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Sustainability and economic policy analysis von Amsberg, Joachim 1993

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SUSTAINABILITY AND ECONOMIC POLICY ANALYSISbyJOACHIM VON AMSBERGM.B.A., The University of British Columbia, 1990Dipl.-Ing. (Wi.Ing.), Technische Universitat Berlin, 1991A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESThe Faculty of Commerce and Business AdministrationPolicy Analysis DivisionWe accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJune 1993© Joachim von Amsberg, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Faculty of Commerce and Business AdministrationDepartment of ^The University of British ColumbiaVancouver, CanadaDate June 28, 1993DE-6 (2/88)SUSTAINABILITY AND ECONOMIC POLICY ANALYSISbyJOACHIM voN AMSBERGABSTRACTThe purpose of this dissertation is to provide a better economic basis for thediscussion on how much natural capital the current generation should be allowed to deplete.Chapter I uses overlapping-generations models to show the effects of different assumptionsabout which generation owns the stock of a natural resource on the distribution ofintergenerational welfare. An increase in the share of the resource stock that is owned by thefirst generation, reduces welfare of later generations. If the first generation owns the fullresource stock, intermediate generations have to be sufficiently wealthy to buy the resourcefrom the first generation and sell it to later generations. This channelling effect can lead toa situation of resource abundance followed by rapidly increasing resource prices and scarcity.Chapter II show that the incompleteness of intergenerational insurance marketsconstitutes a market failure that leads to inefficient intergenerational investment decisions.Risks that increase from generation to generation would be under-insured by the currentgeneration. Examples of excessive reduction of biodiversity, excessive natural resourcedepletion, and inefficiently low protection against global warming are provided.Chapter III analyzes the decision theoretical foundation of environmental choicesunder uncertainty. Since ambiguity and ignorance are important aspects of manyenvironmental problems, subjective expected utility theory (SEU) has significant limitationsas a normative decision making model. The use of SEU leads to a systematic bias againstthe conservation of natural capital. An alternative decision model is suggested based on theDempster-Shafer belief-function theory and Choquet expected utility.iiThe synthesis in chapter IV suggests that the costs of natural capital depletion aresystematically underestimated in conventional analysis. To remedy the biases against futuregenerations and the complete valuation of natural capital, a sustainability constraint on theeconomic activities of the current generation is proposed. This constraint requirescompensation for natural capital depletion through functional substitutes. From thissustainability constraint, an operational sustainable supply rule is derived for determiningshadow prices of natural capital depletion.iiiTABLE OF CONTENTSABSTRACT ^  iiTABLE OF CONTENTS ^  ivLIST OF TABLES ^ ixLIST OF FIGURES^  xACKNOWLEDGEMENT ^  xiINTRODUCTION^  1CHAPTER ITHE DEPLETION OF NATURAL CAPITAL ANDINTERGENERATIONAL WELFARE ^ 131^Introduction ^  132^Natural Resource Endowments and Intergenerational Welfare ^ 152.1 The Endowment Effect ^  182.1.1 An N-Generation Consumption Model ^  202.1.2 The Two-Generation Case ^  222.1.3 An Example With Ten Generations ^  292.1.4 A Two-Generation Model With Production ^ 302.2^The Channelling Effect ^  34iv3^Natural Resource Depletion and Intergenerational Compensation ^ 413.1^Efficiency Limits of Intergenerational Resource Endowments ^ 413.2^Compensation with Hartwick's Rule ^  433.2.1 Compensation with Public Resource Ownership ^ 443.2.2 Compensation with Private Resource Ownership  514^Conclusions ^  54Appendix I-A: Derivation of the Equilibrium for the N-Generation Model^ 58Appendix I-B: Derivation of the Three Generation Equilibrium with K'2 = 0 ^ 60CHAPTER IIINEFFICIENCIES FROM INCOMPLETEINTERGENERATIONAL INSURANCE MARKETS ^ 621^Introduction ^  622^An Intergenerational Investment Model under Risk ^ 672.1^An Under-Insurance Example ^  712.2^An Over-Diversification Example  732.3^The Robustness of the Market Failure from Market Incompleteness ^ 753^Applications of the Under-Insurance Result ^  793.1^A Model of Insurance Investment  803.1.1 The Competitive Solution  813.1.2 The Coordinated Solution ^  823.1.3 Comparison Between Both Equilibria ^ 843.2^Building a Dam to Protect Future Generations  8543.3^Conservation of Biodiversity ^3.4^Consumption of a Non-Renewable Resource ^Conclusions ^8991102CHAPTER IIIENVIRONMENTAL DECISION MAKINGUNDER AMBIGUITY AND IGNORANCE ^ 1051 Introduction ^ 1052 Environmental Decision Making: Models and Reality ^ 1072.1^Building Blocks of Decision Theory ^ 1082.2^Some Aspects of the Reality of Environmental Decision Making . .^•^. 1112.3^Requirements for a Theory of Environmental Decision Making^. .^•^. 1153 Approaches to Decision Making under Uncertainty ^ 1213.1^Subjective Expected Utility Theory 1213.2^Decision Making under Ambiguity ^ 1263.3^Decision Making Under Ignorance 1324 A Model for Environmental Decision Making under Uncertainty ^ 1354.1^Representation of Beliefs ^ 1364.2^Decision Theory 1424.3^An Illustrative Examples 1444.4^An Extension with Incomplete Ordering ^ 1514.5^Implications for Environmental Decision Making ^ 1555 Conclusions ^ 159viCHAPTER IVA SUSTAINABILITY CONSTRAINT:REQUIREMENTS, SPECIFICATIONS, AND APPLICATIONS . . . . 1631^Introduction ^  1632^The Need for a Sustainability Constraint ^  1642.1^Inefficient Market and Shadow Prices of Natural Capital ^ 1652.1.1 Lack of Reliable Market Prices for Natural Capital ^ 1652.1.2 Difficulties with Shadow Pricing Natural Capital  1712.1.3 A Better Default Value for Natural Capital ^ 1752.2^The Intergenerational Problem ^  1782.2.1 Intergenerational Market Failures ^  1792.2.2 The Problem of Intergenerational Welfare Distribution ^ 1803^The Appropriate Sustainability Constraints ^  1843.1^Strong and Weak Sustainability  1873.2^The Geographic Scale of Sustainability  1944^Project and Policy Evaluation Under a Sustainability Constraint ^ 1964.1^The Sustainable Supply Rule ^  1964.2^Application of the Sustainable Supply Rule ^  204vii5^A Stylized Case Study: An Oil Development Project ^ 2095.1^The Issues ^  2105.2^Assumptions  2125.3^Project Evaluation ^  2185.4 Comments ^  2226^Conclusions ^  223REFERENCES ^  228viiiLIST OF TABLESTable 2.1^Consumption in the Competitive and the Coordinated Solution ^ 64Table 3.1^The Ellsberg Paradox ^  125Table 3.2^Belief Function after Observing One Red Ball ^ 147Table 3.3^Summary of Lottery Evaluation ^  149Table 3.4^The Value of Information  151Table 4.1^Carbon Dioxide Reduction Scenario  217Table 4.2^Summary of Depletion Premia ^  221Table 4.3^Adjusted Net Benefits with a Sustainability Premium of $10.2/bbl .^222ixLIST OF FIGURESFigure 1.1^The Structure of the Overlapping-Generation Models ^ 19Figure 1.2^Sensitivity of Welfare to Parameter Changes  26Figure 1.3^Welfare Profile with Ten Generations ^  28Figure 1.4^Equilibrium Changes with Increases in Generation l's Endowment^33Figure 1.5^The Channelling Effect ^  37Figure 1.6 Welfare of Five Consecutive Generations ^  49Figure 1.7^Following or Repudiating Hartwick's Rule  53Figure 2.1^Relative Risk of Generation 2 ^  71Figure 2.2^Under-Insurance with Exogenous Risk ^  73Figure 2.3^Over-Diversification with Endogenous Risk  74Figure 2.4^Insurance Investment and a Second Good's Importance ^ 77Figure 2.5^Damage and Net Marginal Cost of Insurance Investment ^ 81Figure 2.6^Dam Hight as a Function of Costs ^  87Figure 2.7^The Conservation of Species and Welfare of Generation 1 ^ 90Figure 2.8^Offer Curve ^  95Figure 2.9^Resource Depletion with Quadratic Utility ^  99Figure 2.10 Resource Depletion with Log-Utility  100Figure 2.11 Over-Consumption of a Non-renewable Resource ^ 101Figure 3.1^Upper and Lower Probabilities ^  130Figure 3.2^Evaluation under Ambiguity: A Lottery Example ^ 146Figure 4.1^Levels of Sustainability  188Figure 4.2^The Sustainable Supply Curve ^  198Figure 4.3^Resource Price Path Comparison  202Figure 4.4^Costs of Increasing Energy Efficiency  215Figure 4.5^The Carbon Dioxide Reduction Scenario ^  216Figure 4.6^Economic Rate of Return with Sustainability Premium ^ 220ACKNOWLEDGEMENTSI wish to acknowledge the support of all individuals who have contributed to makingmy studies at UBC an enjoyable and productive learning experience of which this thesis isthe tangible result.I am particularly grateful to two individuals who stood out in making my years atUBC a very important and positive personal experience. James A. Brander served as chairof my thesis committee. He was a superb research supervisor who did not curtail creativitybut encouraged me to think through my intuitive ideas analytically and put them into astructured framework. Kenneth R. MacCrimmon served on my thesis committee. Heprovided continuous support and encouragement for my desire to look beyond standardmodeling approaches. Both have contributed significantly to my understanding of the meritsand limitations of economic analysis and decision theory. Both have also provided extensiveand very helpful comments for the improvement of this thesis.The work of Herman Daly was an inspiration for my thinking and motivated manyof the inquiries that are reflected in this thesis. I thank him and Robert Goodland for givingme the opportunity to explore the practical implications of my work in the context of projectevaluation at the World Bank.I wish to thank Scott Taylor who served on my thesis committee. In addition, Ireceived helpful comments on parts of this thesis from, among others, Salah El Serafy,Richard Howarth, Bryan Routledge, and David Wheeler. Useful suggestions for improvingthis work were also made during seminars at UBC and the World Bank. Finally, thanks aredue to several students, faculty, and staff at UBC who provided support in various ways.For work that led into this dissertation, I received financial support from the KillamTrust, the University of British Columbia, and the World Bank. This support is gratefullyacknowledged.xiWe do not inherit the earth from our grandparents.We borrow it from our children.INTRODUCTIONScientists estimate that between 10 and 80 million different species are inhabiting thisplanet. One of these species has experienced a particular remarkable development during thelast two-hundred years. The human species experienced a technological revolution that ledto an explosion of its global population and a dramatic increase of its ability to consume itsenvironment, including other species on the planet. It takes a moment of reflection andcontemplation to realize the extent of the changes to the face of this planet that the activitiesof human beings have brought about during a time span that is minute in the time scales ofnatural history. A large share of the land surface has been converted from wildlands toagriculture or forestry monoculture. As a result of rainforest destruction alone, more than50,000 species annually are estimated to disappear (see Worldwatch Institute 1988, p.101-117). Anthropogenic emissions of carbon dioxide and other trace gases have reached aquantity that has the potential to drastically alter the global climate. The list of significanthuman impacts on the global environment has become rather long. The increasing evidenceof long-term, and often irreversible, global environmental impacts of human economicactivities is summarized, for example, in the Brundtland Report (World Commission onEnvironment and Development 1987) and the annual volumes of the Worldwatch Institute(1984-1993).Human welfare has always depended on the natural environment. Historically,however, this dependency was one of submission and limited human capacity to harness andutilize the forces and resources of nature. The industrial revolution and the populationexplosion has radically changed the type of human dependency on nature. The size of thehuman population and the extent of human activities used to be small in scale compared tothe size of the biosphere. However, human population and technology have grown in anunprecedented way, and phenomena such as the greenhouse effect, ozone layer depletion andocean pollution have made apparent that the scale of human activities has now become12comparable to the scale of services provided by the environment. For example, todayhumans use about 40% of the net primary product of land-based photosynthesis (see Vitouseket al. 1986). Increasingly, signs are emerging that we have moved from an "empty world"to a "full world" (see Daly 1991). Moreover, considering the inequality of consumptionlevels between people in different parts of the world and the desire of the population in manyparts of the world to substantially increase their consumption of material goods, it is likelythat crowding effects with regard