element. That is to say, the raw material input (as well as the fuel and the
desired output) does not need to contain positive amounts of all elements; it is
only assumed to contain at least one element. For instance, the raw material
iron oxide (Fe2 O3 ) contains iron and oxygen, but no carbon. If the element
under consideration in the analysis should be, say, iron, then mrm > 0; but if
the element under consideration should be carbon, then mrm = 0.
The proof is now carried out by showing that either the mass mbp or the
entropy Sbp (or both) of the second output is strictly positive. Note that
not necessarily both the mass and the entropy have to be strictly positive for
an output to exist, since the mass of an output – in our formalization – will
be zero if it does not contain the element under consideration. For carrying
out the proof, we distinguish between the two cases that (i) the statement
of Proposition 5.1 follows already from mass conservation alone and (ii) the
Second Law is essential for the existence of a second output.
Joint production as a consequence of mass conservation
In many instances, the aspect of entropy is not necessary to understand why
there exists a second output besides the main product. So, to start with, let
us focus on the mass aspect of inputs and outputs and neglect their entropic
character. Consider first the extreme case that mp = 0; the complementary
case of mp > 0 will be dealt with later on. Most obviously, if mrm , mf ≥ 0 with
at least one, mrm or mf , strictly positive (according to Properties 5.1 and 5.3
and with a suitable choice of the element under consideration) and mp = 0, it
follows from the mass balance (Equation 5.4) that mbp = mf + mrm > 0. This
means, a joint product cogently exists.11
11
Again, the assumption of mp = 0 should not be interpreted as production resulting in no
desired product at all, or only in an immaterial one. Nor should the assumption of mrm = 0
or mf = 0 be interpreted as saying that these inputs are absent from the production process.
Recall that the mass balance (Equation 5.4) refers to one particular element, say carbon, but
that there are other elements as well, say iron. Then, mp = 0 only means that there is nothing
of the element under consideration contained in the desired output. However, this output may
well contain other elements, e.g. iron. In the example of iron production mentioned above,
let carbon (C) be the element for which the mass balance is considered. Equation (5.4) can
then be read as the mass balance of carbon, that is the mass of carbon in the inputs has to
equal the mass of carbon in the outputs. Of course, there are also mass balances for the other
elements, such as e.g. iron or oxygen, but they are of no interest for the argument. Then,
mf = mf (C) > 0 (the fuel, coke plus oxygen, contains carbon), mrm = mrm (C) = 0 (there is
94
Natural Science Constraints in Environmental and Resource Economics
In general, mass balance considerations make the existence of at least one
joint output necessary as soon as either the raw material or the fuel (or both)
contain an element which is not contained in the desired product. Note that if
such an argument holds for any one material of the, in general, many materials
involved in a production process, then this already suffices to establish the
result.
Joint production as a consequence of the Second Law
It remains to be shown that a second output necessarily exists if mp > 0. In
this case, the Law of Mass Conservation alone may not suffice to prove the
existence of a joint output. For instance, for mp = mf + mrm and mbp = 0 the
mass balance is fulfilled and the existence of a joint output is not immediately
obvious. In this case, however, the Second Law becomes crucial in establishing
the result.
In order to build the argument in this case, let us come back to the entropic
character of inputs and outputs in the reference model of industrial production.
As far as the mass aspect is concerned, we make the assumption that mf = 0.12
The mass balance (Equation 5.4) then becomes
mrm = mp + mbp ,
(5.2)
mrm ≥ mp .
(5.3)
13
from which it follows that
Consider now the entropy balance (Equation 5.5). It can be rearranged into
Sbp = Sf + ΔS + (Srm − Sp ) .
(5.4)
The sign of the term Srm − Sp can be determined from considering the fraction
Srm /Sp which is, according to the definition of specific entropy as σ = S/m,
given by:
mrm σrm
Srm
=
·
>1,
(5.5)
Sp
mp σp
≥1
>1
no carbon in the raw material, iron ore), mp = mp (C) = 0 (the desired main product, pure
iron, does not contain any carbon), and, hence, mbp = mbp (C) > 0. In sum, the existence of
the joint output, carbon dioxide, is necessary because the carbon atoms which are originally
contained in the fuel are not contained in the desired product and cannot disappear either.
12
Again, this should not be seen as restricting the generality of the treatment, since the
mass balance (Equation 5.4) refers to one particular element. So, setting mf = 0 simply
amounts to a suitable choice of the element under consideration. As an illustration, consider
again the example of iron making and take iron (Fe) to be the element under consideration.
Then mf = mf (Fe) = 0 only means that the fuel does not contain any iron.
13
Note that if mrm should be strictly larger than mp , the existence of a second output
with mbp > 0 would follow immediately from mass balance considerations. Therefore, the
interesting case which genuinely requires an entropy balance argument is actually mrm = mp .
Necessity and Inefficiency in the Generation of Waste
95
since mrm ≥ mp (Equation 5.3) and σrm > σp (Property 5.2). Hence,
Srm − Sp > 0 .
(5.6)
From this and Equation (5.4) it follows that
ΔS + (Srm − Sp ) > 0 .
Sbp = Sf + ≥0
≥0
>0
This means that the existence of a joint product with entropy Sbp is cogently
required in order to fulfill the entropy balance. It serves to take up the excess
high entropy, which cannot be contained in the desired low specific entropy
product.
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Natural Science Constraints in Environmental and Resource Economics
6. Optimal Dynamic Scale and Structure of a
Multi-Pollution Economy
with Frank Jöst and Ralph Winkler
6.1
Introduction
The natural environment is being damaged by the stocks of various pollutants,
which are produced in different sectors of the economy, accumulate according
to different dynamic relationships, and damage different environmental goods.
As an example, think of the two economic sectors ‘agriculture’ and ‘industry’. Nitrate and pesticide run-off from agricultural cultivation accumulates in
groundwater and decreases its quality as drinking water (UNEP 2002); carbon
dioxide emissions from fossil fuel combustion in the industrial sector accumulate in the atmosphere and contribute to global climate change (IPCC 2001).
In general, the different pollutants differ in their internal dynamics, i.e. natural
degradation processes, and in their harmfulness. This has implications for the
optimal dynamics of both the scale and structure of the economy. By scale we
mean the overall level of economic activity, measured by total factor input; by
structure we mean the composition of economic activity, measured by relative
factor inputs to different sectors.
In this chapter, we look into these coupled environmental-economic dynamics from a macroeconomic point of view. In particular, we are interested
in the following questions: How should the macroeconomic scale and structure
change over time in response to the dynamics of environmental pollution? Is
this dynamic process monotonic over time, or can a trade-off between long-run
and short-run considerations (e.g. lifetime versus harmfulness of pollutants)
induce a non-monotonic economic dynamics? What is the time scale of economic dynamics (i.e. change of scale and structure), and how is it influenced
by the different time scales and constraints of the economic and environmental
systems? These questions are relevant for the current policy discussion on the
sustainable biophysical scale of the aggregate economy relative to the surrounding natural environment (e.g. Arrow et al. 1995, Daly 1992a, 1996, 1999), and
97
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Natural Science Constraints in Environmental and Resource Economics
how economic policy should promote structural economic change as a response
to changing environmental pressures (e.g. de Bruyn 1997, Winkler 2005).
We address these questions based on a model which comprises two economic sectors, each of which produces one distinct consumption good and, at
the same time, gives rise to one specific pollutant. Both pollutants accumulate to stocks which display different internal dynamics, in the sense that the
respective natural deterioration rates differ, and cause welfare decreasing environmental damage independently of each other. Of course, this relatively simple
model cannot offer detailed policy prescriptions. However, it is detailed enough
to clarify the underlying theoretical issues. In fact, we perform a total analysis
of economy-environment interactions in a twofold manner. First, we analyze
a multi-sector economy, which is fully specified in terms of resource endowment, technology, preferences and environmental quality. Second, we consider
a ‘disaggregate’ natural environment. This goes beyond many contributions
to environmental economics, where either only one (aggregate) pollutant is
considered or different pollutants give rise to the same environmental problem.
Many studies in the extant literature assume that it is the flow of emissions
which causes environmental problems. This neglects stock accumulation and,
thus, an essential dynamic environmental constraint on economic action. Stock
pollution has been taken into account by some authors (e.g. Falk and Mendelsohn 1993, Forster 1973, Luptacik and Schubert 1982, Van der Ploeg and Withagen 1991). This is usually done at a highly aggregated level, such that only
one pollutant is taken into account. The case of several stock pollutants which
all contribute to the same environmental problem (climate change) has been
studied by Michaelis (1992, 1999). He is interested in finding cost-effective climate policy measures in the multi-pollution case for a given structure of the
economy and does not explicitly consider the dynamics of the production side
of the economy. Aaheim (1999) goes beyond Michaelis in that he analyzes
numerically the dynamics of a two-sector economy which gives rise to three
different stock pollutants and which is constrained by an exogenously given
policy target concerning the aggregate level of pollution. Moslener and Requate (2001) challenge the global warming potential as a useful indicator when
there are many interacting greenhouse gases with different dynamic characteristics. Faber and Proops (1998: Chapter 11) and Keeler et al. (1972) explicitly
study the dynamics of different production sectors with pollution, assuming
one single pollutant. Winkler (2005) analyzes optimal structural change of a
two-sector economy characterized by two stock quantities: the capital stock
and the stock of a pollutant which is emitted from the more capital-intense
sector. Baumgärtner and Jöst (2000) study the optimal (static) structure of a
vertically integrated two-sector economy where both sectors produce a specific
by-product. The first sector’s by-product can be used as a secondary resource
in the second sector.
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
99
In this chapter, we determine the optimal dynamic scale and structure of
a multi-pollution economy within an optimal control framework. We use a
linear approximation around the steady-state to obtain analytical results, and
a numerical optimization of the non-approximated system to check for their
robustness. The methodological innovation of our analysis is that we derive a
closed form solution to the intertemporal optimization problem, which includes
explicit expressions for the time scale of economic dynamics and the point in
time where a non-monotonicity may occur. Our analysis shows that along the
optimal time-path (i) the overall scale of economic activity may be less than
maximal; (ii) the time scale of economic dynamics is mainly determined by the
lifetime of pollutants, their harmfulness and the discount rate; and (iii) the
control of economic scale and structure may be non-monotonic.
Although our modeling approach is inspired by Ramsey-type optimal growth
models, which have previously been used to study steady state growth with environmental pollution (e.g. Gradus and Smulders 1993, 1996, Jöst et al. 2004,
Keeler et al. 1972, Plourde 1972, Siebert 2004, Smith 1977, Van der Ploeg and
Withagen 1991), we are essentially concerned with the issue of dynamic change
in both scale and structure of economic activity. Therefore, in this chapter
we do not restrict the analysis to steady states but focus on the explicit timedependence of the solution. Furthermore, we study an economy without any
potential for steady state growth, as this highlights the structural-change-effect,
which may be obscured by growth effects otherwise. The sole genuine generator
of dynamics in our model is the accumulation of pollutant stocks in the natural
environment.
The chapter is organized as follows. In Section 6.2 we present the model.
Section 6.3 is devoted to a formal analysis of the optimal dynamic scale and
structure of the economy, based on a linear approximation around the stationary state. Section 6.4 confirms the analytical results thus obtained by a
numerical optimization of the non-approximated system. Section 6.5 concludes.
6.2
The Model
We study a two sector economy with one scarce non-accumulating factor of
production, say labor, two consumption goods, and two pollutants that accumulate to stocks. Welfare is determined by the amounts consumed of both
consumption goods, as well as by the environmental damage caused by the two
pollutant stocks.
The production of consumption goods in sectors 1 and 2 of the economy is
described by two production functions, yi = P i(li ) for i = 1, 2, where li denotes
the amount of labor allocated in sector i. With index l denoting derivatives
with respect to the sole argument li , Pli ≡ dP i/dli and Plli ≡ d2 P i /dli2 , the
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Natural Science Constraints in Environmental and Resource Economics
production functions are assumed to exhibit the following standard properties:
P i (0) = 0 ,
Pli > 0 ,
lim Pli = +∞ ,
Plli < 0
li →0
(i = 1, 2) .
(6.1)
Since we want to analyze an economy without potential for steady state growth,
we assume a fixed supply of labor, λ > 0. Consumption possibilities are described by
yi = P i (li )
l1 + l2 ≤ λ .
(i = 1, 2) ,
(6.2)
(6.3)
In addition to the consumption good, each sector yields a pollutant which comes
as a joint output in a fixed proportion to the desired output. Without loss of
generality,
ei = yi
(i = 1, 2) .
(6.4)
Both flows of pollutants, e1 and e2 , add to the respective stock of the pollutant,
which deteriorates at the constant rate δi :1
ṡi = ei − δi si
with δi > 0
(i = 1, 2) .
(6.5)
Instantaneous social welfare V depends on consumption of both goods, y1 and
y2 , and on the damage to environmental quality which hinges upon the stocks
of pollutants s1 and s2 . We consider the following welfare function:
σ
σ2 2 1 2
s + s2
(6.6)
with σ1 , σ2 > 0 ,
V (y1 , y2, s1 , s2 ) = U(y1 , y2 ) −
2 1
2
where σi indicates the harmfulness of pollutant i (i = 1, 2) and U represents
welfare gains due to consumption. The function U is assumed to exhibit the
usual property of positive and decreasing marginal welfare in both consumption
goods. In order to have an additively separable welfare function in all four
arguments (y1 , y2 , s1 , s2 ), we assume that neither consumption good influences
marginal welfare of the other. With index i denoting the partial derivative with
respect to argument yi , i.e. Ui ≡ ∂U/∂yi and Uij ≡ ∂ 2 U/∂yi ∂yj with i, j = 1, 2,
the assumptions are:
Ui > 0 ,
lim Ui = +∞ ,
yi →0
Uii < 0 ,
Uij = 0
(i, j = 1, 2 and i = j) .
(6.7)
Both stocks of pollutants exert an increasing marginal damage, which is captured in the welfare function V , for the sake of tractability, by quadratic damage
functions. Furthermore, both stocks decrease welfare independently. This is
1
In general, the decay rate may depend on emissions and the stock: δi = δi (ei , si ). For
analytical tractability, we assume δi to be constant.
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
101
plausible if they damage different environmental goods. Thus, the welfare effect of one additional unit of one pollutant does not depend on the amount of
the other. Note that the overall welfare function V is strictly concave.
Since we are interested in studying questions related to the scale as well
as the structure of economic activity, and in order to simplify the analysis of
corner solutions in the optimization problem, we introduce new dimensionless
variables in the following way:
l1 + l2
l1
.
(6.8)
and
x=
λ
l1 + l2
The variable c stands for the scale of economic activity. It indicates what
fraction of the total available amount of labor is devoted to economic activity,
and may take values between 0 and 1. The remaining fraction 1 − c is left
idle. This can be interpreted as an implicit form of pollution abatement. By
not using all available labor in the production of the consumption goods (and,
consequently, emissions) but leaving part of the labor endowment idle, the
variable c can be thought of as measuring the scale of economic activity in the
sectors producing consumption goods and pollution, whereas the fraction 1−c of
labor may be thought of as being employed in (implicit) pollution abatement.2
The variable x stands for the structure of economic activity. It indicates the
fraction of the total labor employed in production, l1 + l2 , that is allocated to
sector 1, and may take values between 0 and 1. The remaining fraction 1 − x
is allocated to sector 2. The variables l1 and l2 can then be expressed in terms
of c and x:
c=
l1 = l1 (c, x) = cxλ
and
l2 = l2 (c, x) = c(1 − x)λ .
This allows us to replace l1 and l2 in the problem. For notational convenience,
we introduce new production functions F i which depend directly on c and x,
and which are defined in the following way:
F i (c, x) ≡ P i (li (c, x))
for all c, x .
(6.9)
From (6.1) and (6.9) one obtains that the F i have the following properties:
Fc1 = xPl1 λ > 0 ,
Fx1 = cPl1 λ > 0 ,
Fc2 = (1 − x)Pl2 λ > 0 ,
Fx2 = −cPl2 λ < 0 .
2
lim Fc1 (x = 0) = +∞ ,
c→0
(6.10)
(6.11)
lim Fc2 (x = 1) = +∞ ,
c→0
(6.12)
(6.13)
Not taking into account potential abatement activities for the scale of economic activity
is in line with arguments from the ‘green national product’ discussion, according to which
defensive and restorative activities should not be counted as augmenting the net national
product (e.g. Ahmad et al. 1989, World Bank 1997).
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Natural Science Constraints in Environmental and Resource Economics
6.3
Optimal Scale and Structure of the Economy
Taking a social planner’s perspective, we now determine the optimal scale and
structure of the multi-pollution economy described in the previous section. The
control variables are the scale (c) and the structure (x) of economic activity.
In terms of pollution, the choice over c and x is a choice over (i) how much
pollution to emit overall, and (ii) what particular pollutant to emit. These are
the two essential macroeconomic dimensions of every multi-pollution allocation
decision.
6.3.1
Intertemporal Optimization
We maximize the discounted intertemporal welfare over c and x,
∞
σ1
σ2 U(y1 , y2 ) − s21 − s22 e−ρt dt ,
2
2
0
(6.14)
where ρ denotes the discount rate and yi = F i (c, x) (i, j = 1, 2), subject to the
dynamic constraints for the two state variables s1 and s2 which are given by
Equations (6.5):
ṡi = F i (c, x) − δi si
with δi > 0
(i = 1, 2) .
(6.15)
In addition, the following restrictions for the control variables c and x hold:
0≤c≤1
and
0≤x≤1.
(6.16)
Corner solutions with x = 0 or x = 1 cannot be optimal since either case
would imply, due to Assumptions (6.1) and (6.7), that the marginal utility of
one consumption good would go to infinity while the marginal utility of the
other would remain finite. Similarly, a corner solution with c = 0 cannot be
optimal since in that case the marginal utility of both consumption goods would
go to infinity while the marginal damage from environmental pollution would
remain finite. Hence, the only remaining restriction, which we have to control
for explicitly, is:
c≤1.
(6.17)
We introduce two costate variables, p1 and p2 , and a Kuhn-Tucker parameter,
pc . The current value Hamiltonian of the problem then reads
σ1
σ2
H(c, x, s1 , s2 ; p1 , p2 , pc ) = U(F 1 (c, x), F 2(c, x)) − s21 − s22
2
1
2
+ p1 F (c, x) − δ1 s1
+ p2 F 2 (c, x) − δ2 s2
(6.18)
+ pc [1 − c] .
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
103
Since both control variables, c and x, are always strictly positive, the two
state variables, s1 and s2 , are always nonnegative and the Hamiltonian H is
continuously differentiable with respect to c and x, the first order conditions of
the control problem are:
U1 Fc1 + U2 Fc2 + p1 Fc1 + p2 Fc2 − pc
U1 Fx1 + U2 Fx2 + p1 Fx1 + p2 Fx2
σ1 s1 + (δ1 + ρ)p1
σ2 s2 + (δ2 + ρ)p2
pc ≥ 0 , pc (1 − c)
=
=
=
=
=
0,
0,
ṗ1 ,
ṗ2 ,
0,
(6.19)
(6.20)
(6.21)
(6.22)
(6.23)
plus the dynamic constraints (6.15) and the restriction (6.17). These necessary
conditions are also sufficient if, in addition, the transversality conditions
lim pi (t) e−ρt · si (t) = 0
t→∞
(i = 1, 2) ,
(6.24)
hold (see Appendix A6.1). Note that the optimal path is also unique.
6.3.2
Stationary State
Setting ṗ1 = 0, ṗ2 = 0, ṡ1 = 0 and ṡ2 = 0 in the system of first order conditions (6.15), (6.17) and (6.19)–(6.23) yields the necessary and sufficient conditions for an optimal stationary state (c , x , s1 , s2 ), in which neither the scale
nor the structure of economic activity nor the stocks of pollution accumulated
in the environment change over time. From conditions (6.21) and (6.22) one
obtains for the costate variables pi (i = 1, 2):
pi = −
σi si
δi + ρ
(i = 1, 2) .
(6.25)
Inserting (6.25) in (6.19) and (6.20), and rearranging terms, yields the following
necessary and sufficient conditions for an optimal stationary state:
pc Fx2
σ1 s1
=
−
,
δ1 + ρ
Fc1 Fx2 − Fx1 Fc2 −pc Fx1
σ2 s2
=
,
U2 −
δ2 + ρ
Fc1 Fx2 − Fx1 Fc2 U1
(6.26)
(6.27)
where Ui and Fji (i = 1, 2 ; j = c, x) denote functions evaluated at stationary
state values of the argument. From the signs of the Fji and pc stated in (6.10)–
(6.13) and (6.23), it follows that:
Ui
σi si
≥
δi + ρ
(i = 1, 2) ,
(6.28)
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Natural Science Constraints in Environmental and Resource Economics
where the “>” sign indicates a corner solution (c = 1). Furthermore, from the
equations of motion (6.15) one obtains
si
Fi
=
= const.
δi
(i = 1, 2) .
(6.29)
The interpretation of the two conditions (6.28) is that in an interior (corner)
optimal stationary state the scale and structure of economic activity are such
that for each sector the marginal welfare gain due to consumption of that
sector’s output equals (is greater than) the aggregate future marginal damage
from that sector’s current emission which comes as an inevitable by-product
with the consumption good.3
An optimal stationary state exists if the system (6.23), (6.26), (6.27) and
(6.29) of five equations for the five unknowns (c , x , s1 , s2 ) and pc has a solution
with 0 < c ≤ 1 and 0 < x < 1. With the properties of the utility and
production functions assumed here, a unique optimal stationary state always
exists.
Proposition 6.1
(i) There exists a unique stationary state (c , x , s1 , s2 ), which is given as the
solution to (6.23), (6.26), (6.27) and (6.29).
(ii) The optimal stationary state of the economy is an interior solution with
c < 1, if the total available amount of labor λ in the economy is strictly
greater than some threshold value λ̄ = ¯l1 + ¯l2 , where the ¯li are specified by
the following implicit equations:
σi P i (¯li )
Ui (P 1 (¯l1 ), P 2 (¯l2 )) = 2
δi + δi ρ
(i = 1, 2) .
Proof: see Appendix A6.2.
In the following, we shall concentrate on the case of an interior stationary
state with c < 1. Hence, we assume that the total labor amount λ exceeds λ̄
as specified in Proposition 6.1. In order to study the properties of the interior
optimal stationary state (c , x ) some comparative statics can be done with
Conditions (6.26), (6.27) and (6.29). The results are stated in the following
proposition.
3
Note that taking account of discounting and the natural degradation of the respective
pollution stock, the net present value of the accumulated
∞ damage of one marginal unit of
pollution sums up to the right-hand-side of (6.28), as 0 σi si e−(ρ+δi )t dt = σi si /(ρ + δi ) (i =
1, 2) .
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
105
Proposition 6.2
An interior optimal stationary state, if it exists, has the following properties:
dc
>0,
dδ1
dc
<0,
dσ1
dc
>0,
dρ
dx
>0,
dδ1
dx
<0,
dσ1
dx ≥
0 for
dρ <
dc
dx
>0,
<0,
dδ2
dδ2
dc
dx
<0,
>0,
dσ2
dσ2
[U22
δ2 (δ2 + ρ) − σ2 ](δ2 + ρ) ≥ σ2 F 2 Fc1
.
[U11
δ1 (δ1 + ρ) − σ1 ](δ1 + ρ) < σ1 F 1 Fc2 Proof: see Appendix A6.3.
These results can be interpreted as follows. For both pollutants i (i = 1, 2),
the lower is the natural deterioration rate δi and the higher is the harmfulness
σi , the lower is the relative weight of the emitting sector in the total economy
and the lower is the overall scale of economic activity in the stationary state.
An increase in the discount rate ρ increases the optimal stationary scale of
economic activity, c , while its effect on the optimal stationary structure of
economic activity, x , is ambiguous.
6.3.3
Optimal Dynamic Path and Local Stability Analysis
In the following we solve the optimization problem by linearizing the resulting
system of differential equations around the stationary state. Since our model is
characterized by only mild non-linearities,4 we expect the linear approximation
to yield insights which should also hold for the exact problem. In Section 6.4
below, we shall numerically optimize the exact problem, and confirm this expectation.
As we have assumed an interior stationary state, the optimal path will also
be an interior optimal path at least in a neighborhood of the interior stationary
state. Hence, we restrict the analysis to the case of an interior solution, i.e.
c < 1. As shown in Appendix A6.4, the optimal dynamics of the two control
variables c, x and the two state variables s1 , s2 can be described by a system
of four coupled first order autonomous differential equations:
[U1 (δ1 + ρ) − σ1 s1 ]U22 Fx2 − [U2 (δ2 + ρ) − σ2 s2 ]U11 Fx1
,
U11 U22 df
[U2 (δ2 + ρ) − σ2 s2 ]U11 Fc1 − [U1 (δ1 + ρ) − σ1 s1 ]U22 Fc2
ẋ =
,
U11 U22 df
ṡ1 = F 1 − δ1 s1 ,
ṡ2 = F 2 − δ2 s2 ,
ċ =
4
(6.30)
(6.31)
(6.32)
(6.33)
Remember that the welfare function V is additively separable in all four arguments.
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Natural Science Constraints in Environmental and Resource Economics
with df ≡ Fc1 Fx2 − Fx1 Fc2 < 0. Linearizing around the stationary state
(c , x , s1 , s2 ) yields the following approximated dynamic system (see Appendix A6.5):
⎞
⎛
⎞
⎛
c − c
ċ
⎟
⎜
⎜ ẋ ⎟
⎟ ≈ J ⎜ x − x ⎟ with
⎜
(6.34)
⎝ s1 − s1 ⎠
⎝ s˙1 ⎠
s˙2
s2 − s2
⎛
⎜
⎜
J =⎜
⎜
⎝
δ1 Fc1 Fx2 −δ2 Fx1 Fc2
df (δ2 −δ1 )Fc1 Fc2
df 1
Fc
Fc2
ρ+
(δ1 −δ2 )Fx1 Fx2
df δ2 Fc1 Fx2 −δ1 Fx1 Fc2
ρ+
df Fx1
Fx2
2
− Uσ1Fdfx 11
σ1 Fc2
df U11
−δ1
0
σ2 Fx1
df U22
1
− Uσ2Fdfc 22
0
⎞
⎟
⎟
⎟ .
⎟
⎠
−δ2
The Jacobian evaluated at the stationary state, J , has four real eigenvalues
(see Appendix A6.5), two of which are strictly negative (ν1 , ν2 ) and two of
which are strictly positive (ν3 , ν4 ). Hence, the system dynamics exhibits saddlepoint stability, i.e. for all initial stocks of pollutants, s01 and s02 , there exists
a unique optimal path which asymptotically converges towards the stationary state. Because of the transversality conditions (6.24) the optimal path
is restricted to the stable hyperplane, which is spanned by the eigenvectors
associated with the negative eigenvalues. Given the eigenvalues and the eigenvectors, which are calculated in Appendix A6.5, the explicit system dynamics
in a neighborhood around the stationary state is given by:
c(t) =
x(t) =
s1 (t) =
s2 (t) =
Fx2 (ν1 + δ1 )
c +
−
eν1 t −
Fc1 Fx2 − Fx1 Fc2 F 1 (ν2 + δ2 )
eν2 t ,
(s02 − s2 ) 1 x 2 Fc Fx − Fx1 Fc2 Fc2 (ν1 + δ1 )
0
ν1 t
+
x − (s1 − s1 ) 1 2 2 e
1
Fc Fx − Fx Fc
Fc1 (ν2 + δ2 )
0
eν2 t ,
(s2 − s2 ) 1 2 Fc Fx − Fx1 Fc2 s1 + (s01 − s1 ) eν1 t ,
s2 + (s02 − s2 ) eν2 t ,
(s01
s1 )
(6.35)
(6.36)
(6.37)
(6.38)
where s0i = si (0) (i = 1, 2) denote the initial pollutant stocks.
As a measure of the overall rate of convergence of a process z(t) which
asymptotically approaches z , we define the characteristic time scale of convergence τz by
ż(t) −1
,
τz
≡ (6.39)
z(t) − z Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
107
where the horizontal bar denotes the average over time. The greater is the time
scale τz , the slower is the convergence towards z . With this definition, it is
obvious from Equations (6.37) and (6.38) that the pollutant stock si (i = 1, 2)
converges towards its stationary state value si with a characteristic time scale
τsi = 1/ |νi |. As the system approaches the stationary state for t → ∞, the scale
c and structure x (Equations 6.35 and 6.36) converge towards their stationary
state values c and x with a characteristic time scale which is determined by
the eigenvalue with the smaller absolute value, τc = τx = 1/ min{|ν1 |, |ν2|} (see
Appendix A6.6). Proposition 6.3 summarizes these results.
Proposition 6.3
For the linear approximation (6.34) around the stationary state (c , x , s1 , s2 )
the following statements hold:
(i) The stationary state is saddlepoint-stable.
(ii) The explicit system dynamics is given by Equations (6.35)–(6.38).
(iii) The characteristic time scale of convergence towards the stationary state
is given by
• τc = τx = 1/ min{|ν1 |, |ν2 |} for the control variables c and x, and by
• τsi = 1/ |νi | for stock variable si (i = 1, 2).
As shown in Appendix A6.5 the eigenvalues ν1 and ν2 are given by
ν1
ν2
1
=
ρ−
2
1
=
ρ−
2
4σ1
(ρ + 2δ1 )2 − U11
4σ2
(ρ + 2δ2 )2 − U22
!
< 0,
(6.40)
< 0.
(6.41)
!
Hence, the absolute value of νi (time scale of convergence) decreases (increases)
with the discount rate ρ and the curvature of consumption welfare in the stationary state |Uii | (i = 1, 2). It increases (decreases) with the harmfulness σi
and the deterioration rate δi of the pollutant stock.
We now turn to the question of the (non-)monotonicity of the optimal path.
According to Equations (6.37) and (6.38), the stocks of the two pollutants
converge monotonically towards their stationary state values s1 and s2 . In
order to show that the optimal paths for the control variables c and x may be
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Natural Science Constraints in Environmental and Resource Economics
non-monotonic, we differentiate Equations (6.35) and (6.36) with respect to t:
Fx2 (ν1 + δ1 )
ċ(t) =
−
eν1 t −
Fc1 Fx2 − Fx1 Fc2 F 1 (ν2 + δ2 )
eν2 t ,
ν2 (s02 − s2 ) 1 x 2 Fc Fx − Fx1 Fc2 Fc2 (ν1 + δ1 )
0
ν1 t
+
ẋ(t) = −ν1 (s1 − s1 ) 1 2 2 e
1
Fc Fx − Fx Fc
Fc1 (ν2 + δ2 )
0
eν2 t .
ν2 (s2 − s2 ) 1 2 Fc Fx − Fx1 Fc2 ν1 (s01
s1 )
(6.42)
(6.43)
The optimal path is non-monotonic if ċ or ẋ change their sign, i.e. if the paths
c(t) or x(t) exhibit a local extremum for positive times t. According to the signs
of the νi and Fji (i = 1, 2 and j = c, x) and given that ν1 = ν2 , c(t) exhibits a
unique local extremum if sgn(s01 −s1 ) = sgn(s02 −s2 ), and x(t) exhibits a unique
local extremum if sgn(s01 − s1 ) = sgn(s02 − s2 ).5 Solving ċ(t) = 0 and ẋ(t) = 0
for t, using expressions (6.42) and (6.43) for ċ and ẋ, yields:
⎧ 0 1
ν2 (s −s )F (ν2 +δ2 )
⎪
⎨ ln ν1 (s201 −s21 )Fxx2 (ν1 +δ1 ) (ν1 −ν2 )−1 , if sgn(s01 −s1 ) = sgn(s02 −s2 )
.
t̂ =
⎪
⎩ ln ν2 (s020 −s2 )Fc12 (ν2 +δ2 ) (ν1 −ν2 )−1 , if sgn(s0 −s ) = sgn(s0 −s )
1
1
2
2
ν1 (s1 −s1 )Fc (ν1 +δ1 )
(6.44)
According to this equation, it is possible that t̂ may be negative or infinite,
which is meaningless in the context of this analysis. In this case we would
observe monotonic optimal paths for both control variables c and x for times
0 < t < +∞. For instance, t̂ is negative if |s02 − s2 | is sufficiently small, that
is, the second pollutant stock is initially already close to its stationary state
level. Furthermore, t̂ equals (plus or minus) infinity if either |s01 − s1 | = 0
or |ν1 − ν2 | = 0, that is, the first pollutant stock is initially already at its
stationary state level or the eigenvalues are identical. The following proposition
summarizes the behavior of the optimal control path.
Proposition 6.4
In the linear approximation (6.34) around the stationary state (c , x , s1 , s2 ),
the following statements hold for the optimal path:
(i) The stocks of pollutants s1 (t) and s2 (t) converge exponentially, and hence
monotonically, towards their stationary state values s1 and s2 .
(ii) If and only if t̂ as given by Equation (6.44) is strictly positive and finite,
then the optimal control is non-monotonic over time and t̂ denotes the
5
Note that νi + δi < 0, which can easily be verified from Equations (A6.26) and (A6.27).
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
109
time at which the optimal control has a unique local extremum. In particular, if sgn(s01 − s1 ) = sgn(s02 − s2 ), c(t) is non-monotonic and x(t) is
monotonic. If sgn(s01 − s1 ) = sgn(s02 − s2 ), x(t) is non-monotonic and
c(t) is monotonic.
6.4
Numerical Optimization
In this section we illustrate the results derived in Section 6.3 by numerical optimizations of the original, non linearized optimization problem (6.14)–
(6.16). The results thus obtained confirm that the insights from analyzing
the linearized system also hold for the exact solution. All numerical optimizations were carried out with the advanced optimal control software package
MUSCOD-II (Diehl et al. 2001), which exploits the multiple shooting state
discretization (Leineweber et al. 2003).
There are four different qualitative scenarios which have to be examined.
(i) Both stocks of pollutants exhibit the same harmfulness but differ in their
deterioration rates, i.e. σ1 = σ2 , δ1 < δ2 . (ii) The two pollutants differ in their
harmfulness but have equal deterioration rates, i.e. σ1 < σ2 , δ1 = δ2 . (iii)
The pollutants differ in both harmfulness and deterioration rates and the more
harmful pollutant has the higher deterioration rate, i.e. σ1 < σ2 , δ1 < δ2 . (iv)
Both harmfulness and deterioration rates are different, and the more harmful
pollutant has a lower deterioration rate, i.e. σ1 < σ2 , δ1 > δ2 . Furthermore,
each of the four scenarios splits into four subcases, depending on the initial
stocks of pollutants (both initial stocks below, only first stock above, only
second stock above and both stocks above the stationary state levels).
In the following we discuss these four different scenarios. The parameter
values used for the numerical optimization have been chosen so as to illustrate
clearly the different effects, and do not necessarily reflect the characteristics of
real environmental pollution problems. For all numerical examples, the total
labor supply λ has been chosen so as to guarantee an interior stationary state
scale c < 1. As it is not possible to optimize numerically over an infinite time
horizon, the time horizon has been set to 250 years and all parameters have
been chosen in such a way that the system at time t = 250 is very close to the
stationary state. For a more convenient exposition, the figures show the time
paths up to t = 125 only. The parameter values for the numerical optimization
are listed in Appendix A6.7.
In the first scenario (σ1 = σ2 ), both stocks of pollutants exhibit the same
harmfulness but the deterioration rate is smaller for the first pollutant than for
the second. Figure 6.1 shows the result of a numerical optimization of this case.
In this example the initial stocks for both pollutants are above their stationary
110
Natural Science Constraints in Environmental and Resource Economics
state levels (s01 = 30, s02 = 30). The optimal path for the structure exhibits nonmonotonic behavior as expected from Proposition 6.4. Further, we expect that
the optimal stationary state structure x is clearly below 0.5, indicating that
relatively more labor is employed in the second sector, because as the second
stock of pollutant deteriorates at a higher rate the aggregate intertemporal
damage of one unit of emissions is smaller for the second pollutant.6 This
expectation is confirmed by the numerical optimization.
1
35
30
0.8
25
0.6
20
15
0.4
10
0.2
0
5
0
20
40
scale
60
80
100
structure
120
t
0
0
20
40
stock1
60
80
100
120
t
stock2
Figure 6.1 Optimal paths for scale and structure (left) and the two pollutant
stocks (right) for the case σ1 = σ2 , δ1 < δ2 . Parameter values used for the
numerical optimization are given in Appendix A6.7.
In the second scenario (σ1 < σ2 , δ1 = δ2 ), the two stocks of pollutants
are of different harmfulness but the deterioration rate for the two pollutants
are equal. The result of a numerical optimization of this case is presented in
Figure 6.2. In this example the initial stock for the first (second) pollutant
is above (below) their stationary state levels (s1 = 40, s2 = 0). Now, the
optimal path for the scale exhibits a non monotonic behavior as expected from
Proposition 6.4. Further, we expect that the optimal stationary state structure
x is clearly above 0.5, indicating that relatively more labor is employed by
the second sector, because as the second stock of pollutant is less harmful
the aggregate intertemporal damage of one unit of emissions is smaller for the
second pollutant. This expectation is confirmed by the numerical optimization.
The third scenario (σ1 < σ2 , δ1 < δ2 ) – both harmfulness and deterioration
rates are different and the more harmful pollutant has the higher deterioration
rate – is the most interesting as neither of the two pollutants exhibits a priori
more favorable dynamic characteristics for the economy. Hence, we are not
able to predict which production sector will be used to a greater extent in the
stationary state. Furthermore, non monotonic paths – if they occur – are likely
to be more pronounced than in the other cases. Figure 6.3 shows the optimal
paths for a numerical example for all four subcases (initial pollutant stocks
6
Note that both consumption goods are equally valued by the representative consumer,
i.e. μ1 = μ2 (see Appendix A6.7).
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
1
111
45
40
0.8
35
30
0.6
25
20
0.4
15
10
0.2
5
0
0
20
40
60
80
scale
100
structure
120
t
0
0
20
40
stock1
60
80
100
120
t
stock2
Figure 6.2 Optimal paths for scale and structure (left) and the two pollutant
stocks (right) for the case σ1 < σ2 , δ1 = δ2 . Parameter values used for the
numerical optimization are given in Appendix A6.7.
above or below stationary state levels for one and both pollutants). Of course,
the long run stationary state to which the economy converges, is the same
in all four subscenarios, as all parameters are identical except for the initial
stocks of the two pollutants. Nevertheless, the optimal paths and especially
their convergence towards the stationary state is quite different for the four
subcases. As expected from Proposition 6.4, we observe that – if at all – the
optimal path for the structure is non-monotonic if both stocks start above or
below their stationary state levels (subcases a and d) and the optimal path for
the scale is non-monotonic if one initial stock is higher and one is lower than
their stationary state levels (subcase b). We also see that both, structure and
scale, may exhibit monotonic optimal paths (subcase c).
In the fourth scenario (σ1 < σ2 , δ1 > δ2 ), where both pollutants exhibit
different harmfulness and deterioration rates but the second pollutant is more
harmful and has the lower deterioration rate, the first pollutant exhibits clearly
more favorable dynamic properties than the second pollutant. In this case the
economy will nearly exclusively use the first production sector. Although nonmonotonicities in the optimal paths for scale and structure can occur according
to Proposition 6.4, they are not pronounced. As nothing new can be learned
from this case, we do not show a numerical optimization example.
6.5
Conclusion
In this chapter, we have studied the mutual interaction over time between the
scale and structure of economic activity on the one hand, and the dynamics of
multiple environmental pollution stocks on the other hand. We have carried
out a total analysis of a two-sector-economy, in which each sector produces
one distinct consumption good and one specific pollutant. The pollutants of
112
Natural Science Constraints in Environmental and Resource Economics
a) both stocks below stationary state level
1
50
45
0.8
40
35
0.6
30
25
0.4
20
15
0.2
10
5
0
0
20
40
60
scale
80
100
120
t
0
0
20
40
60
80
stock1
structure
100
120
t
stock2
b) first stock above, second stock below stationary state level
1
50
45
0.8
40
35
0.6
30
25
20
0.4
15
10
0.2
5
0
0
20
40
60
scale
80
100
120
t
0
0
20
40
60
80
stock1
structure
100
120
t
stock2
c) first stock below, second stock above stationary state level
1
50
45
0.8
40
35
0.6
30
25
0.4
20
15
0.2
10
5
0
0
20
40
60
scale
80
100
120
t
0
0
20
40
60
80
stock1
structure
100
120
t
stock2
d) both stocks above stationary state level
1
50
45
0.8
40
35
0.6
30
25
0.4
20
15
0.2
10
5
0
0
20
40
scale
60
80
100
structure
120
t
0
0
20
40
stock1
60
80
100
120
t
stock2
Figure 6.3 Optimal paths for scale and structure (left) and the two pollutant
stocks (right) for the case σ1 < σ2 , δ1 < δ2 and all four subscenarios. Parameter
values used for the numerical optimization are given in Appendix A6.7.