to the natural environment will further increase and notdecrease in importance.For a long time, economists have analyzed economic interactions, human behaviour,and human welfare with theoretical models. Most economic theory was developed based onsimplifying assumptions suitable for an empty world, abstracting from the physical realitiesof the natural environment to a very large degree. Only in those instances where the naturalenvironment provided services that were considered scarce, relative to the human capacityto use them, economic subdisciplines have evolved that analyze scarcity and depletion ofindividual components of the natural environment, such as land economics, environmentaleconomics, and resource economics. Beyond these specific sub-disciplines, the environmentwas often not explicitly considered in economic modeling. Undoubtedly, the world haschanged. In a full world, there are many more interactions between economic activity andhuman welfare through the natural environment. Under these changed conditions, differentsimplifying assumptions are appropriate for economic modeling. It will not be appropriateto abstract from the natural environment as often as in the past.Economists are often perceived to be engaged in an inherent conflict withenvironmentalists. This perception is unfortunate. The underlying philosophical objective of(normative) economics is the maximization of human welfare. Human welfare, however,depends to a large, and probably increasing, degree on the health of the natural environment.While there are many differences in thinking, there is no inherent conflict between theobjectives of economists and most environmentalists. However, conflicts are oftenunnecessarily brought about by economic models that, inappropriately, abstract from3important factors of the natural environment. In fact, the economic approach can providemany useful tools for making rational trade-offs and choices between different actions thatimpact on human welfare and give guidance in determining the most efficient way forachieving an objective such as a healthy natural environment.The debate about economic growth illustrates the sometimes unnecessary conflictbetween economists and environmentalists. Environmentalists often accuse economists ofpromoting economic growth at the cost of future living conditions on the planet. They,rightly, point out that the physical resources of the world are finite and physical consumptioncan therefore not grow infinitely. Of course, the objective of economics is not to promotephysical consumption, or economic activity, for its own sake but to maximize humanwelfare, which depends on consumption of physical and nonphysical goods as well as theconditions of the natural environment. Economists have often contributed to the conflict byabstracting from the difference between human welfare and physical consumption ormeasures of economic activity, such as GNP. Clearly, the objective of human welfare wouldinclude the concerns of most (certainly not all) environmentalists. Fortunately, the debate andthe positions of either side are more diverse than portrayed here. Nevertheless, the describedmisunderstanding appears typical.This dissertation is one of many attempts to develop economic models that are moreappropriate for the full world in which we are living today. In particular, this dissertationdeals with two important effects that result from a crowded world. The first effect is thathumans have gained the capacity to influence long-term living conditions on a global scale.As a result, economic activities of the current generation have a significant impact on thewelfare of future generations through the conditions in which we pass on the naturalenvironment to our successors. This means there is a need to systematically examine theintergenerational welfare effects of our dealings with the natural environment. The secondeffect is that in an empty world externalities were the exception. In a full world, however,almost all economic activities are associated with external effects. Consider as an examplea community located at a river. In an empty world, it would have been sensible to dispose4of the community's waste products in the river. Chances would have been that no damagesresulted since the waste products were diluted and little use was made of the river's waterdownstream. If some damage was done, one could deal with this on a case by case basis.This is the typical approach economists have taken in analyzing externalities. In a full world,though, waste products would aggregate from many communities and water would be useddownstream. It is quite likely that any waste disposed of in the river would be delivered atsome point where it causes damage. The case where no externalities would result from apolluting activity would be the exception in a full world. As a result of this pervasivenessof externalities, a different modeling approach would be indicated in many instances.This dissertation is relatively broad in scope and, therefore, draws from the literaturein a large number of areas. The pertinent literature will be briefly reviewed in the respectivechapters. However, a few comments about the development of economic theory in relationto environmental problems will be offered in this introduction. Economists have long beenconcerned with the scarcity of natural resources. Malthus (1798) wrote the most cited earlywork on the conflict between finite natural resources (agricultural land in this case) and thegrowth of the human population. Since then, the debate has led to cyclical interest in theissue of natural resource depletion, depending on whether resource discoveries and increasingsubstitution possibilities were considered to be able to offset consumption growth. The singlemost important contribution to modern resource economics was made by Hotelling (1931)who introduced the user cost concept and compared resource depletion resulting from profitmaximizing owners with welfare maximizing social planners. This theory of non-renewableresources has later been expanded to renewable resources (see Conrad and Clark 1987).While resource economics is primarily concerned with the management of finiteresources that are often privately owned, environmental economics focusses on problems ofexternalities and the design of policies to remedy the market failures resulting fromexternalities. A seminal contribution was made by Pigou (1932), who introduced the conceptof charging a price for an externality that would equal the marginal costs of the externality.Other important contributions were made by Bator (1958), who defined the market failure5resulting from an externality, Coase (1960), who emphasized the possibility of an efficientnegotiation solution when property rights are assigned in cases such as pollution, and Dale(1968), who suggested the introduction of marketable emission permits. A good overviewof the issues is provided in Baumol and Oates (1988). The extensive literature on theevaluation of environmental costs (for example in cost-benefit analysis) can, in part, beviewed as an extension of the work of environmental economists (see, for example, Pearceand Turner 1989).Environmental and resource economists have applied standard economic modelingapproaches to issues involving the natural environment. Recently, criticism has increased thatthis approach is insufficient to deal with the problems of an increasingly full world.Disillusioned with the high degree of abstraction from physical realities in economic modelsand the isolated treatment of individual resources or environmental problems, someeconomists have attempted more fundamental modifications to economic models in order tobetter reflect the nature of current environmental problems. Work following this approachis now often referred to as ecological economics. The development of ecological economicscan be related, among others, to contributions by Georgescu-Roegen (1971), Daly and Cobb(1989), and Daly (1991). A good overview of this branch of research is given in Costanza(1991). Two central concepts of the ecological economics literature are of central importancefor this dissertation. In order to overcome the limitations of the isolated treatment ofindividual resources and environmental problems, the concept of natural capital has beenintroduced. In order to address the inter-temporal, and particular intergenerational, dimensionof environmental and resource problems, the concept of sustainability is now widelydiscussed.The combined capacity of all components of the natural environment to provideservices that can generate economic benefits is defined as natural capital. Natural capitalincludes natural resources, such as fossil fuels, minerals, forestry and fishery, the capacityto absorb wastes from human activities in limited quantities and the overall life supportsystem of the planet including, for example, the atmosphere that provides the air we breathe6and the ozone layer that protects us from harmful ultraviolet radiation. The biosphere as botha source of materials and energy and a sink for waste products represents natural capital.Natural capital is distinct from other, human-made, forms of capital in that it has not beencreated through conscious human efforts. Therefore, benefits from the existence of naturalcapital are often pervasive and its efficient use needs to be ensured through the establishmentof property rights or mechanisms for efficient collective action. On the other hand, naturaland human-made capital are similar in that they can suffer depreciation and depletion throughuse. The term natural capital is used to direct attention to the analogy with human-madecapital: capital represents a stock of assets from which services can be derived. Like human-made capital, natural capital is more than an aggregation of individual assets or separateresource stocks. It is a complicated web of interrelations between assets and processes that,as an aggregate, provides a wide variety of services to humans.Sustainability has become a popular concept in the discussion on inter-temporal andintergenerational welfare distribution. In its most common definition, "...sustainabledevelopment is development that meets the needs of the present without compromising theability of future generations to meet their needs" (World Commission on Environment andDevelopment 1987). While having intuitive appeal, this definition is very far from beingoperational in an economic policy or modeling context. Some attempts have been made totranslate the concept of sustainability into economic terms (see El Serafy 1989 for anapplication to national income accounting and Pezzey 1989 for a variety of growth theoreticaldefinitions of sustainability). In economic terms, sustainability is mostly understood as aconstraint on the current generation's economic activities, requiring a non-declining level ofconsumption or utility for all future generations. However, real life uncertainties make theprecise definition of such a constraint far from obvious. This dissertation contains an attemptto develop and refine a definition of sustainability that would be more operational foreconomic policy making. While sustainable human activities are those which use naturalcapital within its capacity to regenerate, this dissertation deals with activities that go beyondthat limit, deplete natural capital and reduce the earth's capacity to serve as source and sinkfor welfare generating activities in the future.7The purpose of this dissertation is to contribute to a more solid economic basis forthe discussion on how much natural capital the current generation should be allowed todeplete and whether, and how, future generations should be compensated for natural capitaldepletion. This dissertation consists of four chapters. The first chapter analyzes the effectsof different assumptions about which generations own the stock of a natural resource onintergenerational welfare distribution. The second chapter analyzes the incompleteness ofintergenerational insurance markets as a market failure that results in inefficientintergenerational investments under risk. The third chapter reviews and discusses thesuitability of different decision making models for environmental decision making under real-life uncertainties. Finally, the fourth chapter proposes an operational sustainability constrainton the economic activities of the current generation as a robust rule for intergenerationalcompensation under uncertainty. An application of such a sustainability constraint to projectevaluation is presented.This dissertation is theoretical rather than empirical in nature. The purpose of the firsttwo chapters is to show the importance of explicit consideration of intergenerationalproblems in the analysis of natural capital depletion. Both chapters use simple general-equilibrium overlapping-generations models. In these two chapters, a standard economicmodeling approach is applied to new problems with quite interesting results. Themethodology of chapters three and four is different. Their purpose is to explore thelimitations of standard economic modeling approaches and modify more basic assumptionsof standard models rather than apply standard models to a new problem. Chapter threequestions the validity of the standard modeling approach for normative decision makingunder uncertainty and suggests an alternative. Chapter four presents the argument that, inpractice, following a standard efficiency approach for environmental problems leads tosystematic biases that imply sub-optimal decision making.The different methodologies used in different chapters imply a different style ofpresentation as well. The first two chapters are based on the analysis of a problem withmathematical models, and results are presented in a formal manner. The third chapter8questions standard modeling assumptions and focusses on a review of alternative modelingapproaches rather than on the development of new theory. The fourth chapter, finally,contains a synthesis of results from the other three chapter. The focus of the fourth chapteris on the limitations of formal analysis. Subsequently, the exposition is less formal in itselfand contains many conjectural elements. During the work on this dissertation, differentdegrees of progress were made on different aspects of the overall topic of sustainability andeconomic policy analysis. I have chosen to include in this dissertation material at verydifferent stages of development (formal analysis, review, and conjectural synthesis). As aresult, I hope, this dissertation includes a more complete view of the addressed problemsrather than only the isolated presentation of a few polished results.There are very many important and interesting questions relating to the topic of thisdissertation. Naturally, I was neither able to deal with all of them nor in a position to makesignificant progress on more than very few of them. Therefore, many questions that arerelevant to the topic are not addressed in this dissertation. Despite this dissertation's focuson intergenerational interactions, it does not include a thorough discussion or analysis ofintergenerational ethics. Here, a standard economic approach is pursued. The impacts ofcertain actions or institutions on intergenerational welfare distribution are analyzed. Also,institutional arrangements for the implementation of certain value judgements aboutintergenerational equity are suggested. However, very little will be said about the basis forthese intergenerational value judgements. There is, of course, an extensive philosophicaldebate on issues of intergenerational justice. However, this debate itself is considered outsidethe scope of this dissertation and only some references to it are made in the analysis. Beyondthat, appeals to intergenerational justice are based on intuitive arguments.The following paragraphs provide a brief overview and outline of this dissertation.Chapter I emphasizes the importance of distributional considerations between generations inthe depletion of natural resources. The first part of that chapter uses different overlapping-generations general equilibrium models to analyze the effects of different assumptions aboutwhich generation owns the stock of a natural resource on the intergenerational welfare9distribution. In simple consumption and production models, it is shown that increasing theshare of the resource stock that is assumed to be owned by the first generation reduces thewelfare of later generations. If the first generation owns all of the resource stock, subsequentgenerations have to be sufficiently wealthy to buy the resource from the first generation andsell it to later generations. This channelling problem leads to a constraint on resourceconsumption of all future generations that is determined by the wealth of the first twogenerations alone. If ownership of a resource by future generations is desired for justicereasons, but access to the resource by early generations is desired for efficiency reasons,early generations would have to explicitly compensate later generations for resourcedepletion.The second part of chapter I analyzes a continuous time model of inter-generationalresource depletion and suggests a pricing mechanism for the implementation of inter-generational compensation that would bring about constant utility across generations. Everygeneration would be allowed to purchase the resource at an administered, efficient, price.The proceeds would be invested in perpetuity and the returns to this compensatoryinvestment would be used to augment the current generation's endowment. The model showsthat in an inter-generational world, constant welfare across generations can be achievedthrough Hartwick's rule (invest all competitive resource profits) only if the resource pricepath is administered or if all generations have common knowledge about Hartwick's rulebeing observed by all future generations.Chapter II uses an overlapping-generations model to show that the incompleteness ofintergenerational insurance markets constitutes a market failure that leads to inefficientintergenerational investment decisions under risk. Early generations over-diversify if theyface risks that are larger than those of the following generation and could, therefore, beshared with them. On the other hand, if risks are increasing from generation to generation,the current generation would under-insure against those risks. Furthermore, a generation withdecreasing risk aversion would, in many cases, over-consume a natural resource if theresource-stock uncertainty is larger for future generations. The main message of this chapter10is to caution against the belief that markets work efficiently to transmit across generationsthe right signals for economic decisions under risk. The larger the social risks involved, themore reason there is to believe that markets do not bring about efficient decisions by thecurrent generation. The direction of the inefficiency, however, depends on the nature of therisk assumed. Therefore, the policy implications that can be drawn from this chapter dependon the empirical assessment of the risks that current and future generations are facing. Thischapter also provides applications of the general result to environmental problems such asthe reduction of biodiversity, protection against global warming and the depletion of a naturalresource.Chapter III analyzes the decision theoretical foundation of environmental choicesunder uncertainty. Environmental decision making involves choices about the depletion ofnatural capital, i.e., choices about resource depletion, emissions, land use, and other issues.Many of these choices involve large uncertainties that are not well captured by modelingthem as risks. Risk is defined as a situation of uncertainty with a known probabilitydistribution over a well defined set of states of the world. In particular, it is often impossibleto assign a probability distribution over all states of the world, which is the case ofambiguity, or it is impossible to describe all states of the world completely, which is the caseof ignorance. The standard model for decision making under uncertainty is the subjectiveexpected utility model (SEU) which is a theory that models uncertainty as risk. Chapter IIIis motivated by the belief that, by abstracting from ambiguity and ignorance, SEU assumesaway critical aspects of many environmental decision making problems. It is argued that,since SEU is not applicable to situations of ambiguity and ignorance, has significantlimitations as a normative model for environmental decision making.The focus of chapter III is the review of alternative models of decision making underuncertainty and suggestions for a model that is more suitable for normative environmentaldecision making than SEU. The purpose of the discussion is to analyze existing models andcombine elements of those models in order to provide a decision theory that is of practicaluse for environmental decision making. The chapter begins with an analysis of the conceptual11and practical problems of using SEU and reviews several relevant alternative models ofdecision making under ambiguity and ignorance. Then, an alternative decision making modelis suggested based on a combination of the Dempster-Shafer belief-function theory andChoquet expected utility. This proposed theory is illustrated by means of an example. It isshown that, compared to the proposed theory, the use of SEU in decision problems withambiguity leads to a systematic bias against the conservation of natural capital.Chapter IV provides a synthesis of problems dealt with in the other chapters. Thediscussion shows that, in conventional economic analysis, the costs of natural capital aresystematically underestimated due to individual short-term incentives of decision makers, thepresence of externalities and large uncertainties about the functioning of the biosphere. Inaddition, decision making is systematically biased against the interests of future generations.Intergenerational efficiency and justice require that future generations be actually, not onlypotentially, compensated for costs imposed on them by previous generations. To remedy thebiases against proper valuation of natural capital and to reflect a cautious approach towarddepletion of natural capital, it is proposed that the default value of natural capital should bethe cost of providing a sustainable substitute. The rents from depletion of natural capitalshould be shared equally with all future generations who need to be adequately compensatedfor depletion. Following from these considerations, this chapter proposes a sustainabilityconstraint on current economic activities. This constraint would require that the value ofevery group of functionally substituting types of natural and human-made capital be leftintact in the sense that a sustainable stream of services can be derived from it. Thesustainability constraint should be reflected in the shadow prices used to evaluate naturalcapital depletion, for example in cost-benefit analysis.From the sustainability constraint, an operational sustainable supply rule is derivedthat can be applied to the depletion of non-renewable resources and the consumption of thebiosphere's limited capacity to absorb waste products. This rule requires that a sustainableprice, derived from a sustainable supply curve, be used for depletion of natural capital. Thesustainable supply curve is constructed by dividing the economic rents from the depletion of12natural capital into an income and a compensation component. The compensation componentmust be invested into the production of the closest sustainable substitute for the depletingnatural resource. The compensation component should be sufficient to provide the samequantity of the sustainable substitute at the sustainable price forever after depletion. Theincome component could then be sustained after depletion. The sustainable supply rule isapplied in a stylized case study of an oil development project, and the impacts for theevaluation of the project are shown.CHAPTER ITHE DEPLETION OF NATURAL CAPITAL ANDINTERGENERATIONAL WELFARE1^IntroductionResource economists have examined in great depth the question of whether marketsbring about intertemporally efficient resource depletion. The modem stream of inquiry to thisquestion goes back to the seminal paper of Hotelling (1931). The main result is thatcompetitive markets for privately owned resources will lead to efficient resource depletionwith a resource price rising at the rate of interest. The theory of natural resources has sincebeen integrated with macroeconomic growth models describing feasible growth paths in thepresence of natural resources. In addition, it has been recognized that the depletion of naturalresources leads to implications for welfare distribution across time. In general, there aremany efficient resource allocations with very different distributions of intertemporal welfare.Based on the definition of an explicit social welfare function, one can determine the optimalresource allocation, which is the one that maximizes social welfare. Models for choosing theoptimal allocation among all efficient intertemporal resource allocations throughmaximization of a social welfare function have been developed to analyze distributionalconcerns (see Dasgupta and Heal 1974, Solow 1974, Stiglitz 1974, and also Dasgupta andHeal 1979, pp. 255-359, chapters 9 and 10). In this context, particular attention has beenpaid to a maximin social welfare function. Hartwick (1977), Dixit, Hammond and Hodl(1980) and Solow (1986) analyze the conditions under which a maximin welfare path wouldbe obtained. Their main result is "Hartwick's rule" stating that if net-investment equalsresource profits at competitive prices, a constant welfare level will be obtained.1314Most models of optimal resource depletion are based on a managed economy withexogenously determined levels of investment and consumption. Hence, there is little workaddressing the question under which circumstances markets would bring about not onlyefficient but also optimal resource depletion. However, since investment and savingsdecisions are