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
113
both sectors were assumed to differ in their environmental impact in two ways:
(i) with respect to their harmfulness and (ii) with respect to their natural
deterioration rates in the environment.
Most of the results are intuitive. First, it may be optimal not to use all
available labor endowment in the production of consumption goods in order to
avoid excessive environmental damage. Second, under very general conditions
a change in scale and structure of economic activity over time is optimal. Thus,
the optimal economic dynamics is driven by the dynamics of the environmental
pollution stocks. The less harmful is a pollutant, the higher are the relative
importance of the emitting sector and the overall scale of economic activity in
the stationary state. The shorter lived is a pollutant, the higher are the relative
importance of the emitting sector and the overall scale of economic activity in
the stationary state. If emissions differ either in their environmental harmfulness or in their deterioration rates, we should have structural change towards
the sector emitting the less harmful or the shorter-lived pollutant. However,
if the harmfulness and deterioration rates differ and if the environmentally
less harmful emission is also the longer-lived pollutant, no general conclusion
concerning the direction of structural change can be drawn. Third, the characteristic time scale of convergence of scale and structure towards the stationary
state is given by (the inverse of) the eigenvalue with the smaller absolute value.
It increases with the discount rate and the curvature of consumption welfare
in the stationary state; it decreases with the harmfulness and the deterioration
rate of the respective pollutant stock.
Most importantly, our formal analysis as well as the numerical optimizations, show that it is likely that the optimal control paths, i.e. the change in
the scale and structure of the economy, are non-monotonic over time.7 If a
non-monotonic control is optimal, our numerical optimizations suggest that
the local extremum of the control path may be pronounced and that it occurs
at the beginning of the control path.
These results have implications for the design of environmental indicators
and policies. First, the traditional view is that different environmental problems
– such as e.g. acidification of soils and surface waters, groundwater contamination by nitrates or pesticides, and climate change due to anthropogenic greenhouse gas emission – can be regulated by independent environmental policies.
In contrast, our total analysis of a multi-sector economy with several independent environmental pollutants, shows that these problems – even without any
direct physical interaction – interact indirectly because they all affect social
welfare, and the mitigation of all of them is constrained by the available eco7
Non-monotonic optimal control paths, in particular limit-cycles, are known to exist for
control problems with two or more state variables, and for time-lagged and adaptive control
problems, even with one single state variable (e.g. Benhabib and Nishimura 1979, Feichtinger
et al. 1994, Wirl 2000, 2002, Winkler 2004).
114
Natural Science Constraints in Environmental and Resource Economics
nomic resources. As a result, even for non-interacting environmental pollutants
the optimal regulation has to take an encompassing view, taking into account
all of the environmental problems together.
Second, indicators and policies which are solely based on the harmfulness
of environmental pollutants – which is predominant in current environmental
politics – fall short of optimally controlling environmental problems. In a dynamic setting, the lifetime of pollutants is an equally important determinant
of the optimal environmental policy.
Third, the non-monotonicity-result challenges common intuition which suggests that policies should achieve optimal change in a monotonic way. In contrast to this simple intuition, our analysis shows that if pollutants accumulate
on different time scales and if they differ in environmental harmfulness, the optimal policies may be non-monotonic. In particular, the optimal time-path of
structural change towards the stationary state structure may be characterized
by ‘optimal overshooting’; that is, the optimal relative importance of a sector
starts below (above) the stationary state level, increases (decreases) to a point
above (below) the stationary state level, and finally decreases (increases) again.
The same goes for the optimal dynamics of the overall economic scale.
Summing up, in order to develop sustainable solutions to the multiple environmental problems that we face in reality – such as climate change, depletion
of the ozone layer, groundwater contamination, acidification of soil and surface
water, biodiversity loss, etc. – we should adopt an encompassing view and base
policy advice on a total analysis of economy-environment interactions. As our
analysis shows, the resulting optimal policies need to take account of the history, the empirical parameter values and the dynamic relationships of all of the
problems, and these policies might be non-monotonic.
Appendix
A6.1
Concavity of the Optimized Hamiltonian
We show that the Hamiltonian H, without taking into account the restriction
c ≤ 1, i.e. pc = 0, is strictly concave whenever the necessary conditions are
satisfied. Thus, the unique optimal solution is the local extremum of H if we
have an interior solution; it is a corner solution with c = 1 if the local extremum
of H is reached for unfeasible c > 1.
A sufficient condition for strict concavity of the Hamiltonian is that its
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
Hessian H =
∂2H
∂i∂j
115
(i, j = c, x, s1 , s2 ) is negative definite. The Hessian H reads:
⎛
⎞
0
0
Hcc Hcx
⎜ Hxc Hxx 0
0 ⎟
⎟
(A6.1)
H=⎜
⎝ 0
0 −σ1
0 ⎠
0
0
0 −σ2
Due to its diagonal form, H is negative definite if the reduced Hessian
∂2H
H = ∂i∂j
(i, j = c, x) is negative definite, i.e. Hcc , Hxx < 0 and det H > 0.
Hcc = U11 (Fc1 )2 + (U1 + p1 )Fcc1 + U22 (Fc2 )2 + (U2 + p2 )Fcc2 , (A6.2)
1
2
Hxx = U11 (Fx1 )2 + (U1 + p1 )Fxx
+ U22 (Fx2 )2 + (U2 + p2 )Fxx
, (A6.3)
1 1
1
2 2
2
Hcx = U11 Fc Fx + (U1 + p1 )Fcx + U22 Fc Fx + (U2 + p2 )Fcx . (A6.4)
Along the optimal path, the necessary conditions have to be satisfied. In particular, for an interior solution, i.e. c < 1, the necessary and sufficient conditions
(6.19) and (6.20) become:
(U1 + p1 )Fc1 + (U2 + p2 )Fc2 = 0 ,
(U1 + p1 )Fx1 + (U2 + p2 )Fx2 = 0 .
(A6.5)
(A6.6)
Thus, for an interior optimal path the following equations hold:
pi = −Ui
(i = 1, 2) .
(A6.7)
With this, one obtains:
(A6.8)
Hcc = U11 (Fc1 )2 + U22 (Fc2 )2 < 0 ,
1 2
2 2
Hxx = U11 (Fx ) + U22 (Fx ) < 0 ,
(A6.9)
2
det H = Hcc Hxx − Hcx
0.
= U11 U22 (Fc1 )2 (Fx2 )2 + (Fx1 )2 (Fc2 )2 − 2Fc1 Fx1 Fc2 Fx2 > (A6.10)
Hence, whenever H has an extremum it is a maximum. As a consequence, the
necessary conditions (plus the transversality condition 6.24) are also sufficient.
A6.2
Proof of Proposition 6.1
(i) Inserting Equations (6.29) into Equations (6.26) and (6.27), and using the
relationship between F i and P i , as given from Equation (6.9), one obtains:
Ui
pc
σi P i
+
=
δi (δi + ρ) λPli
(i = 1, 2) .
(A6.11)
With the properties for P i , as given by (6.1), and the properties for Ui , as
given by (6.7), the left-hand-side of Equation (A6.11) is strictly decreasing
116
Natural Science Constraints in Environmental and Resource Economics
while the right-hand-side is strictly increasing in li . Thus, there exists at most
one li which satisfies Equation (A6.11). The existence of such a solution is
guaranteed by the properties limli →0 Pli = +∞ and limyi →0 Ui = +∞.
(ii) We derive ¯li by solving (A6.11) for li assuming pc = 0. Thus, ¯li is the
maximal amount of labor which will be assigned to production process i in an
optimal stationary state without taking account for the restriction c ≤ 1. If
¯l1 + ¯l2 ≥ λ the labor supply is short of the optimal labor demand and thus
the stationary state is a corner solution. If, on the other hand, the total labor
supply λ exceeds the sum ¯l1 + ¯l2 , then not all labor will be used for economic
activity and the optimal stationary state will be an interior solution.
A6.3
Proof of Proposition 6.2
Setting pc = 0 in Equation (A6.11) yields for an interior stationary path:
Ui
σi F i
=
δi (δi + ρ)
(i = 1, 2) .
(A6.12)
By implicit differentiation of (A6.12) with respect to δj (j = 1, 2) one obtains:
σj
σj F j (2δj + ρ)
j ∂c
j ∂x
+ Fx
(j = i) ,
Ujj −
Fc
= − 2
∂δj
∂δj
δj (δj + ρ)
δj (δj + ρ)2
σi
i ∂c
i ∂x
+ Fx
= 0
(j =
i) .
Uii −
Fc
∂δj
∂δj
δi (δi + ρ)
Solving for ∂c /∂δj and ∂x /∂δj yields:
σj F j Fxi (2δj + ρ)
∂c
, (A6.13)
=
∂δj
(Fci Fxj − Fcj Fxi )(Ujj
δj (δj + ρ) − σj )δj (δj + ρ)
∂x
σj F j Fci (2δj + ρ)
. (A6.14)
=
∂δj
(Fcj Fxi − Fci Fxj )(Ujj
δj (δj + ρ) − σj )δj (δj + ρ)
From the signs of the Fji (i = 1, 2; j = c, x) it follows that
∂c
>0,
∂δ1
∂c
> 0,
∂δ2
∂x
> 0,
∂δ1
∂x
<0.
∂δ2
(A6.15)
By implicit differentiation of (A6.12) with respect to σj (j = 1, 2) one obtains:
σj
Fj
j ∂c
j ∂x
(j = i) ,
+ Fx
Ujj −
Fc
=
∂σj
∂σj
δj (δj + ρ)
δj (δj + ρ)
σi
i ∂c
i ∂x
+ Fx
Fc
Uii −
= 0
(j =
i) .
∂σj
∂σj
δi (δi + ρ)
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
117
Solving for ∂c /∂σj and ∂x /∂σj yields:
∂c
∂σj
F j Fxi
,
(A6.16)
F j Fci
∂x
.
=
∂σj
(Fci Fxj − Fcj Fxi )(Ujj
δj (δj + ρ) − σj )
(A6.17)
=
(Fcj Fxi − Fci Fxj )(Ujj
δj (δj + ρ) − σj )
From the signs of the Fji (i = 1, 2; j = c, x) it follows that
∂c
<0,
∂σ1
∂c
< 0,
∂σ2
∂x
< 0,
∂σ1
∂x
>0.
∂σ2
(A6.18)
Implicit differentiation of (A6.12) with respect to ρ yields:
∂c
Fc1
∂x
Fx1
+
∂ρ
∂ρ
∂c
∂x
+ Fx2
Fc2
∂ρ
∂ρ
σ1 F 1
= − ,
[U11 δ1 (δ1 + ρ) − σ1 ](δ1 + ρ)
σ2 F 2
= − ,
[U22 δ2 (δ2 + ρ) − σ2 ](δ2 + ρ)
Solving for ∂c /∂ρ and ∂x /∂ρ yields:
∂c 2 1 (Fc Fx − Fc1 Fx2 ) =
∂ρ
σ2 F 2 Fx1
σ1 F 1 Fx2
−
(, A6.19)
=
[U11
δ1 (δ1 + ρ) − σ1 ](δ1 + ρ) [U22
δ2 (δ2 + ρ) − σ2 ](δ2 + ρ)
∂x 2 1 (Fc Fx − Fc1 Fx2 ) =
∂ρ
σ1 F 1 Fc2
σ2 F 2 Fc1
−
(, A6.20)
=
[U22
δ2 (δ2 + ρ) − σ2 ](δ2 + ρ) [U11
δ1 (δ1 + ρ) − σ1 ](δ1 + ρ)
From the signs of the Fji (i = 1, 2; j = c, x) it follows that
∂c
>0,
∂ρ
A6.4
∂x ≥
δ2 (δ2 + ρ) − σ2 ](δ2 + ρ) ≥ σ2 F 2 Fc1
[U22
. (A6.21)
0
⇔
∂ρ <
[U11
δ1 (δ1 + ρ) − σ1 ](δ1 + ρ) < σ1 F 1 Fc2 Derivation of the Differential Equation System
Differentiation of pi = −Ui (Equation A6.7) with respect to time and inserting
into Equations (6.21) and (6.22) yields, together with the equations of motion
(6.15), a system of four differential equations in the four unknowns c, x, s1 and
s2 :
σ1 s1 − U1 (δ1 + ρ) + U11 (Fc1 ċ + Fx1 ẋ)
σ2 s2 − U2 (δ2 + ρ) + U22 (Fc2 ċ + Fx2 ẋ)
s˙1 − F 1 + δ1 s1
s˙2 − F 2 + δ2 s2
=
=
=
=
0
0
0
0
,
,
,
.
(A6.22)
(A6.23)
(A6.24)
(A6.25)
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Natural Science Constraints in Environmental and Resource Economics
The conditions (A6.22)–(A6.25) for an interior optimal solution can be rearranged to yield the system (6.30)–(6.33) of four coupled autonomous differential
equations.
A6.5
Eigenvalues and Eigenvectors of the Jacobian
We obtain the Jacobian J ∗ by differentiating the right-hand-sides of Equations
(6.30)–(6.33) with respect to c, x, s1 and s2 and evaluating them at the stationary state. Taking into account that in the interior stationary state (6.28)
holds with equality, Ui = σi si /(δi + ρ), one obtains for the Jacobian J :
⎛
⎜
⎜
∗
J =⎜
⎜
⎝
δ1 Fc1 Fx2 −δ2 Fx1 Fc2
df (δ2 −δ1 )Fc1 Fc2
df 1
Fc
Fc2
ρ+
(δ1 −δ2 )Fx1 Fx2
df δ2 Fc1 Fx2 −δ1 Fx1 Fc2
ρ+
df Fx1
Fx2
2
− Uσ1Fdfx 11
σ1 Fc2
df U11
−δ1
0
σ2 Fx1
df U22
1
− Uσ2Fdfc 22
0
⎞
⎟
⎟
⎟ .
⎟
⎠
−δ2
The eigenvalues νi and eigenvectors ξi are the solutions of the equation J · ξ =
ν · ξ. The four eigenvalues are:
!
1
4σ
1
ρ − (ρ + 2δ1 )2 − < 0 ,
(A6.26)
ν1 =
2
U11
!
1
4σ
2
ρ − (ρ + 2δ2 )2 − < 0 ,
(A6.27)
ν2 =
2
U22
!
1
4σ
1
ρ + (ρ + 2δ1 )2 − > 0 ,
(A6.28)
ν3 =
2
U11
!
1
4σ
2
ρ + (ρ + 2δ2 )2 − > 0 .
(A6.29)
ν4 =
2
U22
The eigenvectors associated with the negative eigenvalues ν1 and ν2 are:
2
Fx (ν1 + δ1 ) Fc2 (ν1 + δ1 )
ξ1 =
,−
, 1, 0 ,
(A6.30)
df df Fx1 (ν2 + δ2 ) Fc1 (ν2 + δ2 )
ξ2 =
−
,
, 0, 1 .
(A6.31)
df df A6.6
Time Scale of Convergence
Equations (6.35) and (6.36) are of the following type:
z(t) = z + Aeν1 t + Beν2 t
(ν1 , ν2 < 0) ,
(A6.32)
Optimal Dynamic Scale and Structure of a Multi-Pollution Economy
119
with real constants A and B. Without loss of generality assume that |ν1 | <
|ν2 |. Since we are interested in the system dynamics in a neighborhood of the
stationary state, we calculate the characteristic time scale of convergence for z
as t → ∞. According to (6.39), the characteristic time scale of convergence of
z in a neighborhood of the stationary state z is given by:
τz−1
ν1 t + Bν eν2 t (ν2 −ν1 )t Aν
e
Aν
+
Bν
e
1
2
1
2
= lim
= lim
t→∞ A + Be(ν2 −ν1 )t = |ν1 | . (A6.33)
t→∞
Aeν1 t + Beν2 t Hence, for t → ∞ the characteristic time scale of convergence is constant and
given by 1/ min{|ν1 |, |ν2 |}.
A6.7
Parameter Values for the Numerical Optimization
We used a Cobb-Douglas welfare function for the numerical optimizations,
U(y1 , y2) = 0.5 ln(y1 ) + 0.5 ln(y2 ) ,
(A6.34)
and the following production functions:
&
&
P 2 (l2 ) = l2 .
P 1 (l1 ) = l1 ,
(A6.35)
For all numerical optimizations we set λ = 1 and ρ = 0.03. In addition, we
used the following parameter values for the different scenarios:
Figure
1
2
3a
3b
3c
3d
σ1
0.01
0.003
0.002
0.002
0.002
0.002
σ2
0.01
0.03
0.02
0.02
0.02
0.02
δ1
0.02
0.05
0.02
0.02
0.02
0.02
δ2
0.1
0.05
0.1
0.1
0.1
0.1
s1 s2
30 30
40 0
0 0
50 0
0 25
50 25
120
Natural Science Constraints in Environmental and Resource Economics
PART II
Biodiversity
7. Biodiversity as an Economic Good∗
7.1
Introduction
Biological diversity, which has been defined as ‘the variability among living
organisms from all sources [...] and the ecological complexes of which they
are part’ (CBD 1992), is currently being lost at rates that exceed the natural
extinction rates of the past by a factor of somewhere between 100 and 1,000
(Watson et al. 1995b). This is one of the most eminent environmental problems
of our time (Wilson 1988).
Ecologists were among the first to point out this alarming development and
to express concern over its potential negative effect on ecosystems and human
well-being (Ehrlich and Ehrlich 1981, Myers 1979, Soulé 1986, Soulé and Wilcox
1980, Wilson 1988). They vindicated their concern with the important role that
biodiversity plays for ecosystem functioning (Holling et al. 1995, Loreau et al.
2001, Schulze and Mooney 1993, Tilman 1997a) and for providing essential
life-support services to the human existence on planet Earth (Daily 1997b,
Mooney and Ehrlich 1997, Perrings et al. 1995b). Examples of such ecosystem
services include nutrient cycling, control of water runoff, purification of air and
water, soil regeneration, pollination of crops and natural vegetation, or climate
stabilization. The main mechanisms through which biodiversity is currently
being lost were identified to be loss of habitat, overuse of populations, invasion
of non-native species, pollution of ecosystems and climate change (Barbier et
al. 1994, Watson et al. 1995a).
Economists have pointed out the high economic value of biodiversity, which
comprises both use and non-use values (Goulder and Kennedy 1997, Randall 1988, Watson et al. 1995a). Many species have direct use value as food,
fuel, construction material, industrial resources or pharmaceutical substances
(Farnsworth 1988, Plotkin 1988). Beyond that, biodiversity, i.e. the set of all
species, has an important indirect use value in so far as entire ecosystems per∗
Translated and revised from ‘Der ökonomische Wert der biologischen Vielfalt’, in: Bayerische Akademie für Naturschutz und Landschaftspflege (ed.), Grundlagen zum Verständnis
der Artenvielfalt und ihrer Bedeutung und der Maßnahmen, dem Artensterben entgegen zu
wirken (Laufener Seminarbeiträge 2/02), Laufen/Salzach, 2002, pp. 73–90.
123
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Natural Science Constraints in Environmental and Resource Economics
form valuable services as described above. One study went so far as to estimate
the total economic value of all the Planet’s ecosystem goods and services at
US$ 33 trillion per year, a number comparable in order of magnitude to aggregate world GDP (Costanza et al. 1997b). Besides putting a value on what is
currently being lost and what is at risk, economists also provided a number of
explanations for the loss of biodiversity currently being observed (Barbier et al.
1994, Watson et al. 1995a, Moran and Pearce 1997). The fundamental causes
of biodiversity loss include the growth of human population; market failure
because of externalities and the public good character of biological resources;
governance failure in regulating the access to, and use of, biological resources;
and fundamental ignorance pertaining to both individual and social decision
making.
By now, the international community has acknowledged the problem of
biodiversity loss, and the need to enact policies to halt or even reverse this
problem. In June 1992, the Convention on Biological Diversity was signed by
156 states at the United Nations Conference on Environment and Development
in Rio de Janeiro, Brazil (CBD 1992). In the preamble of this convention, the
signatories explicitly declare that biodiversity has – besides ecological, cultural,
spiritual and intrinsic values – also an economic value.
Yet, many ecologists (and even more environmentalists) regard the contribution of economists to the discussion about biodiversity loss and conservation
with great suspicion: ‘Isn’t it the economy, which causes biodiversity loss? And
don’t economists always argue in favor of economic interests?’ While this suspicion is largely based on ignorance about the nature and substance of economics
as a science, there is a corresponding reluctance among professional economists
to engage in the discussion about biodiversity loss and conservation: ‘What
exactly is this thing called “biodiversity”? In what sense is it an economic
good? And how can we discuss its efficient and fair allocation based on standard economic concepts?’ For example, Weimann et al. (2003: 7) express this
wide-spread unease concerning biodiversity among economists, by pointing out
that ‘[b]iodiversity exists, but there is no consensus about what it is. Biodiversity is finite and declining, but there is no consensus about how to measure
it. Biodiversity is important, but there is no consensus about how important
it is, and for whom’ (own translation).
The discussion of biodiversity loss and conservation therefore faces the following challenge. On the one hand, economics seems to have important contributions to make to this discussion. But on the other hand, a number of conceptual questions remain open, which need to be addressed before economists
can apply their standard tools and methods in an analysis:
(i) In what sense can one think of biodiversity as an economic good?
(ii) How can one quantitatively measure biodiversity?
Biodiversity as an Economic Good
125
(iii) In what sense does biodiversity have economic value?
In this and the following chapter, I shall address these questions with the aim of
clarifying the conceptual foundations upon which an ecological-economic analysis of biodiversity loss and conservation is possible. In this chapter, questions (i)
and (iii) are addressed. It is discussed in what sense one can think of biological
diversity as an economic good, and what constitutes its economic value. This
is not to neglect or belittle the importance of the other value dimensions in any
way. Rather, it will be shown that considering the economic value of biological diversity can yield important insights for an encompassing understanding
of biodiversity loss and conservation. Question (ii), of how to quantitatively
measure biodiversity, will then be addressed in Chapter 8. In that chapter, I
will also discuss the intricate relationship between the measurement and the
valuation of biodiversity.
The argument in this chapter proceeds as follows. In Section 7.2, it is
discussed whether, and to what extent, one can consider biodiversity as an
economic good. On this basis, one can then specify what its economic value is
(Section 7.3). Based on the economic value of biodiversity, one can explain the
current loss of biodiversity from an economic perspective, and identify its fundamental causes (Section 7.4). The economic value of biodiversity also offers
a conceptual framework for discussing the question ‘What species and populations should be protected, and to what extent?’ on scientific grounds (Section 7.5). This allows one to prioritize different biodiversity protection goals.
In conclusion, viewing biodiversity as an economic good which has economic
value, makes apparent the potential and limits of economics as an academic
discipline for the analysis of biodiversity loss and conservation (Section 7.6).
7.2
Biological Diversity as an Economic Good
According to a classic definition, ‘economics is the science which studies human
behaviour as a relationship between ends and scarce means which have alternative uses’ (Robbins 1932: 15). In this sense, biological diversity can be thought
of as an economic good (Heal 2000). It satisfies human needs and allows people
to achieve certain ends in a variety of ways. On the other hand, biodiversity is
scarce and can be used in alternative ways. Both aspects will be explained in
detail in the following.
7.2.1
Satisfaction of Human Needs
Biological diversity, and its components, can satisfy human needs and allow
people to achieve certain ends in a variety of ways. The following examples
may illustrate this claim.
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Natural Science Constraints in Environmental and Resource Economics
Food
A large part of our current food supply comes from domesticated plant and
animal species, which have originally been derived from wild species. Of the
240,000 known (vascular) plant species an estimated 25% are edible (Watson et
al. 1995b: 13), that is, 60,000. In the course of human history only 3,000 species
of these have ever served as food, only 150 species have ever been cultivated
on a larger scale, and today less than 20 satisfy more than 90% of total human
food demand (Myers 1989: 54). The largest share is made up by only four
species – wheat, corn, rice and potato – which cover more than 50% of the
whole demand for vegetable food (Plotkin 1988: 107).
Besides specializing on fewer and fewer species, the genetic diversity of
edible plants and animals is being diminished also within individual species, by
using only a few high yield varieties per species. These are being selected by
breeding with respect to certain preferred properties, in particular large and
homogenous amounts of raw product. As a result, in many countries in which
traditionally a large diversity of different varieties have been grown, today only
very few are still being cultivated. For example, the number of rice varieties
grown in Sri Lanka has decreased from 2,000 in 1959 to currently only five
(Swanson 1994: 26f.).
On the one hand, this process of specialization leads to significantly higher
average yields per hectare. But on the other hand, it is accompanied by an
increased susceptibility to diseases, pests or extreme weather conditions. In
order to avoid negative consequences due to these susceptibilities, and in order
to further increase yields for the food demand of a growing world population,
modern agriculture necessarily depends on crossbreeding with genetic material
from wild varieties, which is available in natural ecosystems. These species
develop under largely natural condition and, hence, can permanently develop
new defensive mechanisms against pests and diseases (Ehrlich and Ehrlich 1981:
65). At the same time, they provide the genetic raw material for other desired
properties. For example, the properties of so-called halophytes – plants which
are tolerant against salt – may be transferred to conventional species, which
would mean an enormous gain in the potential area of cultivable land as well
as the potential of irrigating with saltwater (Myers 1983: 54). Wild species in
natural ecosystems therefore provide a reservoir of genetic diversity which is
important for securing long-term food supplies (Heal et al. 2004).
Pharmaceuticals
Biological diversity makes an important contribution to the supply of humankind with pharmaceutical substances. Its particular utility in this respect
stems from the fact that the different organisms in their biotic environment
have developed a number of survival strategies, by developing biologically ac-
Biodiversity as an Economic Good
127
tive chemical substances which have proved successful in the course of evolution.
These chemical substances may also be useful for humans in many instances
because humans have to survive in the same natural system and in interaction with the same other life forms (Swanson 1996: 3). Already today, humans
depend to a large extent on wild organisms in their supply with pharmaceutical substances. Myers (1997: 263) estimates that one quarter of all registered
pharmaceutical substances stem from plants, another quarter stem from animals and microorganisms.
One can distinguish between three different approaches of how plant or
animal species are being used by the pharmaceutical industry (Swanson et al.
1992: 434). First, parts which have been isolated from plants or animals may be
directly used as a therapeutic substance. For example, one can isolate different
substances from snake poison, which inhibit or enhance the coagulation of blood
and may be used for the regulation and diagnostics of various blood diseases
(Hall 1992: 380). Second, parts of plants or animals may be used as a raw
material in the process of synthesizing pharmaceutical substances. Third, parts
of plants or animals can serve as an exemplar for designing and synthesizing
pharmaceutical substances in the lab. The most well-known example is aspirin,
which was originally produced from the leaves of the willow tree, but can be
produced today at much lower costs synthetically.
In 1993, roughly 80% of the 150 most prescribed drugs in the USA were
synthetic drugs which were designed after the exemplar of natural substances,
half-synthetic substances from natural substances or, in a few cases, natural
substances (Watson et al. 1995b: 14), and the worldwide sales of pharmaceuticals on the basis of plant substances was worth 59 billion US dollars (ten
Kate 1995).1 These successful pharmaceutical substances have been identified
although only 5,000 of the estimated 240,000 vascular plants have been researched systematically and thoroughly for their potential as a pharmaceutical
substance (Oldfield 1992: 350). Obviously, biological diversity offers considerable potential for the development of new pharmaceutical substances. This
potential is currently subject to great commercial as well as economic interest
and is being targeted by so-called ‘bio-prospecting’ (Mateo et al. 2000, Polasky
and Solow 1995, Polasky et al. 1993, Rausser and Small 2000, Simpson et al.
1996).
1
The top three pharmaceutical substances in terms of sales, from plants, animals and
microorganisms in 1997, were (WMPQ 1999): (1) Zocor (sales: 3.6 billion US dollars), a
cholesterol-synthesis-inhibitor from Merck & Co., which is produced after the exemplar of
the natural agent lovastatin from the fungus Aspergillus terrestris; (2) Vasotec (sales: 2.5
billion US dollars), an ACE-inhibitor, also from Merck & Co., which was developed from a
peptide in the poison of the fer-de-lance (Bothrops jararaca or athrox); (3) Augmentin (sales:
1.5 billion US dollars), from Smith-Kline-Beecham, with the agent co-amoxiclav, which is a
combination of a beta-lactamase-inhibitor from the bacterium streptomyces lavuligerus and
the half-synthetic antibiotic amoxicillin (penicillium spp. or aspergillus spp.).
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Natural Science Constraints in Environmental and Resource Economics
Industrial resources
Biodiversity makes an important contribution to human welfare in its function
as a supplier of industrial resources, which becomes more and more important
as non-renewable resources (for example fossil fuels and mineral ores) become
scarcer and scarcer. Different kinds of wood, rattan, rubber, fat, oil, wax,
resin, vegetable dye, fibre, and many other resources are extracted from living
organisms and are being used in many instances (Myers 1983: 146ff.). Biological
diversity constitutes a stock of additional promising substances which may be
used as industrial resources in the future. In particular, the chemical industry is
increasingly interested in substances from living organisms. According to some
estimates, this industry obtains more than 10% of its resources from agriculture
and forestry already (Mann 1998: 60). The most important resource is still
crude oil; but in light of the finite supply of fossil fuels, the substitution of this
resource by plant resources is expected to become more and more important
for the chemical industry (Myers 1983: 147).
Bioindicators for science
Biological diversity plays an important role as a source of new scientific insights and as a research model for science. For example, many species can
inform medicinal research about the origin and nature of different human diseases (Myers 1983: 120). Research about hemophilia (bleeding disorder), for
example, has been informed by the study of manatee (dugong), which have
blood with bad coagulation properties. The armadillo (family dasypodidae)
and the mangabey are the only species – besides humans – which can contract
leprosy (Hansen’s disease) and, hence, can yield important insights for research
on this disease.
A special discipline – bionics – exists, which is concerned with systematically transferring problem solutions which have been developed and optimized
over millions of years in nature into the technical domain (Hill 1997, Nader
and Hill 1999). Engineers, for example, have gained insights for aircraft construction from studying the biophysical properties of insects and birds. Here,
biological diversity serves as a role model for technical solutions.
A further use of biological diversity is bio-indication, which is the detection and quantification of anthropogenic environmental change by measuring
changes in organisms and ecosystems (Arndt et al. 1987: 16). Bio-indication
allows one to detect the existence of pollutants in different environmental media (for example air, soil, water), which is possible with technical devices only
at much higher complexity and costs (Hampicke 1991: 30). For example, the
heavy metal content of the atmosphere can be estimated based on the enrichment of heavy metal in mosses (Arndt et al. 1987: 57ff.), and algae can be used
as indicators for the loading of aquatic ecosystems with organic substances and
Biodiversity as an Economic Good
129
heavy metals (Arndt et al. 1987: 277ff.).
Aesthetical satisfaction and recreation
Biodiversity also satisfy human needs under aesthetic criteria. The beauty of
many birds, butterflies, tropical fishes or flowering plants is beyond question
and is certainly capable of satisfying the human need for aesthetic stimulation
and contemplation. This is illustrated by a variety of different leisure activities,
such as nature photography, bird watching, collection of butterflies, or diving
(Ehrlich and Ehrlich 1992: 220). Even little and inconspicuous species are
capable of fascinating the observer by particular properties, their complexity or
unusual behavior. In this respect, it is exactly the diversity and the differences
between species which matter (Ehrlich and Ehrlich 1981: 42).
As an indicator for the actual appreciation of biological diversity in its
aesthetic and recreational function, one can take the increasing expenditures
for eco-tourism. In 1988 approximately 235 million people worldwide took part
in activities of eco-tourism, creating sales of an estimated 233 billion US dollars
(Watson et al. 1995b: 16).
Ecosystem services
Ecosystems generate a number of functions and processes which ultimately
satisfy human needs of consumption and production. The whole range of these
so-called ‘ecosystem services’ (Daily 1997a: 3) can be classified into three main
categories.
First, ecosystem services support human productive activities. For example, different species contribute decisively to the formation of soils, the conservation of the soil’s fertility, and the protection against soil erosion. Thus, they
fulfill important functions for agriculture and forestry. Furthermore, different
species of microorganisms transform the nutrients in the soil (e.g. nitrogen,
sulfur, phosphor etc.) into a form in which they can be processed by higher
plants. These plants then carry out ‘primary production’, that is, by photosynthesis they transform the energy inflow from the sun into energy stored in
chemical compounds, which can then be used as an energy source by animals.
Agriculture also benefits from the control of the vast majority of agricultural
pests by their natural enemies (Naylor and Ehrlich 1997: 151ff.), as well as of
the pollination of agriculturally cultivated and wild flowering plants (Nabhan
and Buchmann 1997: 133ff.).
Second, ecosystems serve as a sink for different wastes of human consumption and production. These are taken up, transformed and, thus, made partially
innoxious or even reusable (Munasinghe 1992: 228). For example, the destruents in the soil decompose organic wastes into simpler inorganic components
which can then serve again as nutrients for plants. Also, the bacteria in aquatic
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Natural Science Constraints in Environmental and Resource Economics
ecosystems are important destruents whose capability of decomposing wastes is
being used today in sewage plants (Ehrlich and Ehrlich 1992: 222). Finally, the
living parts in ecosystems also contribute to the decomposition of pesticides
and air pollutants (McNeely et al. 1990: 32).
Third, ecosystems fulfill essential and irreplaceable life-support functions
without which life on Earth could not exist in its present form (Munasinghe
1992: 228). Among these life supporting ecosystem services are the control of
the gaseous composition of the atmosphere (oxygen,nitrogen and carbon dioxide content;2 existence of the ozone layer which protects from UV radiation),
the transformation (‘primary production’) of solar energy in biomass, in which
form it can be used in the food chain by living beings who do not photosynthesize, regulation of water runoff in watersheds and general water circulation,
regulation of local, regional and global climate, and regulation of nutrient cycles (carbon, oxygen, nitrogen, sulfur, phosphor, etc.) (Ehrlich und Ehrlich
1981: 86, Ehrlich and Ehrlich 1992: 221f., McNeely et al. 1990: 32).
The role of biodiversity for the capability of the ecosystems to generate
all these services and to maintain their functioning, even under environmental
changes, is still subject to scientific research. On the one hand, there are species
the importance of which for the functioning of ecosystems exceeds by far their
relative abundance in the ecosystem, for example the mykorrhiza fungi for the
uptake of nutrients from the soil by plants (van der Heijden et al. 1998).3 The
loss of these so-called ‘keystone species’ (Bond 1993: 237ff.) would necessarily
entail the loss of further species and strongly reduce the functional integrity of
an ecosystem. In contrast, other species are highly redundant in the functions
which they fulfill within the community (Lawton and Brown 1993). In the
literature, these species are often called ‘passenger species’ (Holling et al. 1995:
67). The loss of one of these species can be compensated by another one
(Watson et al. 1995a: 289). According to what we know today, at least in the
short run, a small number of keystone species and physical processes suffice to
guarantee the full functioning of ecosystems (Holling et al. 1995: 67). However,
in the course of time, with changing environmental conditions, species which
are currently passenger species can evolve into keystone species and overtake
2
The oxygen content of the atmosphere has been constant at around 21% for the last
approximately 350 million years due to the existence of green plants (Heintz and Reinhard
1993: 11–16). This is not only important as an essential component of the ‘air to breathe’. A
decrease in this fraction to 15% would imply that even dry wood could not burn any more,
while an increase of this fraction to 25% would imply that even wet tropical forests could
catch fire. This would have far-reaching implications for the development of ecosystems.
3
Mykorrhiza (‘fungus root’) denotes the symbiosis between plants and soil fungi (Strasburger 1991: 229). The fungus fibre penetrates the plant’s roots, such that an exchange of
matter is possible. The plant thus uses the enormous absorptive capability of the fungi in
order to obtain water and nutrients. Conversely, the fungus obtains sugar and carbohydrates
from the plant, which the plant usually has in excess.
Biodiversity as an Economic Good
131
important functions within ecosystems (Barbier et al. 1994: 28). The functional
diversity of species thus contributes to the resilience of ecosystems, that is,
their ability to maintain ecosystem functions under changing environmental
conditions (McCann 2000, Lehman and Tilman 2000).
Summary: Satisfaction of human needs
The examples listed here demonstrate the vast potential for utilizing biodiversity. The large differences between the examples also suggest that there is no
universal criterion according to which one could make an à priori assessment
of which components of biological diversity are of utility for humans and which
are not (Hampicke 1991: 28). While in the past the economic relevance of
the direct consequences of a loss of biological diversity for human consumption
and production have been stressed; the focus, even in the economic research
on biodiversity, is increasingly on the role that the loss of biodiversity has for
the functioning and resilience of ecosystems (Barbier et al. 1994: 17, Perrings
1995c, Perrings et al. 1995b).
7.2.2
Scarcity
What makes biodiversity an economic good is, besides its economic utility, its
scarcity (Lerch 1995: 33). Scarcity means that the provision or conservation of
biodiversity is costly.4 These costs can be monetary expenditures; for example,
for the set up of a nature protection area. The financial resources, which may
be spent on biodiversity conservation or on alternative projects, are scarce.
The by far most important part of the costs of biodiversity conservation are
the opportunity costs which result from the fact that, in order to conserve biodiversity, one cannot use land in alternative forms, for example for agriculture,
developing rivers as water highways, etc.
7.3
The Economic Value of Biological Diversity
From what has been said in Section 7.2 above – namely (i) biodiversity satisfies human needs and (ii) biodiversity is scarce – it follows that biodiversity
can be thought of as an economic good. Hence, one can attribute economic
value to it. Conceptually and practically, the concept of economic value serves
to capture the scarcity of biodiversity with respect to its potential to satisfy
human needs. Valuation facilitates the aggregation of information in complex
4
The economic definition of scarcity, based on (opportunity) costs, is one of relative
scarcity. One may argue that biodiversity – like other natural resources – is not only scarce
in a relative sense, but also in an absolute sense (Baumgärtner et al., in press). This goes
beyond an economic analysis.
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Natural Science Constraints in Environmental and Resource Economics
situations and, thus, is an important prerequisite for making rational decisions
about the efficient allocation of resources (see Section 7.5 below).
Before I shall discuss the economic value of biodiversity in detail in Section 7.3.2 below, I shall first discuss the economic notion of value in general.
This should help to understand the potential, but also the limits, of economic
valuation of a natural resource, such as biodiversity.5
7.3.1
The Notion of Economic Value
When economists speak of a (material or immaterial) good’s ‘value’, in most
cases they mean an instrumental value. That is, the value of this good consists
in it being a useful instrument in order to reach a certain goal. In contrast,
one could ascribe an intrinsic value to a good.6 That is, something could be
valuable in itself, which is independent of it being an instrument in order to
reach a certain goal.7
From the definition and the limitation of economics as the science which
studies human behavior with regard to the satisfaction of human needs from
scarce resources (Section 7.2), it is apparent that the satisfaction of human
needs is the goal for which something should be instrumental, for it to have
economic value. Thus, economic value is – by definition of economics – anthropocentric.
The methodological procedure, which economics follows in order to explain
value, is that of so-called methodological individualism. In this approach, single
individuals and the decisions and actions which result from their individual
preferences and constraints, are taken as the elementary building blocks of explanation. In this perspective, the value of a good is ultimately determined
by the interaction of the subjective valuations and actions of the many individuals in the economy. This means that economic value is determined by the
subjective valuations of individuals in a society – and not, say, by the scientific
5
The ethical and theoretical principles of economic valuation of environmental goods,
services and damages are thoroughly discussed, for example, by Freeman (2003), Hanley and
Spash (1993), Johansson (1987, 1999), Marggraf and Streb (1997).