decentralized at least to some extent in most real life economies, it would beimportant to develop a better understanding of the possibilities to implement optimaldepletion paths in a decentralized economy. This requires a microeconomic modelingcomponent including the savings and investment choices of short-living individuals. In anintergenerational context, this is the natural realm of overlapping generations models aspioneered by Samuelson (1958). However, natural resources were notably absent fromoverlapping generations models until the very recent contributions of Howarth and Norgaard(Howarth and Norgaard 1990, Howarth 1990, Norgaard and Howarth 1991). In a simpletwo-generation framework, Howarth and Norgaard (1990) show the crucial differencebetween efficient and optimal resource depletion paths and the importance of assigningresource property rights to different generations according to the chosen social welfarefunction. They show in an inter-generational context the dependency of welfare and priceson the endowment distribution, which is well understood in a static general equilibriumsetting.The first part of this chapter contains several extensions of the Howarth and Norgaardmodel. The models analyzed and presented here are more general and include production,more than two overlapping generations, the existence of a renewable resource and a varyingshare of expenditures on the natural resource. The extension to a production economy isobviously important to reflect resource use in the real world. The extension to renewableresources would include many urgent environmental problems such as the greenhouse effector ozone layer depletion, which can be modeled as depletion of a renewable absorptioncapacity for emissions. The models are used to analyze the endowment effect, which biaseswelfare in favour of those generations assumed to be endowed with the natural resource. Theextension to more than two generations will prove to make an important difference sincenon-adjacent generations are not able to trade with each other directly. This results in a15channelling effect, which limits the extent to which the demand of distant generations is ableto influence resource prices at earlier times. The second part of this chapter addresses thequestion how a desired welfare distribution can be implemented in a decentralized economy.A continuous time, multiple generations model is used to analyzes the possibilities toimplement a sustainable resource depletion path through a pricing mechanism that requiresearly generations to explicitly compensate later generations for resource depletion.2 Natural Resource Endowments and Intergenerational WelfareMarket economies are based on private ownership and market exchange of resources.The first theorem of welfare economics states that the competitive equilibrium, followingfrom such market exchanges under certain conditions, is efficient (see Varian 1978). Ofcourse, the welfare distribution between individuals depends on the initial distribution ofownership (or the endowments). As a result of unequal endowment distribution, an efficientequilibrium may imply gross welfare inequalities. In this section, these simple ideas areapplied to different generations' ownership of natural resources. Under current institutionalarrangement, a resource is either publicly or individually owned by members of the currentgeneration. This chapter, however, uses the hypothetical construct of explixit ownership ofparts of the resource by future generations in order to provide a model that can be used toanalyse normative questions about depletion rates and their impact on intergenerationalwelfare. The question which institutions could be used to implement the ownership ofresources by future generations is deferred to chapter IV and not addressed in this chapter.With this perspective, I will explicitly analyze the implications of different assumptions aboutwhich generation owns a share of the natural resources for intergenerational welfare.In the real world, there are at least two channels through which a generation passeson assets to its successor generation. First, there are market transactions between overlappinggenerations. For example, a capital stock is accumulated by an early generation throughsavings. At retirement age, these investment assets are sold to a younger generation againstconsumption goods (e.g. by selling stocks out of a pension fund). Second, early generations16bequeath assets to later generations. The models in this chapter focus on market interactionsbetween generations and abstract from intergenerational wealth transmission as the result ofintergenerational altruism. It is assumed that every generation maximizes utility from ownconsumption only. The reason for making this assumption, despite evidence that individualswithin any generation do not maximize utility from own consumption alone but care aboutthe well-being of their off-spring as well, is that it focusses the analysis on the distributionalconflict between generations.In this chapter, optimality is discussed in terms of an intergenerational social welfarefunction. However, the utility function of individual generations in the models does notinclude a bequest motive. This approach may appear paradoxical since it supposes theexistence of a social welfare function that is in this respect independent of the currentgeneration's preferences. Two independent justifications for this approach are provided inthe following paragraphs. The first justification is based on coordination failures and publicgood problems involved in intergenerational altruism that would lead to inefficiently lowindividual provisions for the welfare of future generations. The second justification is basedon the possibility of dual objectives of the current generation: a selfish utility function forindividual choice and a moral welfare function for social planning.There is a significant body of literature examining the question whether individualbequest motives can lead to welfare optima (see Barro 1974, Bernheim 1989, and for asummary Blanchard and Fisher 1989, pp 104-110). The possibility of a welfare optimumdepends critically on the specific way in which intergenerational altruism is modeled (one-sided altruism, two-sided altruism or a bequest motive). Under highly restrictive conditions,a succession of distinct generations behaves like one infinitely lived generations, whoseutility function would then be logically taken as the social welfare function (see Blanchardand Fisher 1989, p.9'7). In most circumstances, however, intergenerational altruism, whetherit is caring about the well-being of own descendants or about future generations in general,does not lead to a welfare optimum due to public good problems and coordination failures.17The public good problem of caring about one•s own descendants has been explainedby Daly (1982). The number of descendants one cares about would rise exponentially withevery generation. Specifically with a constant population, every individual would have 2'descendants in the nth generation. Conversely, every individual in the nth generation wouldhave 2 individuals in generation zero who care about her/his well-being. Unless efficientmechanisms for collective action by co-progenitors are in place, there would be under-investment in the well-being of future generations because of the public good nature of suchinvestment: the well-being of one's great-grand child would depend on the provision of eightco-progenitors. Everyone would try to free-ride on the provisions of seven co-progenitors,and the result would be sub-optimal investment. Coordination mechanisms would beextremely difficult to arrange since, for provisions for distant generations, the individualswith whom one would have to coordinate are not yet identified.The public good problem is even more obvious if individuals in the current generationcared not about their individual descendants but about the well-being of future generationsin general (see Marglin 1963). Again, individuals would inefficiently under-provide resourcesfor future generations. It may be added that some social institutions have traditionally servedto alleviate this public good problem. Such social institutions include the concept ofresponsibility toward the "seventh generation" in North American native ethics as well asEuropean family traditions according to which all family property was passed on to the oldestson instead of being divided between siblings. This tradition can be seen as not onlypreserving family estates but also making it easier for earlier generations to identifythemselves with a single line of descendants and thereby avoiding the discussed public goodproblem. The decline of the importance of extended families shows how some of these socialinstitutions have lost their influence through social changes.In the presence of one of these public good problems, individual action based onintergenerational altruism would be inefficient. Hence, one justification for abstracting fromintergenerational altruism is that the concerns about intergenerational welfare distributioncannot be overcome by individual action based on intergenerational altruism alone. Based18on this justification, this chapter would provide analysis for determining the desiredcollective action to overcome the public good problem of intergenerational altruism. Also,the effect of intergenerational altruism has been explored in similar models before (seeHowarth 1990, pp. 62-76, chapter IV) leading to the discussed public good problem.A second justification for supposing the existence of an intergenerational socialwelfare function, independent of the first generation's preferences, can be found in theextensive literature on the duality of human objectives. Harsanyi (1953), Margolis (1982),Sen (1985), and others describe human objectives as dual in the sense that individuals havemoral objectives, often related to fairness considerations, that are not reflected in dailyindividual choices. However, these moral objectives are used in making long-term choicesabout social institutions and are often delegated to government action (see also Rawls 1971).Based on such dual model of human objectives, the utility functions that determine individualchoices would exclude considerations of intergenerational fairness. The objective ofintergenerational fairness would, however, be reflected in a social welfare function that isused to determine the optimal intergenerational welfare distributions.2.1 The Endowment EffectThe two main effects resulting from different distributions of resource endowmentsacross generations are discussed in different sub-sections on the endowment and thechannelling effect. The present subsection is devoted to the exposition of the endowmenteffect which describes the welfare effect of endowing different generations with the stock ofnatural resources. The endowment effect would be present even if all generations were livingat the same time. The next subsection will address the channelling effect which arisesbecause of the particular intergenerational structure that implies that not all generations cantrade with each other directly.Several similar models will be used to explore the welfare effects of changing theassumptions about which generation owns how much of the resource stock. These models19Figure 1.1 The Structure of the Overlapping-Generation Modelsshould capture the possibility of trade between adjacent generations but also the absence ofmarkets between generations that are further apart in time. Hence, an overlappinggenerations model in which only two adjacent generations overlap at any one time isselected. In order to allow trade and substitution between the natural resource and othergoods, the models include a second generic investment and consumption good which everygeneration is endowed with. In order to be able to explicitly solve for the equilibrium,specific functional forms for utility and production functions are used in the models. In thefirst model, agents consume both the natural resource and the consumption good. Forsimplicity, the rate of return on investment of capital is assumed to be constant. After asolution to the n-generation model is found, the equilibrium welfare of the generations isanalyzed with respect to changes in resource endowment and rates of return. A second modelcaptures the impact of more realistic production possibilities. In that model, agents consume20only one consumption good which can be produced from capital and the natural resourcewith an endogenous rate of return. The overall structure of the models is shown inFigure 1.1 for an example with three generations.2.1.1 An N-Generation Consumption ModelThere are two goods in the economy: K is a generic consumption and investmentgood (the capital good) with a constant rate of return, a-1. Hence, one unit of K investedat time t = i yields a units of K at time t = i+ 1 (normally a > 1). R is a natural resourcewith a constant rate of reproduction 13-1. One unit of R left behind at time t = i yields flunits of R at time t = i+ 1. For a non-renewable, non-perishable resource 13 would equalone. There is a finite number of generations i = 1,2, ...,N, each of which is modeled as onerepresentative agent. Generation i lives at times t = i and t = i+ 1 and consumes R and Kat both times. The utility function of generation i is:U = z Log[4] + (1 -z) Log[Ric] + 8 (z Log[4.1] + (1 -z) Log[Ri+j)^(1)where (Kc1,2, Rc1,2) would denote generation l's consumption at time 2. 