6
For example, Pirscher (1997) argues that biodiversity has an intrinsic value.
7
The distinction between instrumental and intrinsic value correponds to a distinction already made by Immanuel Kant in his Groundwork of the Metaphysics of Morals (Grundlegung
zur Metaphysik der Sitten, 1996[1785]: 84) between ‘price’ and ‘dignity’: ‘In the kingdom of
ends everything has either a price or a dignity (“Würde”). What has a price can be replaced
by something else as its equivalent; what on the other hand is raised above all price and therefore admits of no equivalence has a dignity. What is related to general human inclinations
and needs has a market price; that which, even without presupposing a need, conforms with a
certain taste, that is, with a delight (“Wohlgefallen”) in the mere purposeless (“zwecklosen”)
play of our mental powers, has a fancy price (“Affectionspreis”); but that which constitutes
the condition under which alone something can be an end in itself has not merely a relative
worth, that is, a price, but an inner worth, that is, dignity.’
Biodiversity as an Economic Good
133
judgment of experts.8
From this perspective it becomes obvious that economic value is not an
inherent property of a commodity. Rather, it is attributed to a commodity by
economic agents. What particular economic value is attributed to a commodity,
hence, does not only depend on the objective (e.g. physical or ecological) properties of this thing, but also essentially depends on the whole socio-economic
context in which valuation takes place. For example, when valuing natural resources such as clean drinking water, besides questions such as ‘What is the
utility of clean drinking water?’ it is also important to consider questions such
as: How much clean drinking water is there altogether? How is this amount
distributed spatially and temporally? What are the institutions governing the
access to the resource? What are the alternative uses besides use as drinking
water, and what are the respective institutional constraints? Are there any alternatives to water in its different uses, and what are the respective conditions
of provision?
The currently accepted paradigm of economics is neoclassical value theory.9
According to this theory, value is a marginal concept: what is being valued
are small changes in the state of the world (and not a certain state of the
world), starting from the current state of the world. Thus, economic value
is crucially determined by the current state of the world. This includes the
current level of consumption of all different goods, current preferences, the
current distribution of income and wealth, the current state of the natural
environment, current production technology, and current expectations about
the future – irrespective of whether we would think of this state of the world
as being good or bad. Hence, the value of some good – in the neoclassical view
– is the value of one additional unit of that good, given the amounts that we
already consume of this good and all others. As a result, the value of some
good is not constant, but changes with the amount already consumed of that
good.
All these characteristic properties of the economic value concept, i.e. that
economic value is
• instrumental,
• anthropocentric,
• individual-based and subjective,
• context dependent, and
8
One problem which results from this individualistic approach, in particular when evaluating natural resources and environmental quality, is the aggregation of different subjective
valuations to a social valuation (Seidl and Gowdy 1999: 106).
9
The neoclassical value theory emerged in the so-called ‘marginal revolution’ around 1870,
replacing the classical theory of value (see e.g. Blaug 1996 or Niehans 1990).
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Natural Science Constraints in Environmental and Resource Economics
• marginal and state-dependent,
also apply to the economic value of biodiversity (Goulder and Kennedy 1997,
Hampicke 1993, Nunes and van den Bergh 2001, Nunes et al. 2003, Seidl
and Gowdy 1999, Weimann and Hoffmann 2003). While this focus allows
economists to make clear and strong statements about the allocation of natural resources, its narrowness is a potential problem when linking with other
academic disciplines in an encompassing discussion of biodiversity loss and
conservation.
7.3.2
The Concept of Total Economic Value
In Section 7.2, I have listed a number of examples of how biodiversity satisfies
human needs. Economists have tried to completely classify the different uses of
the resource by the concept of total economic value (Pearce 1993, Pearce and
Turner 1990: 129, Turner 1999b). This concept can be applied to the valuation
of biodiversity (Watson et al. 1995a: 830ff., Geisendorf et al. 1998: 176ff., McNeely 1988: 14ff.). The total economic value of biodiversity, as an encompassing
concept of the different human uses and motives for appreciation of biodiversity, may be classified into use and non-use values. Use values comprise all
those value aspects which stem from actual or potential use of biodiversity. In
contrast, non-use values are completely independent of any actual or potential
use by the valuing individual (Krutilla 1967, Weisbrod 1964). They stem, for
example, from the ethical, spiritual or religious desire to conserve biodiversity
for the future, or for its own sake. On a more detailed level, use values may
be classified into direct use value, indirect use value and option value. Non-use
values may be classified into vicarious use value, bequest value and existence
value.
Direct use value
Biodiversity has a direct use value insofar as different species and organisms, or
parts thereof, directly satisfy human needs. On the one hand, this includes consumptive use, e.g. as food, fuel wood or medicinal plants, as well as productive
use, e.g. as industrial resources, fuel or construction material (see the discussion
in Section 7.2 above). On the other hand, this also includes non-destructive use,
e.g. recreation, tourism, science and education (see the discussion in Section 7.2
above).
Indirect use value
Biodiversity has an indirect use value for humans insofar as biodiversity plays an
important role in maintaining certain ecosystem services (Fromm 2000, Hueting
et al. 1998) which, in turn, directly satisfy human needs or support economic
Biodiversity as an Economic Good
135
processes that ultimately lead to the satisfaction of human needs. Examples
(discussed in Section 7.2 above) include the support of biological productivity
in agro-ecosystems, climate regulation, maintenance of soil fertility, control of
water runoff, and cleansing of water and air.
Option value
Even if humans did not actually use biodiversity today, there is a value in the
option of doing so tomorrow. This constitutes the option value of biodiversity.
For example, the future may bring human diseases or agricultural pests which
are still unknown today. Today’s biodiversity would then have an option value
insofar as the variety of existing plants may already contain a cure against the
yet unknown disease, or a biological control of the yet unknown pest (Heal
et al. 2004, Polasky and Solow 1995, Polasky et al. 1993, Rausser and Small
2000, Simpson et al. 1996, Swanson and Goeschl 2003). In this sense, the
option value of biodiversity conservation corresponds to an insurance premium
(Perrings 1995a, Weitzman 2000), which one is willing to pay today in order
to reduce the potential loss should an adverse event – such as a human disease
or an agricultural pest – occur in the future.10
While, strictly speaking, the conservation of anything has an option value,
it is of particular importance in the case of biodiversity for two reasons. First,
the loss of biodiversity is irreversible. Second, there is still large uncertainty
about the different potential uses of biodiversity, e.g. as a storage of effective
pharmaceutical substances or of desired genetic properties for crop varieties.
Economists have stressed that under uncertainty it may be advantageous to
postpone an irreversible decision, while gathering more information and learning (Arrow and Fisher 1974, Henry 1974).11 In the case of biodiversity, such
an option value clearly exists and may be considerable (Fisher and Haneman
1986, Weikard 2003).
Vicarious use value
The vicarious use value of biodiversity (Watson et al. 1995b: 13) is given by
people’s willingness to pay (or to forgo benefits) to ensure that other members
of the present generation can enjoy the use value of biodiversity or specific
components thereof. This is a form of altruism towards friends, relatives or
strangers.
10
The insurance value of biodiversity is subject to a detailed discussion in Chapters 9 and
10.
11
This part of option value is often called ‘quasi-option’ value. It indicates the value of the
additional information gained from postponing an irreversible decision and learning under
uncertainty (Hanemann 1989).
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Bequest value
The bequest value of biodiversity is given by people’s willingness to pay (or
to forgo benefits) to ensure that future generations can enjoy the use value of
biodiversity or specific components thereof (Pommerehne 1987: 175ff.). This is
a form of altruism towards future generations.
Existence value
The so-called existence value (Krutilla 1967: 781) of biodiversity is given by
people’s willingness to pay (or to forgo benefits) to ensure the continued existence of biodiversity or specific components thereof, irrespective of any actual
or potential use by present or future generations of humans. This expresses
an appreciation of biodiversity which is completely independent of any actual
or potential, present or future use. It stems from a person’s satisfaction from
merely knowing that a particular species or ecosystem exists at all. It may
be seen as a form of altruism towards non-human species or nature in general,
and, in most cases, rests on ethical or religious motives. An indicator of the
high importance of existence values may be the donations collected by nature
conservation organizations for, say, the protection of the Siberian Tiger or the
Panda Bear (Pearce and Turner 1990: 135).12
7.3.3
Methods for Identifying the Total Economic Value
At this point, one may summarize: The total economic value of biological diversity comprises very different components, corresponding to the very different
human needs which are being satisfied by this natural resource. The different
components of total economic value are, in principle, additive; but one needs
to take care to not add mutually exclusive values (Moran and Pearce 1997: 2).
For example, it would be a mistake to add the revenue from selling timber after
clear-cutting a forest with the revenues of other (e.g. recreational) uses of the
forest, because the latter are being destroyed by clear-cutting.
How to determine the total economic value? For goods which are being
traded on markets one can (under certain conditions) take the market price as
expressing the total economic value. For biological diversity, however, like with
most natural resources, there is the problem that the resource is not, or only
partially, being traded on markets. In order to determine its total economic
12
Because the existence value is completely independent of any actual or potential human
use, it does not seem to be an instrumental and, thus, economic value component. Indeed,
the existence value is sometimes classified as an intrinsic value (e.g. by Watson et al. 1995b:
13, Pearce and Turner 1990: 130). However, while the existence value is independent of any
actual or potential human use, it is not independent at all of the valuing economic agent
(Pirscher 1997: 74). Knowing about the existence of a certain species is of utility for that
economic agent. Thus, the existence value is an instrumental value, in so far as the existence
of a certain species is instrumental for the utility for this economic agent.
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137
value, or individual components thereof, one therefore needs to employ (direct
or indirect) methods for non market valuation. These methods can, in principle,
also be used to determine the total economic value of biodiversity (Watson et
al. 1995a: 844–858).13 Examples include the replacement cost method, the
averting expenditure/avoiding costs method, the production function method,
the hedonic pricing method, the travel cost method, or the contingent valuation
method.14
7.4
Economic Causes of Biodiversity Loss
Extinction of species is not a new phenomenon. At all times, ever since life
began on Earth, some species have become extinct and, at the same time, other
species have originated. What is new today, however, is the high rate of species
extinction, which is currently far above the long-term average rate known from
fossil records. According to conservative estimates, the global rate of species
extinction – averaged over all groups of species and ecosystems – currently
varies between 50 and 100 times the natural rate (Watson et al. 1995b: 2).
In tropical rain forests the extinction rate is considerably higher. It currently
exceeds the natural rate of the past by a factor of somewhere between 1,000
to 10,000 (Watson et al. 1995b: 2). The large range of estimates indicates
the considerable uncertainty about the exact number of extant species. The
currently observed loss of biological diversity at all levels – genes, populations,
species, ecosystems – is so dramatic that it may be considered the ‘sixth mass
extinction’ (Wilson 1992: 32, Watson et al. 1995b: 22) in Earth’s history.
The specific mechanisms through which the loss of populations, the extinction of species, and the impairment of ecological communities proceeds, are the
following, listed in the order of their global importance (Watson et al. 1995b:
20):
1. loss, fragmentation, and degradation of habitats,
2. overuse of populations,
3. introduction of non-native species,
4. pollution of soil, water and air,
13
Nunes and van den Bergh (2001) as well as Pearce and Moran (1994: 48) stress the
considerable difficulties which occur when using these methods for the valuation of biological
diversity.
14
A detailed description of these methods would go beyond the scope of this chapter. For
an introduction into these methods see, for example, Bateman et al. (2002), Freeman (2003),
Hanley and Spash (1993), Pommerehne (1987) or Smith (1996).
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5. climate change.
While in continental ecosystems the loss, fragmentation, and degradation of
habitats is the most important mechanism, in oceanic ecosystems the overuse
by fisheries and pollution are the most important factors. Coral reefs, which are
a hot spot of biological diversity, are particularly affected by climate change.
On islands, the introduction of non-native species and habitat loss are equally
important.
The five specific mechanisms listed above are proximate causes of biodiversity loss. The underlying primary causes can be analyzed based on the concept
of total economic value, which has been introduced in Section 7.3.2 above. Such
an economic analysis is based on the identification of incentive structures which
govern individual and social behavior in concrete situations. From such an economic perspective, the following four primary causes of biodiversity loss can be
identified (Watson et al. 1995a: 830–832, Moran and Pearce 1997: 83–89):
1. population growth,
2. market failure,
3. governance failure,
4. fundamental ignorance.
These are discussed in detail in the following.
7.4.1
Population Growth
One cause of biodiversity loss, which seems most obvious, is the continuous population growth – though with decreasing rates – in developing countries and
the continuous growth of the economy in the industrialized countries (Ehrlich
and Holdren 1971, Holdren and Ehrlich 1974, Smith et al. 1995). Both developments imply an increasing demand for biological resources, and an ever
increasing pressure on development of land as industrial space, for infrastructure (housing, highways, airports, etc.), or as agricultural land.
It seems inevitable that population growth and economic growth lead to
a loss of biological diversity. The reason is the fundamental competition in
land use: land can either be left in a natural state, thus serving as habitat
for populations of wild species, or it can be developed for economic use, which
means a loss of habitat for the originally living populations and, thus, their
extinction. Since the land area on this planet is limited, population growth
and economic growth necessarily mean – everything else being constant – that
the economic use of land is being attributed a higher value due to increased
demand, while its value as a natural habitat remains constant. As a result,
Biodiversity as an Economic Good
139
more and more land is developed for economic use, which implies a continuous
loss of biodiversity.
But this development is not as necessary as it may, at first, appear (Swanson 1995b). For, in the trade-off between the two fundamental alternatives –
conservation of biological diversity versus economic development – there are
many distortions, with the result that this trade-off is systematically biased in
favor of economic development and against conservation of natural habitats.
One may argue that the current loss of biodiversity is not primarily caused by
population growth or economic growth, but rather by such distortions, which
are now discussed in detail.
7.4.2
Market Failure
One standard result of economic theory is that the equilibrium on a competitive
market, under certain conditions, is socially optimal in the sense that it is not
possible to improve one individual’s well-being without worsening some other
individual’s. One of the conditions under which this result holds is the absence
of externalities. This means, all consequences of the transaction are mutually
agreed upon by market participants and are being reflected in the market price.
In contrast, if there are externalities, that is, if the market price does not
reflect all consequences of a transaction, then markets may fail: the market
price of a good does not reflect the total economic value of the good and the
market equilibrium is not socially optimal. Externalities are ubiquitous in the
allocation of biological resources.
Externalities due to incompletely specified property rights and
missing markets
An externality arises if property rights for biological resources are only incompletely defined, or not defined at all (Lerch 1996, 1998). In the case of
completely missing property rights or utilization rules, for populations of fish
beyond the national coastal zones, for example, there is open access to a resource. It should be apparent that a resource which is useful and scarce but
can be accessed without limit will, as a general rule, be overused. The case
is similar for a resource which is being utilized by a number of individuals as
a group, for example a village community, without any mandatory utilization
rules. This has been described by Hardin (1968) as ‘the tragedy of the commons’. Individual users of such a resource have an incentive to overuse the
resource, because the benefits completely accrue to the individual while the
problems stemming from overuse have to be carried by the whole group of
users, and thus, only to a certain fraction by the individual user.
Market failure stemming from missing or incompletely specified property
rights may be cured, in principle, by defining and enforcing property rights
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(Swanson 1994). This is the logic behind the principle, expressed in the UN
Convention on Biological Diversity (CBD 1992), that biological resources are
the ‘property’ of the country in which these resources are located, and a similar
proposal by the World Trade Organization (WTO) to introduce and enforce
property rights in the form of patents (‘intellectual property rights’) on animal
and plant genes (Sedjo and Simpson 1995, Swanson and Goeschl 2000b). The
hope is that the loss of genetic diversity may be halted by attributing an adequate value to hitherto freely accessible, and thus undervalued resources, and
to make the new resource owners manage these resources appropriately in their
own interest. An example which illustrates the workings of this mechanism,
is the agreement signed in 1991 between Costa Rica’s National Institute for
Biodiversity and the U.S. pharmaceutical company Merck Inc. (see Sedjo and
Simpson 1995: 84ff., Lerch 1998: 292f.),15 which has stimulated a host of similar
agreements.
Character as public good
While biological resources have partly the character of a normal private economic good (as food or industrial resource, for example), in many other important respects they have the character of a public good. This means that
(i) the use of the resource by one individual does not restrict or diminish the
possibilities of use by another individual (non-rivalry), and (ii) no individual
may be excluded from utilizing the resource (non-excludability). These two
properties hold, in particular, for the important function of biodiversity in the
provision of life supporting ecosystem services for humans, such as regulation of
atmospheric composition or control of water circulation. The fact that one individual uses the constant oxygen fraction of the atmosphere for breathing does
not restrict the possibility of other individuals to do the same. Furthermore, it
is not possible to exclude individuals from using the atmosphere’s oxygen for
breathing.
The allocation of a public good on a competitive market is generally suboptimal (Varian 1992: Chapter 23), that is, there is market failure. The reason
is that because of non-rivalry in consumption every single individual has an
15
The Instituto Nacional de Biodidiversidad (INBio) of Costa Rica – a private nonprofit
organization, which was founded following a recommendation of the government of Costa
Rica – aims at conserving Costa Rica’s biological wealth by promoting its intellectual and
economic use. The agreement signed in 1991 and prolonged in 1994 and 1996 between INBio
and the U.S. pharmaceutical company Merck Inc. states that Merck, at the beginning of
each two-year term, makes a one time payment of 1 million U.S. dollars to INBio to support
the Institute’s work and the conservation of the natural rain forest, and obtains a certain
number of plant specimens from the forest in exchange. In addition, Merck pays as a royalty
to INBio a certain percentage of its sales from products which have been developed from
these genetic resources. The contracting parties have stipulated non-disclosure of the level
of these royalties. It is estimated that they are between 1% and 5%.
Biodiversity as an Economic Good
141
incentive to ‘free ride’, that is, to not reveal his or her true preferences for
the consumption of the public good but to consume the amount of the good
provided by other individuals without contributing to financing its provision.
As a result, there is an under-provision with the public good on a competitive
market compared to the social optimum. This particular form of market failure
cannot be cured by defining private property rights in the public good because,
due to the special character of these goods, in particular the property of nonexcludability, the definition of property rights is, as a matter of principle, not
possible.
Intragenerational spatial externalities: Global values vs. local
markets
Many of the value components of biodiversity identified in Section 7.3.2 are
global, for instance the vicarious use value, the bequest value, the existence
value, but also the indirect use value which stems, for example, from the complex ecosystem of the Amazonian rain forest regulating weather and climate
patterns on a global scale. This means, a large part of these values is appreciated by humans who do not live in the place where the resource is and, hence,
do not take part in local decisions, as about land use in the Amazonian rain forest and its potential transformation into agricultural land. Put the other way
round, the total economic value of biodiversity is not fully taken into account
in local decisions about land use. In a local decision, the value of agricultural
land use is compared with the value of land use as primary rain forest such
as it is perceived by the local population. The latter is certainly much lower
than the globally aggregated total economic value of primary rain forest. The
externality consists in the fact that in a potential decision to clear-cut primary
rain forest, the valuations of many of those who are affected by the transaction, namely the non-local users of the global public good, are not taken into
account. Compared with the global total economic value, the value of biological diversity considered in the local decision is too low. As a result, too large
a share of primary rain forest is transformed into agricultural land.
Intergenerational externalities: Present vs. future costs and benefits
A similar argument applies to the discrepancy of present and future costs and
benefits of biodiversity. Today’s markets only take into account today’s decision
makers’ (expectations of) costs and benefits of transactions. Hence they neglect
the part of total economic value which is due to future users who cannot take
part in current decision processes.
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Summary: Market failure
In the case of biodiversity a great number of different externalities act in the
same direction. Today’s market prices for essential components of biological diversity are considerably below their socially optimal value, which is given by the
(globally and intertemporally aggregated) total economic value and comprises,
besides the direct use value, also indirect use values and non-use values. In
some cases, the (implicit) market prices are even zero. As a result, the private
(opportunity) costs of, say, transforming primary rain forest into agricultural
land, are far below the true (opportunity) costs which accrue to society at large.
In turn, the private profits which can be made from transforming primary rain
forest into agricultural land, are far above the profit for society at large. As a
result of many externalities, unregulated markets lead to a rate of biodiversity
loss which is too high compared with a social optimum (Swanson 1994).
7.4.3
Governance Failure
Many of the problems that fall under the heading ‘market failure’ could be
cured, in principle, by adequate regulation of market processes. For example,
one could propose that the global costs of loss of primary rain forest, which
are not considered in local decisions, can be compensated for by a Pigouvian
tax on tropical timber. The tax rate should be such that it covers the social
costs of deforestation which are not currently included in market prices. The
responsibility for such regulation is with the sovereign political bodies at all
levels of organization – from the community level to the level of states and the
international community of states. Failure to regulate in order to correct for
market failure, is a form of governance failure.
Not only is governance failure widespread in current environmental problems, because regulative corrections of market failure are largely absent or not
carried out appropriately, but some countries even reinforce market failure by
policies which make market prices deviate even further from socially optimal
prices. Examples include the subsidies for ‘land cultivation’ paid in Brazil for
the clear-cutting of primary rainforest (Binswanger 1991) and the subsidies for
offshore fisheries by the European Union.
An additional cause of biodiversity loss is the extreme inequality of income
and wealth between the industrialized OECD countries of the North and the
developing countries of the South, or more precisely, the poverty in rural areas
in poor developing countries (Dasgupta 1993, 1995, Munasinghe 1992, Myers
1995, Swanson and Goeschl 2000a). The by far largest part of currently known
biological diversity is found in the poorest countries of this world, namely in
the equatorial regions of South America, Africa and Southeast Asia.16 While
16
Tropical rain forests host an estimated 50%, or even more, of all existing species on only
6% of the land area of the Earth (Myers 1995: 111). Tropical forests are currently destroyed
Biodiversity as an Economic Good
143
biodiversity protection in the OECD countries, such as by the establishment of
nature protection areas, means an only moderate renunciation of (agricultural
or industrial) economic use compared with total economic activity, for the rural
population in the poorest countries of the world there is simply no decision
problem ‘nature protection versus economic use’. A renunciation of agricultural
use, which is the sole source of income and food, would amount to starvation
and, thus, represents no option at all. In so far as one considers it to be a task
of responsible governance to establish international distributional justice, there
is also a form of governance failure here.
7.4.4
Fundamental Ignorance
So far in this section, I have discussed causes of biodiversity loss as if the
total economic value of biodiversity was perfectly known. In fact, however, it
is not exactly known. Even if there should be scientific progress in revealing
different components of this value, fundamental ignorance will remain because
total economic value depends, inter alia, on the potential future use and on the
indirect use of the resource. Particularly, with respect to these two potential
uses of biodiversity a fundamental, ‘irreducible’ ignorance exists (Faber and
Proops 1998: Chapter 7). It is still a largely open question, for example, what
the exact role of biodiversity is for the stability of ecosystems and for the
generation of different ecosystem services of human value (e.g. Holling et al.
1995, Hooper et al. 2005, Kinzig et al. 2002, Loreau et al. 2001, 2002b, Schulze
and Mooney 1993, Tilman 1997a). Due to the high complexity of ecosystems
one can safely assume that this ignorance cannot be reduced, even by intensive
research, so much that more accurate predictions are possible about how human
interference with biodiversity influences the functioning of ecosystems and the
provision of ecosystem services. But this would be needed for a proper valuation
of biodiversity.
This raises doubts about the relevance of the concept of total economic
value as a basis for decision making, if individual components are subject to
fundamental ignorance. Is this concept more than just a taxonomy? Indeed,
acknowledging the fundamental ignorance about the exact quantity of biodiversity’s total economic value could lead to the conclusion that society – rather
than seeking the ‘optimal’ allocation of biodiversity – should follow a policy of
‘safe minimum standards’ (Ciriacy-Wantrup 1965), that is, set limits to habitat destruction so as to avoid the irreversible loss of critical biodiversity and
ecosystem services.
at a higher rate than any other large-scale biome.
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7.5
Relevance of Economic Valuation for the
Protection of Biodiversity
As far as the pace and dimension of current biodiversity loss is concerned, the
Global Biodiversity Assessment – a report from the United Nations Environment Programme – reaches the following conclusion: ‘Because of the world-wide
loss or conversion of habitats that has already taken place, tens of thousands
of species are already committed to extinction. It is not possible to take preventive action to save all of them’ (Watson et al. 1995b: 2). This conclusion
contrasts markedly with naive ideas – which govern the thinking of many environmentalists and, to some extent, have influenced environmental legislation,17
– that nature and species protection should aim at protecting all endangered
species. But if it is not possible to protect all species which are at the brink of
extinction today, the question is: what part of biodiversity should be protected?
This covers two aspects which require trade-offs at different levels:
1. How important is the protection of one endangered species compared to
another one?
2. How important is the protection of biological diversity compared with
other societal goals?
These questions can be answered only unsatisfactorily, because any answer
implies that a certain fraction of the currently existing biological diversity will
become irreversibly extinct. But decisions have to be made, which means that
these questions have to be answered – one way or the other. The economic
value of biodiversity (cf. Section 7.3 above) offers a conceptual framework for
discussing and answering these questions in a rational manner, based on the
full set of available scientific knowledge.
7.5.1
Relative Importance of Different Biodiversity Protection
Goals
So far, economic considerations only play a minor role in nature protection. The
legislation on nature and species protection requires enforcement authorities to
make recourse mainly to ecological and natural science concepts and evidence
when classifying species as endangered – a decision which automatically entails
efforts to protect these species. Economic considerations were not taken into
account until recently.18 As a result, for the listing of species as endangered,
and thus worthy of being protected, it does not matter at all that different
17
For example, this idea is explicated in the U.S. Endangered Species Act of 1973. See
Brown and Shogren (1998) for an economic analysis of this legislation and its implementation.
18
When U.S. Congress enacted the first version of the Endangered Species Act in 1973,
they expressed explicitly that economic criteria are neither relevant for classifying species
Biodiversity as an Economic Good
145
species have different value for humans, and that the effective protection of
different species comes at different costs. A species the protection of which is
very expensive and which nevertheless has only moderate value, is treated on
the same footing as a species with high economic value and relatively low costs
of protection.
While the official rhetoric of nature and species protection starts from the
premise that all endangered species are to be protected and, hence, does not
include any explicit prioritization, time and budget constraints of the relevant enforcement authorities nevertheless require the setting of such priorities.
Often, this is done only implicitly. Metrick and Weitzman (1996, 1998) studied, what criteria have actually determined the decisions of the U.S. Office of
Endangered Species to classify a species as endangered, as well as public expenditures between 1989 and 1993 for the protection of individual species.19 They
found that the most important explanatory variables for listing a species as endangered were the degree of endangeredness, the distinctiveness of the species
from other species (that is, its taxonomic uniqueness) and its size. Accordingly, mammals and fish ranged ahead of amphibians and reptiles. Spiders and
insects, which make up the highest number of endangered species, are almost
not represented on the list at all.
Metrick and Weitzman (1996, 1998) also found that, in contrast to listing
a species as endangered, the expenditure for species protection measures correlates negatively (!) and significantly with the degree of the endangeredness of a
species. There is a positive and significant correlation, however, with the body
size of the animal. Expenditures on protection are also positively correlated
with the status of the species as mammal or bird, and negatively correlated
with the status as amphibian or reptile. Metrick and Weitzman (1998: 32)
explain these results, in particular the surprising negative correlation of expenditures and endangeredness, by pointing to unobservable charismatic factors
which are negatively correlated with the endangeredness, and which were not
taken into account in their study. In this context, they speak of ‘charismatic
megafauna’ – large and popular animals – which is obviously a decisive criterion for the willingness to spend money for the protection of the species. More
than 50% of expenditures have been made for only 10 species (among which are
the grizzly bear and the heraldic animal of the USA, the bald eagle). 95% of
expenditures were in favor of vertebrate species. These numbers suggest that
as endangered, nor for setting up critical habitats (Brown and Shogren 1998: 4). The U.S.
Supreme Court confirmed this view in 1978 in a leading decision (Tennessee Valley Authority
v. Hill, 437 U.S. 187, 184 (1978)): ‘the value of endangered species is incalculable’ and ‘it is
clear from the Act’s legislative history that Congress intended to halt and reverse the trend
toward species extinction – whatever the cost’.
19
Over these five years, a total of 914 million U.S. dollars has been spent on the protection
of 229 vertebrate species (Metrick and Weitzman 1998: 28). The analysis of Metrick and
Weitzman only studied expenditures which could be attributed to individual species.
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an emotional identification with certain animals is actually more important for
the protection decision than rational considerations based on scientific evidence
and transparent criteria.
But if the goal should be to trade-off different alternatives and to set priorities based on scientific evidence and transparent criteria, the tool box of
economics can be very helpful. Recall (from Section 7.2) that economics studies the optimal allocation of scarce resources from the point of view of society.
Economic valuation of (ecological and economic) costs and benefits of different
alternatives is one tool to assess the relative desiredness of different alternatives
from the point of view of society. Economic valuation can therefore contribute
to placing decisions on biodiversity protection on a rational and transparent
basis (Dasgupta 2000, Weikard 1998b).
In particular, the total economic value of a species, which is broadly defined
and covers, in principle, also the ecological functions of the species within an
ecosystem, could help set up a rank ordering of species to be protected. Besides
the value of the species, a prioritization should, of course, also be based on the
costs of the protection measures for the species, and the increase of survival
probability by this measure. Weitzman (1998) and Metrick and Weitzman
(1996, 1998) have suggested – based on such reasoning and a formal economic
analysis – the following simple criterion for calculating the rank of a species.
Let Vi be the total economic value of species i, ΔPi the increase in the species’
survival probability by a protection measure, and Ci the measure’s cost. Then
a rank ordering in which different species are ranked according to the value
of Ri = Vi · ΔPi /Ci is optimal from an economic point of view.20 According
to this criterion, the protection priority of the species is higher, the higher its
total economic value, the more its survival probability can be increased by a
protection measure, and the lower the costs of protection.
Of course, employing such a simple economic criterion for drawing up a rank
ordering of species raises a number of questions. As already discussed above,
to quantitatively determine the total economic value of a species is fraught
with great difficulties. Also, one needs to be aware that this simple criterion is
based on the assumption that species’ survival probabilities are independent,
which is clearly wrong when species interact in an ecosystem (Baumgärtner
2004c).21 But the criterion allows one to enter a rational discussion about what
information to use, and how, in order to prioritize among protection measures
for different species. This is superior to the wide-spread current practice which
20
Instead of total economic value Vi , Weitzman (1998: 1280) and Metrick and Weitzman
(1998: 26) employ the sum Di + Ui of direct utility Ui and distinctiveness/uniqueness Di of
species i compared with other species. In so far as the latter gives rise to an indirect use
value or an option/insurance value, both components are part of total economic value.
21
How species interaction in an ecosystem influences optimal investment in species protection measures will be studied in detail in Chapter 11 below.
Biodiversity as an Economic Good
147
prioritizes only implicitly.
7.5.2
Relative Importance of Nature and Species Conservation
Compared with Other Societal Goals
Concerning the question of the importance of the conservation of biodiversity
compared with other societal goals, the essential economic idea is that biodiversity conservation requires the protection of natural habitats, that is, area of
land. Land can be used for alternative purposes, for example as agricultural
area, industrial space, or infrastructure, and is limited. It is therefore necessary to decide upon what share of land should be set aside for nature protection
and what share is made available for economic development. The same goes for
public or private financial resources. Put provocatively, the question is, how
important is the conservation of biodiversity compared with social security,
health care, education, etc.?
Even if one presupposes that the current loss of biodiversity is so dramatic
that society is willing, based on the total economic value of biodiversity, to
employ additional means for its protection, it is also obvious that, in principle,
this trade-off can lead to the opposite result. That is, it could be that, at some
point, society is not willing to sacrifice additional means for the protection
of biodiversity because other purposes are considered to be more important.
This means, an economic cost-benefit analysis can lead to the result that it
is optimal to not protect a certain fraction of biodiversity, but let it become
extinct, and employ the resources thus saved for achieving other societal goals.
7.5.3
The Design of Biodiversity Conservation Measures
Once a decision has been made upon (i) how important the protection of one
endangered species is compared to another one, and (ii) how important the
protection of biological diversity is compared with other societal goals, there is
still the question of how to conserve biodiversity. This is the question of what
instruments to apply, and how to design biodiversity conservation measures in
order to reach a certain conservation goal. Any policy-relevant answer to this
question must also rely on economics and economic valuation (Klauer 2001,
Shogren et al. 1999).
With respect to in-situ conservation of species, the selection and design of
reserve sites is the primary problem and, hence, has drawn most attention in
the literature. Traditionally, the optimal selection and design of reserve sites is
a domain of ecology and, in particular, its sub-field conservation biology (e.g.
Margules et al. 1988, Soulé 1986). However, as Ando et al. (1998) and Polasky
et al. (2001) have shown, cost savings of up to 80% could be achieved by
integrating economic costs (i.e. land prices) into previously ecologically based
selection algorithms for reserve sites. This demonstrates that economics and
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Natural Science Constraints in Environmental and Resource Economics
economic valuation are important for the design of cost effective and efficient
measures of nature conservation. As a result, economists have turned to this
issue and made a number of important contributions (e.g. Costello and Polasky
2004, Johst et al. 2002, Polasky et al. 1993, Solow et al. 1993, Weitzman 1993,
Metrick and Weitzman 1996, 1998, Wu and Boggess 1999).
7.6
Conclusion: What can Economics Contribute to
Biodiversity Conservation?
Economics is – like any scientific discipline – limited by its research interest
and its methodology. It views biodiversity as a resource for the satisfaction
of various human needs, and studies its efficient allocation for that sake (see
Section 7.2). This also defines the economic value of biodiversity. In particular, the economic value notion is instrumental, anthropocentric, individualbased/subjective, context dependent, marginal and state-dependent (see Section 7.3). In spite of these limitations, economics can make important contributions to the study of biodiversity loss and conservation (Brown and Shogren
1998: 15–19).
First, the economic perspective gives a detailed and good understanding of
the specific mechanisms and fundamental causes of the current dramatic loss of
biodiversity (see Section 7.4). Besides human population growth, it is mainly
different forms of market failure which make market prices on today’s markets for biological resources deviate from their optimal levels. This results in
a suboptimally high rate of biodiversity loss. Governance failure in regulating
the access to, and use of, biological resources in a fair, efficient and sustainable manner is an equally important reason for this development. Another
major reason is the fundamental ignorance about potential future uses of biological diversity and about the role that biodiversity plays for the functioning
of ecosystems.
Second, the question of what species are threatened by extinction or will
be in the near future, is not only an ecological question but also an economic
question. For, besides ecological constraints, economic developments influence
the extinction probability of a species as well. This extinction probability is
higher for species which rival with economic development (e.g. highway construction leading to habitat fragmentation). It is lower for species which are
under intensive protective care. Since the decision between economic development and nature protection is crucially determined by economic considerations,
so is the probability of extinction of different species. Endangeredness, thus, is
not a purely ecological quality, which could be determined solely from natural
science research, but is crucially determined by economic factors as well.
Biodiversity as an Economic Good
149
Third, since not all species that are threatened by extinction today can be
saved (Watson et al. 1995b: 2), the question arises ‘What species and populations should be protected, and to what extent?’ Economics can provide a
methodological framework to discuss this question rationally. It can thereby
help base decisions about biodiversity protection on scientific evidence and
transparent criteria (see Section 7.5). This framework essentially builds on
valuation and the comparison of alternative options based on their respective
(ecological and economic) costs and benefits. Economics therefore allows one
to prioritize among different protection goals.
Fourth, economics is indispensable when a given protection goal is to be
achieved in a cost effective manner. This means, after a certain protection goal
has been set (for example, the conservation of an endangered population in a
certain region), one chooses among all protection measures that are suitable
to actually achieve the goal (for example, displacing the highway, building
bridges over the highway, establishment of a compensation habitat in a different
location, etc.) the one with minimal costs.
As far as the first and the last points are concerned, economics is in its traditional domain and can make powerful contributions, even to the solution of a
problem such as biodiversity loss, which is primarily defined in biological terms.
As far as the second and third points are concerned, economic research is still in
its infancy. While there are already a number of promising contributions, it has
also become obvious that – in order to develop meaningful analyses and relevant
policy recommendations – one needs an interdisciplinary cooperation between
ecologists and economists, leading to fully integrated ecological-economic analyses of biodiversity loss and conservation (Wätzold et al. 2005).
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8. Ecological and Economic Measures of
Biodiversity
8.1
Introduction
For analyses of how biodiversity contributes to ecosystem functioning, how it
enhances human well-being, and how these services are currently being lost,
a quantitative measurement of biodiversity is crucial. Ecologists, for that
sake, have traditionally employed different concepts such as species richness,
Shannon-Wiener-entropy, Simpson’s index, or the Berger-Parker-index (e.g.
Magurran 2004, Pielou 1975, Purvis and Hector 2000). Recently, economists
have added to that list measures of biodiversity that are based on pairwise
dissimilarity between species (Solow et al. 1993, Weikard 1998a, 1999, 2002,
Weitzman 1992, 1993, 1998) or, more generally, weighted attributes of species
(Nehring and Puppe 2002, 2004).
The full information about the biodiversity of an ecological system is only
available in the full description of the system in terms of the number of different entities (i.e. genes, species, or ecological functions), their abundances and
characteristic features. Such a full description comes in different and complex
statistical distributions. For the purpose of comparing two systems, or describing the system’s evolution over time, both of which is essential for policy
guidance, it seems therefore necessary to condense this information into easyto-calculate and easy-to-interpret numbers, although that certainly means a
loss of information. Most often, all the relevant information about the diversity of a system is condensed into a single real number, commonly called a
‘measure of diversity’ or ‘diversity index’. As there are virtually infinitely many
ways of calculating such a diversity index from the complex and multifarious
information about the system under study, it is crucial to be aware of which
aspects of information are being stressed in calculating the index and which
aspects are being downplayed, or even neglected altogether. Not surprisingly
then, the purpose for which a particular index is calculated and used is crucial
for understanding how it is prepared.
In this chapter, I give a conceptual comparison of the two broad classes
151
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Natural Science Constraints in Environmental and Resource Economics
of biodiversity measures currently used, the ecological ones and the economic
ones. It will turn out that there are systematic differences between these two
classes, concerning exactly what information is being used to construct the
index. For example, while ecological indices consider abundances, the economic
ones deliberately do not take abundances into account. In order to explain these
differences, I will argue that the two types of measures aim at characterizing
two very different aspects of the ecosystem. One important observation is that
the rationale for, and basic conceptualization of, the economic measures of
biodiversity stems from the economic idea of product diversity and is intimately
related to the idea of choice between different products which can, in principle,
be produced in any given number. These measures characterize an abstract
commodity/species space, rather than a real allocation of commodities/species.
This raises a number of questions about the applicability of these concepts in
ecology. I will conclude that the measurement of biodiversity requires prior
value judgments as to what purpose biodiversity serves in ecological-economic
systems.
The chapter is organized as follows. Section 8.2 introduces a formal and
abstract description of ecosystems. This framework allows the rigorous definition and comparison of different biodiversity indices later on. Section 8.3 then
addresses the question of how to quantitatively measure the biodiversity of an
ecosystem by surveying different ecological and economic measures of diversity.
Section 8.4 compares the different measures at the conceptual level, identifies
essential differences between them, and critically discusses these differences.
Section 8.5 concludes.
8.2
Species and Ecosystems
‘Biodiversity’ has been defined as ‘the variability among living organisms from
all sources [...] and the ecological complexes of which they are part’ (CBD
1992), which encompasses a wide spectrum of biotic scales, from genetic variation within species to biome distribution on the planet (Gaston 1996, Groombridge 1992, Purvis and Hector 2000, Wilson 1992). In this chapter, I shall
only be concerned with the level of species, as this is the level of organization
which is currently being given most attention in the discussion of biodiversity
conservation policies.1 That is, biodiversity is here considered in the sense of
species diversity.