6 is the utilitydiscount factor and z (0 < z < 1) represents the relative importance of the capital goodcompared to the natural resource. This specific form of the utility function is chosen becauseit allows derivation of simple explicit equilibrium solutions. Also, log-utility can be obtainedfrom the more common Cobb-Douglas form, (IC, yozeiyi-z), by a simple monotonictransformation. Generation i receives an endowment (Kei, Re) at time i. IC; can be thoughtof as labour endowment with labour being able to produce K on a one to one basis. The totalinitial stock of the resource, So, is fully distributed as endowments. Hence:N Re- sopi-i(2)Generation i invests K, and Rii at time i. At every time i, the old generation, i-1, and theyoung generation, i, can trade R for K at price pi (K is the numeraire good with prices in21current terms). Generations are assumed to behave competitively. There is no uncertainty,and generations have perfect foresight.Generation i maximizes its utility, U, by choice of consumption,^Re,,,,and Rci,,+i, and investment, IC', and Ri1, subject to budget constraints at time i and i+1:Kie -Kii^-kic) = 0^(3)a Kii -K1÷Pi+1(PRii^=Investments cannot be negative (IC',^0 and R',^0). Since these non-negativity constraintsare not included in the maximization problem, they have to be verified once a solution isfound. Corner solutions to the problem will be discussed in a separate section. The modelcan be solved with one market clearing condition for every time t = 2,...,N:^0 RI-1+ Rte = Rzc+ki+Ri^ (4)The derivation of demands and prices is shown in Appendix I-A.' For the interior solution(IC1; > 0 and R', > 0), prices are:Pi -N^ Nai -1 (1 - z)E Ki(1-Z)Ej=1 r`i^ 1=1 Npi-lzE L30j=(5)and demands are:1 The solutions to the maximization problems in chapters I and II were obtained usingMathematica, Version 2.0, as described in Wolfram (1991).22^(K; +piRie)z^_ (Kie +piRie)(1_z)1+ 6 P,(1 + 6)^(6)c^a 8(K; +piRie)z^DC^f3 8 (Kie +piRie)(1 -z)^1+6 Pi(1 +8)Equation (5) implies p1+1 = (a/13) pi, which for a non-renewable resource (0=1) is the wellknown Hotelling price path. For interior solutions, the price path in equilibrium dependsonly on the present value of total endowments and not on the distribution of the endowments.As a result of the assumed log-utility, the demands reflect a fixed share of expenditures oneach of the four consumption goods with demand changing over time at the rate oiS and afor the capital good and the resource respectively.Proposition 1.1:Proof: If, in equilibrium, all generations i invest a strictly positive amountof capital and resource at time i, the competitive intertemporalresource allocation is efficient. Hence in an interior solution, theabsence of markets between non-adjacent generations does notintroduce an inefficiency.All market clearing conditions taken together imply that the resource is completelyused up. Full depletion of the resource and the Hotelling price path are sufficient conditionsfor intertemporal efficiency of resource use in the simple consumption economy of thismodel (see Dasgupta and Heal 1979). •2.1.2 The Two-Generation CaseSeveral important observations can already be made in the simplest possible settingof a two-generation model (N=2). With two-generations, trade takes place only at t=2between generations 1 and 2. The first result was shown before by Howarth and Norgaard(1990) for a = 0, f3 = 1 and z = 0.5. In general form it is:23A shift of resource endowments from generation 2 to generation 1unambiguously increases welfare of generation 1 and reduceswelfare of generation 2, and vice versa.Proposition 1.2:Proof: Since the resource stock is fully allocated as endowments, Re2^(S0-W1).Substituting the price into demands and demands into utilities, welfare effects of changingresource endowments can be calculated. With 0 > z > 1 and So Re1 the comparativestatic results are: (1+ 8)(a Kie + K2e) (1- z)>0^(7)ditie^(1 - z) a Kie Rie + (1— z)K2e Rie +a Kie Sozd U2^(14-8)(aK1e+K27)(z-1)ale^(1 -z)(S0^a^+ (S0 —(1 — Z)Rie)K2eand<0^(8)•From the comparison with a static general equilibrium model, proposition 1.2 shouldbe fairly obvious. Propositions 1.1 and 1.2 together imply that for every possible distributionof resource endowments, there is a different resource depletion path. All of these depletionpaths are efficient; however, they result in different welfare distributions betweengenerations. Often, intertemporal efficiency is incorrectly taken as a sufficient condition foroptimal resource depletion. However, an efficient depletion path is optimal only in anequilibrium that is based on the desired endowment distribution. The market mechanism onlyassures intertemporal efficiency. The market cannot make the decision on which generationshould own how much of the resource. In the context of natural resources, this decision isparticularly critical since it is irreversible (once an early generation has assumed ownershipof the resource and consumed it, it cannot be consumed by later generations) and is imposedon later generations by the earlier generations. Following proposition 1.2, the first generation24has an incentive to assume ownership of the full resource stock. Figure 1.2 (a) demonstratesthe significance of the decision on who owns the resources by means of an example for anon-renewable resource (0 = 1) with z = 0.5, a = 2, 5 = 0.9, IC, = Ke2 = 1, and So =1. Utilities of generations 1 and 2 are plotted as a function of Rel.The extent to which assumptions about which generations own the resource affect thewelfare distribution depends on the relative importance of the resource in the economy. Inthis model, relative insignificance of the resource is measured by the parameter z thatrepresents the expenditure share on the capital good K. Figure 1.2 (b) shows the utility ofboth generations as a function of z for Re, = So (with = 1, a = 2, .5 = 0.9, Kei = Ke2= 1, and So = 1). At one extreme, z = 0, K does not enter the utility function. Sincegeneration 2 is endowed with K only, its endowment is worthless and its utility minusinfinity. At the other extreme, z = 1, the resource is worthless and its distribution does notmatter. Since trade cannot take place with only one good of value, and both generations areendowed with the same quantity of K, utility across generations is equal. As a result, thewelfare relevance of resource endowments depends on the assumption about the relativeimportance of natural resources in the economy. As indicated in the introduction, this inquiryis motivated by the belief that natural capital depletion is widespread and that the relativeimportance of natural capital is higher than generally assumed if not only source but alsosink resources, such as the environments capacity to absorb waste products, are taken intoconsideration.To analyze the impacts of modifying the rate of reproduction of the natural resource,(3, on welfare, the derivatives of equilibrium welfare with respect to (3 are evaluated:25dUl^8 (1 - z) >0d13^13 (9)dU2^(1 +2d)(1 -z)^0di3Hence, an increase in the rate of resource reproduction unambiguously benefits bothgenerations under any endowment distribution. This is not surprising since an increase in 13increases the sum of resource endowments when So is kept constant. Also, the impacts of theearly generation's assumption about the ownership of the resource depend on the rate ofreproduction of the resource, (3. With increasing (3, the endowment distribution required toachieve equal utility of both generations shifts toward the first generation. Figure 1.2 (c)depicts the utility profiles at Re1 = 2/3-S0. Increased reproduction of the resource benefitsgeneration 2 through a slower increase in resource prices as well as an increased endowmentof the resource when Re1 is kept constant. This result highlights that distributional concernsare particularly important for non-renewable resources and renewable resources with a lowrate of reproduction.The possible impacts of varying the rate of return on capital investment are surprisingat first. With the additional assumptions Re1 = So (the first generation assumes ownershipof the full resource stock) and Kel = Ke2, the welfare derivatives with respect to the rate ofreturn, a, are:dU > 0da6> 1 -za (2 + a -z)dU2<0daifa -z) 1 <z+2az-a8 >  a (1-z) z+2az - aif zfz(10)Hence, the second generation's utility can fall with increasing productivity of capital. To seethat the restrictions on the utility discount factor would be satisfied for a wide range ofparameter values, consider the example of a = 2 and z = 0.5. The comparative static1.6^2.2^2.8^3.4^426Sensitivity of Welfare to Parameter Changes(a) Increase in First Generation's^(b) Decreasing ResourceEndowmentUl, U2-0.5-1ImportanceU1, U2-1.5-2-2.5-30^0.2^0.4^0.6^0.8^1Re1/So(c) Increasing ResourceProductivityU1, U20(d) Increasing CapitalProductivity1^1.6^2.2^2.8^3.4^4Welfare of Gen. 1, U1^Welfare of Gen. 2, U2Figure 1.2 Sensitivity of Welfare to Parameter Changes27results in Equation (10) would then hold for any (3 with 0.071 < ô < 2. A positive rate ofpure time preference implies < 1. Figure 1.2 (d) shows an example in which welfare ofgeneration 2 declines with a (0 = 1, = 0.9, K°1 = Ke2 = 1, and So = Re, = 1).This result is important since it implies that if the early generation assumes ownershipof a large share of the resource, technological progress that increases the rate of return oncapital investment can make the second generation worse off. If generation 1 could choosea, for example through technological innovations, it would have an incentive to increase aat the expense of generation 2. The intuition as to why generation 2's welfare declines withincreasing a is that generation 2 needs to buy the resource from generation 1 in exchangefor capital. An increase in a increases the total amount of capital available and increases theresource price in terms of capital. This increase of the resource price increases the relativeincome of (the resource-owning) generation 1. Generation 1 would increase consumption ofboth goods and leave behind less of the natural resource. Unless the increased rate of returnon generation 2's own capital endowment offsets the effect of an increased resource price(which, with most parameter values, is the case only at negative rates of time preference),generation 2 is made worse off. Clearly, this effect is limited to the specific type oftechnological progress that increases productivity of capital but does not change productivityof the natural resource. This effect appears to have practical relevance since the public goodnature of many natural resources would lead to little incentives to invest in technologicalprogress that would increase resource productivity. Hence, technological progress wouldoften only increase the productivity of human-made capital, which would lead to increasedresource depletion and reduced welfare of future generations.Proposition 1.3: An increase in the rate of reproduction of the natural resource, fi ,increases the welfare of both generations. Under the conditions in(10), with equal capital endowment of both generations and withall of the resource owned by generation 1, an increase in the rateof return on capital investment, a, increases welfare of generation1 but reduces welfare of generation 2 and vice versa.28Welfare Profiles in a Ten Generation Model-10-50(a) Resource Owned byFirst Generation Only(b) Resource Ownedat Equal QuantitiesUl5Ul50-5-10-15  ^-151 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10Generation^ Generation(c) Resource Ownedat Equal Market ValueU l ---15  2 3 4 5 6 7 8 9 10Generation(d) Constant ResourcePriceUi50-5-101- l5 3 4^5^6^7 8^9 10Generation50-5-10Figure 1.3 Welfare Profile with Ten Generations29Proof: Follows from equations (9) and (10). •The result that an increased return to capital investment can reduce the welfare ofgeneration 2 is related to other models that show how imbalanced technological progress canhurt agents through price effects. In the international trade literature such a terms of tradeeffects has become known as immiserizing growth (see Bhagwati 1958 and 1968).Immiserizing growth occurs when technological progress in the export sector leads to aworsening of the terms of trade such that the negative price effect exceeds the positive effectfrom an increased quantity of the exported good. Similarly, in the above model, an increaseof the capital good, which is "exported" by generation 2, leads to a decrease of its marketprice that can more than offset the positive quantity effect. Corresponding to the conditionsgiving rise to immiserizing growth, the negative welfare effect for generation 2 depends onthis generation's competitive