In order to describe the species diversity of an ecosystem, and to compare
two systems in terms of their diversity, one can consider different structural
1
Ceballos and Ehrlich (2002) have pointed out that the loss of populations is a more
accurate indicator for the loss of ‘biological capital’ than the extinction of species.
Ecological and Economic Measures of Biodiversity
153
characteristics of the system(s) under study:
• the number of different species in the system,
• the characteristic features of the different species, e.g. their functional
traits, and
• the relative abundances with which individuals are distributed over different species.
Intuitively, it seems plausible to say that a system is more diverse than another
one if it comprises a higher number of different species, if the species in the
system are more dissimilar from each other, and if individuals are more evenly
distributed over the different species. A simple example can illustrate this idea
Figure 8.1 Two samples of species, which may be compared in terms of their
diversity based on different criteria: species number, species abundances and
species features. Figure taken from Purvis and Hector (2000: 212).
(Figure 8.1). Consider two systems, A and B, which both consist of eight
individuals of insect species: system A comprises six monarch butterflies, one
dragonfly and one ladybug; system B comprises four swallowtail butterflies and
four ants. Obviously, according to the first criterion (species number), system
A has a higher diversity (three different species) than system B (two different
species). But according to the third criterion (evenness of relative abundance)
one may as well say that system B has a higher diversity than system A, because
there is less chance in system B that two randomly chosen individuals will be
of the same species. And as far as the second criterion goes (characteristic
species features), one would have to start by saying what the characteristic
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species features actually are, which can then be used to assess the aggregate
dissimilarity of both systems.
Before discussing these ideas in detail, let me first introduce a formal and
abstract description of the ecosystem whose species diversity is of interest.
Let n be the total number of different species existent in the system and let
S = {s1 , . . . , sn } be the set of these species. Each si (with i = 1, . . . , n)
represents one distinct species. In the example illustrated in Figure 8.1, S A
= {monarch butterfly, ladybug, dragonfly } and S B = {swallowtail butterfly,
ant}. In the following, n ≥ 2 is always assumed.
Let m be the total number of different relevant features, according to which
one can distinguish between species, and let F = {f1 , . . . , fm } be the list of
these features. Each fj (with j = 1, . . . , m) represents one distinct feature. For
example, possible features could include the following:
• being a mammal/bird/fish/. . .,
• being a herbivore/carnivore/omnivore,
• unit biomass consumption/production,
• being a ’cute little animal’,
• etc.
Then one can characterize each species si (with i = 1, . . . , n) in terms of all
features fj (with j = 1, . . . , m). Let xij be the description of species si in terms
of feature fj , so that x = {xij }i=1,...,n; j=1,...,m is the complete characterization
of all species in terms of all relevant features.
The abundance of different species in the ecosystem is described by the
distribution of absolute abundances of individuals over different species. Let
ai be the absolute abundance of individuals of species si (with i = 1, . . . , n).
If the system under study contains only species of the same or a very similar
kind, the abundance of a species in an ecosystem may be measured by simply
counting the number of individuals of that species. In the example illustrated
in Figure 8.1, aA = (6, 1, 1) and aB = (4, 4), i.e. in system A there are six individuals of species 1 (monarch butterfly), one individual of species 2 (ladybug),
one individual of species 3 (dragonfly), and in system B there are four individuals of species 1 (swallowtail butterfly) and four individuals of species 2 (ant).
However, if the system comprises species which are very different in size, e.g.
deer, birds, butterflies, ants and protozoa, it makes very little sense to measure
their respective absolute abundances by just counting individuals (Begon et al.
1986: 594). Their enormous disparity in size would make a simple count of
individuals very misleading. In that case, the absolute abundance of a species
may be measured by the total biomass stored in all individuals of that species.
Ecological and Economic Measures of Biodiversity
155
Typically, it is more interesting to consider not the absolute abundance ai of
species si , but its relative abundance in relation to all
nthe other species. The rel(p1 , . . . , pn )
ative abundance of species si is given by pi = ai / i=1 ai . Let p = be the vector of relative abundances. By construction of pi , one has ni=1 pi = 1
and 0 ≤ pi ≤ 1, where pi = 0 means that species i is absent from the system and
pi = 1 (implying pj = 0 for all j = i) means that species i is the only species in
the system. In the example illustrated in Figure 8.1, pA = (0.75, 0.125, 0.125)
and pB = (0.5, 0.5). If species abundances are measured by counting individuals
of that species, the relative abundance pi indicates the probability of obtaining an individual of species si in a random draw from all individuals in the
system. When abundances are measured in biomass, the relative abundance pi
indicates the relative share of the ecosystem’s biomass stored in individuals of
species si . Without loss of generality, assume that p1 ≥ . . . ≥ pn , i.e. species
are numbered in the sequence of decreasing relative abundance, such that s1
denotes the most common species in the system whereas sn denotes the rarest
species.
Altogether, the formal description of an actual or potential ecosystem state
Ω comprises the specification of n, S, m, F , p and x, which completely describes
the composition of the ecosystem from different species as well as all species in
terms of their characteristic features. In the following, a biodiversity measure
of the ecosystem Ω means a mapping D of all these data on a real number:
D : Ω → IR with Ω = {n, S, m, F, p, x} .
(8.1)
That is, I consider only biodiversity measures which characterize the species
diversity of an ecosystem by a single number (‘biodiversity index’).2 The various measures differ in what information about the ecosystem state Ω they take
into account and how they aggregate this information to an index.
8.3
8.3.1
Different Measures of Biodiversity
Species Richness
The simplest measure of biodiversity of an ecosystem Ω is just the total number
n of different species found in that system. This is often referred to as species
richness (following McIntosh 1967):
D R (Ω) = n.
2
(8.2)
Note that the focus on biodiversity indices constitutes a considerable reduction in generality and has a significant economic bias. The desire to characterize a set of objects by
a single number – instead of, say, by the distribution of properties or abundances – can be
vindicated by the aim of establishing a rank ordering among different sets, which is necessary
in order to choose the best – in the sense of: most diverse – set (e.g. Weitzman 1992, 1998).
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Natural Science Constraints in Environmental and Resource Economics
Species richness is widely used in ecology as a measure of species diversity.
One example is the long-standing and recently revitalized diversity-stability
debate, i.e. the question whether more diverse ecosystems are more stable and
productive than less diverse systems (Elton 1958, Odum 1953, Hooper et al.
2002, Loreau et al. 2001, 2002, MacArthur 1955, May 1972, 1974, McCann
2000).3 Another example are the so-called species-area relationships,4 which
are important for the present biodiversity conservation debate because they
are virtually the only tool to estimate the number of species that go extinct
due to large-scale habitat destruction (Gaston 2000, Kinzig and Harte 2000,
MacArthur and Wilson 1967, May et al. 1995, Rosenzweig 1995, Whitmore
and Sayer 1992). Species richness is also the biodiversity indicator implicitly
used in the public discussion, which often reduces biodiversity loss to species
extinction.
In the species richness index (8.2), all species that exist in an ecosystem
count equally. However, one might argue that not all species should contribute
equally to an index of species diversity. Two different strands have evolved in
the literature both of which develop indices in which different species are given
different weight. The first strand, which has evolved mainly in ecology, weighs
different species according to their relative abundance in the system. This
is vindicated by the observation that the functional role of species may vary
with their abundance in the system. These biodiversity indices are discussed
in Section 8.3.2 below. The other strand, which has been contributed to the
discussion of biodiversity mainly by economists, stresses that different species
should be given different weight in the index due to the characteristic features
they possess. These biodiversity indices are discussed in Section 8.3.3 below.
8.3.2
Indices Based on Relative Abundances
Ecologists have tackled the problem of incorporating the functional role of
species in a measure of species diversity by formulating diversity indices in
which the contribution of each species is weighted by its relative abundance
in the ecosystem (Begon et al. 1998, Magurran 1988, Pielou 1975, Purvis and
Hector 2000, Ricklefs and Miller 2000). Intuitively, rare species should contribute less than common species to the biodiversity – in the sense of ‘effective’
species richness – of an ecosystem.5 A general measure for the effective number
ν of species, which uses the information about pure species number n and the
3
The diversity-stability relationship is tantamount for the insurance value of biodiversity
(Chapter 9). See the detailed discussion on this relationship in Section 9.2.
4
The well established species-area-relationships state that species richness n increases
with the area l of land as n ∼ lz , where z (with 0 < z < 1) is a characteristic constant for
the type of ecosystem.
5
Rao (1982) equates species richness and distribution of relative abundances with community size and shape respectively.
Ecological and Economic Measures of Biodiversity
157
distribution of relative abundances p = (p1 , . . . , pn ) to build on this intuition,
is the following (Hill 1973):6
(1/(1−α)
' n
να (n, p) =
pαi
with α ≥ 0 .
(8.3)
i=1
This measure has a number of desired properties, which have made it the
foundation for various biodiversity indices in ecology:
1. The measure (8.3) – more exactly: its logarithm Hα = log να – is well
known from information theory where it has been introduced by Rényi
(1961) as a generalized entropy (‘entropy of order α of the probability
distribution p’). Its properties are well studied and understood (Aczél
and Daróczy 1975).
2. The maximal value of να (n, p) increases with the number n of different
species.
3. For given n, the measure να takes on values between 1 and n, depending on p. Technically, it is an inverse generalized mean relative abundance: να gives the equivalent number of equally abundant (hypothetical)
species that would reproduce the entropy value Hα of the actual system
of n species with unevenly distributed relative abundances p (Whittaker
1972). Thus, να can be interpreted as an effective species number in a
system of n unevenly distributed species.
4. For given n, the measure να (n, p) assumes its maximal value – that is,
pure species richness n – when all species have equal relative abundance,
i.e. pi = 1/n for all i = 1, . . . , n. In this case of an absolutely even
distribution of relative abundances, the effective number να (n, p) simply
equals the total number n of different species in the system.
The measure να (n, p) decreases with increasing unevenness of the distribution of relative abundances p. This means, dominance of a few species,
or, more generally, an uneven distribution of relative abundances, brings
down the index of effective species number να (n, p). The index assumes
its minimal value when a system is dominated by one single species, with
6
Neither Rényi (1961) nor Hill (1973), who introduced this measure to information theory
and to ecology respectively, restrict the range of α to non-negative real numbers. Indeed,
Equation (8.3) is well defined for all −∞ ≤ α ≤ +∞. However, for α < 0, να (n, p) yields
values greater than n, which means that rarer species are given greater weight than more
common species in the measure of effective species number. This contradicts the intuition
that the effective species number should be smaller than the pure number, depending on heterogeneity. Therefore, when it comes to measuring species diversity it seems to be reasonable
to constrain the parameter α to non-negative values.
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Natural Science Constraints in Environmental and Resource Economics
all other species having negligible relative abundances, i.e. pi ≈ 0 for all
i = 1, . . . , n except for i = i∗ , where i∗ denotes the dominant species, and
pi∗ ≈ 1. In this case, να (n, p) ≈ 1, which means that the effective number
of different species is approximately one.
5. The parameter α ≥ 0 weighs the influence of evenness of the distribution
of relative abundances p against the influence of pure species number
n on the effective species number ν. For α = 0, the evenness of the
distribution of relative abundances p is completely irrelevant, and the
effective species number ν is simply given by the pure species number n.
The larger α, the higher is the weight of the evenness in the calculation
of the effective species number (8.3). For α = +∞, the pure species
number n is completely irrelevant, and the effective species number ν
is exclusively determined by how (un)evenly the relative abundances of
species are distributed.
6. For different values of the parameter α one can recover from expression
(8.3) different species diversity indices that are well-established in ecology
(Hill 1973). They can thus be considered as special cases of the general
measure (8.3):
Species richness index
With α = 0 one obtains the species richness index already discussed in Section 8.3.1 above:
D R (Ω) = ν0 (n, p) = n .
(8.4)
That is, to zeroth order effective species number is just pure species richness.
Shannon-Wiener index
With α = 1 one obtains7 the Shannon-Wiener-index
D
SW
(Ω) = ν1 (n, p) = exp H
with H = −
n
pi log pi ,
(8.5)
i=1
where H is well known from statistics and information theory as the ShannonWiener expression for entropy (Shannon 1948, Wiener 1948).8
7
This is not immediately obvious. See Hill (1973: Appendix) for a proof.
The expression H (Equation 8.5) has been proposed independently by Claude Shannon
(1948) and Norbert Wiener (1948). It is sometimes referred to as Shannon-Weaver-entropy
because it has been popularized by Shannon and Weaver (1949). In information theory the
base of the logarithm is usually taken to be 2, consistent with an interpretation in terms of
‘bits’. In ecology the tendency is to employ natural logarithm’s, i.e. a base of e, although
some use a base of 10. There is, of course, no natural reason to prefer one base over the
8
Ecological and Economic Measures of Biodiversity
159
Shannon-Wiener entropy, and the index built from it, does not have a
straightforward, let alone ecologically meaningful, interpretation as Simpson’s
index has (see below). Being a logarithmic measure, it is also more difficult to
calculate than Simpson’s index. Nevertheless, it is a popular measure of heterogeneity and effective species number. This is especially due to the logic of
its development within statistical physics (Balian 1991) and information theory
(Krippendorff 1986), and its formal elegance and consistency. For example, of
all the measures defined by the general expression (8.3) for 0 ≤ α ≤ +∞, only
Shannon-Wiener entropy (α = 1) allows consistent aggregation of heterogeneity over different hierarchical levels of a system: upper level Shannon-Wiener
entropy of a system of individuals clustered in lower level subsystems can be
additively decomposed to show the contributions from heterogeneity within
and between lower level subsystems.
Simpson’s index
With α = 2 one obtains Simpson’s index (Simpson 1949):
D S (Ω) = ν2 (n, p) = 1/
n
p2i .
(8.6)
i=1
Simpson’s index has been, and still is, fairly popular among ecologists. The
reasons include the ease of calculating the index, the bounded properties of
2
the expression
p
i , and – not the least – the ecological meaningfulness of
its interpretation:
p2i is the probability that any two individuals drawn at
random from an infinitely large ecosystem belong to different species.9 The
inverse of this expression is taken to form the biodiversity index, so that D S
increases with the evenness of the distribution of relative abundances. This
makes sense as an index of effective species number when viewing ecosystems
as functional relationships, e.g. based on predator-prey-relations, parasite-hostrelations, etc.
Berger-Parker index
With α = +∞ one obtains the Berger-Parker-index (Berger and Parker 1970,
May 1975) as
DBP (Ω) = ν+∞ (n, p) = 1/p1 ,
(8.7)
other, but care should be taken when comparing results from different studies in terms of H,
which might have been obtained using different bases. Yet, the choice of a particular base
does not have any influence on ν (as long as one chooses the same base for the logarithm
and the exponential function in Equation 8.5.)
9
The appropriate formula for a finite community is [n
i (ni − 1)/(N (N − 1)], where ni
n
is the number of individuals in the ith species and N =
i=1 ni is the total number of
individuals.
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Natural Science Constraints in Environmental and Resource Economics
that is, the inverse relative abundance of the most common species in the system. It can be interpreted as an effective species number in the sense that 1/p1
gives the equivalent number of equally abundant (hypothetical) species with
the same relative abundance as the most abundant species in the system. Obviously, the Berger-Parker-index only takes into account the relative dominance
of the most common species in the system, neglecting all other species.
7. One of the properties of the biodiversity measure (8.3) is that for given
n and p the value of να (n, p) decreases with α. As the most widely used
diversity indices can all be expressed as special cases of Equation (8.3) for
different values of a, it becomes evident that the results for the effective
species number in a given system yielded by these indices are related in
the following way:
n = D R ≥ D SW ≥ D S ≥ D BP > 1 .
(8.8)
Illustration
Table 8.1 illustrates the working of the various indices in comparison for different hypothetical communities. The first observation is that for all communities,
species si
s1
s2
s3
s4
s5
s6
s7
n
(α = 0)
SW
D
(α = 1)
S
D
(α = 2)
D BP (α = +∞)
relative abundance pi in
Ω1
Ω2
Ω3
Ω4
0.25 0.20 0.24 0.249
0.25 0.20 0.24 0.249
0.25 0.20 0.24 0.249
0.25 0.20 0.24 0.249
0.20 0.04 0.004
4
5
5
5
4.00 5.00 4.48 4.08
4.00 5.00 4.31 4.03
4.00 5.00 4.17 4.02
community
Ω5
Ω6
0.50 0.50
0.30 0.30
0.10 0.10
0.07 0.07
0.03 0.01
0.01
0.01
5
7
3.42 3.53
2.81 2.82
2.00 2.00
Table 8.1 Comparison of different diversity indices for hypothetical communities Ωj (j = 1, . . . , 6) of four, five or seven different species with relative
abundances pi . The parameter α pertains to the general definition (8.3), n is
the species richness of the respective community (Equation 8.4), D SW is the
Shannon-Wiener index (Equation 8.5), D S is Simpson’s index (Equation 8.6),
D BP is the Berger-Parker index (Equation 8.7).
α = 0 yields species richness n as the effective species number of that community. Second, the Berger-Parker index, as the limit case of α = +∞, gives
Ecological and Economic Measures of Biodiversity
161
the number of equally abundant (hypothetical) species with the same relative
abundance as the most abundant species in the community, 1/p1 . If, for example, the most common species has a relative abundance of p1 = 0.5, with the
other species in that community having smaller relative abundances, then the
effective number of species in that community would be D BP = 1/0.5 = 2, irrespective of the number and relative abundances of the other species (Table 8.1,
columns 6 and 7). Third, all indices – i.e. all values of α – yield species richness n as the effective species number if the community consists of absolutely
evenly distributed species, for example in communities Ω1 and Ω2 (Table 8.1,
columns 2 and 3). In this case, in index value is the higher the higher n.
Fourth, for a given number of species, e.g. n = 5, all indices assume their
maximal value – species richness n – if species are absolutely evenly distributed,
for example in community Ω2 as compared to Ω3 , Ω4 and Ω5 (Table 8.1,
columns 3 through 6). Conversely, the value of the index decreases if species
are distributed more unevenly. The higher α, the stronger the index value decreases with unevenness. For example, comparing the five-species-communities
Ω2 and Ω5 (Table 8.1, columns 3 and 6) shows that the index value drops from
5 to 2 for α = +∞, while it only drops to 3.42 for α = 1 and remains at the
level of 5 for α = 0. Communities Ω3 and Ω4 (Table 8.1, columns 4 and 5)
illustrate that with n − 1 species of equal relative abundance and one species,
s5 , with much lower relative abundance, the Simpson, Shannon-Wiener and
Berger-Parker indices will be only slightly greater than n − 1. The smaller p5 ,
the closer they approach n − 1.
Sixth, a comparison of communities Ω2 and Ω6 (Table 8.1, columns 3 and 7)
shows that the effective species number as measured by να can actually decrease
although species richness, n, increases between two communities. This is due
to the increase in unevenness outweighing the increase in species richness.10
Seventh, the higher α, the more weight an index puts on the more abundant
species in the community while being less sensitive to differences in small relative abundances and in total species richness, as can be seen from comparing
communities Ω5 and Ω6 (Table 8.1, columns 6 and 7). These two communities only differ in the number and composition of very rare species. The
Berger-Parker index (α = +∞), which takes into account only the most abundant species, is completely insensitive to this difference. Even Simpson’s index
(α = 2) is hardly sensitive to this difference. The Shannon-Wiener index
(α = 1) is more sensitive to differences in small relative abundances than
Simpson’s index,11 but only species richness (α = 0) fully takes into account
the higher number of very rare species in community Ω5 compared to Ω6 .
10
May (1975) has shown that for n > 10 the underlying species abundance distribution
makes a crucial difference for how, and even whether at all, DS increases with n.
11
On the other hand, it is less sensitive to small differences in large relative abundances,
whereas Simpson’s index responds more substantially to these differences.
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Coming back to the simple example of comparing the two communities
described in Section 8.2 (Figure 8.1), the different biodiversity indices discussed
in this section yield the results shown in Table 8.2. As one can see from the
species si
s1
s2
s3
n
(α = 0)
SW
D
(α = 1)
S
D
(α = 2)
D BP (α = +∞)
relative abundance pi in
system A
system B
0.75
0.5
0.125
0.5
0.125
—
3.00
2.00
2.09
2.00
1.68
2.00
1.25
2.00
Table 8.2 Comparison of different diversity indices for the two systems described in Section 8.2 (Figure 8.1). The parameter α pertains to the general
definition (8.3), n is the species richness of the respective community (Equation 8.4), D SW is the Shannon-Wiener index (Equation 8.5), D S is Simpson’s
index (Equation 8.6), D BP is the Berger-Parker index (Equation 8.7).
table, which system is rated to be ‘more diverse’ depends on the parameter
α, i.e. on how one weighs pure species richness against evenness of relative
abundances: for small values of α sample A is found to be more diverse, while
system B turns out to be more diverse for large values of α.
8.3.3
Indices Based on Characteristic Features
The biodiversity indices discussed in Section 8.3.2 all take the species richness
of an ecosystem, properly adjusted by the distribution of relative abundances
so that rare species are given less weight than common species, to be a measure
of diversity. According to these indices, systems with more, and more evenly
distributed, species are found to have a higher biodiversity than systems with
less, and less evenly distributed, species. This procedure has been criticized
for not taking into account the (dis)similarity between species. For example, a
system with 100 individuals of some plant species, 80 individuals of a different
plant species, and 50 individuals of yet another plant species will be found to
have exactly the same biodiversity, according to these indices, than a system
with 100 individuals of some plant species, 80 individuals of a mammal species,
and 50 individuals of some insect species. Yet, intuitively one would say that
the latter has a higher biodiversity. This intuition is based on the (dis)similarity
between the various species.12
12
The richness-and-abundance based indices discussed in Section 8.3.2 implicitly assume
that all species are pairwise equally (dis)similar.
Ecological and Economic Measures of Biodiversity
163
In order to account for the (dis)similarity of species when measuring biodiversity, one needs a formal representation of the characteristic features of
species. Based on these characteristic features, the (dis)similarity of species
can be measured and taken into account when constructing a biodiversity index. Two different approaches exist so far. One has been initiated by ecologists
(May 1990, Erwin 1991, Vane-Wright et al. 1991, Crozier 1992) and put on a
rigorous axiomatic basis, enhanced and popularized by Weitzman (1992, 1993,
1998). I shall therefore call it the Weitzman-approach.13 It builds on the
concept of a distance function to measure the pairwise dissimilarity between
species. The diversity of a set of species, in this approach, is then taken to be an
aggregate measure of the dissimilarity between species. This approach is most
appealing when applied to phylogenetic diversity. The other approach, developed by Nehring and Puppe (2002, 2004), generalizes the Weitzman-approach.
It builds directly on the characteristic features of species and their relative
weights. Both approaches are now discussed in detail.
Weitzman index
Weitzman (1992) defines a diversity measure, D(S), of a set S of species based
on the fundamental idea that the diversity of a set of species should be an
aggregate measure of the pairwise dissimilarity between species. The dissimilarity between two species, si and sj , is conceptualized by a distance function,
d : S × S → IR+ . In general, a distance function has the following properties.
It is non-negative and symmetric, i.e. d(si, sj ) = d(sj , si ) > 0 for all si , sj ∈ S
and si = sj . Furthermore, d(si, si ) = 0 for all si ∈ S, which expresses the
very nature of what one means by ‘dissimilarity’: a species compared to itself
does not have any dissimilarity.14 The pairwise distances of all species are the
elementary data upon which the diversity measure builds. Weitzman (1992,
1993) suggests the use of phylogenetic information to determine the pairwise
distances between species, but also states that any other quantifiable trait of
species could be used for that purpose as well, e.g. morphological or functional
features. A distance function can, of course, also be meaningfully defined when
species differ in more than one feature, for instance, as a weighted sum of
differences in different features.
With given pairwise distances between all species, Weitzman’s (1992) di-
13
Solow et al. (1993) and Weikard (1998a, 1999, 2002) have developed biodiversity indices
that follow a very similar logic.
14
Sometimes the so-called triangle inequality, d(si , sj ) ≤ d(si , sk ) + d(sk , sj ) for all
si , sj , sk ∈ S, is invoked in addition to obtain a metric distance measure (e.g. Weikard
1998a, 1999, 2002). This is not necessary for developing the Weitzman index.
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versity index D(S) of a set S of species is then defined recursively by
D W (Q ∪ {si }) = D W (Q) + δ(si , Q) for all si ∈ S\Q and 0 ⊂ Q ⊂ S(8.9)
where D W ({sj }) = D0 ∈ IR+ for all sj ∈ S
and
δ(si , Q) = min d(si , sj ) for all si ∈ S\Q .
sj ∈Q
This means, the calculation of the index starts from an arbitrarily chosen start
value D0 ∈ IR+ assigned to the set that contains only one species, irrespective
of what species sj that is. Depending on the particular application, D0 may be
chosen to be zero or a very large number. One then calculates the biodiversity
index of an enlarged subset Q of S (with 0 ⊂ Q ⊆ S) that one obtains when
adding species si ∈ S\Q to the set Q, Q = Q ∪ {si }, by adding the increase
in diversity δ(si , Q) which species si adds to the diversity of the subset Q.
This increase in diversity is calculated as the minimal distance between the
added species si and any of the species sj in the subset Q.15 So, the recursive
algorithm (8.9) allows one to calculate the diversity of a set S of species, starting
from the arbitrarily chosen diversity value of a single species set, D0 , and then
adding one species after the other until the whole set S is complete.
In general, the recursive algorithm (8.9) is path dependent, i.e. the value
calculated for D W depends on the particular sequence in which species are
added when constructing the full set S. Therefore, the diversity function D W
as defined by Equation (8.9) is, in general, not unique.16 However, Weitzman’s
diversity measure (8.9) is unique, and therefore most appealing, for the special case when the feature space is ultrametric.17 Ultrametric spaces have an
interesting geometric property which is also ecologically relevant: A set S of
species characterized by ultrametric distances can be represented graphically
by a hierarchical (e.g. phylogenetic) tree, and any hierarchical (phylogenetic)
tree can be represented by ultramteric distances. Figure 8.2 shows an example
of such a phylogenetic tree. In a phylogenetic tree, the distance d(si , sj ), which
indicates the dissimilarity between species si and sj , is given by the vertical
distance to the last common ancestor of si and sj , and the diversity D W (S) of
the set S of all species is given by the summed vertical length of all branches
of the tree.
15
This corresponds to the standard topological definition of the distance between a point
and a set of points
16
By imposing a condition called ‘monotonicity in species’, Weitzman (1992) can show
that the class of, in general, path dependent diversity indices (8.9) reduces to a unique, path
independent index which is given by D(S) = maxsi ∈S [D(S\{si }) + δ(si , S\{si })].
17
A space is called ultrametric if the pairwise distances between any three
points in space have the property that the two greatest distances are equal:
max{d(si , sj ), d(sj , sk ), d(si , sk )} = mid{d(si , sj ), d(sj , sk ), d(si , sk )} for all si , sj , sk ∈ S.
Ecological and Economic Measures of Biodiversity
1
2
3
4
5
165
6
Figure 8.2 Phylogenetic tree representation of a set of six species with ultrametric distances (from Weitzman 1992: 370).
Nehring-Puppe index
Even more general than Weitzman’s distance-function-approach is the so-called
‘multi-attribute approach’ proposed by Nehring and Puppe (2002, 2004). Like
Weitzman, they base a measure of species diversity on the characteristic features of species. In contrast to Weitzman, the elementary data are not the
pairwise dissimilarities between species, but the characteristic features f themselves. From the different features f and their relative weights λf ≥ 0, which
may be derived from the individuals’ or society’s preferences, Nehring and
Puppe construct a diversity index as follows:
λf .
(8.10)
D N P (Ω) =
f ∈F : ∃si ∈S with ‘si possesses feature f ’
In words, the diversity index for a set S of species is the sum of weights λf
of all features f that are represented by at least one species si in the system.
Each feature shows up in the sum at most once. In particular, each species si
contributes to the diversity of the set S exactly the relative weight of all those
features which are possessed by si and not already possessed by any other
species in the set.
Nehring and Puppe also show that under certain conditions the characterization of an ecosystem by its diversity D N P uniquely determines the relative
weights λf of the different features. This means, in assigning a certain diversity
to an ecosystem one automatically reveals an (implicit) value judgement about
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Natural Science Constraints in Environmental and Resource Economics
the relevant features according to which one distinguishes between species and
one describes an ecosystem as more or less diverse.
8.4
8.4.1
Conceptual Comparison and Assessment
Diversity Indices and Value Judgments
All the diversity indices discussed here build – more or less obviously – on
prior value judgements at different levels (Baumgärtner 2006). Technically,
these value judgements enter the construction of the index as some parameter or underlying metric that determines how much weight is given to what
information in calculating the index.
As for the ecological indices (cf. Section 8.3.2), the parameter α plays such a
role: it determines how much weight is given to the unevenness of the distribution of relative abundances as compared to the weight of pure species number.
The value of α is a priori arbitrary. In particular, it cannot be inferred from any
ecological information about the system to be studied. So, there is no ‘true’ or
‘correct’ value of α, but its value has to be chosen by the scientist describing
and analyzing a system for a particular purpose. For example, such a purpose
may be to assess the development over time of a nature reserve in terms of its
biodiversity and its associated potential to sustain ecosystem functioning; or
it may be to compare two patches of rainforest in terms of their biodiversity
of potentially useful pharmaceutical substances. Of course, with a particular
purpose in mind, some values of α may be found to be better suited than others. But this choice of α then reflects individual or social preferences about
why biodiversity is useful.
Similarly, in the Nehring-Puppe measure (cf. Section 8.3.3) it is the parameters λf which play this role: they determine the relative weight that the
different features have in constructing the biodiversity index. For example, if
pharmaceutical effectiveness is held to be a very important property of species,
and their complementarity in the use of ecological resources is a less important
property, a sample of species that differ mainly in pharmaceutical respect will
be found to be more diverse than a sample of species differing mainly in ecological respect. And the result will be exactly the opposite if pharmaceutical
effectiveness is taken to be a less important property than ecological complementarity. Again, the construction of the index, and the result it yields in
terms of the level of biodiversity, depends on individual or social value judgments about why biodiversity matters.
This also applies to Weizman’s index (cf. Section 8.3.3). Here, the weighting
based on value judgments lies in the choice of a particular metric that is used
for specifying the pairwise distances between species. For example, if species
Ecological and Economic Measures of Biodiversity
167
differ in two (or more) features, say pharmaceutical effectiveness and ecological
complementarity, then it is the underlying metric that determines how the two
dimensional description of species translates into one dimensional distances between them, which are then used to calculate the index. Weitzman claims that
the pairwise distances between species are basic ecological information which
comes, in principle, from ecological research. But since the metric introduces a
weighting of different features, it involves individual or social value judgments
and, therefore, is not purely ecological information. Like all other measures of
biodiversity, the Weitzman index thus depends on prior judgments about why
biodiversity is valuable.
8.4.2
Information Used and Not Used
Comparing the ecological and economic biodiversity indices (cf. Section 8.3)
at the conceptual level, it is obvious that the two classes are distinct by the
information they use for constructing a diversity index (Figure 8.3). While
Information about species . . .
. . . abundances p
?
. . . number n
. . . features f
Q
Q
Q
Q
Q
Q
QQ
?
+
s
• Shannon-Wiener
• Simpson
• Berger-Parker
ecological indices
• species richness
•
•
•
•
?
Weitzman
Solow et al.
Weikard
Nehring-Puppe
economic indices
Figure 8.3 Biodiversity indices differ by the information on species and ecosystem composition they use.
the ecological measures (Section 8.3.2) use the number n of different species
in a system as well as their relative abundances p, the economic ones (Section 8.3.3) use the number n of different species as well as their characteristic
features f . In a sense, the indices discussed in Section 8.3.2 above measure
‘heterogeneity’ rather than ‘diversity’ (Good 1953, Hurlbert 1971, Peet 1974),
as they are based on richness and abundances but completely miss out features.
The indices discussed in in Section 8.3.3 above measure ‘dissimilarity’ rather
than ‘diversity’, as they are based on richness and dissimilarity but completely
miss out abundances. Both kinds of indices contain pure species richness as a
special case.
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Natural Science Constraints in Environmental and Resource Economics
Up to now, there do not exist any encompassing diversity indices based on
all ecological information considered here – species richness n, abundances a,
and features f . A logical next step at this point could be to construct a general
diversity index based on species richness, abundances and features, which contains the existing indices as special cases. However, one should not jump to this
conclusion too quickly. It is important to note that the ecological and economic
diversity indices have come out of very different modes of thinking. They have
been developed for different purposes and are based on fundamentally different
value systems. Therefore, they may not even be compatible. This point is
addressed in detail in the following.
8.4.3
Diversity of What? – The Relevance of Abundances and
Features
From an economic point of view, relative abundances are usually considered
irrelevant for the measurement of diversity. The reason is that in economics
the diversity issue is usually framed as a choice problem. Diversity is then a
property of the choice set, i.e. the set of feasible alternatives to choose from.
Individuals facing a situation of choice should consider only the list of possible
alternatives (say, the menu in a restaurant), rather than the actual allocation
which has been realized as the result of other people’s earlier choices (say, the
dishes on the other tables in a restaurant). Furthermore, when economists talk
about product diversity, relative abundances are irrelevant since there is the
possibility of production.18 If all people in a restaurant order the same dish
from the menu, then this dish will be produced in the quantity demanded; and
if all people order different dishes, then different dishes are produced. In any
case, the diversity of the choice set is determined by the diversity of the order
list (the menu), and not by the actual allocation of products (the dishes on the
tables).
This argument has influenced economists view on biodiversity as well. Economists consider biological diversity as a form of product diversity, i.e. a diverse
resource pool from which one can choose the most preferred option(s) (cf. Section 7.2). And this diversity is essentially determined by the choice set, i.e.
the list S of species existent in an ecosystem (e.g. Weitzman 1992, 1993, 1998).
The actual abundances of individuals of different species, in that view, do not
matter.
Ecologists, in contrast, often argue that biological species living in natural
ecosystems – even when considered merely as a resource pool to choose from –
are different from normal economic goods for a number of reasons (e.g. Begon
et al. 1998, Ricklefs and Miller 2000). First, individuals of a particular species
18
While the scarcity of production factors may limit the absolute abundances of the produced products, all possible relative abundances can be produced without restriction.
Ecological and Economic Measures of Biodiversity
169
cannot simply be produced; at least not so easily, not for any species, and not
in any given number. Second, there are direct interactions between individuals
and species within ecosystems, which heavily influence survival probabilities
and dynamics in an ecosystem. And for that sake, relative abundances matter.
And third, while some potential systems (in the sense of: relative abundance
distributions) are viable in situ, others are not. For example, a community
with very high relative abundance of predator species and very low relative
abundance of prey species will go extinct altogether once the prey has been
completely exhausted.
Hence, it becomes apparent that the two types of biodiversity measures –
the ecological ones and the economic ones – aim at characterizing two very
different aspects of the ecosystem. While the ecological measures describe
the actual, and potentially unevenly distributed allocation Ω of species, the
economic measures characterize the abstract list S of species existent in the
system. In a sense, the two are not different measures of the same concept, but
measures of two different concepts.
8.4.4
Diversity for What Purpose? – Different Philosophical
Perspectives on Diversity
The underlying reason for this difference between the ecological and economic
measures of biodiversity can be found in the philosophically distinct perspective
on diversity between ecologists and economists. Ecologists traditionally view
diversity more or less in what may be called a ‘conservative’ perspective, while
economists predominantly have what may be called a ‘liberal’ perspective on
diversity (Kirchhoff and Trepl 2001).
In the conservative view, which goes back to Gottfried Wilhelm Leibniz
(1646–1716) and Immanuel Kant (1724–1804), diversity is an expression of
unity. By viewing a system as diverse, one stresses the integrity and functioning of the entire system. The ultimate concern is with the system at large.
In this view, diversity may have an indirect value in that it contributes to
certain overall system properties, such as stability, productivity or resilience
at the system level. In contrast, in the liberal view, which goes back to René
Descartes (1596–1650), John Locke (1632–1704) and David Hume (1711–1776),
diversity enables the freedom of choice for autonomous individuals who choose
from a set of diverse alternatives. The ultimate concern is with the well-being
of individuals. In this view, diversity of a choice set has a direct value in that it
allows individuals to make a choice that better satisfies their individual subjective preferences. Once one alternative has been chosen, the other alternatives,
and the diversity of the choice set, are no longer relevant.
Of course, the integrity and functioning of the entire system will also be
important for the well being of autonomous individuals who simply want to
choose from a set of diverse alternatives. For example, today’s choice may
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Natural Science Constraints in Environmental and Resource Economics
impede the system’s ability to properly work in the future and, therefore, to
provide diversity to choose from in the future. This is an intertemporal argument, which combines (i) an argument about diversity’s importance at a given
point in time for individuals, who want to make an optimal choice at this point
in time, and (ii) an argument about diversity’s role for system functioning and
evolution over time. From an analytical point of view, one should distinguish
these two arguments. This underlies the distinction between the conservative
and the liberal perspective, which is analytical to start with.
These two distinct perspectives on diversity – the conservative one and
the liberal one – correspond to some extent with the two types of biodiversity
measures considered here (Section 8.3): the ecological measures that take into
account relative abundances, and the economic measures that deliberately do
not take into account relative abundances. The ecological measures are based
on a conservative perspective in that their main interest is to represent biodiversity as an indicator of ecosystem integrity and functioning. With that concern,
the distribution of relative abundances is an essential ingredient in constructing a biodiversity index. In contrast, the economic measures are based on a
liberal perspective in that their main interest is to represent biodiversity as a
property of the choice set from which economic agents – individuals, firms or
society – can choose to best satisfy their preferences. With that concern, it
seems plausible that the actual distribution of relative abundances is not taken
into account when constructing a biodiversity index.
8.5
Summary and Conclusion
I have reviewed the two broad classes of biodiversity measures currently being used, the ecological ones and the economic ones, and compared them at
a conceptual level. It has turned out that the two classes are distinct by the
information they use for constructing a diversity index. While the ecological
measures use the number of different species in a system as well as their relative abundances, the economic ones use the number of different species as well
as their characteristic features. Thereby, the two types of measures aim at
characterizing two very different aspects of the ecosystem. The economic measures characterize the abstract list of species existent in the system, while the
ecological measures describe the actual, and potentially unevenly distributed
allocation of species.
I have argued that the underlying reason for this difference is in the philosophically distinct perspective on diversity between ecologists and economists.
Ecologists traditionally view diversity more or less in what may be called a
conservative perspective, while economists predominantly adopt what may be
called a liberal perspective on diversity (Kirchhoff and Trepl 2001). In the
Ecological and Economic Measures of Biodiversity
171
former, the ultimate concern is with the integrity and functioning of a diverse
system at large, while in the latter, the ultimate concern is with the well-being
of individuals who want to make an optimal choice from a diverse resource
base.
This difference in the philosophical perspective on diversity leads to using
different information when constructing a measure of diversity. In the conservative perspective, the aim is to represent biodiversity as an indicator of
ecosystem integrity and functioning. For tat purpose, the relative abundances
of species are an important ingredient into a measure of biodiversity. In contrast, in the liberal perspective the aim is to represent biodiversity as a property
of the choice set from which economic agents can choose to best satisfy their
preferences. For that purpose, the characteristic features of species are very
important, but relative abundances are not.
Hence, the question of how to measure biodiversity is intimately linked to
the question of what is biodiversity good for (Baumgärtner 2006). This is not a
purely descriptive question, but also a normative one. There are many possible
answers, but in any case an answer requires value judgements. Do we consider
biodiversity as valuable because it contributes to overall ecosystem functioning
– either out of a concern for conserving the working basis of natural evolution, or
out of a concern for conserving certain essential and life-supporting ecosystem
services, such as oxygen production, climate stabilization, soil regeneration,
and nutrient cycling (Barbier et al. 1994, Perrings et al. 1995a, Daily 1997b,
Millennium Ecosystem Assessment 2005)? Or do we consider biodiversity as
valuable because it allows individuals to make an optimal choice from a diverse
resource base, e.g. when choosing certain desired genetic properties in plants for
developing pharmaceutical substances (Polasky and Solow 1995, Polasky et al.
1993, Simpson et al. 1996, Rausser and Small 2000), or breeding or genetically
engineering new food plants (Myers 1983, 1989, Plotkin 1988)?
These are examples for different value statements about biodiversity which
are made on the basis of different fundamental value judgements: in the former
case dominates the conservative perspective, in the latter the liberal one. As I
have shown here, these two perspectives lead to different measures of biodiversity, the ecological measures and the economic measures. Of course, there is a
continuous spectrum in between these two extreme views on why biodiversity
is valuable and how to measure it. But in any case, one is lead to conclude, the
measurement of biodiversity requires prior value judgments as to what purpose
biodiversity serves in ecological-economic systems.
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9. The Insurance Value of Biodiversity in the
Provision of Ecosystem Services∗
9.1
Introduction
In the face of uncertainty, diversity provides insurance for risk averse economic
agents. For example, investors in financial markets diversify their asset portfolio in order to hedge their risk; firms diversify their activities, products or
services when facing an uncertain market environment; farmers traditionally
grow a variety of crops in order to decrease the adverse impact of uncertain
environmental and market conditions. In this chapter, I argue that biological
diversity plays a similar role: it can be interpreted as an insurance against the
uncertain provision of ecosystem services, such as biomass production, control
of water run-off, pollination, control of pests and diseases, nitrogen fixation,
soil regeneration etc. Such ecosystem services are generated by ecosystems and
are used by utility-maximizing and risk averse economic agents (Daily 1997b,
Millennium Ecosystem Assessment 2005).1
In order to explore the hypothesis that biodiversity has an insurance value
in the provision of ecosystem services, I take an interdisciplinary approach
and study a conceptual ecological-economic model that combines (i) current
results from ecology about the relationships between biodiversity, ecosystem
functioning, and the provision of ecosystem services with (ii) economic methods
to study decision-making of risk averse agents under uncertainty. The focus
here is on how to model the ecology-economy-interface. Relevant economic
and policy questions that arise from this view on biodiversity are only briefly
sketched and are discussed in more detail elsewhere (Baumgärtner and Quaas
2005, Quaas and Baumgärtner 2006, Quaas et al. 2004).
Although ecologists usually stress the large extent of ignorance about the
detailed mechanisms of ecosystem functioning (e.g. Holling et al. 1995, Loreau
et al. 2001, Schulze and Mooney 1993), there now seems to be a consensus about
some of the basic mechanisms through which biodiversity influences ecosystem
∗
1
Forthcoming in Natural Resource Modeling.
See the detailed discussion in Chapter 7 (p. 129) on biodiversity and ecosystem services.
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Natural Science Constraints in Environmental and Resource Economics
functioning and the provision of ecosystem services (Hooper et al. 2005, Kinzig
et al. 2002, Loreau et al. 2001, 2002b). Among other insights, it has become
clear that biodiversity may decrease the variability of these services. This result
has lead economists to suggest that biodiversity may be seen as a form of insurance, for instance in agriculture or medicine (Perrings 1995a, Schläpfer et al.
2002, Swanson and Goeschl 2003, Weitzman 2000). On the other hand, availability of financial insurance against the over- or under-provision with ecosystem services, or other financial products that allow the hedging of income risk,
may be seen as substitutes for the natural insurance provided by biodiversity
(Quaas and Baumgärtner 2006, Ehrlich and Becker 1972). The implications of
this idea for both economic well-being and the state of ecosystems in terms of
biodiversity, however, have hardly been explored so far.
One notable exception is to be found in the field of agricultural economics.
A number of studies have analyzed the contribution of crop diversity to the
mean and variance of agricultural yields (Smale et al. 1998, Schläpfer et al. 2002,
Widawsky and Rozelle 1998, Zhu et al. 2000) and to the mean and variance of
farm income (Di Falco and Perrings 2003, 2005, Di Falco et al. 2005). It has
been conjectured that risk averse farmers use crop diversity in order to hedge
their income risk (Birol et al. 2005a, 2005b, Di Falco and Perrings 2003) and
that this may be affected by agricultural policies such as subsidized crop yield
insurance or direct financial assistance (Di Falco and Perrings, 2005).2
With this analysis, I want to look into these issues in greater generality and
with a particular focus on modeling the ecology-economy interface. In order to
study the role of biodiversity as a form of natural insurance I employ a conceptual model that captures the relevant ecological and economic relationships in
a stylized way. While such a simple model cannot offer any quantitative predictions or detailed policy prescriptions, it can clarify the underlying theoretical
structure of the problem: The ecosystem generates a valuable ecosystem service
at a level that is uncertain because of environmental stochasticity. Its probability distribution is influenced by the level of biodiversity, which is measured by
a suitable index. In line with evidence from ecology, I posit a monotonically increasing and concave relationship between biodiversity and the mean absolute
level of the ecosystem service provided by the ecosystem, and a monotonically decreasing and convex relationship between biodiversity and the variance
of ecosystem service. The ecosystem service is being used by an ecosystem
manager, say, a farmer, who is assumed to be a risk averse expected utility
maximizer. Protection of biodiversity is costly. There exists a financial form
of insurance against over- or under-provision with the ecosystem service. The
ecosystem manager decides upon (i) the level of biodiversity and (ii) the level
2
In this respect, biodiversity plays a similar role for farmers as other risk changing production factors, such as e.g. nitrogen fertilizer or pesticides (Horowitz and Lichtenberg 1993,
1994a, b).
The Insurance Value of Biodiversity
175
of financial insurance coverage.
In this framework, I analyze the optimal allocation of biodiversity as a
choice of endogenous environmental risk in mean-variance space.3 In particular,
I
• determine the insurance value of biodiversity, i.e. the marginal value of
biodiversity in its function to reduce the risk premium of the ecosystem
manager’s income risk from using ecosystem services under uncertainty,
• study the optimal allocation of funds in the trade-off between investing
into natural capital, that is, biodiversity protection, and the purchase of
financial insurance, and
• analyze the effect of different institutional regimes in the market for financial insurance (e.g. availability, transaction costs and profitability of
financial insurance) on biodiversity protection.
I conclude that biodiversity acts as a form of natural insurance for risk averse
ecosystem managers against the over- or under-provision with ecosystem services. Therefore, biodiversity has an insurance value, which is a value component in addition to the usual value arguments (such as direct or indirect use
or non-use values, or existence values)4 which hold in a world of certainty. In
this respect, biodiversity and financial insurance are substitutes. Hence, the
availability, and the exact institutional design, of financial insurance, influence
the level of biodiversity protection.
The chapter is organized as follows. Section 9.2 discusses the ecological
background and surveys the relevant literature. Section 9.3 introduces a formal ecological-economic model. The model analysis and results are presented
in Section 9.4, with all formal derivations and proofs given in the Appendix.
Section 9.5 critically discusses the limitations and the generality of the results,
and Section 9.6 concludes.
9.2
Ecological Background: Biodiversity and the
Provision of Ecosystem Services
Over the past fifteen years, there has been intensive research in ecology on the
role of biodiversity for ecosystem functioning and the provision of ecosystem
3
This procedure has been inspired by Crocker and Shogren (1999, 2001, 2003) and Shogren
and Crocker (1999). It is also employed by Baumgärtner and Quaas (2005) and Quaas and
Baumgärtner (2006).
4
See the detailed discussion on the economic value of biodiversity in Section 7.3.
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Natural Science Constraints in Environmental and Resource Economics
services. ‘Biodiversity’ has been defined as ‘the variability among living organisms from all sources [...] and the ecological complexes of which they are
part’ (CBD 1992), which encompasses a wide spectrum of biotic scales, from
genetic variation within species to biome distribution on the planet (Gaston
1996, Purvis and Hector 2000, Wilson 1992). Biodiversity can be described in
terms of numbers of entities (e.g. genotypes, species, or ecosystems), the evenness of their distribution, the differences in their functional traits, and their
interactions. The simplest measure of biodiversity at, say, the species level is
therefore simply the number of different species (‘species richness’). Much of
ecological research has relied on this measure when quantifying ‘biodiversity’,
although more encompassing information has also been employed.5
Research on the role of biodiversity for ecosystem functioning and the provision of ecosystem services built on (i) observations of existing ecosystems,
(ii) controlled experiments both in the laboratory and in the field (‘pots and
plots’) and (iii) theory and model analysis. While the discussion of results has
been, at times, heated and controversial, there now seems to be a consensus
over some of the basic results from this research (Hooper et al. 2005, Kinzig et
al. 2002, Loreau et al. 2001, 2002b).6 Among other insights two ‘stylized facts’
about biodiversity and ecosystem functioning emerged which are of crucial importance for the issue studied here:
1. Biodiversity may enhance ecosystem productivity. In many instances, an
increase in the level of biodiversity monotonically increases the mean absolute level at which certain ecosystem services (e.g. biomass production
or nutrient retention) are provided. This effect decreases in magnitude
with the level of biodiversity.
2. Biodiversity may enhance ecosystem stability. In many instances, an increase in the level of biodiversity monotonically decreases the temporal
variability of the level at which these ecosystem services are provided
under changing environmental conditions. This effect decreases in magnitude with the level of biodiversity.
These two stylized facts are now discussed in turn.7
5
The question of how to construct an aggregate and encompassing measure of biodiversity
has been extensively discussed and is still subject to on-going research (Baumgärtner 2004b,
Crozier 1992, Magurran 2004, May 1990, Nehring and Puppe 2004, Peet 1974, Purvis and
Hector 2000, Vane-Wright 1991, Weitzman 1992, 1998, Whittaker 1972). See the detailed
discussion of this issue in Chapter 8.
6
The article by Hooper et al. (2005) is a committee report commissioned by the Governing
Board of the Ecological Society of America. Some of its authors have previously been on
opposite sides of the debate. This report surveys the relevant literature, identifies a consensus
of current knowledge as well as open questions, and can be taken to represent the best
currently available ecological knowledge about biodiversity and ecosystem functioning.
7
This discussion is compiled from the report of Hooper et al. (2005: Sections II.A and
The Insurance Value of Biodiversity
9.2.1
177
Biodiversity May Enhance Ecosystem Productivity
The absolute level of a certain ecosystem service (e.g. biomass production, carbon sequestration or nitrogen fixation) may be influenced by species or functional diversity in several ways.8 Indeed, more than 50 potential response
patterns have been proposed (Loreau 1998a, Naeem 2002). There are two primary mechanisms through which biodiversity may increase the mean absolute
level at which certain ecosystem services are provided (Figure 9.1):
(i) Only one or a few species might have a large effect on any given ecosystem
service. Increasing species richness, i.e. the number of different species,
increases the likelihood that those key species would be present in the
system (Aarssen 1997, Huston 1997, Loreau 2000, Tilman et al. 1997b).
This is known as the ‘sampling effect’ or the ‘selection probability effect’
(Figure 9.1A).9
(ii) Species or functional richness could increase the level of ecosystem services through complementarity – i.e. species use different resources, or
the same resources but at different times or different points in space –
and facilitation – i.e. positive interactions among species so that e.g. certain species alleviate harsh environmental conditions or provide a critical
resource for other species. Both complementarity and facilitation lead
to an ‘overyielding effect’ (Figure 9.1B), in which biomass production in
mixtures exceeds expectations based on monoculture yields (Ewel 1986,
Harper 1977, Hector et al. 1999, Loreau 1998b, Trenbath 1974, Vandermeer 1989).
Complementarity, facilitation and sampling effects will all lead to a saturating average impact of species richness on the level of some ecosystem service
(Figure 9.1A, B).
Experiments have confirmed the important role of these two primary mechanisms through which biodiversity may increase the mean absolute level of
certain ecosystem services. This holds, in particular, for experiments with
herbaceous plants, in which average primary production and nutrient retention
were found to increase with increasing plant species or functional richness, at
II.B), with large parts being original quotes from this report. For a more detailed and
encompassing discussion see Hooper et al. (2005).
8
The patterns depend on the degree of dominance of the species lost or gained, the strength
of their interactions with other species, the order in which species are lost, the functional
traits of both the species lost and those remaining, and the relative amount of biotic and
abiotic control over process rates (Lawton 1994, Naeem 1998, Naeem et al. 1995, Sala et al.
1996, Vitousek and Hooper 1993).
9
There is still disagreement over whether sampling effects are relevant to natural ecosystems, or whether they only occur in artificially assembled systems (Huston 1997, Loreau
2000, Mouquet et al. 2002, Schläpfer et al. 2005, Tilman et al. 1997b, Wardle 1999).
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Natural Science Constraints in Environmental and Resource Economics
Figure 9.1 Ecological theory has suggested two basic mechanisms of how biodiversity could increase the mean absolute level of ecosystem services: sampling or
selection probability effect (A), and complementarity or facilitation (B). Points
show individual treatments, and lines show the average response. (Figures are
taken from Tilman 1997b, as compiled by Hooper et al. 2005.)
least within the range of species richness tested and over the relatively short
duration of the experiments (Fridley 2003, Hector et al. 1999, Loreau and Hector 2001, Niklaus et al. 2001, Reich et al. 2001, Tilman et al. 1996, 1997a, 2001,
2002).10 In these experiments, the responses to changing diversity are strongest
at low levels of species richness and generally saturate at 5–10 species. It has
also become evident that complementarity, facilitation and sampling/selection
effects are all relevant and can be observed in experiments.11 They are not
necessarily mutually exclusive, but they may be simultaneously or sequentially
at work in one system. The strength of species complementarity and interspecific facilitation and, thus, the quantitative response in the level of ecosystem
services to changes in species richness varies with both the functional characteristics of the species involved and the biotic as well as abiotic environmental
context.
These general findings need to be qualified in a number of respects:
• Experiments have shown that the exact response of ecosystem services on
changes in biodiversity is determined at least as much by differences in
10
Much of the experimental work has focused on the effect of plant diversity on primary
production and nutrient retention. Recently, evidence for ecosystem services other than
biomass production and from ecosystems other than grasslands has begun to accumulate as
well. Important insights come from research on intercropping and agroforestry (Ewel 1986,
Fridley 2001, Harper 1977, Hector et al. 2002, Loreau 1998b, Smale et al. 1998, Trenbath
1974, Vandermeer 1990, Zhu et al. 2000).
11
Identifying the exact mechanisms by which experimental manipulation of species leads to
increased levels of ecosystem processes has led to substantial debate (Aarssen 1997, Garnier
et al. 1997, Hector et al. 2000, Huston 1997, Huston and McBride 2002, Huston et al. 2000,
Schmid et al. 2002, van der Heijden et al. 1999, Wardle 1999), as many experiments were
originally designed to test general patterns, rather than to test the underlying mechanisms.
The Insurance Value of Biodiversity
179
species composition, i.e. which species and functional traits are lost and
remain behind, as by species richness, i.e. how many species are lost.
• Patterns of response to experimental manipulation of species richness
vary for different ecosystem processes and services, different ecosystems,
and even different compartments within ecosystems.
• Varying multitrophic diversity and composition, i.e. the diversity and
composition of an ecological community at more than one trophic level,
can lead to more idiosyncratic behavior than varying diversity of primary
producers alone.
The different patterns found under experimental conditions may or may not
reflect actual patterns seen for a particular ecosystem under a particular scenario of species loss or invasion, which will depend not only on the functional
traits of the species involved, but also on the exact pattern of environmental
change and the species traits that determine how species respond to changes
in environmental conditions (Lavorel and Garnier 2002, Schläpfer et al. 2005,
Symstad and Tilman 2001).
9.2.2
Biodiversity May Enhance Ecosystem Stability
The debate about whether (or not) biodiversity enhances ecosystem stability,
i.e. whether (or not) ecosystem properties are more stable in response to environmental fluctuations as diversity increases, has a long tradition in ecology
(McCann 2000). This so-called ‘diversity-stability-debate’ has been initiated
in the 1950s by observations from natural ecosystems which were found to be
more productive and more stable when more diverse (Elton 1958, Odum 1953,
MacArthur 1955). This early diversity-stability-hypothesis has been shaken in
the early 1970s by computer simulations of ecosystems which demonstrated
that these systems were more unstable when more diverse (May 1972, 1974).
However, because the simulated model systems were randomly and purely fictional, the diversity-stability-question for real ecosystems remained open.12 In
the 1990s, the debate gained new momentum and research was organized and
discussed more systematically, with results coming from controlled laboratory
experiments, field studies and theoretical analysis.
The diversity-stability-debate is generally clouded by inconsistent terminology, as ‘stability’ is an umbrella term that denotes a large number of potential
12
The simulated model systems in the analysis of May (1972, 1974) were randomly constructed by putting together a given number of system elements (species) and, in particular,
linking them by randomly assigned interaction strengths which were taken from a uniform
distribution over all possible interaction strengths. This is in contrast to recent empirical
evidence that in real ecosystems the vast majority of pairwise interactions are weak (Paine
1992, Wootton 1997, McCann et al. 1998).
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Natural Science Constraints in Environmental and Resource Economics
phenomena, including, but not limited to, resistance to disturbance, resilience
to disturbance, temporal variability in response to fluctuating abiotic conditions, and spatial variability in response to differences in either abiotic conditions or the biotic community (Chesson 2000, Chesson et al. 2002, Cottingham
et al. 2001, Grimm and Wissel 1997, Holling 1986, Lehmann and Tilman 2000,
Loreau et al. 2002a, May 1974, McNaughton 1993, Peterson et al. 1998, Pimm
1984). Most research so far has focused on temporal variability, but some of
the results may also apply to other measures of ecosystem stability.
Theory, both via simple ecological reasoning and via mathematical models,
has lead to the understanding that a diversity of species with different sensitivities to a suite of environmental conditions should lead to greater stability of
ecosystem properties. The basic idea is that with increasing number of functionally different species, the probability increases that some of these species
can react in a functionally differentiated manner to external disturbance of the
system and changing environmental conditions. In addition, the probability
increases that some species are functionally redundant, such that one species
can take over the role of another species when the latter goes extinct. This
is what ecologists have been calling an ‘insurance effect’ of biodiversity in carrying out ecological processes (Borrvall et al. 2000, Elton 1958, Chapin and
Shaver 1985, Hooper et al. 2002, Lawton and Brown 1993, MacArthur 1955,
Naeem 1998, Naeem and Li 1997, Petchey et al. 1999, Trenbath 1999, Walker
1992, Walker et al. 1999, Yachi and Loreau 1999).13 With this logic, processes
that are carried out by a relatively small number of species are hypothesized to
be most sensitive to changes in diversity (Hooper et al. 1995). Also, the number of species or functional traits necessary to maintain ecosystem processes
under changing environmental conditions increases with spatial and temporal
scales (Casperson and Pacala 2001, Chesson et al. 2002, Field 1995, Pacala and
Deutschman 1995).
Several mathematical models generally support these hypotheses (see McCann 2000, Cottingham et al. 2001, Loreau et al. 2002a for reviews) and highlight the role of statistical averaging – the so-called ‘portfolio effect’ – for the
result (Doak et al. 1998, Tilman et al. 1998): if species abundances are negatively correlated or vary randomly and independently from one another, then
overall ecosystem properties are likely to vary less in more diverse communities
than in species-poor communities.14 The strength of the modeled effects of di13
In such cases, there is compensation among species: as some species do worse, others do
better due to differences in their functional traits. As a result, unstable individual populations stabilize properties of the ecosystem as a whole. Hence, instability of the community
composition is no contradiction to, but may actually support stability of ecosystem processes
(Ernest and Brown 2001, Hughes and Roughgarden 1998, Ives et al. 1999, Landsberg 1999,
Lehman and Tilman 2000, McNaughton 1977, Tilman 1996, 1999, Walker et al. 1999).
14
This is similar to the effect of diversifying a portfolio of financial assets, e.g. stocks.
The Insurance Value of Biodiversity
181
versity depends on many parameters, including the degree of correlation among
different species’ responses (Chesson et al. 2002, Doak et al. 1998, Lehman and
Tilman 2000, Tilman 1999, Tilman et al. 1998, Yachi and Loreau 1999), the
evenness of distribution among species’ abundances (Doak et al. 1998), and the
extent to which the variability in abundances scales with the mean (Cottingham
et al. 2001, Tilman 1999, Yachi and Loreau 1999).15
While theory is well developed and predicts that increased diversity will
lead to lower variability of ecosystem properties under those conditions in which
species respond in a differentiated manner to variations in environmental conditions, it cannot tell us how important the underlying basic mechanisms are
in the real world or whether they saturate at high or low levels of species
richness. This requires experimental investigations. However, controlled experiments are very difficult to carry out, because one needs to make sure that
the effect of species diversity is not confounded by other variables, such as
e.g. soil fertility or disturbance regime. Nevertheless, considerable evidence exists from field studies in a variety of ecosystems that in diverse communities,
redundancy of functional traits and compensation among species can buffer
ecosystem processes in response to changing conditions and species loss. Examples include studies of arctic tundra (Chapin and Shaver 1985), Minnesota
grasslands (Tilman 1996, 1999, Tilman et al. 2002), deserts (Ernest and Brown
2001), lakes (Frost et al. 1995, Schindler et al. 1986), and soil ecosystems (de
Ruiter et al. 2002, Griffiths et al. 2000, Ingham et al. 1985, Liiri et al. 2002).
As an example, Figure 9.2 shows experimental results for aboveground plant
biomass production in response to climatic variability in a Minnesota grassland (Figure 9.2A), and net ecosystem CO2 flux in a microbial microcosm
(Figure 9.2B). While the overall stability patterns found are as predicted from
theory, the experiments so far give little insights about the underlying basic
mechanisms. Also, mechanisms other than compensation among species can
affect stability in response to changing species richness.
Several experiments that manipulate diversity in the field and in microcosms generally support theoretical predictions that increasing species richness increases stability of ecosystem properties. For instance, stability of plant
production, as measured by resistance and/or resilience to nutrient additions,
drought and grazing, increased with the Shannon-Wiener index of diversity16
in a variety of successional and herbivore-dominated grasslands (McNaughton
1977, 1985, 1993). And in Minnesota grasslands, resistance to loss of plant productivity to drought increased with increasing plant species richness (Tilman
15
It is generally acknowledged that the underlying assumptions of the mathematical models
as to these parameters need further investigation and more experimental confirmation. Also,
the role of the stability measures used and other mechanisms built into the models (such as
e.g. overyielding) need further clarification.
16
See Footnote 5.
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Natural Science Constraints in Environmental and Resource Economics
Figure 9.2 Ecological experiments found that species richness may decrease
the variability of ecosystem services, such as e.g. aboveground plant biomass
production in response to climatic variability in a Minnesota grassland (A), or
net ecosystem CO2 flux in a microbial microcosm (B). (Figures are taken from
Tilman 1999 [A] and McGrady-Steed et al. 1997 [B], as compiled by Hooper et
al. 2005 and Loreau et al. 2001.)
and Downing 1994). However, results of these experiments may be confounded
by a variety of variables other than species richness or diversity, which has
raised considerable controversy over the interpretation of these results (e.g.
Givnish 1994, Grime 1997, Grime et al. 2000, Huston 1997, Huston et al. 2000,
Pfisterer and Schmid 2002). Experiments in microcosms and grasslands suggest that increased species richness, either in terms of numbers of different
functional groups, or numbers of species within trophic functional groups, can
lead to decreased temporal variability in ecosystem properties (Emmerson et
al. 2001, McGrady-Steed et al. 1997, Naeem and Li 1997, Petchey et al. 1999,
Pfisterer et al. 2004; but see also Pfisterer and Schmid 2002). But while species
richness or the Shannon-Wiener-index of species diversity was statistically significant in all these experiments, species composition (where investigated) had
an at least equally strong effect on stability.
In sum, the experimental work provides qualified support for the hypothesis that species richness can increase the stability of ecosystem processes and
services, although the underlying mechanisms can differ from theoretical predictions and in many cases still need to be fully resolved (Loreau et al. 2001).
9.3
Ecological-Economic Model
In order to study the economic implications of the insights from ecology about
how biodiversity affects ecosystem functioning and the provision of ecosystem
services, I shall cast them into a simple and stylized ecological-economic model.
The Insurance Value of Biodiversity
9.3.1
183
Biodiversity and the Provision of Ecosystem Services
For notational simplicity, consider only one ecosystem service and let s be the
amount generated of that service. As an example, think of insects providing
pollination service to an orchard farmer. Because of environmental stochasticity
the level s, at which the ecosystem service is provided, is a random variable.
Assume, for analytical simplicity (and lack of specific ecological evidence on this
point), that s is normally distributed with mean μs and standard deviation σs .
As discussed in the previous section, ecological research provides evidence
that the level of biodiversity affects the statistical distribution of the ecosystem
service. Let v ∈ [0, ∞] be an appropriate index of biodiversity.17 The two
stylized facts about the relationship between biodiversity and the provision of
ecosystem services, which emerged from ecological research (cf. Section 9.2),
can then formally be expressed as:
μs = μs (v)
σs = σs (v)
with
with
μs (v) > 0 ,
σs (v) < 0 ,
μs (v) ≤ 0 ,
σs (v) ≥ 0 ,
(9.1)
(9.2)
where the prime denotes a derivative. That is, the mean level of ecosystem
service increases and the standard deviation decreases with the level of biodiversity. Both effects decrease in magnitude with the level of biodiversity.
While biodiversity, thus, is beneficial in a twofold manner – by increasing the
mean level, at which the ecosystem service is being provided, and by decreasing
its standard deviation – its provision is costly. Assume that the (direct and
opportunity) costs of biodiversity are given by a cost function
C(v) with C (v) > 0,
C (v) ≥ 0 .
(9.3)
In the example of an orchard farmer using insects’ pollination services, the costs
of biodiversity could result from setting aside land from agricultural cultivation
and leaving it in a natural state, so that hedges and wetlands can provide
habitat for insects.18
17
According to the discussion in the previous section, ‘biodiversity’ could in many instances
simply be measured by the number of different species (‘species richness’). However, the
discussion in the previous section also suggests that in some instances it should be measured
by a more sophisticated index which takes into account the functional traits and relative
abundances of different species as well as their interactions (see Footnote 5).
18
According to the well established species-area-relationships, the level of biodiversity v
increases with the area l of land as v ∼ lz , where z (with 0 < z < 1) is a characteristic
constant for the type of ecosystem (MacArthur and Wilson 1967, Rosenzweig 1995, Gaston
2000). Assuming constant per-hectare-costs of land, this leads to a strictly convex cost
function.
184
9.3.2
Natural Science Constraints in Environmental and Resource Economics
Ecosystem Manager
The ecosystem manager, who manages the system for the services s it provides,
chooses the level of biodiversity v ∈ [0, ∞].19 On the one hand, the choice of
v implies costs as given by Equation (9.3). On the other hand, biodiversity is
essential for ecosystem functioning and the provision of ecosystem services. The
ecosystem manager has benefits from ecosystem services, B(s). For simplicity,
assume that:
B(s) = s .
(9.4)
Since ecosystem services s are a random variable (normally distributed with
mean μs and standard deviation σs ) and the level of biodiversity v determines
the distribution of this random variable according to (9.1) and (9.2), the benefits are also a random variable normally distributed with mean μs (v) and
standard deviation σs (v). The ecosystem manager’s net income y is then given
by
y = B(s) − C(v) = s − C(v) ,
(9.5)
which is a random variable normally distributed with mean μy and standard
deviation σy :
μy (v) = μs (v) − C(v) and
σy (v) = σs (v) .
(9.6)
(9.7)
Hence, by choosing the level of biodiversity v, the ecosystem manager chooses
a particular (normal) distribution N(μy (v), σy (v)) of net income. That is, he
chooses a particular income ‘lottery’ (Crocker and Shogren 2001).
The ecosystem manager’s preferences over his uncertain net income y are
represented by a von Neumann-Morgenstern expected utility function
U = E[u(y)],
(9.8)
where E is the expectancy operator and u(y) is a Bernoulli utility function
which is assumed to be increasing (u > 0) and strictly concave (u < 0), i.e.
the ecosystem manager is non-satiated and risk averse.20 In order to obtain
19
Of course, it is a major simplification to assume that one can directly choose a certain
level of biodiversity. Actually, one would choose some instrumental variable, such as area of
protected land, or investment in some species protection/recovery plan, which then results
in a certain level of biodiversity. Chapter 11 deals in more detail with the question of how
to attain a certain level of biodiversity.
20
While risk-aversion is a natural and standard assumption for farm households (Besley
1995, Dasgupta 1993: Chapter 8), it appears as an induced property in the behavior of
(farm) companies which are fundamentally risk neutral but act as if they were risk averse
when facing e.g. external financing constraints or bankruptcy costs (Caillaud et al. 2000,
Mayers and Smith 1990).
The Insurance Value of Biodiversity
185
simple closed-form solutions, assume that u(y) is the constant absolute risk
aversion Bernoulli utility function
u(y) = −e−ρ y ,
(9.9)
where ρ > 0 is a parameter describing the ecosystem manager’s Arrow-Pratt
measure of risk aversion (Arrow 1965, Pratt 1964). The ecosystem manager’s
von Neumann-Morgenstern expected utility function (9.8) is then given by (see
Appendix A9.1)
ρ
U = μy − σy2 ,
(9.10)
2
which is the simplest expected utility function of the mean-variance type.
9.3.3
Financial Insurance
In order to analyze the influence of availability of financial insurance products
on the ecosystem manager’s choice of biodiversity (in Section 9.4.4), financial
insurance is introduced in a simple and stylized way.21 I assume that the
manager does or does not have the option of buying financial insurance under
the following contract:
• The insurant chooses the fraction a ∈ [0, 1] of insurance coverage.
• He receives (pays)
a (s − μs )
(9.11)
from (to) the insurance company as an actuarially fair indemnification
benefit (risk premium) if his realized income is below (above) the mean
income.22
• In addition, he pays a mark-up for the transaction costs of insurance and
the insurance company’s profit:
δ 2 2
a σs ,
2
(9.12)
where δ ≥ 0 is a parameter describing how actuarially unfair is the insurance contract. Thus, the costs of insurance over and above the actuarially
21
This stylized insurance institution is a special case of the one studied by Quaas and
Baumgärtner (2006).
22
This benefit/premium-scheme is actuarially fair, because the insurance company has
an expected net payment stream of E[a(s − s̄)] = 0. To the insurant, this actuarially fair
benefit/premium-scheme does not come at any real costs, as E[a(s − s̄)] = 0. It is fully
equivalent to the traditional model of insurance (e.g. Ehrlich and Becker 1972: 627) where
losses compared with the maximum income are insured against and one pays a constant
insurance premium irrespective of actual income.
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Natural Science Constraints in Environmental and Resource Economics
fair risk premium, which are a measure of the ‘real’ costs of insurance to
the insurant,23 are assumed to follow a quadratic cost function.
This is a highly idealized form of financial insurance which captures in the
most simple way the essence of financial insurance with an actuarially fair risk
premium and some mark-up (due to transaction costs and profits) on top. The
higher the insurance coverage a, the lower the effective income risk; and the
effective income risk can be continuously reduced down to zero by increasing a
to one. This follows the ‘Venetian Merchant’-model of insurance: there exists
an insurance company (the ‘Venetian Merchant’) which is ready to (fully or
partially) take over the income risk from the insurant. In order to abstract from
any problems related to informational asymmetry I assume that the statistical
distribution N(μs , σs ) and actual level s of ecosystem service are observable to
both insurant and insurance company.
9.4
Analysis and Results
When analyzing the insurance value of biodiversity (Section 9.4.2), the optimal
allocation of biodiversity (Section 9.4.3), and the effect of different institutional
settings in the market for financial insurance products on biodiversity protection (Section 9.4.4), the idea is to treat the level of biodiversity v as the choice
variable and to analyze the choice of biodiversity as the choice of an income
lottery.
9.4.1
The Choice Set
To start with, neglect the option to buy financial insurance and focus on biodiversity as the natural insurance. Financial insurance will be taken into account
in Section 9.4.4. As v can range from zero to infinity, the resulting feasible
and efficient distributions of net income y (Equation 9.5) in μy -σy -space can
be depicted by an income-possibility-frontier as in Figure 9.3. Income distributions above the income-possibility-frontier are not feasible; income distributions
below the income-possibility-frontier may be feasible, but are not efficient.
The right hand end of the curve corresponds to very low levels of biodiversity v: the standard deviation σy of income is high. As v increases, one
moves left along the curve: the standard deviation of income is reduced due
to the stabilizing effect of biodiversity (Equations 9.2 and 9.7) and the mean
income increases, because the mean level of ecosystem service increases with
23
Since the actuarially fair risk premium does not cause any expected payoff/costs to the
insurant, only the price component over and above the actuarially fair risk premium (the
so-called ‘loading’ of the premium) constitutes real costs of insurance to the insurant (Ehrlich
and Becker 1972: 626-627).
The Insurance Value of Biodiversity
187
μy 6
v→∞
v=0
?-
σy
Figure 9.3 Feasible and efficient distributions of net income y (Equation 9.5)
in μy -σy -space are represented by the income possibility frontier (solid line).
The vertical line separates the domain with a trade-off between mean and standard deviation of income (left) from the domain without such a trade off (right).
biodiversity while the costs of biodiversity are not too important at low levels of biodiversity (Equations 9.1, 9.3 and 9.6). As the level v of biodiversity
increases further, i.e. moving left along the curve even further, the additional
mean benefits from additional ecosystem service become smaller and smaller
(Equation 9.1) while the additional costs of biodiversity become greater and
greater (Equation 9.3), thus eventually causing additional mean net benefits
y from biodiversity to become negative. This corresponds to the left hand
end of the curve: as biodiversity v increases (i.e. moving left along the curve)
the standard deviation σy of income still decreases while the mean income μy
decreases.
Overall, the income possibility frontier in μy -σy -space has two parts: in
the left hand part (corresponding to high levels v of biodiversity) the mean
income μy increases with increasing standard deviation σy ; in the right hand
part (corresponding to low levels v of biodiversity) the mean income μy decreases with increasing standard deviation σy . Given the ecosystem manager’s
expected utility function (9.10), according to which a high mean income and a
low standard deviation of income are desirable, this means that for low levels
of biodiversity there does not exist any economic problem. For, increasing the
level of biodiversity at low v (right hand part of the curve) has a double desirable effect: it increases the mean income and it reduces the standard deviation
of income. In contrast, for high levels of biodiversity (left hand part of the
curve) when (opportunity) costs of biodiversity become important, the ecosystem manager faces a trade-off: increasing the level of biodiversity reduces the
standard deviation of income, but reduces mean income, too.
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Natural Science Constraints in Environmental and Resource Economics
It is the left part of the curve which suggests the interpretation that biodiversity provides insurance. As with buying financial insurance, increasing
the level of biodiversity reduces the standard deviation of income, but reduces
mean income, too. In this domain, a choice has to be made in order to optimally balance the two opposing goals of a high mean income and a low standard
deviation of income.
9.4.2
The Insurance Value of Biodiversity
In order to precisely define the insurance value of biodiversity, let me come back
to the idea that the ecosystem can be seen as an infinite set of lotteries (Crocker
and Shogren 2001). By choosing the level of biodiversity v, the ecosystem manager determines the distribution N(μs (v), σs (v)) of ecosystem service (Equations 9.1 and 9.2), which then determines the distribution N(μy (v), σy (v)) of
income (Equations 9.6 and 9.7). Thus, by choosing the level of biodiversity
v, he chooses a particular income lottery. In the model employed here, this
lottery is uniquely characterized by the level of biodiversity v. Therefore, one
may speak of ‘the lottery v’.
One standard method of how to value the riskiness of a lottery to a decision
maker is to calculate the risk premium R of the lottery, which is defined by
(e.g. Kreps 1990, Varian 1992: 181)24
u (E[y] − R) = E [u(y)] .
(9.13)
The risk premium R is the amount of money that leaves a decision maker
equally well-off, in terms of utility, between the two situations of (1) receiving
for sure the expected pay-off from the lottery E[y] minus the risk premium R,
and (2) playing the risky lottery with random pay-off y.25 In general, if the
utility function u characterizes a risk averse (risk neutral, risk loving) decision
maker, the risk premium R is positive (zero, negative).
In the model employed here the risk premium of the lottery v depends on
the level of biodiversity and is given by (see Appendix A9.2)
ρ
R(v) = σs2 (v) .
(9.14)
2
The insurance value of biodiversity can now be defined based on the risk premium of the lottery v (Baumgärtner and Quaas 2005).
24
By Equation (9.13), E[y] − R is the certainty equivalent of lottery v, as it yields the
expected utility E [u(y)]. According to Equations (9.3) and (9.5) y ∈ Y with Y as an interval
of IR, and according to Equation (9.9) u is continuous and strictly increasing, so that a risk
premium R uniquely exists for every lottery v (Kreps 1990: 84).
25
In the simple model employed here, the risk premium is equivalent to the so-called ‘option
price’ of risk reduction, that is, the amount of money that a decision maker would be willing
to pay for getting the expected pay-off from the lottery, E[y], for sure instead of playing the
risky lottery with random pay-off y.
The Insurance Value of Biodiversity
189
Definition 9.1
The insurance value V of biodiversity v is given by the change of the risk
premium R of the lottery v due to a marginal change in the level of biodiversity
v:
V (v) := −R (v) .
(9.15)
Thus, the insurance value of biodiversity is the marginal value of biodiversity in its function to reduce the risk premium of the ecosystem manager’s
income risk from using ecosystem services under uncertainty. Being a marginal
value, it depends on the existing level of biodiversity v. The minus sign in
the defining Equation (9.15) serves to express biodiversity’s ability to reduce
the risk premium of the lottery v as a positive value. Applying Definition 9.1
to Equation (9.14), one obtains the following result for the insurance value of
biodiversity in this model.
Proposition 9.1
The insurance value V (v) of biodiversity is given by
V (v) = −ρ σs (v) σs (v) > 0 .
(9.16)
From this equation it is apparent that the insurance value of biodiversity has
an objective and a subjective dimension. The objective dimension is captured
by the sensitivity of the standard deviation of ecosystem services to changes in
biodiversity, σs and σs ; the subjective dimension is captured by the ecosystem
manager’s degree of risk aversion, ρ. The insurance value V increases
• with the degree ρ of the ecosystem manager’s risk aversion and
• with the sensitivity of the standard deviation of ecosystem services to
changes in biodiversity, σs and |σs2 |.
As a function of biodiversity v, the insurance value V (v) decreases (Figure 9.4):
as biodiversity becomes more abundant (scarcer), its insurance value decreases
(increases).
9.4.3
The Optimal Level of Biodiversity
In order to study how the ecosystem manager will make use of the insurance
function of biodiversity, consider first the situation in which there is no financial
insurance available. The ecosystem manager chooses a level of biodiversity v
such as to maximize his expected utility (9.10):
max U(v) .
v
(9.17)
With no financial insurance available, income y is given by Equation (9.5), such
that the mean income μy and the standard deviation of income σy are given
by Equations (9.6) and (9.7). The following proposition states the properties
of the optimal solution to problem (9.17).
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Natural Science Constraints in Environmental and Resource Economics
V 6
-
v
Figure 9.4 The insurance value V of biodiversity (Equation 9.16) as a function
of existent biodiversity v.
Proposition 9.2
(i) The optimal level of biodiversity v , which solves the ecosystem manager’s
optimization problem (9.17), is characterized by the necessary and sufficient condition
μs (v ) + V (v ) = C (v ) .
(9.18)
(ii) The higher the ecosystem manager’s degree of risk aversion ρ, the higher
the optimal level of biodiversity v :
dv >0.
dρ
(9.19)
Proof: see Appendix A9.3.
Condition (9.18) states that the optimal level of biodiversity v is chosen
such that the marginal benefits of biodiversity equal its marginal costs. The
marginal benefits here are composed of two additive components: the marginal
gain in the mean level of ecosystem service and the insurance value V (v ) of
biodiversity. Hence, the insurance value of biodiversity is a value component in
addition to the usual value arguments (such as direct or indirect use or non-use
values, or existence values)26 which hold in a world of certainty. It leads to
choosing a higher level of biodiversity than without taking the insurance value
into account.
The second part of the proposition states that the higher the degree of risk
aversion ρ, the higher the optimal level of biodiversity v . This is intuitively
obvious, and confirms the idea that biodiversity is being used by a risk averse
ecosystem manager as a form of natural insurance.