behaviour despite its market power for the capital good.2.1.3 An Example With Ten GenerationsIn order to gain an intuitive understanding of the endowment effect with more thantwo generations, a numerical example with ten generations is presented in this subsection.Throughout the example, an interior solution is assumed. The example is based on z = 0.5,= 1, ô = 0.9 and IC; = 1 (for all ten generations). Figure 1.3 (a), (b), and (c) depict theutility profiles across generations with different distributions of the initial resource stock of10 and a = 2. Figure 1.3 (a) shows the utility profile when all 10 units of the resource areallocated to the first generation. Hence, generation 1 has a higher valued endowment whilethe market value of the endowment of generations 2-10 is equal. The declining utility fromgenerations 2 to 10 stems from the increasing resource price (following the Hotelling path).Figure 1.3 (b) shows the utility profile resulting from distribution of the resource in equalquantities to all ten generations (Rei = 1). The U-shaped utility profile results from theinteraction between the income effect (higher market value of later generations' endowments)and the price effect. Figure 1.3 (c) shows the isolated price effect. This case is based on adistribution in which the market value of resource endowments has been equalized across all30ten generations by distributing the resource in inverse proportions to the current resourceprice. With equal endowment value for every generation no trade occurs since the ratiobetween the market value of both goods' endowments is equal. Utility decreases strictly dueto the price effect. Figure 1.3 (d), finally, reflects a constant resource price (achieved byassuming a = 1) and equal resource distribution, resulting in no trade and equal utilityamong all generations.2.1.4 A Two-Generation Model With ProductionFor the consumption model in the preceding section, a very simple structure wasdeliberately chosen in order to be able to derive simple equilibria for cases with manygenerations. Some features of the model, however, are excessively simplistic. In particular,it could be suspected that the welfare results would be weakened in a model that allows theuse of the resource not only as a consumption good but also as an input to production.Intuitively, one might argue that if the resource can be made productive, its ownership anduse by earlier generations may actually benefit later generations as well. Hence, it would beimportant to verify the robustness of some of the previous results in a model that wouldinclude production of a consumption good, using resources and capital as input, with anendogenous rate of return. This section examines a two-generation model that incorporatesthese features.There are two goods in the economy, the investment and consumption good, K, anda non-renewable resource, R. Output of K is produced from K and R with a log productionfunction:fi[Rin,Kin] = I Log[l +K'] +Log[l +where fi is output of K at time t = i+ 1 and Kr', and Rnt are inputs of capital and the naturalresource, respectively, at time t = i. There are only 2 generations. Generation 1 lives attimes 1 and 2 and generation 2 at times 2 and 3. Generation i receives the endowment (ICei,Re) at time t = i (the early time of its life). For simplicity, generation i consumes only K31and only at time t = i+1 (the late time of its life). The parameter I in the productionfunction is chosen such that the marginal rate of return on investment of K always exceeds1. Hence, K is always invested and never stored. At time 2, generations 1 and 2 can tradeR for K at price p (K is the numeraire good). Both generations behave competitively.Generations maximize their consumption by choice of the amounts traded (generation 1 sellsRs, of the resource; generation 2 buys Rb2 of the resource) subject to their respective budgetconstraints. Since both generations consume only one good at one point in time, there is noneed to define a utility function. Given the endowments, the trade choices determineinvestments. There is no uncertainty and generations have perfect foresight.The maximization problem of generation 1 is:max CI = 1Log[1 + K1eJ + Log[l + Rie — Ris] + pRis^(12)Generation 2's problem is:max C2 — ILogrl + K2e -p R211 + Log[l + R2e +R211R2bThe resulting trade choices are:b^1 + K2e - lp (1 + R2e)Ris = 1 -1^e+ R1^R2 - P p(1 +1)The equilibrium price can be obtained from the market clearing condition Rs, = Rb2;K2e +1 +2(15)1 +Rie +1(2 + Rie + R2e)(13)(14)With Re2 =Re1- So, the resulting equilibrium consumptions of both generations are:32^ + Log[l +Kie] + Log1-1 + IC2e —1•50+ Rie (1 + I +K2e)^ 1+21+15 +Re0 1 +21+IS0+R1eC2 = Log +K;(2+50)+24-412 +I+K;^+ I LogProposition 1.4:Proof: In the two-generation production economy, a shift of resourceendowment toward generation 1 increases consumption ofgeneration 1 and decreases consumption of generation 2. Also, theprice of the resource depends on the distribution of resourceendowments between generations. The price decreases withincreasing distribution of the resource toward generation 1.With W2 =^So and Re1 LC. S0, the comparative statics results confirm the resultsfor the consumption economy that welfare of generation 1 increases and welfare ofgeneration 2 decreases with increased distribution of resource endowment toward generation1:dCi^12(2+;)+1(6+2K2 e +3S0+K2e S0)+1+4 0dRie (1 + Rie +1(2+ S0))2^(17)dC2^12(2 +.50)+/(6 +2K; +350 +K;So)+1^< 0dRie^- (1+ Rie + 1(2 + So)) (3 + 2 K2e + So (2 + K2e) - Rie)Also, in this model, the resource price, p, depends on resource endowment distribution:dp _ _^2 +I+K; < 0^(18)dRie^(1+ Rie +1(2 + So))22 + I + K2(16)I1(3 +K (2 +So) +24-R1e)1+1(2 +S0) +Rie•Generation 1Generation 2.■1■1■,Price at Time 2•••■■(a) Consumption as a Function ofGeneration l's EndowmentC1,C230.540^0.2^0.4^0.6^0.8^1^0^0.2^0.4^0.6^0.8^1Resource Endowment of Gen.1, Rel Resource Endowment of Generation 1, Re1(b) Price as a Function ofGeneration l's Endowment2. 1.4 Equilibrium Changes with Increases in Generation 1 's EndowmentThese impacts of increasing resource distribution to generation 1 are depicted inFigure 1.4 for a numerical example with Kel = Ke2 = 1, So = 1, and I = 3. The model,analyzed in this section for two generations, can also be solved for three and moregenerations. The main results of the discussed consumption model hold then as well. Inaddition, it can be shown that with three or more generations, the discount rate (the rate atwhich the resource price rises) decreases with increased distribution of resource endowmentstoward earlier generations. Since the discount rate rises with changes of the resourcedistribution toward the earlier generation, there is an additional bias in such distribution thatwas not apparent in an economy with a constant rate of return on capital investment. Inaddition to the income effect from a lesser endowment of generation 2, there is the discountrate effect that leads to a faster rise in the resource price, benefiting earlier generations. Inaddition to the log-production function, other similar models were analyzed; but the results34are not reported here in detail. For example, with a Cobb-Douglas production function, themodel can be solved for two generations with the main results being the same.2.2 The Channelling EffectIn an intergenerational world, not all generations can trade with each other directly.Generations, far apart in time, can trade with each other only indirectly through the channelof intermediate generations. Specifically, the volume of trade between distant generation islimited by the wealth of intermediate generations since an intermediate generation's abilityto buy goods from an early generation and sell them to a later generation is restricted by itsown wealth. This section seeks to explore the implications of this channelling effect. First,consider the general consumption model introduced in section 2.1.1. In a two-generationmodel, all generations overlap and can trade with each other directly. A three-generationmodel is the simplest one possible to explore the impacts of missing markets between non-adjacent generations since generations 1 and 3 would not overlap and could not trade witheach other directly.Proposition 1.1 has already established that efficient resource depletion is achievedif the N generation equilibrium implies strictly positive investment of K and R of allgenerations. Hence, the interesting aspects of a three-generation model are its cornersolutions. The case of Ri2 = 0 will be discussed later. The discussion here focusses on thecorner solution with le2 = 0; hence, the intermediate generation invests only in resourcesand not capital. This case would occur if most of the resource is owned by generation 1, andgeneration 2 has to buy resources from generation 1 at time 2 in order to sell it to generation3 at time 3. Capital investment of generation 2 at time t = 2 can be expressed as:= a (K:^+ K2` -Klca-K2ca^ (19)After demands and prices are substituted into equation (19), the conditions can be determinedunder which this corner solution (Ki2^0) will be obtained (with z = 0.5, 13 = 1 and WI=^:351C0 if Ke >21 +^Ke(20)a^3= 0 if K2e 1 + K3ea 8As expected, the corner solution arises if generation 2 is 'poor' compared to generation 3.The equilibrium for this corner solution is derived in Appendix I-B. Demands of generation1 and 3 remain as in equation (6). The demands of generation 2 and the prices, however,change.Proposition 1.5:^If, in equilibrium. generation 2 does not undertake capitalinvestment, the resource price changes with the distribution ofresource endowments between generations In particular, a shift ofendowment (from generations 2 and 3, equally) to generation 1leads to a decrease in p2 and an increase in p3.Proof: In case of the corner solution, prices are (from Appendix I-B):(1 +8)alCie+(1 -2 8)K2eP2 —^(1 + 8)Rie + R2e(21)al + 8) a Kit + (1 +2 8) K2e)1C3eP3 - IC2e RI` + a OICIeR2e +2 8K2eR2e +(1 + 6) a KieR3e +(1 +2 8) K2 e R3eThese prices depend on the resource distribution. Setting W., = Re3^(S0—Re1)/2, the priceeffects of shifting the resource endowment toward generation 1 can be assessed:36dp2^-2(1 +2001 + 8)Kie + (1 +2 OKI') <0dRie^((1 +2 OR: +S0)2^(22)dp3^2(1 +2 8)(a Kle + K2e)((l + OK: + (1 +28)K2e)K3e >0dRie^+ 28)(-Ki e Rie -K2 e Rie +1Ci e So) + (l +4 8)1C2eS0)2•The corner solution arises when most of the resource is owned by generation 1because generation 2 has to be sufficiently wealthy to buy the resource from generation 1 andsell it to generation 3. Positive investment by generation 2 (K'2 > 0) requires the capitalendowment of the second generation to exceed an amount proportional to the thirdgeneration's capital endowment. If the intermediate generation's endowment is insufficient,investment is zero and the Hotelling price path for the resource will no longer hold betweentimes 2 to 3. The price of the resource rises faster than the rate of return on capitalinvestment. Normally, this would induce generation 2 to purchase more of the resource fromgeneration 1 and sell it to generation 3. However, generation 2 can offer generation 1 atmost Ke2 for acquiring the natural resource demanded by generations 2 and 3, regardless ofgeneration 3's wealth. Note that this channelling effect is not due to myopia of generation2 or some market failure. It is the result of the corner solution in which generation 2 alreadyspends all its endowment on buying the resource from generation 1 (beside own consumptionat time 2). Since the wealth of the intermediate generations is a technical constraint in themodel, the corner solution equilibrium is not inefficient. A Pareto improvement could onlybe achieved if later generations were able to transfer capital (discounted at the rate of return)to intermediate generations. This would be technically infeasible if Ke is thought of as anendowment that is not physically present before its owner-generation is born (i.e. labourendowment). Even though it does not imply an inefficiency, the result emphasizes thedistributional concerns about the first generation's ownership of the resource stock sincefuture generations' access to the resource would be limited not only by their own wealth butalso by the wealth of intermediate generations. The channelling effect implies that the impact(a) Welfare of Generation 3^(b) Resource Price at Time 3U3^ P3-5  1210864200^0.4^0.8^1.2^1.6^2Capital Endowment of Generation 2, Ke2Corner Solution interior Solution Market EquilibriumNOM-120^0.4^0.8^1.2^1.6^2Capital Endowment of Generation 2, Ke2Corner Solution Interior Solution Market EquilibriumGINN.37that the demand of distant generations can have on current markets is limited by theintermediate generations' wealth.Figure 1.5 The Channelling EffectFigure 1.5 depicts a numerical example (Kei = 1, Ke3 = 1, So =R^3, .5 = 0.9,a = 2) for the impact of the wealth of generation 2 on the resource price at time 3 and theutility of generation 3. At these parameter values, the corner solution is obtained at any r,< 1.055. Hence, the market equilibrium for Ke2 > 1.055 corresponds