26
See the detailed discussion on the economic value of biodiversity in Section 7.3.
The Insurance Value of Biodiversity
9.4.4
191
The Effect of Financial Insurance
Consider now the situation in which there is financial insurance available. As
an example, think again of the orchard farmer, who crucially depends on the
pollination service provided by insects and who can manage his agro-ecosystem
by choosing the level of biodiversity, e.g. by setting aside land for hedges and
wetlands. As we have seen above, this farmer can manage his income risk from
the random level of ecosystem service by choosing the level of biodiversity.
On the other hand, the farmer may also have access to commercial crop yield
insurance. Hence, his risk management now comprises two instruments. The
ecosystem manager chooses a level of biodiversity v and a fraction of financial
insurance coverage a such as to maximize his expected utility (9.10):
max U(v, a) ,
(9.20)
v, a
Income y is now given by
y = s − C(v) − a(s − μs (v)) −
δ 2 2
a σs (v) .
2
(9.21)
The first two components represent the benefits and costs of ecosystem management (Equation 9.5), the third component is the actuarially fair insurance
premium/indemnification benefit (Equation 9.11) and the fourth component
are the real costs of financial insurance (Equation 9.12). While the real costs
of both ecosystem management and financial insurance (i.e. the second and
fourth component) are certain, the benefits (i.e. the first and third component)
are random. As a result, the mean and standard deviation of income are given
by
μy (v, a) = μs (v) − C(v) −
σy (v, a) = (1 − a) σs (v) .
δ 2 2
a σs (v) and
2
(9.22)
(9.23)
Since the actuarially fair insurance premium/indemnification benefit corresponds to an expected payment of exactly zero, the mean income (Equation 9.22) is given by the mean benefits of ecosystem service minus the real
costs of ecosystem management and financial insurance. The standard deviation of income (Equation 9.23) is given by the standard deviation of ecosystem
service, reduced by a factor of 0 ≤ (1 − a) ≤ 1. This should be compared to
the case without financial insurance, where the standard deviation of income
is given by the full standard deviation of ecosystem service (Equation 9.7).
Equation (9.23) expresses the fact that the ecosystem manager can reduce the
standard deviation of his income, besides by increasing the level of biodiversity
v and thus lowering σs (v), by increasing the fraction a of financial insurance
coverage. In the extreme, with full coverage by financial insurance (a = 1) the
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Natural Science Constraints in Environmental and Resource Economics
standard deviation of income vanishes. With (9.22) and (9.23), the expected
utility (9.10) is given by
ρ
U(v, a) = μy (v, a) − σy2 (v, a)
2
δ
ρ
= μs (v) − C(v) − a2 σs2 (v) − (1 − a)2 σs2 (v) .
2
2
(9.24)
The following proposition states the properties of the optimal solution to problem (9.20).
Proposition 9.3
(i) The optimal level of biodiversity v̂ and the optimal fraction of financial
insurance coverage â, which solve the ecosystem manager’s optimization
problem (9.20), are characterized by the necessary and sufficient conditions
μs (v̂) +
â =
δ
V (v̂) = C (v̂)
ρ+δ
and
ρ
ρ+δ
(9.25)
(9.26)
(ii) The higher the real costs of financial insurance, as measured by δ, the
lower the optimal fraction â of coverage by financial insurance and the
higher the optimal level v̂ of biodiversity:
dâ
<0
dδ
and
dv̂
>0.
dδ
(9.27)
(iii) A risk averse ecosystem manager chooses
– full coverage by financial insurance (â = 1) if δ = 0,
– partial coverage by financial insurance (0 < â < 1) if 0 < δ < +∞,
and
– no coverage by financial insurance (â → 0) if δ → +∞.
(iv) A risk averse ecosystem manager chooses v̂ < v .
Proof: see Appendix A9.4.
The optimal allocation of biodiversity v̂ and financial insurance coverage
â is characterized by Conditions (9.25) and (9.26). Condition (9.25) states
– similarly to Condition (9.18) in the absence of financial insurance – that
the optimal level of biodiversity v̂ is chosen such that the marginal benefits
of biodiversity equal its marginal costs. The marginal benefits, again, are
The Insurance Value of Biodiversity
193
composed of two additive components: the marginal gain in the mean level of
ecosystem service and the natural insurance value V (v̂) of biodiversity, which is,
however, not fully taken into account but only to a fraction δ/(ρ+δ) < 1. That
is, biodiversity’s natural insurance function is only partly taken into account
when determining the optimal allocation.
The reason is that, of course, part of the income risk is now covered by
financial insurance. Condition (9.26) specifies the optimal level of financial
insurance coverage. It is obvious that â and the factor in front of V (v̂) in
Condition (9.25) add up to one. This means, biodiversity as the natural form
of insurance and financial insurance together provide the optimal coverage of
income risk.27 Indeed, the two forms of insurance are substitutes: whatever
part of the risk is not covered by biodiversity is covered by financial insurance.
And what part of the risk is covered by financial insurance is determined by the
real costs of financial insurance. Part (ii) of the proposition details this result:
the higher the real costs of financial insurance, i.e. costs over and above the
actuarially fair risk premium, the lower is the fraction of income risk covered
by financial insurance and the higher is the fraction covered by the natural
insurance, i.e. biodiversity.
Part (iii) of the proposition describes this in more detail. A risk averse
ecosystem manager (ρ > 0) chooses full coverage by financial insurance (â = 1)
if it is available at actuarially fair conditions (δ = 0); he chooses only partial
coverage by financial insurance (0 < â < 1) if financial insurance comes at additional costs over and above the actuarially fair risk premium (0 < δ < +∞);
and he chooses no coverage by financial insurance (â → 0) if financial insurance
becomes infinitely costly (δ → +∞). These three cases imply, respectively, that
the fraction of biodiversity’s insurance value V (v̂) which is taken into account
according to condition (9.25), which is also the fraction of income risk covered
by the natural insurance of biodiversity, is zero if financial insurance is available at actuarially fair conditions; it is in between zero and one if financial
insurance is available at actuarially unfair conditions; and it goes to one for
infinitely unfair financial insurance.
Part (iv) of the proposition states that in any case, a risk averse ecosystem
manager chooses a lower level of biodiversity if financial insurance is available
compared to a situation where no financial insurance is available: v̂ < v . That
is, financial insurance crowds out biodiversity as the natural from of insurance.
27
Note that this does not necessarily mean that in the optimal allocation there is no more
income risk, i.e. σy2 (v̂, â) = 0. It only means that the overall amount of income variance that
the decision maker wishes to avoid in the optimum is covered by both natural and financial
insurance. This may still leave the decision maker with some positive income risk in the
optimum, i.e. σy2 (v̂, â) > 0.
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Natural Science Constraints in Environmental and Resource Economics
9.5
Discussion
Although the results have been derived from a very simple and specific model,
they are robust to a fair amount of generalization. For instance, while the
choice of the preference representation (9.9) served to obtain simple closedform solutions, all results thus obtained are qualitatively robust to generalizations to expected utility functions of the type U(μy , σy2 ) with ∂U/∂μy > 0
and ∂U/∂σy2 < 0. Also, while the specific form of financial insurance contract
assumed here (Section 9.3.3) served to obtain simple closed-form solutions, all
results thus obtained are qualitatively robust to generalizations to more general
financial insurance contracts with an actuarially fair insurance premium plus a
transaction costs/profit mark-up on top (Quaas and Baumgärtner 2006). And
while I have assumed for simplicity that the level of biodiversity is the only determinant of the statistical distribution of the ecosystem service (Equations 9.1,
9.2), one could easily generalize the analysis so that the stochastic production
of the ecosystem service depends also on inputs other than biodiversity, say labor, capital, fertilizer or chemical pest control. This could be formalized with
the help of a Just-Pope-production function (Just and Pope 1978, 1979), which
is well suited for mean-variance-analysis of stochastic production, and would
not qualitatively alter the basic results about the role of biodiversity for income
risk.
Of particular importance are Assumptions (9.1), (9.2) and (9.3) about the
benefits and costs of biodiversity. While Assumptions (9.1) and (9.2) represent
the best available ecological knowledge and describe a relevant problem, it is
an interesting question whether these assumptions are actually necessary in order to arrive at the main result (i.e. biodiversity’s insurance value) or whether
this result holds under more general conditions. It turns out that the crucial
assumption is σs < 0, while μs > 0 is not necessary. If biodiversity did depress
the mean level of ecosystem services (μs < 0) then this could be considered as
costs of biodiversity and could be included in the function C(v). This assumption would therefore not lead to a different result. If, however, biodiversity did
increase the variance of ecosystem services (σs > 0) then it would obviously not
have any insurance value. Clearly, this would fundamentally alter the main results of the paper. As for the assumption on second derivatives (μs ≤ 0, σs ≥ 0,
C ≥ 0), their role is mainly technical, making sure that second order conditions
are fulfilled and that one has an interior solution. Without these assumptions,
the main results would not change fundamentally, but would require a more
elaborate formulation and proof of results.
So, the crucial assumptions which ultimately limit the generality of results
are the following:
• The ecosystem manager is risk averse and maximizes his expected utility
The Insurance Value of Biodiversity
195
from an uncertain income which is determined by the random level of
some ecosystem service.
• The level of biodiversity determines the probability distribution of the
ecosystem service and, thus, of income. Taking into account the (direct or
opportunity) costs of biodiversity, there is a positive correlation between
expected income and standard deviation of income in the relevant range
of feasible income distributions.
• A financial insurance contract specifies only the state dependent redemption payment and the corresponding risk premium. In particular, it is
not explicitly contingent on the particular level of biodiversity chosen by
the ecosystem manager.28
• Both insurant and insurance company have the same ex ante knowledge
about the probability distribution of ecosystem services. Both can observe ex post the actual state of nature.
While these assumptions limit the generality of the results obtained here, they
describe – in a very stylized way – a realistic scenario of managing stochastic
ecosystems under uncertainty for the ecosystem services they provide. Hence,
this analysis yields relevant insights into the issue.
9.6
Conclusion
I have presented a conceptual ecological-economic model that combines (i)
ecological results about the relationships between biodiversity, ecosystem functioning, and the provision of ecosystem services with (ii) economic methods to
study decision-making under uncertainty. In this framework I have (1) determined the insurance value of biodiversity, (2) studied the optimal allocation
of funds in the trade-off between investing into biodiversity protection and the
28
This gives rise to what is known in the insurance economics literature as ‘moral hazard’
(Kreps 1990). As long as the behavior of the economic agent (here: the level of biodiversity
chosen be the ecosystem manager) cannot be observed by the insurance company, but only
the resulting outcome can be observed (here: the provision of some ecosystem service), the
existence of insurance will induce the insurant to choose a riskier behavior than if insurance
was not available. Moral hazard is a problem for many insurance markets, e.g. health insurance or car insurance, and has been identified as a major reason for the absence of private
insurance markets for most agricultural risks (Chambers 1989). Because of the moral hazard
problem, most insurance contracts intentionally do not allow for full coverage at actuarially
fair premiums, but contain deductibles or upper limits in either the degree of coverage or
the amount to be insured. Other insurance policies try to include a specification of the insurant’s behavior (or observable proxies thereof) into the contract. These mechanisms serve
to diminish the moral hazard problem, yet they cannot eliminate it completely.
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Natural Science Constraints in Environmental and Resource Economics
purchase of financial insurance, and (3) analyzed the effect of different institutional settings in the market for financial insurance on biodiversity protection.
The focus was on how to model the ecology-economy-interface. Relevant economic and policy questions that arise from this view on biodiversity – e.g.
the public good character of the problem, the dynamics of the problem or implications for environmental and development policies – are discussed in more
detail elsewhere (Baumgärtner and Quaas 2005, Quaas and Baumgärtner 2006,
Quaas et al. 2004).
The conclusion from this analysis is that biodiversity can be interpreted as
a form of natural insurance for risk averse ecosystem managers against the overor under-provision with ecosystem services, such as biomass production, control
of water run-off, pollination, control of pests and diseases, nitrogen fixation, soil
regeneration etc. Thus, biodiversity has an insurance value, which is a value
component in addition to the usual value arguments (such as direct or indirect
use or non-use values, or existence values) holding in a world of certainty. This
insurance value should be taken into account when deciding upon how much
to invest into biodiversity protection. It leads to choosing a higher level of
biodiversity than without taking the insurance value into account, with a higher
degree of risk aversion leading to a higher optimal level of biodiversity. As far
as the insurance function is concerned, biodiversity and financial insurance
against income risk, e.g. crop yield insurance, may be seen as substitutes. If
financial insurance is available, a risk averse ecosystem manager, say, a farmer,
will partially or fully substitute biodiversity’s insurance function by financial
insurance, with the extent of substitution depending on the costs of financial
insurance. Hence, the availability, and exact institutional design, of financial
insurance influence the level of biodiversity protection.
Appendix
A9.1
With
Expected Utility Function (9.10)
2
(y−μy )
−
1
2
2σy
f (y) = &
e
(A9.1)
2πσy2
as the probability density function of the normal distribution of income y with
mean μy and variance σy2 , the von (Neumann-Morgenstern) expected utility
from the (Bernoulli) utility function (9.9) is
ρ 2
(A9.2)
Ũ = E[u(y)] = − e−ρ y f (y)dy = −e−ρ [μy − 2 σy ] .
Using a simple monotonic transformation of Ũ , one obtains the expected utility
function U (Equation 9.10).
The Insurance Value of Biodiversity
A9.2
197
Risk Premium (9.14)
The risk premium R has been defined in Equation (9.13) as
u (E[y] − R) = E [u(y)] .
(A9.3)
With the Bernoulli utility function (9.9) and E[y] = μy the left hand side of
this equation is given by
u (E[y] − R) = −e−ρ [μy −R] ,
(A9.4)
and the right hand side is given by Equation (A9.2). Hence, we have
ρ
2
−e−ρ [μy −R] = −e−ρ [μy − 2 σy ] .
(A9.5)
Rearranging, and observing that σy2 = σs2 (Equation 9.7), yields the result
stated in Equation (9.14).
A9.3
Proof of Proposition 9.2
ad (i). In Problem (9.17), the objective function to be maximized over v is
ρ
U(v) = μs (v) − C(v) − σs2 (v) ,
2
(A9.6)
such that the first order condition for a solution v is
μs (v ) − C (v ) − ρ σs (v ) σs (v ) = 0 .
(A9.7)
Observing that −ρ σs (v ) σs (v ) = V (v ) (Equation 9.16) yields Equation (9.18).
With Assumptions (9.1), (9.2) and (9.3) about the curvature of these functions,
the second order condition for a maximum,
2
μs (v ) − C (v ) − ρ (σs (v )) − ρ σs (v ) σs (v ) < 0 ,
(A9.8)
is satisfied, such that the necessary first order condition is also sufficient.
ad (ii). The total derivative of first order condition (9.18) with respect to
ρ is
dv dv dv − C +V
− σs σs = 0 .
(A9.9)
μs
dρ
dρ
dρ
This can be rearranged into
σs σs
dv = >0,
dρ
μs − C + V (A9.10)
which is strictly positive due to Assumptions (9.1), (9.2), (9.3) and V < 0
(Proposition 9.1).
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Natural Science Constraints in Environmental and Resource Economics
A9.4
Proof of Proposition 9.3
ad (i). In Problem (9.20), the objective function to be maximized over v and
a is
U(v, a) = μs (v) − C(v) −
δ 2 2
ρ
a σs (v) − (1 − a)2 σs2 (v) ,
2
2
(A9.11)
such that the first order conditions for a solution (v̂, â) are
0,
Uv (v̂, â) = μs (v̂) − C (v̂) − δâ2 σs (v̂)σs (v̂) − ρ(1−â)2 σs (v̂)σs (v̂) =(A9.12)
2
2
(A9.13)
Ua (v̂, â) = −δâσs (v̂) + ρ(1 − â)σs (v̂) = 0 .
As σs2 (v) > 0 for all v, Condition (A9.13) can be solved to yield
â =
ρ
,
ρ+δ
(A9.14)
which is the result stated in the proposition (Equation 9.26). This can be
inserted into Condition (A9.12), which yields, after rearranging,
μs (v̂) +
δ
(−ρ σs (v̂) σs (v̂)) = C (v̂) .
ρ+δ
(A9.15)
Observing that −ρ σs (v̂) σs (v̂) = V (v̂) (Equation 9.16) yields Equation (9.25).
As for the second order condition, note that
2
Uvv (v̂, â) = μs (v̂) − C (v̂) − δ â2 (σs (v̂)) − δ â σs (v̂) σs (v̂)
2
−ρ (1 − â)2 (σs (v̂)) − ρ (1 − â)2 σs (v̂) σs (v̂) < 0 ,(A9.16)
Uaa (v̂, â) = −(δ + ρ)σs2 (v̂) < 0 ,
(A9.17)
(A9.18)
Uva (v̂, â) = −2δâ σs (v̂) σs (v̂) + 2ρ(1 − â) σs (v̂) σs (v̂) = 0 ,
where the last equality follows from using first order condition (A9.14). Hence,
2
Uvv Uaa −Uva
> 0, so that the second order condition for a maximum is satisfied
and the necessary first order conditions are also sufficient.
ad (ii). The total derivative of first order condition (9.25) with respect to
δ is
ρ
δ
dv̂
dv̂
dv̂
μs
+
V
− C =0,
(A9.19)
V +
2
dδ (ρ + δ)
ρ+δ
dδ
dδ
which can be rearranged into
ρ
V
dv̂
(ρ+δ)2
= − >0,
δ
dδ
μs + ρ+δ V − C (A9.20)
which is strictly positive due to Assumptions (9.1), (9.3) and V < 0 (Proposition 9.1). The result about dâ/dδ follows immediately from Condition (9.26).
The Insurance Value of Biodiversity
199
Part (iii) of the proposition follows immediately from Condition (9.26).
ad (iv). Compare Conditions (9.18) and (9.25) for v and v̂ respectively in
a slightly rearranged version:
μs (v ) − C (v ) =
μs (v̂) − C (v̂)
=
−V (v ) ,
δ
(−V (v )) .
ρ+δ
(A9.21)
(A9.22)
From Assumptions (9.1) and (9.3) it follows that μs (v) − C (v) is a decreasing
function of v, while it follows from Proposition 9.1 that −V (v) is an increasing function of v, so that v and v̂ are determined by the intersection of the
decreasing curve representing the left-hand-side and the increasing curve representing the right-hand side of Conditions (A9.21) and (A9.22) respectively.
The difference between these two conditions is that for every v the function on
the right-hand side of Condition (A9.22) yields smaller values than the one in
Condition (A9.21), as 0 < δ/(ρ + δ) < 1, so that the intersection determining the optimal v in Condition (A9.22) is further to the left than the one in
Condition (A9.21), i.e. v̂ < v .
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Natural Science Constraints in Environmental and Resource Economics
10. Insurance and Sustainability through
Ecosystem Management∗
with Christian Becker, Karin Frank, Birgit Müller
and Martin Quaas
10.1
Introduction
There is a widely held belief that individual myopic optimization is at odds
with long-term sustainability of an ecological-economic system. In this paper,
we want to take a fresh look at this position. We show that for typical ecosystems and under plausible and standard assumptions about individual decision
making, myopic optimization may lead to sustainable outcomes. In particular,
in order to explain the sustainable use of ecosystems, it is not necessary to
assume preferences for sustainability – or any special concern for the distant
future – on the part of the decision maker; it suffices to assume that a myopic
decision maker is sufficiently risk averse.
The ecological-economic system under study here is grazing in semi-arid
rangelands. Semi-arid regions cover one third of the Earth’s land surface.
They are characterized by low and highly variable precipitation. Their utilization in livestock farming provides the livelihood for a large part of the local
populations. Yet, over-utilization and non-adapted grazing strategies lead to
environmental problems such as desertification.
Grazing in semi-arid rangelands is a prime object of study for ecological
economics, as the ecological and economic systems are tightly coupled (e.g.
Beukes et al. 2002, Heady 1999, Janssen et al. 2004, Perrings 1997, Perrings
and Walker 1997, 2004, Westoby et al. 1989). The grass biomass is directly
used as forage for livestock, which is the main source of income; and the grazing
pressure from livestock farming directly influences the ecological dynamics. The
crucial link is the grazing management.
∗
Forthcoming in Ecological Economics as ‘Uncertainty and sustainability in the management of rangelands’.
201
202
Natural Science Constraints in Environmental and Resource Economics
The ecological dynamics, and thus, a farmer’s income, essentially depend on
the low and highly variable rainfall. The choice of a properly adapted grazing
management strategy is crucial in two respects: first, to maintain the rangeland
system as an income base, that is, to prevent desertification; and second, to
smooth out income fluctuations, in particular, to avoid high losses in the face
of droughts.
Assuming that the farmer is non-satiated in income and risk averse, we
analyze the choice of a grazing management strategy from two perspectives.
In a first step we determine a myopic farmer’s optimal grazing management
strategy. We show that a risk averse farmer chooses a strategy in order to obtain
‘insurance’ from the ecosystem (Baumgärtner and Quaas 2005). That is, the
optimal strategy reduces income variability, but yields less mean income than
possible. In a second step we analyze the long-term ecological and economic
impact of different strategies. We conclude that the more risk averse a myopic
farmer is, the more conservative is his optimal grazing management strategy.
If he is sufficiently risk averse, the optimal strategy is conservative enough to
be sustainable.
Following the literature on grazing management under uncertainty, we analyze the choice of a stocking rate of livestock, as this is the most important
aspect of rangeland management (e.g. Hein and Weikard 2004, Karp and Pope
1984, McArthur and Dillon 1971, Perrings 1997, Rodriguez and Taylor 1988,
Torell et al. 1991, Westoby et al. 1989). The innovative analytical approach
taken here is to consider the choice of a grazing management strategy, which is
a rule about the stocking rate to apply in any given year depending on the rainfall in that year. This is inspired by empirical observations in Southern Africa.
Rule-based grazing management has the twofold advantage that a farmer has
to make a choice (concerning the rule) only once, and yet, keeps a certain flexibility and scope for adaptive management (concerning the stocking rate). The
flexibility thus obtained is the decisive advantage of choosing a constant rule
over choosing a constant stocking rate.
The paper is organized as follows. In Section 10.2, we discuss grazing management in semi-arid rangelands in more detail and describe one particular
‘good practice’-example: the Gamis Farm, Namibia. In Section 10.3, we develop a dynamic and stochastic ecological-economic model, which captures the
essential aspects and principles of grazing management in semi-arid rangelands,
and features the key aspect of the Gamis-strategy. Our results are presented
in Section 10.4, with all derivations and proofs given in the Appendix. Section 10.5 concludes.
Insurance and Sustainability through Ecosystem Management
10.2
203
Grazing Management in Semi-Arid Rangelands:
The Gamis Farm, Namibia
The dynamics of ecosystems in semi-arid regions are essentially driven by low
and highly variable precipitation (Behnke et al. 1993, Sullivan and Rhode 2002,
Westoby et al. 1989).1 Sustainable economic use of these ecosystems requires
an adequate adaption to this environment. The only sensible economic use,
which is indeed predominant (Mendelsohn et al. 2002), is by extensive livestock farming. However, over-utilization and inadequate management lead to
pasture degradation and desertification. Rangeland scientists have proposed
different types of grazing management strategies in order to solve these problems. A low constant stocking rate was recommended by Lamprey (1983) and
Dean and Mac Donald (1994), who assumed that grazing pressure is the main
driving force for vegetation change and that rangeland systems reach an equilibrium state. Other authors considered the highly variable rainfall to be the
major driving force and claimed that grazing has only marginal influence on
vegetation dynamics (Behnke et al. 1993, Scoones 1994, Sandford 1994, Westoby et al. 1989). They recommend an ‘opportunistic’ strategy which matches
the stocking rate with the available forage in every year. Thus, the stocking
rate should be high in years with sufficient rainfall, and low when there is little forage in dry years (Beukes et al. 2002: 238). Recent studies have shown
that both grazing and variable rainfall are essential for the vegetation dynamics on different temporal and spatial scales (Cowling 2000, Briske et al. 2003,
Fuhlendorf and Engle 2001, Illius and O’Connor 1999, 2000, Vetter 2005).
One example of a sophisticated and particularly successful grazing management system has been employed for forty years at the Gamis Farm, Namibia
(Müller et al. forthcoming, Stephan et al. 1996, 1998a, 1998b). The Gamis
Farm is located 250 km southwest of Windhoek in Namibia (24◦ 05 S 16◦ 30 E)
close to the Naukluft mountains at an altitude of 1,250 m. The climate of this
arid region is characterized by low mean annual precipitation (177 mm/y) and
high variability (variation coefficient: 56 %). The vegetation type is dwarf shrub
savanna (Giess 1998); the grass layer is dominated by the perennial grasses Stipagrostis uniplumis, Eragrostis nindensis and Triraphis ramosissima (Maurer
1995).
Karakul sheep (race Swakara) are bred on an area of 30,000 hectares. The
primary source of revenue is from the sale of lamb pelts. Additionally, the
wool of the sheep is sold. In good years, up to 3,000 sheep are kept on the
farm. An adaptive grazing management strategy is employed to cope with the
variability in forage. The basis of the strategy is a rotational grazing scheme:
1
Another important driver of ecological dynamics in semi-arid rangelands is the stochastic
occurrence of fire (Janssen et al. 2004, Perrings and Walker 1997, 2004). In our case, fire
plays only a minor role, but the stochasticity of rainfall is crucial (Müller et al. forthcoming).
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Natural Science Constraints in Environmental and Resource Economics
the pasture land is divided into 98 paddocks, each of which is grazed for a short
period (about 14 days) until the palatable biomass on that paddock is used up
completely, and then is rested for a minimum of two months. This system puts
high pressure on the vegetation for a short time to prevent selective grazing
(Batabyal and Beladi 2002, Batabyal et al. 2001, Heady 1999). While such a
rotational grazing scheme is fairly standard throughout semi-arid regions, the
farmer on the Gamis Farm has introduced an additional resting: in years with
sufficient precipitation one third of the paddocks are given a rest during the
growth period (September – May). In years with insufficient rainfall this rest
period is reduced or completely omitted. Once a year, at the end of the rainy
season (April), the farmer determines – based on actual rainfall and available
forage – how many paddocks will be rested and, thus, how many lambs can be
reared. This strategy is a particular example of what has been called ‘rotational
resting’ (Heady 1970, 1999, Stuth and Maraschin 2000, Quirk 2002).
The grazing management system employed at the Gamis Farm has been
successful over decades, both in ecological and economic terms. It, therefore,
represents a model for commercial farming in semi-arid rangelands.
10.3
The Model
Our analysis is based on an integrated dynamic and stochastic ecologicaleconomic model, which captures essential aspects and principles of grazing
management in semi-arid regions. It represents a dynamic ecosystem, which
is driven by stochastic precipitation, and a risk averse farmer, who rationally
chooses a grazing management strategy under uncertainty.
10.3.1
Precipitation
Uncertainty is introduced into the model by the stochasticity of rainfall, which
is assumed to be an independent and identically distributed (iid) random variable. For semi-arid areas, a log-normal distribution of rainfall r(t) is an adequate description (Sandford 1982).2 The log-normal distribution, with probability density function f (r) (Equation A10.1), is determined by the mean μr
and standard deviation σr of precipitation. Here, we measure precipitation in
terms of ‘ecologically effective rain events’, i.e. the number of rain events during
rainy season with a sufficient amount of rainfall to be ecologically productive
(Müller et al. forthcoming).
2
While the distribution of rainfall r(t) is exogenous, all other random variables in the
model follow an induced distribution.
Insurance and Sustainability through Ecosystem Management
10.3.2
205
Grazing Management Strategies
The farm is divided into a number I ∈ IN of identical paddocks, numbered by
i ∈ {1, . . . , I}. In modeling grazing management strategies, we focus on the
aspect of additional resting during the growth period, which is the innovative
element in the Gamis grazing system. That is, we analyze rotational resting of
paddocks from year to year, but do not explicitly consider rotational grazing
during the year (cf. Section 10.2). The strategy is applied in each year, after
observing the actual rainfall at the end of the rainy season. Its key feature is
that in dry years all paddocks are used, while in years with sufficient rainfall a
pre-specified fraction of paddocks is rested. Whether resting takes place, and
to what extent, are the defining elements of what we call the farmer’s grazing
management strategy:
Definition 10.1
A grazing management strategy (α, r) is a rule of how many paddocks are
not grazed in a particular year given the actual rainfall in that year, where
α ∈ [0, 1] is the fraction of paddocks rested if rainfall exceeds the threshold
value r ∈ [0, ∞).3
Thus, when deciding on the grazing management strategy, the farmer decides on two variables: the rain threshold r and the fraction α of rested paddocks. While the rule is constant (i.e. α = const., r = const.) its application
may yield a different stocking with livestock in any given year depending on
actual rainfall in that year.
In the resource economics literature, this type of strategy is called ‘proportional threshold harvesting’ (Lande et al. 2003). This is a form of adaptive
management: the (constant) rule adapts the fraction of fallow paddocks and
the number of livestock kept on the farm as actual rainfall changes. Note that
the ‘opportunistic’ strategy (e.g. Beukes et al. 2002: 238) is the special case
without resting, i.e. α = 0.
10.3.3
Ecosystem Dynamics
Both the stochastic rainfall and grazing pressure are major determinants of
the ecological dynamics. Following Stephan et al. (1998a), we consider two
quantities to describe the state of the vegetation in each paddock i at time t:
the green biomass Gi (t) and the reserve biomass Ri (t) of a representative grass
species,4 both of which are random variables, since they depend on the random
3
We assume that the number I of paddocks is so large that we can treat α as a real
number.
4
We assume that selective grazing is completely prevented, i.e. there is no competitive
disadvantage for more palatable grasses (see e.g. Beukes et al. 2002). Hence, we consider a
single, representative species of grass.
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Natural Science Constraints in Environmental and Resource Economics
variable rainfall. The green biomass captures all photosynthetic (‘green’) parts
of the plants, while the reserve biomass captures the non-photosynthetic reserve
organs (‘brown’ parts) of the plants below or above ground (Noy-Meir 1982).
The green biomass grows during the growing season in each year and dies
almost completely in the course of the dry season. The amount Gi (t) of green
biomass available on paddock i in year t after the end of the growing season
depends on rainfall r(t) in the current year, on the reserve biomass Ri (t) on
that paddock, and on a growth parameter wG :
Gi (t) = wG · r(t) · Ri (t).
(10.1)
As the green biomass in the current year does not directly depend on the green
biomass in past years, it is a flow variable rather than a stock.
In contrast, the reserve biomass Ri (t) on paddock i in year t is a stock
variable. That is, the reserve biomass parts of the grass survive several years
(‘perennial grass’). Thereby, the dynamics of the vegetation is not only influenced by the current precipitation, but also depends on the precipitation of
preceding years (O’Connor and Everson 1998). Growth of the reserve biomass
from the current year to the next one is
Ri (t)
Ri (t)
i
i
R (t+1)−R (t) = −d·R (t)· 1 +
+wR ·(1−c·x (t))·G (t)· 1 −
,
K
K
(10.2)
where wR is a growth parameter and d is a constant death rate of the reserve
biomass, which we assume to be sufficiently small, i.e. d < wR wG μr . A density
dependence of reserve biomass growth is captured by the factors containing the
capacity limits K: The higher the reserve biomass on paddock i, the slower
it grows. The status variable xi captures the impact of grazing on the reserve
biomass of paddock i. If paddock i is grazed in year t, we set xi (t) = 1, if it
is rested, we set xi (t) = 0. The parameter c (with 0 ≤ c ≤ 1) describes the
amount by which reserve biomass growth is reduced due to grazing pressure.
For simplicity, we assume that the initial (t = 1) stock of reserve biomass of all
paddocks is equal,
(10.3)
Ri (1) = R for all i = 1, . . . , I.
i
10.3.4
i
i
Livestock and Income
As for the dynamics of livestock, the herd size S(t), that can be kept on the
farm at time t, is limited by total available forage. We normalize the units of
green biomass in such a way that one unit of green biomass equals the need of
one
unit per year. Thus, total available green biomass on the farm,
I livestock
i
i=1 G (t), determines the ‘carrying capacity’, i.e. the maximum number of
Insurance and Sustainability through Ecosystem Management
207
livestock that can be held on the farm in the period under consideration.5 In
general, the farmer will not stock up to this carrying capacity in every year.
Rather, the herd size kept on the farm in period t is given by
S(t) =
I
xi (t) · Gi (t) .
(10.4)
i=1
That is, the herd size in year t is determined by the total green biomass of
the paddocks used for grazing (i.e., not rested) in that year. For the sake
of the analysis, we assume that the farmer annually rents his livestock on a
perfect rental market for livestock.6 This allows the farmer to exactly adapt the
actual herd size to the available forage and to his chosen grazing management
strategy.7
The herd size S(t) kept on the farm in year t determines the farmer’s income
y(t). We assume that the quantity of marketable products from livestock, e.g.
lamb furs and wool, is proportional to the herd size. Normalizing product
units in an appropriate way, the numerical value of output equals livestock
S(t). The farmer sells his products on a world market at a given price and
takes the annual rental rate of livestock as given. The difference between the
two is the net revenue per livestock unit, p. Assuming that farming is profitable,
i.e. p > 0, the farmer’s income y(t) is
y(t) = p · S(t).
(10.5)
Since the herd size S(t) is a random variable, income y(t) is a random variable,
too.8 In order to simplify the notation in the subsequent analysis, we normalize
p ≡ (wG · I · R)−1 .
(10.6)
This means, from now on we measure net revenue per livestock unit in units of
total forage per unit of precipitation. As a result, income is measured in units
of precipitation.
5
In contrast to the capacity limit K of reserve biomass, the carrying capacity of livestock is
not a constant, but it depends on rainfall and the stock of reserve biomass (cf. Equation 10.1),
and, therefore, will change over time.
6
If the farmer owns a constant herd of size S0 , he would rent a number S(t) − S0 if
S(t) > S0 or rent out a number S0 − S(t) if S(t) < S0 . Without loss of generality, we set
S0 = 0.
7
Hence, the herd size S(t) does not follow its own dynamics, but it is determined by
precipitation and the chosen strategy.
8
In our analysis, we neglect uncertainty of prices. Including a price stochasticity uncorrelated to rainfall would not alter our results. Including a price stochasticity with a negative
correlation to rainfall would most likely reinforce our central result that a risk averse farmer
chooses a conservative grazing management strategy, since high stocking rates in good rainy
years become less valuable (as indicated by Hein and Weikard 2004).
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Natural Science Constraints in Environmental and Resource Economics
For the subsequent analysis of a myopic farmer’s decision, first and second
year income are of particular interest. Given the actual rainfall r(1) in the
first grazing period, the initial reserve biomass (Equation 10.3) and a grazing
management rule (α, r), the herd size S(1) is determined by Equation (10.4).
Inserting Equation (10.1) and using Assumption (10.3), as well as normalization (10.6), the farmer’s income y(1) in the first grazing period is given by
Equation (10.5) as
)
I
1 i
r(1)
if r(1) ≤ r
y(1) =
x (1) · r(1) =
.
(10.7)
(1 − α) · r(1) if r(1) > r
I
i=1
Given the probability density distribution f (r) of rainfall, the mean μy(1) (α, r)
and the standard deviation σy(1) (α, r) of the first period’s income are (see Appendix A10.1)
∞
(10.8)
μy(1) (α, r) = μr − α r f (r) dr
r
*
⎤2
⎡ ∞
+
∞
∞
+
+ 2
σy(1) (α, r) = ,σr + 2 α μr
r f (r) dr − α2 ⎣ r f (r) dr ⎦ − α (2 − α) r 2 f (r) dr,
r
r
r
(10.9)
where μr and σr are the mean and the standard deviation of rainfall.
The model implies that resting in the first period has a positive impact on
reserve biomass and, thus, on future income. In particular, if the farmer applies
a grazing management strategy (α, r) with α > 0 and r < ∞, rather than full
stocking, he can gain an extra income in the second year. Given the actual
rainfall r(1) in the first year, the additional reserve biomass in the second year
is (cf. Equations 10.1, 10.2 and 10.3)
)
R
0 if r(1) ≤ r
ΔR = wR · wG · I · R · 1 −
.
(10.10)
· r(1) ·
α if r(1) > r
K
This additional reserve biomass gives rise to extra green biomass growth, and,
hence, to additional income in the second year (cf. Equations 10.1, 10.4, 10.5
and 10.10):
)
)
R
1
if r(2) ≤ r
0 if r(1) ≤ r
Δy(2) = wG ·r(2)·
.
·wR · 1 −
·r(1)·
1 − α if r(2) > r
α if r(1) > r
K
(10.11)
This means, the reserve biomass can be used as a buffer: by applying a grazing
strategy with resting, the farmer can shift income to the next year. For a risk
averse farmer, this extra income is particularly valuable if the second year is a
dry year.
Insurance and Sustainability through Ecosystem Management
10.3.5
209
Farmer’s Choice of Grazing Management Strategy
We assume that the farmer’s utility only depends on income y, and that he is
a non-satiated and risk averse expected utility maximizer. Let
∞
Et u(y(t))
U=
(1 + δ)t−1
t=1
(10.12)
be his von Neumann-Morgenstern intertemporal expected utility function, where
δ is the discount rate, the Bernoulli utility function u(·) is a strictly concave
function of income y, and Et is the expectancy operator at time t. In particular,
we employ a utility function with constant relative risk aversion,
y 1−ρ − 1
u(y) =
,
1−ρ
(10.13)
where ρ > 0 is the constant parameter which measures the degree of relative
risk aversion (Gollier 2001).
The farmer will choose the grazing management strategy which maximizes
his von Neumann-Morgenstern intertemporal expected utility function (10.12).
The basic idea is to regard the choice of a grazing management strategy as
the choice of a ‘lottery’ (Baumgärtner and Quaas 2005). Each possible lottery
is characterized by the probability distribution of pay-off, where the pay-off is
given by the farmer’s income. Given the ecological dynamics, both the mean
income and the standard deviation solely depend on the grazing management
strategy applied. Thus, choosing a grazing management strategy implies choosing a particular distribution of income.
We assume that the farmer initially, i.e. at t = 0 prior to the first grazing
period, chooses a grazing management strategy (α, r), which is then applied
in all subsequent years. When choosing the strategy, the farmer does not
know which amount of rainfall will actually occur, but he knows the probability distribution of rainfall. As a result, he knows the probability distribution
of his income for any possible grazing management strategy. A far-sighted
farmer would choose the grazing management strategy that maximizes his intertemporal utility (10.12), taking into account the effect of the strategy on the
ecosystem dynamics, as given by Equations (10.1) and (10.2). In particular, he
would account for the effect that resting improves the reserve biomass in the
long run, compared to a strategy with full stocking. However, our aim is to
show that a sufficiently risk averse farmer will choose a conservative strategy,
even if he does not consider the long-term benefits. For this sake, we assume
that the farmer is myopic in the following sense (Kurz 1987):
Definition 10.2
A myopic farmer neglects the long-term effects of his grazing management strategy on the ecosystem: (i) He assumes that reserve biomass remains constant at
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Natural Science Constraints in Environmental and Resource Economics
the initial level R on all paddocks, irrespective of the chosen strategy, with the
exception that (ii) he takes into account the extra income Δy (Equation 10.11)
in a year after resting.
This means, a myopic farmer bases his decision on a very limited consideration
of ecosystem dynamics: he only takes into account the short-term buffering
function of reserve biomass, while neglecting all long-term ecological impact of
the grazing management strategy chosen. Such a myopic farmer considers his
income in year t ≥ 2 to be
1
if r(t) ≤ r
·
y(t) = r(t) ·
1 − α if r(t) > r
-
)
R
0 if r(t − 1) ≤ r
. (10.14)
· r(t − 1) ·
· 1 + wR · wG · 1 −
α if r(t − 1) > r
K
)
Since the myopic farmer neglects the long-term ecological impact of his grazing
strategy, the functional form of how annual income y(t) (Equation 10.14) depends on actual rainfall and on the chosen strategy, remains constant over time.
Furthermore, since precipitation is independent and identically distributed in
each year, and the strategy is constant, the mean μy(t) and standard deviation
σy(t) of the annual income y(t) for t ≥ 2 are also constant over time.
In order to be able to express the expected instantaneous utility in any year
t in terms of the mean and the standard deviation of that year’s income, we
approximate the probability density function of annual income by a log-normal
distribution with the same mean and standard deviation. Using the specification (10.13) of the Bernoulli utility function u(y), expected instantaneous
utility is given by the following explicit expression (see Appendix A10.2):
E u(y(t)) =
.
/−ρ (1−ρ)/2
2
2
μ1−ρ
/μ
−1
1
+
σ
y(t)
y(t)
y(t)
1−ρ
.
(10.15)
The indifference curves of the farmer’s expected instantaneous utility function
can be drawn in the mean–standard deviation space. Figure 10.1 shows such a
set of indifference curves for a given degree ρ of relative risk aversion. The indifference curves are increasing and convex if the standard deviation is sufficiently
small compared to the mean, i.e. for (μy /σy )2 > 1 + ρ (see Appendix A10.3).
The slope of the indifference curves is increasing in the degree of relative risk
aversion ρ (see Appendix A10.3). In particular, the indifference curves are
horizontal lines for risk-neutral farmers, i.e. for ρ = 0.
Formally, the decision problem to be solved by a myopic farmer is to choose
a grazing management strategy (α, r) such as to maximize U (Equation 10.12)
subject to Conditions (10.7), (10.14), (10.15). In the context of semi-arid
mean income μy
Insurance and Sustainability through Ecosystem Management
211
μ̂0y
μ̃0y
μ0y
standard deviation σy
Figure 10.1 A set of indifference curves of the risk averse farmer in the meanstandard deviation space for log-normally distributed incomes and constant relative risk aversion ρ = 1.
rangelands, the growth rate of reserve biomass is small, i.e. wR 1. In Appendix A10.4 we show that under this condition the farmer’s decision problem
effectively becomes
0
1−ρ/2
,
max μy (α, r) · 1 + σy2 (α, r)/μ2y (α, r)
(10.16)
(α,r)
where the effective mean and standard deviation of income are
⎡
μy (α, r) = ⎣μr − α
∞
⎤ ⎡
r f (r) dr ⎦ · ⎣1 + α ω
r
∞
⎤
r f (r) dr ⎦
(10.17)
r
*
⎤2
⎡ ∞
+
∞
∞
+
+ 2
2
σy (α, r) = ,σr + 2 α μr
r f (r) dr − α ⎣ r f (r) dr ⎦ − α (2 − α) r 2 f (r) dr
r
r
r
*
+
∞
+
+
· ,1 + 2 α ω r f (r) dr , (10.18)
r
with ω = wR · wG · (1 − R/K)/(1 + δ). We analyze this decision problem in the
following.
212
10.4
Natural Science Constraints in Environmental and Resource Economics
Results
The analysis proceeds in three steps (Results 10.1, 10.2 and 10.3 below): First,
we analyze the optimization problem of a risk averse myopic farmer who faces
a trade-off between strategies which yield a high mean income at a high standard deviation, and strategies which yield a low mean income at a low standard
deviation. Second, we analyze the long-term consequences of different grazing
management strategies on the ecological-economic system. In particular, we
study how the long-term development of the mean reserve biomass and the
mean income depend on the strategy. Finally, we put the two parts of the
analysis together and derive conclusions about how the long-term sustainability of the short-term optimal strategy depends on the farmer’s degree of risk
aversion.
10.4.1
Feasible Strategies and Income Possibility Set
To start with, we define the income possibility set as the set of all effective
mean incomes and standard deviations of income (μy (α, r), σy (α, r)) ∈ (0, ∞)×
[0, ∞), which are attainable by applying a feasible management rule (α, r) ∈
[0, 1] × [0, ∞). These are given by Equations (10.17) and (10.18). Figure 10.2
shows the income possibility set for particular parameter values.
mean income μy
μr
0
σr
0
standard deviation σy
Figure 10.2 The set of all means μy and standard deviations σy of the farmer’s
income y, each point denoting a separate strategy, as well as the income possibility frontier (thick line). Parameter values are μr = 1.2, σr = 0.7 and
ω = 0.14.
The figure provides one important observation: there exist inefficient strategies, i.e. feasible strategies that yield the same mean income, but with a higher
Insurance and Sustainability through Ecosystem Management
213
standard deviation (or: the same standard deviation, but with a lower mean)
than others. These strategies can be excluded from the set of strategies from
which the optimum is chosen by a risk averse and non-satiated decision maker.
In the following, we thus focus on the efficient strategies, which generate the
income possibility frontier (Figure 10.2, thick line):
Definition 10.3
The income possibility frontier is the set of expected values μy and standard
deviations σy of income for which the following conditions hold:
1. (μy , σy ) is in the income possibility set, i.e. it is feasible.
2. There is no (μy , σy ) = (μy , σy ) in the income possibility set with μy ≥ μy
and σy ≤ σy .
The question at this point is, ‘What are the grazing management strategies
(α, r) that generate the income possibility frontier?’ We call these strategies
efficient.
Lemma 10.1
The set of efficient strategies has the following properties.
• Each point on the income possibility frontier is generated by exactly one
(efficient) strategy.
• There exists Ω ⊆ [0, ∞), such that the set of efficient strategies is given
by (α∗ (r), r) with
∞
∗
α (r) =
r (r − r) f (r) dr
r
∞
r (r − r/2) f (r) dr
for all r ∈ Ω.
(10.19)
r
• α∗ (r) has the following properties:
∗
α (0) = 1,
∗
lim α (r) = 0,
r→∞
and
dα∗ (r)
< 0 for all r ∈ Ω.
dr
Proof: see Appendix A10.5.
Figure 10.3 illustrates the lemma. Whereas the set of feasible strategies is
the two-dimensional area bounded by r = 0, α = 0, α = 1, the set of efficient
strategies, as given by Equation (10.19), is a one-dimensional curve. Thus,
the efficient strategies are described by only one parameter, r, while the other
parameter α is determined by α = α∗ (r) (Equation 10.19). Alternatively, the
214
Natural Science Constraints in Environmental and Resource Economics
1
α
0.8
0.6
0.4
α∗ (r)
0.2
0
0
2
4
6
8
10
r
Figure 10.3 The set of feasible strategies is given by the whole area α ∈ [0, 1],
r ∈ [0, ∞). The set of efficient strategies for parameters μr = 1.2 and σr = 0.7
is the curve.
inverse function of Equation (10.19) – which exists by Lemma 10.1 – specifies
the efficient rain threshold r as a function of the fraction α of resting. The
curve α∗ (r) is downward sloping: With a higher rain threshold r, i.e. if resting
only takes place in years with higher precipitation, the efficient fraction α∗ (r)
of rested paddocks is smaller. In other words, for efficient strategies, a higher
rain threshold r does not only mean that the condition for resting is less likely
to be fulfilled, but also that a smaller fraction α∗ of paddocks is rested if resting
takes place. Hence, if an efficient strategy is characterized by a smaller r, and,
consequently, by a larger α∗ (r), we call it more conservative.
Knowledge of the efficient strategies allows us to characterize the income
possibility frontier, and to establish a relationship between efficient grazing
management strategies and the resulting means and standard deviations of
income.
Lemma 10.2
The farmer’s expected income in the first grazing period, μy (α, r) (Equation 10.17),
is increasing in r for all efficient strategies:
d μy (α∗ (r), r)
>0
dr
for all r ∈ Ω.
The extreme strategies, r = 0 and r → ∞, lead to expected incomes of μy (α∗ (0), 0) =
0 and lim μy (α∗ (r), r) = μr .
r→∞
Proof: see Appendix A10.6.
Insurance and Sustainability through Ecosystem Management
215
For all efficient strategies a higher rain threshold r for resting, i.e. a less
conservative strategy, implies a higher mean income. Whereas no resting, r →
∞ (opportunistic strategy), leads to the maximum possible mean income of μr ,
the opposite extreme strategy, r = 0 (no grazing at all), leads to the minimum
possible income of zero. Overall, a change in the grazing management strategy
affects both, the mean income and the standard deviation of income.
Lemma 10.3
The income possibility frontier has the following properties:
• The income possibility frontier has two corners:
– The southwest corner is at σy = 0 and μy = 0. At this point, the
income possibility frontier is increasing with slope μr /σr .
– The northeast corner is at σy = σr and μy = μr . At this point, the
income possibility frontier has a maximum and its slope is zero.
• In between the two corners, the income possibility frontier is increasing
and located above the straight line from one corner to the other. It is
S-shaped, i.e. from southwest to northeast there is first a convex segment
and then a concave segment.
Proof: see Appendix A10.7.
Figure 10.2 illustrates the lemma. With no resting at all (northeast corner
of the income possibility frontier), the farmer obtains the highest possible mean
income (μy = μr ), but also faces the full environmental risk (σy = σr ). Conversely, with the most conservative strategy, i.e. no grazing at all (southwest
corner of the income possibility frontier), the farmer can completely eliminate
his income risk (σy = 0), but also cannot expect any income (μy = 0). The
property, that the income possibility frontier is increasing, suggests that resting
acts like an insurance for the farmer. This means, by choosing a more conservative grazing management strategy, the farmer can continuously decrease his
risk (standard deviation) of income, but only at the price of a decreased mean
income. Thus, there is an insurance value associated with choosing a more
conservative strategy (Baumgärtner and Quaas 2005).
10.4.2
Optimal Myopic Strategy
The optimal myopic strategy is obtained by solving Problem (10.16), and results from both the farmer’s preferences (Figure 10.1) and the income possibility
frontier (Figure 10.2). In mean–standard deviation space, it is determined by
the mean μ∗y and the standard deviation σy∗ , at which the indifference curve is
tangential to the income possibility frontier (Figure 10.4). It turns out that
the optimal strategy is uniquely determined.
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Natural Science Constraints in Environmental and Resource Economics
μr
mean income μy
μ∗y
∗
+
μ0y
σy∗
σr
standard deviation σy
Figure 10.4 The optimum for a risk averse farmer (ρ = 5.5, denoted by ∗)
and a risk-neutral farmer (ρ = 0, denoted by +).
Lemma 10.4
(i) If (μr /σr )2 > 1 + ρ, the optimum (μ∗y , σy∗ ) is unique.9
(ii) For ρ > 0, the optimum is an interior solution with 0 < μ∗y < μr and
0 < σy∗ < σr . For ρ = 0, the optimum is a corner solution with μ∗y = μr
and σy∗ = σr .
Proof: see Appendix A10.8.
The optimal myopic strategy crucially depends on the degree of risk aversion. In the particular case of a risk-neutral farmer (ρ = 0), the strategy that
yields the maximum mean, irrespective of the standard deviation associated
with it, is chosen. The optimal grazing management strategy of such a riskneutral farmer is the strategy without resting, i.e. with r = ∞ (and, therefore,
α = 0). That is, he employs an opportunistic strategy.
If the farmer is risk averse, he faces a trade-off between expected income and
variability of the income, because strategies that yield a higher mean income
also display a higher variability of income. This leads to the following result,
which is illustrated in Figures 10.4 and 10.5.
Result 10.1
A unique interior solution (α∗ (r ∗ ), r ∗ ) to the farmer’s decision problem (10.16),
if it exists (see Lemma 10.4), has the following properties:
9
This is a sufficient condition which is quite restrictive. A unique optimum exists for a
much larger range of parameter values.
Insurance and Sustainability through Ecosystem Management
217
fraction of resting α∗ (r ∗ )
0.6
0.5
0.4
0.3
0.2
1
2
3
4
5
6
7
risk-aversion ρ
Figure 10.5 The rain threshold r ∗ of the optimal strategy as a function of
the farmer’s degree of risk aversion ρ. Parameter values are the same as in
Figure 10.2.
(i) The more risk averse the farmer, the smaller are the mean μ∗y and the
standard deviation σy∗ of his income.
(ii) The more risk averse the farmer, the more conservative is his grazing
management strategy:
dr ∗
<0
dρ
and
dα∗
> 0.
dρ
(10.20)
Proof: see Appendix A10.9.
This means, a risk averse farmer chooses a grazing management strategy
such as to obtain insurance from the ecosystem: by choosing a particular grazing management strategy the farmer will reduce his income risk, and carry the
associated opportunity costs in terms of mean income foregone (the ‘insurance
premium’), to the extent that is optimal according to his degree of risk aversion.
10.4.3
Long-Term Impact of Grazing Management Strategies
To study the long-term ecological and economic impact of the grazing management strategy chosen on the basis of myopic optimization (Problem 10.16), we
assume that the farmer continues to apply this strategy in every subsequent period. We compute the resulting probability distribution of income and reserve
biomass over several decades in the future. This calculation covers all efficient
strategies (α∗ (r), r). The results of the numerical computation10 are shown
10
Numerical details are given in Müller et al. (forthcoming).
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Natural Science Constraints in Environmental and Resource Economics
1.2
1
mean income μy (t > 0)
mean reserve biomass μR (t > 0)
in Figure 10.6, which enables the comparison of the long-term impacts, both
in ecological and economic terms, of the different strategies that are efficient
from the viewpoint of a myopic farmer. In this figure, the mean values μR (t)
0.8
0.6
0.4
0.2
0
0.4
t=1
t = 10
t = 40
t = 70
t = 100
0.5 0.6 0.7 0.8 0.9
fraction of resting α (r)
1
t=1
t = 10
t = 40
t = 70
t = 100
1
0.8
0.6
0.4
0.2
0
0.4
0.5 0.6 0.7 0.8 0.9
fraction of resting α (r)
a
b
Figure 10.6 Relation between the grazing management strategy (given by the
efficient fraction of resting α (r)) and (a) future mean reserve biomass μR (t >
0) (in units of initial reserve biomass), as well as (b) future mean income
μy (t > 0) for different strategies on the income possibility frontier. Parameter
values are μr = 1.2, σr = 0.7, I · K = 8000, d = 0.15, wG = 1.2, wR = 0.2,
c = 0.5, I · R = 2400.
of reserve biomass and μy (t) of income at different times t are plotted against
the efficient fraction α (r) of resting for different rain thresholds r ∈ Ω. The
higher α (r) is, the more conservative is the respective strategy. Interpreting
Figure 10.6 leads to the following result (see Appendix A10.10 for a sensitivity
analysis).
Result 10.2
For parameter values which characterize typical semi-arid rangelands (i.e. wG ,
wR , μr are small and c, σr are large) the long-term ecological and economic
impact of a strategy (α (r), r) is as follows:
(i) The more conservative the strategy, the higher the mean reserve biomass
μR (t) in the future:
d μR(t)
< 0 and
dr
d μR (t)
> 0 for all t > 1.
dα
(ii) For high rain thresholds r > r̂, the following holds: The more conservative
the strategy, the higher the mean income μy (t) in the long-term future for
t > t̂:
d μy (t)
< 0 and
dr
d μy (t)
> 0 for all t > t̂
dα
and r > r̂, α < α (r̂).
1
Insurance and Sustainability through Ecosystem Management
219
Result 10.2 states that the slope of the curves in Figure 10.6 is positive throughout, as far as reserve biomass is concerned; and is positive for small α (r), i.e.
for α (r) < α (r̂), and t > t̂, as far as income is concerned. The higher the
fraction α (r) of paddocks rested, i.e. the more conservative the strategy, the
higher is the mean reserve biomass, if the same strategy is applied over the
whole period. This effect is in line with intuition: the more conservative the
strategy, the better is the state of the rangeland in the future. As far as income
is concerned, the argument is less straightforward. In particular, the mean income in the first period is increasing in r, i.e. decreasing in α (r) (Lemma 10.2).
A less conservative strategy yields a higher mean income in this period, since
more livestock is kept on the rangeland. This holds for several periods in the
near future (cf. the line for t = 10 in Figure 10.6b). However, in the long run
(for t > t̂ ≈ 40), the strong grazing pressure on the pasture leads to reduced
reserve biomass growth and less forage production in the long-term future,
compared to a more conservative strategy. As a result, mean income is smaller.
This can be seen in Figure 10.6b: the curves are upward-sloping for sufficiently
high t ≥ t̂ and sufficiently small α (r). As can be seen in the figure, this effect
becomes stronger in the long-term future: the curves are steeper for higher t.
Result 10.2 holds if the growth rates of the green and reserve biomass are
low, the impact of grazing on the growth of the reserve biomass is high, and
rainfall is low and highly variable. This is just the range of parameter values
which is adequate for semi-arid rangelands, because these are fragile ecosystems
which are highly susceptible to degradation if grazing pressure is high. For very
robust ecosystems or very low stochasticity of rainfall, however, the result is
not valid.
For a large fraction of resting, i.e. α (r) > α (r̂), a more conservative strategy (i.e. a larger α (r)) leads to a lower mean income, not only in the first
period (Lemma 10.2), but also in the future. In this domain of strategies, resting is already so high that the future gains in reserve biomass from additional
resting do not outweigh the losses from lower stocking.
While Result 10.2 describes the dynamic long-term impact of different grazing management strategies, the following lemma analytically extends this result
by specifying the steady-state mean values of reserve biomass and income. The
steady-state mean value of reserve biomass is determined as the fixed point
of the mean vegetation dynamics (according to Equations 10.1 and 10.2). The
steady-state mean value of reserve biomass, in turn, determines the steady-state
mean value of income.11
11
These steady-state mean values represent the trend of the stochastic dynamics, but not
the purely random part of the dynamics. The latter could lead, by chance, to irreversible
extinction of the reserve biomass in the long-run even when a very conservative strategy is
applied.
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Natural Science Constraints in Environmental and Resource Economics
Lemma 10.5
1. For an efficient strategy (α (r), r) the steady-state mean value of reserve
biomass is
)
wG wR (μR − c μy(1) (α (r), r)) − d
stst
,0 ,
(10.21)
μR = max K
wG wR (μR − c μy(1) (α (r), r)) + d
and the steady-state mean value of income is
μstst
y
μstst
= R μy(1) (α (r), r) ,
R
(10.22)
where μy(1) (α (r), r) is given by Equation (10.8), and R is the initial value
of reserve biomass.
2. μstst
R is monotonically decreasing in r,
dμstst
R
<0,
dr
(10.23)
assumes a maximum value at r̂ > 0, such that
while μstst
y
dμstst
y
< 0 for r > r̂ .
dr
(10.24)
Proof: see Appendix A10.11.
For r > r̂, we thus have established the following result: The more conservative the strategy, i.e. the lower r and the higher α (r), the higher the
steady-state mean reserve biomass and income in the long run.
As the final step in our analysis, we now relate this insight to the issue of
sustainability of grazing management strategies. For the sake of this analysis,
we understand sustainability in the following way.
Definition 10.4
A grazing management strategy (α, r) is called sustainable, if and only if it
leads to strictly positive steady-state mean values of both reserve biomass and
stst
> 0.
income, μstst
R > 0 and μy
The notion of sustainability, while expressing an idea which seems obvious
and clear at first glance, is notoriously difficult to define in an operational way.
As a result, there are a multitude of different definitions of ‘sustainability’,
which reveal different aspects and, at bottom, fundamentally different understandings of the term (see e.g. Klauer 1999, Neumayer 2003 and Pezzey 1992 for
a detailed discussion). In the framework of our model, Definition 10.4 captures
essential aspects of what has been called ‘strong sustainability’ (Pearce et al.
Insurance and Sustainability through Ecosystem Management
221
1990, Neumayer 2003). It comprises an ecological as well as an economic dimension, with mean reserve biomass as an ecological indicator and mean income
as an economic indicator. It expresses the aspect of long-term conservation of
an ecological-economic system in the sense that the steady-state mean values
of both reserve biomass and income are strictly positive.12 In contrast, an unsustainable strategy is one that leads to the collapse of the ecological-economic
system, in the sense that the steady-state mean value of either reserve biomass
or income (or both) is zero. Definition 10.4 constitutes a rather weak criterion
of strong sustainability, by setting the minimum requirements with respect to
the steady-state mean values of both reserve biomass and income at zero.13
Yet, it enables a clear and unambiguous distinction between sustainable and
unsustainable strategies in the following manner.
Lemma 10.6
If c > 1 − d/(wG wR μr ), a strategy (α (r ), r ) exists, such that all efficient
strategies which are less conservative (i.e. r > r and α (r) < α (r )) are
unsustainable and all efficient strategies with r > 0 that are more conservative
(i.e. r < r and α (r) > α (r )) are sustainable.
Proof: see Appendix A10.12
If the impact of grazing on reserve biomass growth is very small, i.e. if
c < 1 − d/(wG wR μr ), all strategies are sustainable. Long-term degradation
of the pasture is only a problem at all when the impact of grazing on the
vegetation is high. In this case, there is a clear and unambiguous threshold
between strategies that are conservative enough to be sustainable and strategies
which are not. From Result 10.1, we know that the more risk averse a farmer
is, the more conservative is his optimal myopic strategy. Combining this result
with Lemma 10.6, we can now make a statement about the relation between a
risk averse farmer’s myopic decision and its long-term implications in terms of
sustainability.
Result 10.3
If the uncertainty of rainfall, σr , is large and the impact of grazing c is not too
large, a sufficiently risk averse myopic farmer will choose a sustainable grazing
management strategy.
Proof: see Appendix A10.13.
12
Under uncertainty, positive steady-state mean values do not mean that a sustainable
strategy will actually yield positive values of reserve biomass and income. For, by chance,
a sequence of rain events may occur which drives the reserve biomass to extinction. See
Footnote 11.
13
As an alternative, one could set minimum requirements at strictly positive values, representing e.g. the levels of ‘critical natural capital’ and ‘subsistence income’. We have chosen
zero for the sake of analytical clarity.
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Natural Science Constraints in Environmental and Resource Economics
Result 10.3 sheds new light on the question ‘How can one explain that
people do behave in a sustainable way?’ For, Result 10.3 suggests the following
potential explanation. That a farmer A manages an ecosystem in a sustainable
manner, while another farmer B does not, may be explained simply by a higher
risk aversion of farmer A. In particular, it is not necessary to assume that
farmer A has any kind of stronger preferences for future income or sustainability
than farmer B. This result holds if (i) uncertainty is large and (ii) the impact of
grazing is not too large. If uncertainty were small, it would only play a minor
role in the decision making of the farmer. Hence, even a large risk aversion
would not induce a myopic farmer to choose a conservative strategy. If, on the
other hand, the impact of grazing were very high, the optimal strategy of even
a very risk averse myopic farmer would not be conservative enough to ensure
sustainability.
For a large standard deviation of rainfall and not too large grazing impacts, the model predicts a critical degree ρ of risk aversion which separates
the myopic farmers choosing a sustainable strategy from those choosing an
unsustainable one. This critical degree of risk-aversion characterizes precisely
that myopic farmer who chooses the strategy (α (r ), r ), which separates sustainable from unsustainable strategies (Result 10.1(ii) and Lemma 10.6). For
the parameter values used in our numerical simulations (see the caption of Figure 10.6), this critical degree of risk aversion is ρ = 1.85, which is well within
the range of degrees of risk aversion commonly considered as reasonable (i.e.
ρ ≤ 4; see e.g. Gollier 2001).14
10.5
Conclusions and Discussion
We have developed an integrated dynamic and stochastic ecological-economic
model of grazing management in semi-arid rangelands. Within this, we have
analyzed the choice of grazing management strategies of a risk averse farmer,
and the long-term ecological and economic impact of different strategies. We
have shown that a myopic farmer who is sufficiently risk averse will choose a
sustainable strategy, although he does not take into account long-term ecological and economic benefits of conservative strategies. The intuition behind this
result is that a conservative strategy provides natural insurance for a risk averse
farmer. In years with good rainfall the farmer does not fully exploit the carrying capacity of the farm. Due to the buffering function of the reserve biomass
of vegetation he thereby can shift income to the next year with possibly worse
conditions. The more risk averse the farmer is, the higher is the benefit from
14
If the standard deviation of rainfall is small, or the grazing impact is very large, the
threshold value of risk aversion exceeds this range of reasonable degrees of risk aversion.
Insurance and Sustainability through Ecosystem Management
223
this insurance function and the more conservative is his optimal strategy. A
sufficiently risk averse farmer chooses a strategy which is conservative enough
to be sustainable.
However, one should not conclude from our analysis that risk aversion is
sufficient to ensure a sustainable development in semi-arid areas. This issue
requires a variety of further considerations. First, one could adopt a more demanding sustainability criterion than we have used (cf. Definition 10.4). Second, we have focused on environmental risk resulting from the uncertainty of
rainfall. Other forms of risk, e.g. uncertainty concerning property-rights, or the
stability of social and economic relations in general, might generate a tendency
in the opposite direction, and promote a less conservative and less sustainable
management of the ecosystem (e.g. Bohn and Deacon 2000). Hence, in the face
of different uncertainties, the net effect is not clear and has to be analyzed in
detail. Third, additional sources of income (say from tourism) or the availability of financial services (such as savings, credits, or commercial insurance),
constitute possibilities for hedging income risk. For farmers, all these are substitutes for obtaining natural insurance by conservative ecosystem management
and, thus, may induce farmers to choose less conservative and less sustainable
grazing management strategies (Quaas and Baumgärtner 2006). This becomes
relevant as farmers in semi-arid regions are more and more embedded in world
trade and have better access to global commodity and financial markets.
Our analysis addressed the context of grazing management in semi-arid
rangelands. This system is characterized by a strong interrelation between
ecology and economic use, which drives the results. While this is a specific ecological-economic system, the underlying principles and mechanisms of
ecosystem functioning and economic management are fairly general. Hence,
we believe that there are similar types of ecosystems managed for the services
they provide, e.g. fisheries or other agro-ecosystems, to which our results should
essentially carry over.
Appendix
A10.1
Mean and Standard Deviation of the First Year’s Income
The rainfall r is log-normally distributed, i.e. the probability density function
is
(
'
1
(ln r − mr )2
f (r) = &
.
(A10.1)
exp −
2s2r
r 2πs2r
The two parameters mr and sr can be expressed in terms of the mean μr and
standard deviation σr , mr = ln μr − 12 ln (1 + σr2 /μ2r ) and s2r = ln (1 + σr2 /μ2r ).
224
Natural Science Constraints in Environmental and Resource Economics
Using the probability density function (A10.1) of rainfall and Equation (10.7)
for the farmer’s first year income, the expected value and the variance of the
first year’s income are easily calculated. The expected value is
∞
μy(1) (α, r) =
r
y(1) f (r) dr =
0
∞
r f (r) dr = μr −α
r f (r) dr+(1−α)
0
∞
r
r f (r) dr.
r
The variance is
∞
2
σy(1)
(α, r)
=
0
y(1) − μy(1)
12
f (r) dr =
0
= σr2 + 2 α μr
∞
⎡
r f (r) dr − α2 ⎣
r
A10.2
−μ2y
r
r f (r) dr + (1 − α)
+
0
∞
2
∞
r 2 f (r) dr
r
⎤2
r f (r) dr ⎦ − α (2 − α)
r
2
∞
r 2 f (r) dr.
r
Expected Utility Function
With the specification (10.13) of the farmer’s Bernoulli utility function u(y),
and the assumption that income
is log-normally
distributed
we
1
1 get (using the
0
0
1
2
2
2
2
2
notation my = ln μy − 2 ln 1 + σy /μy and sy = ln 1 + σy /μy ):
'
(
∞ 1−ρ
1
(ln y − my )2
y
−1
2
exp −
E u(y) =
dy
1 − ρ y 2πs2 )
2s2y
y
0
⎡
⎤
(
'
∞
2
1 ⎣ 1
(z − my )
z=ln y
&
dz − 1⎦
=
exp ((1 − ρ) z) exp −
2
1−ρ
2s2y
2πsy
0
=
A10.3
0
−∞
exp (1 − ρ) my +
1−ρ
1−ρ
2
s2y
11
−1
0
1−ρ (1−ρ)/2
1 + σy2 /μ2y
μ1−ρ
−1
y
=
.
1−ρ
Properties of the Indifference Curves
Each indifference curve intersects the μy -axis at σy = 0. The point of intersection, μ0 , is the certainty equivalent of all lotteries on that indifference curve.
Hence, the indifference curve is the set of all (μy , σy ) ∈ IR+ × IR+ for which
0
1−ρ/2
μy 1 + σy2 /μ2y
= μ0 .
(A10.2)
The slope of the indifference curve is obtained by differentiating Equation (A10.2)
with respect to σy (considering μy as a function of σy ) and rearranging:
ρ σy μy
dμy
=
> 0.
dσy
(1 + ρ) σy2 + μ2y
(A10.3)
Insurance and Sustainability through Ecosystem Management
225
The curvature is obtained by differentiating this equation with respect to σy ,
inserting dμy /dσy again and rearranging
ρ μy (μ2y − (1 + ρ) σy2 )(σy2 + μ2y )
d2 μy
d dμy
=
=
,
0
13
dσy2
dσy dσy
(1 + ρ) σy2 + μ2y
(A10.4)
which is positive, if and only if μ2y > (1 + ρ) σy2 . Furthermore, the slope of the
indifference curves increases with rising risk aversion,
0
1
σy μy σy2 + μ2y
d dμy
=0
12 > 0.
dρ dσy
(1 + ρ) σy2 + μ2y
A10.4
Effective Decision Problem
1−t
= 1/δ,
Using a monotonic transformation of U and employing ∞
t=2 (1 + δ)
the decision problem becomes
0
1−ρ/2 1
0
1−ρ/2
2
2
max μy(1) 1 + σy(1)
/μ2y(1)
+ μy(t) 1 + σy(t)
/μ2y(t)
.
(A10.5)
(α,r)
δ
Since rainfall is independent and identically distributed in each year, we find
from Equation (10.14) that mean income is
⎡
⎤
∞
R
μy(t) = μy(1) ⎣1 + α wR wG 1 −
r f (r) dr ⎦ .
(A10.6)
K
r
When calculating the variance, we neglect terms of second order in wR , since
this is a very small number (see also the specification of parameters in the
caption of Figure 10.6). With this simplification the variance is
*
+
∞
+
R
+
r f (r) dr .
(A10.7)
σy(t) = σy(1) ,1 + 2 α wR wG 1 −
K
r
Plugging this into the decision problem (A10.5) and again dropping terms of
second order in the growth rate of the reserve biomass, we find
0
1−ρ/2 1
0
1−ρ/2
2
2
/μ2y(1)
+ μy(t) 1 + σy(t)
/μ2y(t)
μy(1) 1 + σy(1)
⎡
⎡δ
⎤⎤
∞
0
1−ρ/2
2
⎣ 1 + 1 ⎣ 1 + α wR wG 1 − R
= μy(1) 1 + σy(1)
/μ2y(1)
r f (r) dr ⎦⎦
δ
K
r
⎤
⎡
∞
0
1
1+δ
−ρ/2
2
⎣ 1 + α wR wG 1 − R
=
μy(1) 1 + σy(1)
/μ2y(1)
r f (r) dr ⎦
δ
1+δ
K
r
(A10.8)
226
Natural Science Constraints in Environmental and Resource Economics
Using the abbreviation ω = wR wG (1 − R/K)/(1 + δ), a monotonic transformation of the objective function, i.e. multiplication by δ/(1 + δ), and, once
more, the approximation of dropping second-order terms in ω, one obtains the
proposed result.
A10.5
Proof of Lemma 10.1
To find the efficient strategies, we first determine the strategies which minimize the standard deviation of income given the mean income. Out of these
strategies those are efficient which maximize the mean income for a given standard deviation. Each point on the income possibility frontier is generated by
exactly one efficient strategy, since the solution of the corresponding minimization problem is unique.
Equivalent to minimizing the standard deviation, we minimize the variance
for a given mean income,
min σy2
α,r
s.t. μy ≥ μ̄y , α ∈ [0, 1], r ∈ [0, ∞).
(A10.9)
For a more convenient notation, we use the abbreviations
∞
R1 (r) =
∞
r f (r) dr and R2 (r) =
r
r 2 f (r) dr.
(A10.10)
r
The Lagrangian for the minimization problem (A10.9) is
L = σy2 (α, r) + λ [μy (α, r) − μ̄y ]
= σr2 + 2 α μr R1 (r) − α2 R12 (r) − α (2 − α) R2 (r) · [1 + 2 α ω R1 (r)]
+λ [[μr − α R1 (r)] · [1 + α ω R1 (r)] − μ̄y ] .
The first order condition with respect to r is
α r f (r) [−2 (μr − α R1 (r)) + (2 − α) r] · [1 + 2 α ω R1 (r)]
− σr2 + 2 α μr R1 (r) − α2 R12 (r) − α (2 − α) R2 (r) · 2 ω α r f (r)
= −λ α r f (r) · [1 + α ω R1 (r)] + λ [μr − α R1 (r)] ω α r f (r). (A10.11)
The first order condition with respect to α is
[2 R1 (r) (μr − α R1 (r)) − 2 (1 − α) R2 (r)] · [1 + 2 α ω R2 (r)]
+ σr2 + 2 α μr R1 (r) − α2 R12 (r) − α (2 − α) R2 (r) · 2 ω R1 (r)
= λ R1 (r) · [1 + α ω R2 (r)] − λ [μr − α R1 (r)] ω R1 (r). (A10.12)
Insurance and Sustainability through Ecosystem Management
227
Canceling the common terms α r f (r) in Equation (A10.11), and plugging the
result into (A10.12) leads, with some rearranging, to
R1 (r) (2 − α) r = 2 (1 − α) R2 (r)
α∗ (r) =
⇔
R2 (r) − r R1 (r)
.
R2 (r) − 12 r R1 (r)
Re-inserting (A10.10) leads to (10.19), which is the unique solution of the first
order conditions. σy (α∗ (r), r) is the minimum, since σy (α, r) is maximum at
the corners α = 1 (with ρ > 0), or ρ = 0 (with α < 1), as can verified easily.
Equation (10.19) determines the set of strategies, which generate the minimum standard deviation for any given mean income. This set may include
different strategies which lead to the same standard deviation, but different
mean incomes. In such a case, we drop the strategy associated with the lower
mean income, which is determined by α∗ (r, r), where r is chosen from the
appropriate subset Ω ⊆ [0, ∞) of feasible rain thresholds.
Turning to the properties of α∗ (r), for r = 0 the numerator and denominator
of (10.19) are equal, hence α∗ (0) = 1. For r → ∞, we have, using L’Hospital’s
rule repeatedly, lim α∗ (r) = 0. Numerical computations for a wide range of
r→∞
parameters (μr , σr ) resulted in qualitatively the same curves α∗ (r) as shown in
Figure 10.3.
A10.6
Proof of Lemma 10.2
Plugging Equations (10.19) and (A10.10) into (10.17) and differentiating with
respect to r yields:
dα∗ (r)
d μy (α∗ (r), r)
∗
= −
R1 (r) + α (r) r f (r) · [1 + 2 α ω R1 − ω μr ]
dr
r
(A10.13)
2
2 R1 (r) R2 (r)
=
+ α∗ 2 (r) r f (r) · [1 + 2 α ω R1 − ω μr ] > 0,
(2 R2 (r) − r R1 (r))2
since, by assumption, ω μr < 1.
For r → 0, we have lim R1 (r) = μr , lim R2 (r) = σ22 + μ2r , and α∗ (0) = 1.
r→0
r→0
Inserting into equations (10.17) and (10.18) yields lim μy (α∗ (r), r) = 0 and
r→0
lim σy (α∗ (r), r) = 0.
r→0
For r → ∞, we have lim R1 (r) = 0 and lim R2 (r) = 0, and lim α∗ (r) = 0.
r→∞
r→∞
∗
r→∞
Inserting into equations (10.17) and (10.18) yields lim μy (α (r), r) = μr and
∗
lim σy (α (r), r) = σr .
r→∞
r→∞
228
Natural Science Constraints in Environmental and Resource Economics
A10.7
Proof of Lemma 10.3
As shown in Appendix A10.6, lim μy (α∗ (r), r) = μr and lim σy (α∗ (r), r) = σr .
r→∞
r→∞
This is the northeast corner of the income possibility frontier, since μy = μr is
the maximum possible mean income (cf. Lemma 10.2). The slope of the income
possibility frontier is
dμipf
dμy (α∗ (r), r)/dr
dμy (α∗ (r), r)/dr
y
∗
=
=
2
σ
(α
(r),
r)
.
y
dσy
dσy (α∗ (r), r)/dr
dσy2 (α∗ (r), r)/dr
From Appendix A10.5 we derive
dσy2
dμy
= −λ
dr
dr
(A10.14)
where λ is the costate-variable of the optimization problem (A10.9), which is
determined by Equations (A10.11) and (A10.12),
−λ =
−2μy (α∗ (r), r) + (2 − α (r)) r (1 + 2 α (r) ω R1 (r)) −
Thus, we have
σy2 (α∗ (r),r) 2 ω
1+2 α (r) ω R1 (r)
1 + 2 α (r) ω R1 (r) − ω μr
dμipf
2 σy (α∗ (r), r)
y
=
.
dσy
−λ
In particular for r → ∞, it is lim (−λ) =
r→∞
−2 μr +2 r−σr2 2 ω
1−ω μr
.
(A10.15)
= ∞. Hence,
dμipf
y
lim
= 0.
r→∞ dσy
For r → 0 both the mean income μy (α∗ (r, r)) and the standard deviation of
income σy (α∗ (r, r) vanish (cf. Appendix A10.6). Since both cannot be negative,
this is the southwest corner of the income possibility frontier. At this point,
the slope of the income possibility frontier is
dμipf
μy (α∗ (r, r))
y
= lim
= lim
r→0 dσy
r→0 σy (α∗ (r), r)
r→0
lim
μr
(1 − α∗ (r))2 μ2r (1 + α∗ (r) ω μr )2
=
,
σr2 (1 − α∗ (r))2 (1 + 2 α∗(r) ω μr )
σr
neglecting terms of second order in ω. For r = 0, and any given α, we have
μy (α, 0) = [μr − αR1 (0)] [1 + α ω R1 (0)] = (1 − α) μr (1 + α ω μr )
σy2 (α, 0) = σr2 + 2 α μr R1 (0) − α2 R12 (0) − α (2 − α) R2 (0) [1 + 2 α ω R1 (0)]
= (1 − α)2 σr2 (1 + 2 α ω μr ),
Insurance and Sustainability through Ecosystem Management
229
i.e., for small ω, the straight line between (μy , σy ) = (0, 0) (α = 1) and
(μy , σy ) = (μr , σr ) (α = 0) is always within the income possibility set. Since
for r = 0 the standard deviation is maximum for given mean income (cf. Appendix A10.5), the income possibility frontier is located above this straight
line.
We have numerically determined the income possibility frontier for a large
variety of parameters μr , σr , and ω. The results have provided strong evidence
that under any set of parameters the income possibility frontier is divided into
two domains: a convex domain for small σy and a concave domain for large
σy . For very small σr , these two domains maybe separated by a jump in the
income possibility frontier, such taht, in these extreme cases, the left borders
of the respective income possibility sets inwardly curved to the right.
A10.8
Proof of Lemma 10.4
To prove part (i), we show that (a) the optimal indifference curve is convex
over the whole range σy ∈ [0, σr ], and (b) the optimum is within the concave
domain of the income possibility frontier.
Ad (a). Rearranging Equation (A10.2) yields the following expression for
the optimal indifference curve (where μ∗0 is the certainty equivalent for the
optimum)
2 2/ρ
μy
σy
=
− 1.
(A10.16)
μy
μ∗0
Inserting in the condition for the convexity of the indifference curve yields
2
2/ρ
μy
2+ρ
μy
>1+ρ ⇔
<
.
(A10.17)
σy
μ∗0
1+ρ
By assumption, this condition is fulfilled for μy = μr on the indifference curve
which intersects (μr , σr ), i.e. which is below the optimal one. Since μy ≤ μr
for all efficient strategies, this condition is fulfilled for all μy on the optimal
indifference curve.