to the interiorsolution while, for Ke2 < 1.055, it corresponds to the corner solution. In case of an interiorsolution, increasing the wealth of generation 2 bids up the price of the resource for allgenerations, leading to declining utility of generation 3, whose endowment value remainsunchanged. In case of a corner solution, the amount of the resource offered to generation 3is constrained by Ke2 and the resource price at time 3 rises as Ke2 is lowered with adversewelfare impacts for generation 3. The resource price has its minimum and utility ofgeneration 3 its maximum at Ke2 = 1.055. The practical relevance of the channelling effectis highlighted by the fact that the corner solution is obtained for a wide range of reasonable38parameter values. In the numerical example above, the channelling effect prevails alreadyfor equal capital endowments of all generations (Ke, = 1).As the number of generations in the model is increased, the channelling effectbecomes more severe since intermediate generations have to channel the resource not onlyto one but to all future generations. The wealth of intermediate generations limits the amountby which all future generations together can effectively bid on resources in the present. Now,consider the extent to which the channelling effect limits the amount of resources that willbe left behind for all future generations if the first generation assumes ownership of the fullresource stock in an N-generation model.Proposition 1.6:Proof: Since non-adjacent generations can trade with each other onlythrough the channel of adjacent generations, the amount by whichfuture generations' demand can be effective on resource marketsbefore their lifetimes, is limited by intermediate generations'wealth. If the first generation assumes ownership of the full stockof a non-renewable resource, the total amount of the resource thatcan be obtained by any number of future generations will belimited to a fraction of the resource stock that is determined by thecapital endowment of generations 1 and 2 only.The payment that generation 1 receives from generation 2 for resources sold for theuse of all future generations cannot exceed the total capital endowment of generation 2,hence:p2 (So -^- K2) IC2e^(23)Substituting the demands of generation 1 (from Equation (38) in Appendix I-A) into thisequation, it can be solved for p2:+Ric,i+1 1 - a I Cie + K2ea Kle +2 K2e1i=2 ■So (26)39a I Cie + 2 K;SoSubstituting p2 back into demands of generation 1 yields:a + K2eRa+ Rica ^a Kei + 2 K2`The resource consumption of all future generations is, hence, bounded from above as afraction of initial resource stock that is determined by the wealth of the first two generationsonly:P2(24)(25)For example, with Kei = Ke2 = 1 and a = 2, the first generation will consume at least 3/4of the resource stock while all future generations together will consume at most 1/4 of theresource stock. •Models, different from the one above, can be constructed that would avoid thechannelling effect. However, the effect is of relevance even if some of the assumptions madeabove are relaxed. The channelling effect occurs as well if the production model in section2.1.4 is extended to three or more generations. In principle, the channelling effect remainsin effect even in a more realistic setting with more than two generations overlapping at anyone time. Suppose, the first x generations own the resource stock. There are y moregenerations that overlap with at least one of the first x generations. Then, the amount of theresource that would be available to all future generations (x + 00) is always less or equalto the amount of the resource that generations 1 to x are willing to sell against payment ofgenerations' x +1 to x +y aggregate endowment and is independent of the wealth of any latergeneration (x+y+1,..., 00).40The channelling effect makes transparent the way in which future generations areexcluded from current resource markets. It increases the concerns about intergenerationalequity if early generations assume ownership of the resource stock. Also, it shows that ifearly generations want to benefit future generations, they have to do so in a way that benefitsall generations. For example, predictable catastrophic events that directly affect only onegeneration could, in fact, drastically reduce welfare of all subsequent generations, since itwould lead to excessive resource depletion by the generations before the one impoverishedgeneration. Moreover, the channelling effect shows the limited value of resource prices asan indicator of long-term resource scarcity. It would be consistent with this model to observea period of resource abundance and low resource prices over several generations followedby generations with drastic resource price increases, resource scarcity and declining welfare.It appears plausible that the channelling effect has real life relevance, particularly fordeveloping countries that face a constraint on foreign exchange borrowing. The resourceowning (old) generation would rapidly deplete the country's natural resources since theyoung generation is poor and unable to borrow internationally in order to buy the resourceand later sell it to the next (richer) generation. Hence, the channelling effect would beparticularly severe in currently poor countries with high long-term growth expectations. Ifenvironmental amenities are considered a luxury good whose expenditure share rises withincreasing income, the effect would be exacerbated. Countries like Brazil or Indonesia withrapid depletion of their natural capital would be candidates for an emprical investigationwhether they are in a channelling situation. Note that the frictions in international capitalmarkets (for example a country's foreign borrowing constraint) bring about a real marketinefficiency that, through the channelling effect, leads to excessive depletion of naturalresources.413^Natural Resource Depletion and Intergenerational CompensationThe discussion in section 2 has shown that the efficiency of resource markets alonedoes not guarantee an optimal allocation of resources to different generations. Therefore, itis important to consider the instruments that could be used to achieve such an optimalresource allocation. In this section, different institutional arrangements for implementing adesired welfare distribution between generations are discussed. The modeling approach insection 2 would suggest the explicit assignment of resource endowments to differentgeneration. This approach was taken by Howarth and Norgaard (1990). Inefficiencies thatcan result from using this approach will be discussed. An alternative approach would beexplicit compensation of future generations for resource depletion. This was suggested byHartwick (1977). Two different approaches for implementing Hartwick's suggestion arediscussed in this section.3.1^Efficiency Limits of Intergenerational Resource EndowmentsHowarth and Norgaard (1990) suggest that the optimal, as opposed to merelyefficient, allocation of resources can be arrived at by maximizing a social welfare functionby choice of intergenerational resource endowments. They solve their two-generation modelfor the optimal resource endowments under a maximin and an additive intergenerationalwelfare function. However, it can be shown that this approach can lead to inefficiencies ina model with more than two generations. Consider the three generation model discussed insection 2.2. If resource endowments are distributed primarily toward generation 3, a situationcan arise in which the demand for the resource at times 1 and 2 exceeds the resourceendowments of generations 1 and 2. In this case, generation 2 would want to buy resourcesfrom generation 3 at time 3 but use it already at time 2 (i.e. invest a negative amount of theresource). However, resource investment are required to be non-negative and a cornersolution with zero resource investment would arise. This corner solution with zero resourceinvestment is the opposite of the previously discussed corner solution with zero capitalinvestment that gives rise to the channelling effect.42If generation 2's optimal resource investment declines to zero, and the corner solutionarises, the Hotelling price path would be violated by the resource price rising at less than therate of return on capital investment, a. Such an equilibrium would be inefficient since, if theresource is assumed to be physically existent already before its owner-generation is born,there exists a feasible redistribution of endowments that could lead to an actual Paretoimprovement. Specifically, in a three-generation model zero resource investment bygeneration 2 would lead to p3 < ap2. Then, a feasible redistribution of resource endowmentswould involve shifting a marginal share of generation 3's resource endowment, AR, togeneration 2. The marginal loss for generation 3 would be p3AR. At time 2, generation 2could provide capital investment of p2AR as compensation for generation 3 and be indifferentat the margin. Generation 3 would receive ap2AR from this investment and would be madestrictly better off since ap2 > p3. Hence an actual Pareto improvement would be achieved.Thus, the initial resource distribution leads to an inefficiency that is caused by incompletemarkets. The market incompleteness consists of generation 3's inability to trade its resourceendowments before time 3. Since generation 3 cannot trade before time 3, the endowmentof generation 3 can never be consumed before time 3, even if trade could increase thewelfare of all generations.Numerical analysis shows that this corner solution arises in a model with more thanthree generations for a variety of parameter values with equal capital endowments if resourceendowments are chosen to equalize utility across generations. In these cases, futuregenerations are given ownership over a share of the resources for distributional reasons.However, for efficiency reasons, earlier generations should be able to use the resource andhence, need to be able to buy the resource from later generations. Since they cannot directlytrade with non-adjacent generations, an inefficiency arises. Subsequently, a desiredintergenerational welfare distribution may not be achievable through efficient redistributionof resource endowments. Hence, it would be desirable to find an alternative mechanism bywhich the desired intergenerational welfare distribution could be achieved efficiently. Sucha mechanism, in which a resource consuming generation would explicitly compensate future43generations for resource depletion with capital investment, will be discussed in the nextsubsection.3.2 Compensation with Hartwiek's RuleAbove, I have shown that a desired intergenerational welfare distribution may not beachievable through an efficient redistribution of resource endowments. In this subsection, Iwill analyze the possibility to achieve the desired intergenerational welfare distributionthrough a direct compensation mechanism. In this subsection it is assumed that the desiredintergenerational welfare distribution is determined by a maximin welfare function (hence,a social planner would maximize the welfare of the worst-off generation). The use of amaxitnin welfare function does not mean to imply a normative statement. However, thisparticular welfare function is chosen since it focusses the analysis on intergenerationalwelfare distribution.Hartwick (1977) shows in a continuous time model with Cobb-Douglas productionthat welfare is constant across time if, first, the resource is priced competitively (the priceincreases at the rate of interest, and the resource depletes asymptotically), and second, netcapital investments equal competitive resource profits. Zero-population growth is anadditional assumption of the model. Dixit, Hammond and Hoel (1980) have obtained thesame result with general functional forms and showed the conditions under which theconstant welfare path is the maximin welfare path. The investing of competitive resourceprofits will be called Hartwick's rule (HR). The intuition of Hartwick's rule is that ifcompetitive resource profits are invested in perpetuity and only returns to this investment areconsumed, the returns to this investment would increase with time and compensate, inwelfare terms, precisely for the rising resource price. Hartwick (1977) and Dixit, Hammondand Hoe (1980) present Hartwick's rule as an investment rule that leads to constant welfare.However, use of the rule is not modeled in an explicitly intergenerationa1 framework. Theydo not discuss conditions that would lead to investment according to Hartwick's rule in acompetitive economy. Also, they do not discuss institutional arrangements that could be used44to implement Hartwick's rule. These questions will be discussed throughout the remainderof this chapter.Hartwick's rule could be implemented in a socially managed economy by determiningconsumption and investment plans that follow Hartwick's rule. The following analysis ismotivated by the belief that a socially managed economy may not be desirable. Therefore,it is important to know which interventions