Ad (b). The minimum slope of the income possibility frontier in the convex
domain (i.e. at the southwest border) is μr /σr (Lemma 10.3). The slope of the
indifference curve at the optimum (μ∗y , σy∗ ), however, is smaller,
2
ρ σy∗ μ∗y
μr
μr μ∗y
ρ
μr μ∗y
μr
1+ρ <
<
⇒
<
⇔
< ,
∗2
2
2
∗
∗
∗
∗
σ
σr
σr σy
σr σy
μy + (1 + ρ) σy
σr
1 + (1 + ρ) μy∗ 2
y
where the inequality μr /σr < μ∗y /σy∗ holds as a consequence of Lemma 10.3,
and the expression on the left hand side of the last inequality is the slope of the
indifference curve at the optimum (cf. Equation A10.3). Hence, the optimum
cannot be in the convex domain of the income possibility frontier.
230
Natural Science Constraints in Environmental and Resource Economics
Ad (ii). For ρ = 0, the indifference curves are horizontal lines. Hence, the
maximum of the income possibility frontier, which is at the corner (μy , σy ) =
(μr , σr ), is the optimum.
For ρ > 0 corner solutions are excluded. At the corner (μy , σy ) = (μr , σr )
the slope of the income possibility frontier is zero (Lemma 10.3), whereas
the indifference curves have a positive slope, provided ρ > 0. At the corner (μy , σy ) = (0, 0), the income possibility frontier is increasing with a slope
μr /σr (Lemma 10.3), but the slope of the indifference curves is zero for σr = 0
(cf. Appendix A10.3).
A10.9
Proof of Result 10.1
We have shown that the unique optimum is in the concave domain of the
income possibility frontier (Appendix A10.8), and that the slope of the farmer’s
indifference curves increases with ρ (Appendix A10.3). Thus, the optimal mean
income μ∗y decreases if ρ increases. Since for efficient strategies the mean μ∗y
is increasing in r, the rain threshold r ∗ of the optimal strategy decreases if ρ
increases.
A10.10
Sensitivity Analysis of Result 10.2
The aim of this Appendix is to show in a sensitivity analysis how the qualitative
results shown in Figure 10.6 and stated in Result 10.2 depend on the parameters of the model. The sensitivity analysis was performed using a Monte Carlo
approach, repeating the computations with multiple randomly selected parameter sets. We focused on three parameters, namely the growth parameter of
green biomass wG , the influence c of grazing on the growth of reserve biomass,
and the standard deviation σr of rainfall. The other parameters either affect
the outcomes in the same direction as the selected parameters (this is the case
for the growth parameter of the reserve biomass wR and the expected value of
rainfall μr ), or in the inverse direction (this is the case for the death rate of the
reserve biomass d).15 Hence their variation enables no further insights.
A sample size of N = 20 parameter sets was created according to the Latin
Hypercube sampling method (Saltelli et al. 2000).16 The three parameters were
assumed to be independent uniformly distributed, with 0 ≤ wG ≤ 5, 0 ≤ σr ≤
2.4 and 0 ≤ c ≤ 1, the upper bounds for wG and σr are guesses which proved
to be suitable. The respective simulation results were compared to the results
shown in Figure 10.6. The following types of long-term dynamics of mean
15
For the two parameters K and R, no substantial influence is to be expected: they just
rescale the problem.
16
This method, by stratifying the parameter space into N strata, ensures that each parameter has all proportions of its distribution represented in the sample parameter sets.
Insurance and Sustainability through Ecosystem Management
231
1.2
1
mean income μy (t > 0)
mean reserve biomass μR (t > 0)
reserve biomass and mean income (distinct from those stated in Result 10.2)
were found:17
(i) If the growth parameter of the green biomass wG is very low, i.e. if
wG · wR < d, the reserve biomass is not able to persist at all. Keeping livestock
is not possible, independent of the chosen grazing management strategy.
0.8
0.6
0.4
0.2
0
0.4
t=1
t = 10
t = 40
t = 70
t = 100
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
1
t=1
t = 10
t = 40
t = 70
t = 100
1
0.8
0.6
0.4
0.2
0
0.4
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
1
Figure A10.1 Parameter values are as in Figure 10.6, except for c = 0.9.
1.2
1
mean income μy (t > 0)
mean reserve biomass μR (t > 0)
(ii) If the impact c of grazing on the growth of the reserve biomass is very
high, the mean reserve biomass declines to zero in finite time, unless the grazing
management strategy is very conservative. This is illustrated in Figure A10.1,
where we have chosen c = 0.9.
0.8
0.6
0.4
0.2
0
0.4
t=1
t = 10
t = 40
t = 70
t = 100
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
Figure A10.2
1
t=1
t = 10
t = 40
t = 70
t = 100
1
0.8
0.6
0.4
0.2
0
0.4
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
Parameter values are as in Figure 10.6, except for wG = 4.
(iii) If the growth parameter of the green biomass is very high or the impact
of grazing on the growth of the reserve biomass is very low, the future mean
income is the higher the less conservative the strategy is, i.e. resting is not
required to preserve the ecosystem. This is illustrated in Figure A10.2 for a
very high growth rate of the biomass, wG = 4. Qualitatively the same outcome
arises for very low c (see also Müller et al. 2004).
17
To illustrate them, additional calculations were done, where one parameter was chosen
differently from the original parameter set of Figure 10.6 in each case.
1
Natural Science Constraints in Environmental and Resource Economics
1.2
1
mean income μy (t > 0)
mean reserve biomass μR (t > 0)
232
0.8
0.6
0.4
0.2
0
0.4
t=1
t = 10
t = 40
t = 70
t = 100
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
1
t=1
t = 10
t = 40
t = 70
t = 100
1
0.8
0.6
0.4
0.2
0
0.4
0.5 0.6 0.7 0.8 0.9
fraction of resting α∗ (r)
Figure A10.3 Parameter values are as in Figure 10.6, except for σr = 0.05.
(iv) If the standard deviation of rainfall σr is very small, resting is almost
deterministic: for r > μr , resting will take place in hardly any year, such that
mean reserve biomass μR and mean income μy are independent of the strategy.
For r < μr , resting will take place in almost every year, i.e. the fraction α∗ (r)
of rested paddocks determines the outcome, as illustrated in Figure A10.3 for
σr = 0.05.
A10.11
Proof of Lemma 10.5
In order to determine the steady-state mean value Rstst of the reserve biomass,
we plug Equation (10.1) into Equation (10.2) an take the expected value on
both sides of the resulting equation. In the long-term, the expectation value of
i
Rti and Rt+1
are the same and equal to Rstst . Given that in the long-term each
camp will be rested with equal probability, we derive
1
Rstst
Rstst 0
stst
stst
dR
1+
1−
= wR wG R
μr − c μy(1) (α (r, r) .
K
K
This equation is solved by Rstst = 0 and by
Rstst = K
wG wR (μR − c μy(1) (α (r), r)) − d
.
wG wR (μR − c μy(1) (α (r), r)) + d
(A10.18)
If it is positive, the last expression is the solution; otherwise Rstst = 0 is
the solution, since the reserve biomass cannot become negative. It is easily
confirmed that Rstst is monotonically decreasing in μy(1) . With a very similar
argument as in Lemma 10.2, it is shown that μy(1) is monotonically increasing
in r. Hence, Rstst is monotonically decreasing in r.
Income in each year is given by y(t) = R(t)/(I R) Ii=1 xi r. Given that
each camp is equally likely to be rested in the long-term, the long-term expected
value of income is
μstst
R
μy(1) (α (r), r).
μstst
=
(A10.19)
y
R
1
Insurance and Sustainability through Ecosystem Management
The unique interior extremum for which Rstst > 0 is given by
&
wG wR μr + d − 2 d (wG wR μr + d)
μ̂y(1) =
.
c wG wR
233
(A10.20)
Is is a maximum, since for both corners μy(1) = 0 and μy(1) = μr we have
= 0. Since μy(1) is monotonically increasing in r, a unique r̂ exists, for
μstst
y
which μy(1) = μ̂y(1) .
A10.12
Proof of Lemma 10.6
If c > 1 − d/(wG wR μr ), μstst
= 0 for r → ∞, by Lemma 10.5. That is, a
R
strategy without resting is unsustainable. If, however, r → 0, μy(1) = 0 (by
Lemma 10.2). Hence, as d < wG wR μr , the strategy with complete resting is
sustainable. By Lemma 10.2 μy(1) is monotonically increasing with r, which
concludes the proof.
A10.13
Proof of Result 10.3
By Lemma 10.6, all strategies are sustainable if c ≤ c = wwGGwwRRμrμ−d
. Hence,
r
even the strategy chosen by risk-neutral farmers is sustainable. The interesting
case is c > c. In that case, the strategy chosen by a risk-neutral farmer is
unsustainable. What remains to be shown is that for sufficiently large σr and
sufficiently small c, a ρ exists, such that all farmers with risk aversion ρ > ρ
will choose a sustainable strategy. A necessary and sufficient condition for this
statement is that
lim μy (α (r (ρ)), r (ρ)) <
ρ→∞
wG wR μr − d
,
c wG wR
(A10.21)
where ((α (r (ρ)), r (ρ)) is the optimal strategy for a myopic farmer with risk
aversion ρ. For, if Condition (A10.21) holds, the strategy chosen by an infinitely
risk averse farmer is sustainable (cf. Lemma 10.6). Condition (A10.21) is fulfilled, if (i) the right hand side is large enough and (ii) the left hand side is small
enough. The right hand side is large, if c and d are small. The right hand side
is small, if σr is large compared to μr . This has been shown in Appendix A10.7:
if σr is large, the income-possibility frontier is very flat in its concave domain.
Hence, the optimal μy is only slightly smaller than μr , and Condition (A10.21)
is violated, unless c is very small. In Figure A10.4, the threshold degree of
risk aversion is plotted against σr (left hand side) and c (right hand side). For
both, low σr and high c, this threshold value exceeds plausible values of ρ. But
for high σr and comparatively low c, the threshold value ρ lies well within the
range of degrees of risk aversion which are commonly considered as reasonable
(ρ ≤ 4; see, e.g., Gollier 2001).
Natural Science Constraints in Environmental and Resource Economics
7
6
5
4
3
2
1
0
0.4
0.6
0.8
1
standard deviation of rainfall σr
1.2
threshold risk-aversion ρ
threshold risk-aversion ρ
234
9
8
7
6
5
4
3
2
1
0
c
0.5
0.525
impact of grazing c
Figure A10.4 The threshold value of risk aversion, above which a myopic
farmer chooses a sustainable strategy. On the left hand side plotted against
the standard deviation σr of rainfall, on the right hand side plotted against the
impact of grazing on vegetation. For c ≤ c = wwGGwwRRμrμ−d
, all strategies are susr
tainable (Lemma 10.6). The remaining parameter values are as in Figure 10.6.
0.55
11. Optimal Investment in Multi-Species
Protection: Interacting Species and
Ecosystem Health∗
11.1
Introduction
The global loss of biodiversity currently proceeds at rates exceeding the natural
rate of species extinction by a factor of 100 to 1000, mainly due to human
disturbance of natural ecosystems (Watson et al. 1995b). As a response, in
the past decades there have been an increasing number of policies targeted
at the protection of endangered species, such as the U.S. Endangered Species
Act (Brown and Shogren 1998). Only recently have these conservation policies
come under scrutiny not only for their conservational effectiveness (Hoekstra
et al. 2002, Shouse 2002) but also for their economic efficiency (Cullen et al.
2001, Dawson and Shogren 2001, Metrick and Weitzman 1996, 1998).
Under the U.S. Endangered Species Act the U.S. Fish and Wildlife Service, a division of the Department of Interior, lists species as endangered in the
United States after (i) they have been suggested for listing by some individual
or organization, public or private, (ii) scientific studies support the proposed
listing, and (iii) no serious reasons against a listing emerge during a 60-day
period for public comments. Listed species enjoy special protection from harm
and must have official recovery plans created by the Fish and Wildlife Service.
They are eligible for public spending on the federal and state levels. In 1995,
there were 957 species listed as endangered in the United States and expenditures by federal and state agencies for all species recovery plans totalled US$
280 million (Dawson and Shogren 2001).
As of the mid 1990s almost all endangered species had official recovery
plans, but the expenditures were distributed rather unevenly among the different plans. Nearly 95% of the total reported spending by federal agencies
were spent on about 200 vertebrate species, and only 5% were spent on about
800 invertebrate and plant species (Dawson and Shogren 2001). This has lead
∗
Previously published in EcoHealth, 1(1), 101-110 (2004).
235
236
Natural Science Constraints in Environmental and Resource Economics
to the suggestion that the status of a species as ‘charismatic megafauna’ is a
major factor in explaining the amount of funding for a recovery plan (Metrick
and Weitzman 1996, 1998). While the number of species listed as endangered
has almost doubled over the past decade – from 554 in 1989 up to 957 in 1995
– and total expenditure on species recovery programs has increased by a factor of almost seven – from US$ 44 million in 1989 to US$ 280 million in 1995
(Dawson and Shogren 2001) – only 13 species have actually recovered enough
to warrant removal from the list (Shouse 2002).
One reason for the obviously poor performance of species recovery plans
may be our poor understanding of the functioning of the natural ecosystems in
which the target species live. The design of species recovery plans requires extensive knowledge of the species’ life history and ecology (Bowles and Whelan
1994, MacMahon 1997). Yet, recent ecological surveys stress the large extent
of uncertainty about the functioning of ecosystems (Brown et al. 2001, Holling
et al. 1995, Loreau et al. 2001, Tilman 1997). Given this large uncertainty it is
understandable that species recovery plans under the Endangered Species Act
traditionally target single species, with the respective expenditure being highly
species specific. Likewise, influential economic studies on optimal species protection plans for multi-species ecosystems assume that species are independent
(Solow et al. 1993, Weitzman 1993, 1998).
However, considering species interactions is potentially important for the
design of multi-species protection plans and to ensure the efficient allocation of
limited conservation budgets (Wu and Bogess 1999). Here I show that taking
species interaction into account makes a crucial difference for how to optimally allocate a given conservation budget. I conclude that effective species
protection should go beyond targeting individual species, and consider species
relations within whole ecosystems as well as overall ecosystem functioning. To
make this conclusion operational I suggest to look at indicators of ecosystem
health, which is a necessary prerequisite for successful species protection in
situ.
11.2
Ecosystems and Species Extinction Risk
The formal framework used here follows and expands the one of Solow et al.
(1993) and Weitzman (1998). Consider an ecosystem of n ∈ IN different species.
Each of them may be subject to stochastic extinction. Let pi (with i = 1, . . . , n)
denote species i’s survival probability, i.e. the probability that species i still
Optimal Investment in Multi-Species Protection
237
exists after a time-period of T years.1 , 2
Let Ei (with i = 1, . . . , n) be the status variable indicating whether species i
will still be in existence after T years or whether it will have gone extinct:
)
1
if species i survives,
(11.1)
Ei =
0
if species i becomes extinct.
Due to the stochastic nature of extinction the variable Ei is a random variable. In general, the different Ei are not independent. The existence of certain species will influence the survival probabilities of others. This is most
obvious for species that interact directly, for instance through a mutualistic
relation (positive correlation between survival probabilities), competition for
a common resource (negative correlation between survival probabilities), or
a predator-prey relation. More generally, the relations among species in an
ecosystem can be analyzed in terms of a trophic network. Such a food-web
depicts the flow of food (measured in biomass) between the different species.
The normalized flow between two species may be taken as a measure of the
interaction strength between the two (Paine 1992). Food-web analysis permits
to identify indirect interactions among species which are coupled through a
food chain that comprises one or more intermediate nodes. Food-web analysis
reveals the high degree of connectance and a complex pattern of species interactions even when looking at only a limited number of species in relatively few
trophic groups (Elton 1927).
11.3
Human Appreciation of Species and Ecosystem
Services
Individual species as well as entire ecosystems are valuable for humans for a
number of reasons. Many species have direct use value as food, fuel, construction material, industrial resource or pharmaceutical substance (Farnsworth
1988, Plotkin 1988). More recently, it has been stressed that biodiversity, i.e.
the set of all species, also has an important indirect use value in so far as entire
ecosystems perform valuable services such as nutrient cycling, control of water
1
The concepts of extinction risk of a population of species i, 1 − pi , and its survival
probability, pi , are equivalent measures of population viability (Burgman et al. 1993). Here,
survival probabilities are used for the ease of interpretation. Another equivalent measure of
population viability is its expected lifetime, τi . It is related to the survival probability pi over
a time-period T via the equation pi = exp(−T /τi ) (Wissel et al. 1994). T is typically taken
to be 10, 50 or 100 years in population viability analysis.
2
On a more fundamental level the survival probabilities are determined by a number of
factors, such as the species’ population size, geographic range, age structure and spatial
distribution (Lande 1993).
238
Natural Science Constraints in Environmental and Resource Economics
runoff, purification of air and water, soil regeneration, pollination of crops and
natural vegetation, or partial climate stabilization (Daily 1997a, Mooney and
Ehrlich 1997, Perrings et al. 1995b). These ecosystem services are essential to
support the human existence on Earth. They can only be provided by more or
less intact ecosystems and result from the complex – and, up to now, not well
understood – interplay of many different species in these ecosystems (Holling
et al. 1995, Tilman 1997).
Following Weitzman (1998) and Metrick and Weitzman (1998), the utility
gained directly and indirectly from a multi-species ecosystem can be written
as a sum of the direct utilities of all individual species, Ui (i = 1, . . . , n), and
the utility gained indirectly from the entire ecosystem through the ecosystem
services provided collectively by all species, UES . In general, the utility of
ecosystem services will be a function of the existence or non-existence of all
species, UES = UES (E1 , . . . , En ). Hence:
U = UES (E1 , . . . , En ) +
n
Ui .
(11.2)
i=1
For example, Weitzman (1998) specifies UES as the diversity of the set of all
actually existing species. His diversity function provides an aggregate measure
of the diversity of a set of species based on the pairwise dissimilarities among
them (Weitzman 1992). This is in line with the idea that biodiversity may
be taken as as a proxy for an ecosystem’s capability of providing the valuable
services described above (Holling et al. 1995, Loreau et al. 2001, Perrings et al.
1995b, Tilman 1997).
Because of the stochastic risk of species extinction a decision maker will
consider not the utility, U, but the expected utility, E[U]. With pi as species i’s
survival probability the expected direct utility of that species is given by pi Ui .
Hence,
n
E[U] = E[UES (E1 , . . . , En )] +
pi Ui .
(11.3)
i=1
Specification of the function UES would require a detailed ecological model
of how all the species in an ecosystems collectively provide certain ecosystem
services. In order to keep matters simple I shall assume that the ecosystem
provides all its services at full scale if, and only if, species 1 exists. The level of
utility derived from ecosystem services then only depends on whether species 1
exists or not. Furthermore, in order to focus on species interaction in the
ecosystem (instead of trade-offs on the utility side) I assume that all species
have vanishing direct utility: U1 = . . . = Un = 0. The value of all the different
species, thus, is an indirect one and consists of their contribution to ecosystem
functioning, and, in particular, of their support of species 1. Hence, the relevant
Optimal Investment in Multi-Species Protection
239
objective function for making conservation decisions is
E[U] = p1 UES ,
(11.4)
where UES is a positive constant. While species 1 thus plays a prime role, all
the other species are potentially important, too, as their existence or absence
may influence species 1’s survival probability, p1 . I will address this latter point
explicitly in Section 11.5 below.
11.4
Species Protection Plans and Optimal
Allocation of a Conservation Budget
Consider now the economic decision problem of how to allocate a conservation budget among different species protection plans. The time structure of
the problem is as follows. The decision about how to allocate the conservation budget is made today, and the corresponding species protection plans are
enacted immediately. The result in terms of actual species survival or extinction is observed tomorrow (which means, more precisely, after the course of T
years). The actual ecosystem situation tomorrow yields a certain utility, the
expectation of which is the basis for today’s decision.
For the moment, I will neglect species interaction, as it is done in the
existing economic literature (Solow et al. 1993, Weitzman 1993, 1998). That
is, in this section I will introduce the economic decision framework for the case
that all n species are independent. I will then introduce species interaction in
Section 11.5 below.
Following Weitzman (1998) assume that investment in some protection plan
aimed at species i can enhance that species’ survival probability pi within
certain limits:
pi ≤ pi ≤ pi
with
pi ≥ 0 and pi ≤ 1.
(11.5)
The probability pi gives the ‘down-risk’ for species i’s survival. This is the survival probability if no investment in protection is made. On the other hand, pi
indicates the ‘up-risk’ for species i. This is the maximum survival probability
amenable for species i through the particular protection plan under consideration. In the extreme, pi = 0 and pi = 1. That is, without protection species i
will become extinct for sure, but undertaking the protection plan at full scale
will save it for sure. Any protection plan can also be undertaken at any level
in between not-at-all and full-scale, leading to survival probabilities pi which
are on a continuum pi ≤ pi ≤ pi .
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Natural Science Constraints in Environmental and Resource Economics
Species protection plans are also costly. Suppose that out of an exogenously
given and fixed budget b > 0 an amount bi ≥ 0 is spent on protecting species i.
Then the following budget constraint holds:
n
bi ≤ b .
(11.6)
i=1
Investment bi in protecting species i will enhance the species’ survival probability pi according to a ‘survival probability enhancement function’, or ‘enhancement function’ for short:
pi = Pi (bi )
with Pi (0) = pi , Pi (bi ) ≤ pi for all bi , Pi ≥ 0.
(11.7)
The qualifying properties state that without any investment species i’s survival
probability will stay at the lower bound, pi . On the other hand, species i’s
survival probability cannot exceed its upper bound, pi , no matter how much
is invested in its protection. Generally, the more money is spent to enhance
species i’s survival probability the higher will pi actually turn out to be. For
example, Weitzman (1998) uses linear enhancement functions with
)
Pi (bi ) = min
1
bi 0
pi − pi + pi , pi
ci
,
(11.8)
where the parameter ci > 0 indicates the costs of enhancing the survival probability all the way from its lower bound pi to its upper bound pi .
The economic decision problem can then be stated as follows: choose a
budget allocation such as to maximize the expected utility function (11.3) subject to the budget constraint (11.6) and the feasible possibilities for survival
probability enhancement as described by (11.7). Formally:
maximize{bi }i=1,...,n E[U] s.t.
n
bi ≤ b and
i=1
pi = Pi (bi ) for all i = 1, . . . , n.
(11.9)
This is a typical stochastic programming problem which is continuous in the
bi .
Weitzman (1998) has characterized the solution to problem (11.9) under the
assumptions that (i) E[UES (E1 , . . . , En )] is specified as the expected diversity of
the set of all species, (ii) all species are independent and (iii) the enhancement
functions are linear and given by (11.8). Obviously, with the simple objective
function (11.4) the optimal solution, {b∗i }i=1,...,n , is that the entire conservation
Optimal Investment in Multi-Species Protection
241
budget is spent on species 1:3
b∗1 = b and b∗i = 0 for i = 2, . . . , n.
11.5
(11.10)
Species Interaction
The formalization of problem (11.9) above, as well as the properties of its solution (11.10), rely heavily on the simplifying assumption of independent species.
To illustrate how one could construct a more general framework for the case of
interacting species let me introduce the effect of species interaction for a simple
model ecosystem that consists of just two species (n = 2), and in which the existence of species 2 influences the survival probability of species 1 but not vice
versa. For example, one could think of species 2 as a potential prey for species 1
(positive interaction), or as a predator of it (negative interaction). Focusing
on n = 2 is not as restrictive as it may appear at first sight. For species 2
may be interpreted as ‘all the rest of the ecosystem’ besides species 1. In this
interpretation, it then also appears plausible to assume that while species 2
influences species 1’s survival probability, the reverse influence is negligible.
In this case the survival probability of species 1 (‘target species’) depends on
the existence of species 2 (‘support species’). Let p1|E2 denote the conditional
survival probability of species 1 given the existence or non-existence of species 2.
In particular, p1|1 is the survival probability of species 1 if species 2 exists
(E2 = 1) and p1|0 is the survival probability of species 1 if species 2 does not
exist (E2 = 0). The (unconditional) survival probability of species 1, taking
into account that species 2 exists with probability p2 , is then given by
p1 = p1|1 p2 + p1|0 (1 − p2 ).
(11.11)
An investment in a protection plan for species 1 will increase the conditional
survival probability p1|E2 :
)
1
if species 2 exists,
where E2 =
(11.12)
p1|E2 = P1|E2 (b1 ),
0
if species 2 is extinct.
In particular, the existence of species 2 may be thought of as having an influence on the up and down risk for species 1, which also become conditional
probabilities: p1|E2 and p1|E2 . If the existence of species 2 has a positive
influence on species 1 it seems natural to assume that
p1|1 ≥ p1|0
3
and p1|1 ≥ p1|0
(11.13)
For c1 < b the budget will not be completely exhausted by funding a full-scale protection
plan for species 1. Since spending money on protecting other species would not increase
utility under the objective function (11.3) the remaining budget, b − b1 , could either be left
idle or allocated randomly among the other species.
242
Natural Science Constraints in Environmental and Resource Economics
with at least one inequality holding as a strict inequality. This is illustrated in
Figure 11.1 which shows the feasible range of species 1’s survival probability,
p1|E2 , conditional on the the absence (E2 = 0) or existence (E2 = 1) of species 2.
The effect of a positive species interaction essentially is that it shifts the upp1|E2 6
1
p1|1
p1|0
p1|1
p1|0
-
0
0
1
E2
Figure 11.1 Feasible range of species 1’s survival probability conditional on the
the existence or absence of species 2, p1|E2 in the case of a positive influence.
risk and the down-risk for species 1, and therefore the entire feasible range of
survival probabilities, upward.
Note that there is actually a number of ways, all consistent with condition (11.13), in which the existence of species 2 may have a positive influence
on species 1’s range of survival probabilities (Figure 11.2). One possibility (Figb
a
p1|E2 6
1
0
c
p1|E2 6
1
-
0 1 E2
0
d
p1|E2 6
1
-
0 1 E2
0
p1|E2 6
1
-
0 1 E2
0
-
0 1 E2
Figure 11.2 Different possibilities (a–d)of how the existence of species 2 may
have a positive influence on species 1’s feasible range of conditional survival
probabilities.
ure 11.2a) is that under the positive influence of species 2 (E2 = 1) the upper
bound for species 1’s conditional survival probability increases while the lower
bound is not altered compared with a situation in which species 2 is absent
(E2 = 0). Or the lower bound for the conditional survival probabilities may
Optimal Investment in Multi-Species Protection
243
increase while the upper bound is not altered (Figure 11.2b). Another possibility is that both the lower and upper bound increase such that the entire range
of feasible conditional survival probabilities shifts upward (Figure 11.2c, d).
This may happen in such a way that the range with and without existence of
species 2 overlap (Figure 11.2c) or such that they do not overlap (Figure 11.2d).
The latter case may be particularly relevant for evolutionary old ecosystems
in which target species have coevolved with, and are well adapted in a special
way to, their support species and ecosystem. For well adapted and specialized
target species the existence of supporting species and ecosystems may have a
larger effect on the species’ survival probability than any protection plan aimed
directly at that species.
If the existence of species 2 has a negative influence on species 1 one has:
p1|1 ≤ p1|0
and p1|1 ≤ p1|0 ,
(11.14)
where at least one inequality holds as a strict inequality. Like in the case of
positive interaction, condition (11.14) can be fulfilled in a variety of ways. And
if the existence of species 2 does not have any influence on species 1 one has
p1|1 = p1|0
and p1|1 = p1|0 .
(11.15)
In this formal framework the economic decision problem of how to allocate a
conservation budget among interacting species, now reads as follows:
maximize{b1 ,b2 } E[U] s.t. b1 + b2 ≤ b,
p1|E2 = P1|E2 (b1 ) and p2 = P2 (b2 ). (11.16)
11.6
Species Interaction and Optimal Allocation of
the Conservation Budget
Species interaction can make a big difference for how to optimally allocate a
conservation budget among different species protection plans. This is illustrated in this section by the example of a concrete parameterization of species
interaction based on the formal framework developed in the previous section.
According to the objective function (11.4), if species 1 exists the utility
is UES , and it is zero otherwise. Assume that the feasible range of survival
probabilities for species 2 comprises the entire interval [0, 1], i.e. p2 = 0 and
p2 = 1. The feasible range of survival probabilities for species 1 is contingent
upon the existence of species 2 and, furthermore, depends on the type and
strength of influence of species 2 on species 1:
1
1
(2 + 2κ) ≤ p1|1 ≤ (3 + 2κ),
with species 2 (E2 = 1):
5
5
2
3
without species 2 (E2 = 0):
≤ p1|0 ≤ ,
(11.17)
5
5
244
Natural Science Constraints in Environmental and Resource Economics
where κ ∈ [−1, +1] parameterizes the influence of species 2 on species 1’s
survival probability conditional on the existence of species 2. With κ = 0 the
two are independent and the existence of species 2 does not make any difference
for the range of survival probabilities of species 1. With κ > 0 (< 0) species 2
has a positive (negative) influence on species 1’s survival probability. The
entire range of feasible survival probabilities is shifted upwards (downwards).
Figure 11.3 shows the feasible range of survival probabilities for species 1 with
and without existence of species 2 depending on the interaction strength κ.
p1|E2
6
!
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!
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!
!
1/5
!!
!
!
1
−1
0
+1
with species 2 (E2 = 1)
without species 2 (E2 = 0)
-
κ
Figure 11.3 Feasible range of species 1’s survival probability conditional on
the existence or non-existence of species 2, p1|E2 , depending on the interaction
strength κ.
Assume that the total conservation budget is b = 1 and the enhancement
functions for both species are as follows:
) 1√
b + 25
without species 2 (E2 = 0)
5√ 1
P1|E2 (b1 ) =
,(11.18)
1
2
b
+
(1
+
κ)
with
species 2 (E2 = 1)
1
5
5
&
P2 (b2 ) =
b2 .
(11.19)
Note that the enhancement functions for both species exhibit strictly decreasing
returns. With a budget of b = 1 and enhancement possibilities as specified here
the economically viable survival probabilities for both species are within the
feasible range described by (11.17). The budget of b = 1 allows either a full
scale conservation project for species 1, or a full scale project for species 2, or
projects for both of them at less than full scale. Spending the entire budget
on species 1 allows to increase its survival probability, for given interaction
strength κ and contingent on the existence or non-existence of species 2, from
its lower bound to its upper bound. Similarly, spending the entire budget on
species 2 allows to increase its survival probability from its lower bound to its
upper bound. As these bounds for species 2 are given by 0 and 1, the size of
Optimal Investment in Multi-Species Protection
245
the budget (b = 1) and the particular form of enhancement function (11.19)
allows full control over species 2. With b2 = 0, species 2 will be extinct for
sure; with b2 = 1, it will exist
√ for sure; and for all levels 0 < b2 < 1, it will
exist with probability p2 = b2 . This simple setting focuses on the influence
of the interaction between the two species on how to split up the total budget
between the two in order to maximize species 1’s expected survival probability.
With b = 1 and b2 as the expenditure on species 2, the remaining budget
of b1 = 1 − b2 can be spent on species 1. The expected utility is given as the
survival probability of species 1 times the utility derived from it. With (11.4)
and (11.11):
E[U] = p1 UES = p1|1 p2 + p1|0 (1 − p2 ) UES ,
(11.20)
where p1|1 , p1|0 , and p2 depend on the expenditures b1 and b2 according to the
enhancement functions (11.18) and (11.19). With b1 = 1 − b2 one has
&
&
&
& 1
1
2
2
1 − b2 + (1 + κ)
b2 +
1 − b2 +
E[U] =
(1 − b2 ) UES
5
5
5
5
2 2 &
1&
+ κ b2 +
=
1 − b2 UES .
(11.21)
5 5
5
The term in brackets is the survival probability for species 1 in terms of b2 .
Maximizing this expression over 0 ≤ b2 ≤ 1 yields the following optimal conservation expenditures b∗2 and b∗1 = 1 − b∗2 :
3
3
0
; κ<0
1
; κ<0
∗
∗
2
(11.22)
b1 =
and
b2 =
1
4κ
; κ≥0
; κ≥0
2
1 + 4κ2
1 + 4κ
Figure 11.4 illustrates the result. It shows how the optimal allocation of the
conservation budget depends on the interaction strength κ. As long as species 2
has a negative (κ < 0) or neutral (κ = 0) influence on the target species 1,
the optimal allocation of the conservation budget is to entirely devote it to
protection of species 1.4 , 5 Obviously, spending money on conserving species 2
which then negatively impacts species 1 will not be optimal if, in the end, all
utility derives from species 1. But if the support species 2 has a positive (κ > 0)
influence on the target species 1, it is optimal to allocate a certain fraction of
4
If species 2 has a negative influence on the desired target species and no direct utility
in itself, it may even be optimal to not only not invest in its protection, but to invest in its
reduction. For example, species 2 may be a pest or parasite for species 1 and, for the sake
of protecting species 1, it may seem desirable to eliminate this pest or parasite. However,
in the formal framework employed here I only consider species protection plans, i.e. one can
only invest into enhancing a species’ survival probability.
5
Note that for vanishing interaction strength, κ = 0, solution (11.22) reduces to the
solution (11.10) obtained in Section 11.4 above for the case of independent species.
246
Natural Science Constraints in Environmental and Resource Economics
b, b∗1 , b∗2
6
b=1
b∗1
b∗2
-
−1
0
+1
κ
Figure 11.4 Optimal allocation of the conservation budget (b = 1) among the
target species (b∗1 ) and the support species (b∗2 ), depending on the interaction
strength κ. The curve shows b∗2 as a function of the interaction parameter κ,
with the distance between the curve and b = 1 corresponding to b∗1 .
the allocation budget to the protection of the support species as well. This
fraction grows as the positive interspecific influence (κ) grows in strength.
For κ = +1 the optimal allocation of the conservation budget is b∗1 =
0.2, b∗2 = 0.8. In this case the positive influence from species 2 on species 1
is so strong that by spending the largest part of the budget on protecting
species 2, one obtains a higher survival probability of species 1 than any direct
investment into that species would produce. The reason for this result is in the
assumption, illustrated in Figure 11.3, that for κ = +1 the entire feasible range
of conditional survival probabilities for species 1 with species 2 in existence,
[4/5, 1], is higher than in the absence of species 2, [2/5, 3/5]. As argued above
(Figure 11.2d), this corresponds to an evolutionary old ecosystem with a high
degree of mutual adaptation among species. Existence of the support species
can then provide a better service to the survival of the target species than
any direct investment into protecting the target species could possibly achieve.
Hence, spending money on increasing the support species’ survival probability,
thus indirectly also increasing the target species’ survival probability, is more
cost-effective than spending the entire budget directly on the target species.
The result, thus, is that species interaction can completely reverse the optimal allocation of a conservation budget. In the example studied here, while the
entire conservation budget would be allocated to species 1 without any interaction, a strongly positive interaction will make it optimal to allocate almost
the entire budget to conservation of species 2.
If one substitutes result (11.22) back into expression (11.21) for the uncon-
Optimal Investment in Multi-Species Protection
ditional survival probability of species 1 one obtains
√
2 2 & ∗ 1&
1 + 4κ2 + 4κ2 + 1
2
∗
√
+ κ b2 +
p1 =
1 − b∗2 =
.
5 5
5
5 1 + 4κ2
247
(11.23)
Figure 11.5 illustrates this result. It shows how the optimal survival probability
∗
p , p1|E2
61
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!!
!
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!!
!
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!
!
1/5
!!
!
!
1
−1
0
+1
with species 2 (E2 = 1)
without species 2 (E2 = 0)
-
κ
Figure 11.5 Optimal survival probability p∗1 of the target species (thick curve),
depending on the interaction strength κ between support and target species.
of the target species, p∗1 , increases with the interaction strength for κ ≥ 0.
11.7
Summary and Discussion of Results
This analysis has shown that taking into account species interactions in an
ecosystem is crucial for the optimal allocation of a conservation budget. Compared with policy recommendations obtained under the assumption of independent species, interactions in an ecosystem can reverse the rank ordering of
spending priorities among species conservation projects. Hence, an approach
to species protection that is efficient in terms of both species conservation and
budget resources should be based on a multi-species framework and should take
into account the basic underlying ecological relations. Another interesting result is that even if biological conservation decisions are exclusively derived from
a utilitarian framework, with species interaction it may be optimal to invest in
the protection of species that do not directly contribute to human well-being.
This is due to their role for overall ecosystem functioning and for safeguarding
the existence of those species that are the ultimate target of environmental
policy.
For practical purposes, however, one is confronted with a large extent of
uncertainty about the functioning of ecosystems, including fundamental uncertainty about the exact nature of species interaction (Brown et al. 2001, Holling
248
Natural Science Constraints in Environmental and Resource Economics
et al. 1995, Loreau et al. 2001, Tilman 1997). In many cases, it is not even
known whether two species have a positive or negative interaction. In terms
of the model outlined above, this means that neither the exact value of κ nor
its sign are known. It may be due to this large ecological uncertainty that the
multi-species recovery plans, which have become more and more important in
the U.S. Fish and Wildlife Service’s approach to protecting endangered species,
turned out to be even less successful in terms of species recovery than the more
traditional single-species plans (Clark and Harvey 2002).
The description of species interactions proposed here is very simple since it
takes into account species and their interaction only on a discrete basis (species i
exists/does not exist). A more realistic picture would involve population size
and population dynamics for each species. Yet, this would not alter the qualitative results obtained here. The description of species protection plans is
equally simple, as it is assumed that each plan affects only the very species at
which it is directed. In practice, however, every species protection plan is likely
to affect other species in the ecosystem as well.
The analysis here was mainly based on the illustrating example of a twospecies-ecosystem with one-way interaction. The absence of feedbacks excludes
any kind of complex dynamics among the species. While this is a very simple
and special setting, it can be generalized. With n different species, all of which
are potentially interacting, there are n(n − 1) pairwise directed interactions,
leading to indirect interactions among species as well as positive and negative
feedback loops. This number rises very fast as n becomes large. Empirical
evidence suggests, however, that the vast majority of pairwise interactions in
real ecosystems are weak (McCann et al. 1998, Paine 1992, Wootton 1997).
The hope may thus be that in applied studies of how to allocate a conservation
budget one can safely neglect a whole many interactions, except for the few
strong ones for each species, and that there are considerably less than n(n − 1)
interactions to be taken into account.
However, empirical evidence also suggests that even the weak interactions
are important: the complex interdependence of species survival probabilities,
together with the existence of extinction thresholds (Lande 1987, Muradian
2001), is known to give rise to so-called extinction cascades (Borrvall et al.
2000, Lundberg et al. 2000). This means that extinction of one species could
entail a cascade of further extinctions. Thus, the extinction of some species
may threaten even the existence of other species that are only very weakly
linked to the former.
Optimal Investment in Multi-Species Protection
11.8
249
Conclusion: Managing for Ecosystem Health
This discussion suggests that, for conservation purposes, not only are interactions among individual species important, but also the functioning of ecosystems at large is tantamount. Indeed, conservationists have been arguing for
years that effective species protection should go beyond targeting individual
species, and aim at whole ecosystems or landscapes.6 This analysis suggests
how such a claim can be made more substantial and operational.
In the multi-species-interaction approach taken here, systemic properties of
an ecosystem, such as e.g. their structural and functional organization, their
productivity, their resilience under disturbances, and their ability to mitigate
the impact of various stresses, underly and influence the survival probabilities of
individual species. Thus, individual species’ survival depends on, and is determined by, what has been called ‘ecosystem health’. The concept of ecosystem
health is a complex one, as it involves considerations from the natural, social
and health sciences (Rapport et al. 1998). Although difficult to measure and
operationalize (Mageau et al. 1995), the notion of ecosystem health reminds
one that species conservation in situ ultimately depends on certain properties
of the entire system in which the target species lives.
As an encompassing and detailed analysis of the myriad of mutual interactions on the species level in an ecosystem may generally not be possible
for a particular species protection plan, a useful alternative and complement
can be to take a system approach and manage ecosystems for their functions
and health.7 Ecosystem functioning and health is a necessary prerequisite for
species conservation in situ.
6
Individual species may nevertheless be of crucial importance for devising, assessing and
marketing such a more holistic approach, for instance as so-called ‘keystone’, ‘flagship’ or
‘umbrella’ species (Simberloff 1998).
7
Mageau et al. (1995), among others, suggest an operational and quantifiable definition
of ‘ecosystem health’ in terms of ecosystem functions.
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