would be necessary for implementation ofHartwick's rule in a decentralized, competitive economy. In a competitive model of identicaland selfish generations with positive marginal utility of consumption, Hartwick's rule couldnot be realized since generations would always consume or sell their capital stock, but neverbequeath it to the next generation. Two different interventions for implementing Hartwick'srule will be discussed. Section 3.2.1 suggests a mechanism based on public resourceownership. Section 3.2.2 discusses the implementation of Hartwick's rule with privateresource ownership. The latter discussion is also applicable to a situation in whichgenerations are not selfish and voluntarily bequeath some capital to their successors.3.2.1 Compensation with Public Resource OwnershipIn this section, I will apply Hartwick's rule to an intergenerational context andsuggest a mechanism that would lead to net-zero investment (change in capital stock minusresource depletion), satisfying Hartwick's rule. No single generation would own theresource. Instead, every generation would have the right to buy the resource against apayment that is invested in perpetuity. The price at which every generation can buy theresource is administered such that it follows the Hotelling path and leads to asymptoticaldepletion of the resource. Hence, the resource is allocated efficiently and trade betweengeneration would not take place. (Simultaneously living generations could buy the resourceat the same price. Since the Hotelling price path is administered, there are no arbitrageprofits that could be made through trade.) The instantaneous returns to accumulatingcompensatory investment are added to the current generation's endowment. FollowingHartwick's rule, a constant welfare path would be obtained if net investment at all times45equalled resource profits. Since all resource profits are invested by the suggested mechanism,no additional investment or borrowing may take place in the economy for Hartwick's ruleto hold. No additional investment would be undertaken if either each generation's lifetimewas infinitely short, or all individual generations' optimal savings were zero, or netinvestment at all times was zero because saving and dissaving of overlapping generationscancel out perfectly at any time. Under any one of these three conditions, the constantwelfare path is obtained through administration of the efficient price path. The followingmodel will formally show how such an institutional arrangement would bring about aconstant welfare path.The model is similar in structure to the consumption model in the first section of thischapter. However, Dasgupta and Mitra (1983) have shown that Hartwick's rule isincompatible with competitive pricing in a discrete time model. A continuous time modelwould be closer to the realities of intergenerational trade and resource depletion. Since itwould be unwise to use a modeling abstraction which is known to distort the results in thiscontext, a continuous time model is used. Under the suggested pricing mechanism, there areno trade opportunities between generations. Hence, for simplicity, the generations in thismodel do not actually overlap but simply replace each other. As before, there are twoconsumption goods, a non-perishable, non-renewable resource, R, with the initial stock, So,and capital, K. The initial capital stock is zero and the instantaneous return to investedcapital is a. There is an infinite number of generations i = 0,1,2, ..., 00• Generation i livesin the time interval [i,i+1]. The consumption streams are denoted kc(t) and P(t). Since onlyone generation lives at any time, t identifies time as well as the currently living generation.Generation i receives constant instantaneous endowments at the rate kei. This endowmentconsists on the original endowment for all generations, keo, and returns to compensatinginvestment accumulated by all previous generations. Hence:46^= koe + a f p(t) r (t) dt^ (27)Generation i's utility function is:t+1111 = f e 4") (zLog[k c(t)] + (1 -z) Log[r '(t)])dt^(28)where 6 is the instantaneous utility discount rate. Generation i maximizes its utility by choiceof consumption, kc(t) and I'M, subject to the budget constraint:i(t) = k + a (X(t) + C(t)) - k (t) - p(t) r (t)^(29)where X(t) is the stock of investments of generation i's own capital (with i -= Integer[t]).The initial and terminal conditions for X(t) are X(i) = 0 and X(i+1) = 0. C(t) iscompensatory investment accumulated during the lifetime of this generation i, with:^C(t) = f p(t)r (t) dt^ (30)Integer(t)A generation receives the instantaneous returns to accumulating compensatoryinvestment, aC(t), as an addition to their endowment. This is reflected in their budgetconstraint, (29). They do not treat C(t) as a function of their choice variable I'M. Thisreflects the assumption of competitive resource depletion which is a component of Hartwick'srule. To find the demands, the maximization problem is solved for the optimal consumptionstreams using the Hamiltonian approach (for an exposition of the Hamiltonian approach tocontinuous time optimization see, for example, Conrad and Clark 1987). The Hamiltonianis:^H = e -8 (t-t)[z LogR^+ (1 -z) Lo g[r^+ (t)(kie k (t)^- p(t)r c (t) + a (X(t) + C(t)))1 (31)The first order conditions are:47dH^dH-o   -odkc(t) dr '(t)^d ^i(t) _  dH ^dX(t) dA(t)(32)With the border conditions X(i) = 0 and X(i+1) = 0, the system of first order conditionscan be solved for the demands:^e(a-6*-i)zkie^e (a -6)(t-i)(1—Z)kiekc(t) - ^ ,^rc(t) - ^q q p(t)withq - a(-6e+6e6+cceaz-8e6z-ae"+6z+8e"6z) 8 (8 - a)(ea+8 - e6)(33)In order to fulfil Hartwick's rule, investment in addition to compensatory investmentfor resource depletion must be zero at all times (X(t) = 0). With X(t) = 0, instantaneousutility within one generation would be constant. Substituting demands into utility and settingthe derivative of instantaneous utility with respect to time equal to zero, gives 6 = az (theutility discount rate equals the rate of reproduction of goods weighted by their expenditureshare). This is a restriction on the utility function that will be imposed as an assumptionabout preferences, in order to ensures constant instantaneous utility within each generation.The significance of this assumption will be explored below. Now, the model can be used toshow the following:Proposition 1.7: With (5 = az and an administered resource price, p, rising at therate a such that the resource depletes asymptotically, constantwelfare across generations will be obtained by selling the resourceto any generation at the administered price, investing the proceedsin perpetuity and returning the instantaneous returns to thiscompensatory investment to the currently living generation asaugmentation of their endowment.48Proof: Substituting (5 = az into equation (33) gives q = 1. Since every generations'instantaneous utility is constant within their lifetimes, it is sufficient to compare instantaneousutility of generations i at t = i with instantaneous utility of generation i +1 at t = i+1. Theresource price is administered as the Hotelling price path: p(t) = p(0) eat. Substitutingresource demand (33) into (27) leads to the following relation between endowments ofconsecutive generations:= kie ea-"^ (34)Also with p(i +1) = ea p(i), 1-'(i+1) = rc(i) e-az. Denoting instantaneous utility of generationi+1, u(i+1):u(i + 1) = z Log[k` (i + 1)] + (1 -z) Log[r (i + 1)]= z Log[k (i) e az] + (1 -z) Log[r c(i)e -"z]= zLog[lcc(i)] +az -z2a +(1-z)Log[rc(i)] + z2a - az^(35)= zLog[k c(i)] + ( 1 -z) Log[r 'OA= u(i)With 6 = az, instantaneous utility within any one generation is constant; equation(35) shows that instantaneous utilities at t = i and t = i +1 are equal across generations;hence, utility is constant across generations i and i +1, and hence across all generations.Therefore, administration of the Hotelling price path, at which the resource does not depletein finite time, is sufficient to obtain constant utility across generations. •It turns out that, with 6 = az, le(t) and I'M are continuous across generations. Theadministration of p(t) leads to the same price, consumption and instantaneous utility pathsthat would be obtained through competitive markets with only one infinitely livinggeneration. With 6 = az, p(0) is easily calculated from re(t) = ke0(1-z)/(ep(0)) by equatingUtility of Consecutive Generationsui0.5543Timelnst Utility with d=az lnst Utility with d>az111. Mmi^Mis •0.50.450.40.350.30.25 049resource consumption until infinity with the initial resource stock So at the time at whichaccumulation of compensatory investment begins:S -^ke (1-z) dt43 0 e'p(0)14 (1 -z)p(0) -a z S0(36)Figure 1.6 Welfare of Five Consecutive Generations50It remains to examine the significance of the assumption (5 = az. With 5 az, everygeneration would shift parts of its capital endowment across time through saving orborrowing. Subsequently, net investment in the economy would not equal resource profits.The continuity of consumption across generations would break down and, hence, thebehaviour of one infinitely living generation and many short living generations would nolonger be identical. For a numerical example (p(0) = 1, a = 1, leo = 5, z = 0.5),Figure 1.6 shows instantaneous utility of five consecutive generations. With 6 = 0.5 =az), instantaneous utility is constant within and across generations. With (5 = 0.75 (6 > az),every generation moves consumption toward earlier times (borrowing from the owngeneration's endowment, X(t) < 0, is allowed in this model). Also, total utility driftsdownward from generation to generation. Hartwick's rule is violated with X(t) 0 and theconstant utility path does not hold any more. With 5 < az, instantaneous utility wouldincrease within the lifetime of any generation, and utility across generations would driftupward.In this model of discrete generations, b az leads to non-zero investment due to thechanges in the population's age distribution with time. In a more realistic model of acontinuum of overlapping generations, the population's age distribution would be uniformand constant. Then, X(t) 0 if 6 O az still holds for each individual generation. However,since net savings aggregated over one generation's life are zero, net savings aggregated overa population with uniform age distribution would be zero at any one point in time as well.Hence, X(t) would be constant across time, net investment would again equal resourceprofits, and the constant welfare path would be obtained without any restrictions on theutility discount rate 6. Hence, the assumption (5 = az could be dropped in a more realisticmodel with uniform age distribution.513.2.2 Compensation with Private Resource OwnershipIn the previous section it was assumed that resources are publicly owned and sold tothe current generation at an administered price. This section addresses the question, underwhich conditions a constant welfare path through Hartwick's rule can be obtained ifresources are privately owned. It is assumed that all resources are owned by the firstgeneration. The government could tax hundred percent of resource profits and invest theproceeds in perpetuity. Under this policy, private resource ownership would be essentiallymeaningless. Owners would not receive any benefits from selling their resources. Hence,they would have no incentive to optimize resource depletion across time, and, in order toachieve efficient depletion, resource prices would have to be administered as under publicownership.Alternatively, the government could observe resource profits and raise revenues ofthe same amount through an endowment tax that is invested in perpetuity. Under thismechanism, resource owners retain the incentive for efficient depletion. Also, if parts of thecapital stock are bequeathed to the next generation, as it can often be observed in the realworld, governments can reduce their compensating investment accordingly. The firstgeneration would sell a part of the resource stock to the second generation, the secondgeneration would sell a part to the third and so on. However, the resource profits gained bythe first generation from selling part of the stock to the second generation would be fullyrecovered as an endowment tax, invested by the government, and thus fully returned to thesecond generation as their endowment. The same procedure would take place between thesecond and third and all following generations. The net effect is that the resource stock isprivately owned at all times, every generation invests the resource profits from ownconsumption in perpetuity, and the endowment effect of resource ownership by the firstgeneration is eliminated through the endowment tax set equal to resource profits. If allgenerations followed this mechanism, and the resource price under this mechanism wasefficient, it would meet all conditions of Hartwick's rule and, hence, a constant welfare pathwould be the result.52If it was common knowledge that every generation will follow Hartwick's rule, theresulting resource price must be efficient since arbitrage profits could be made otherwise.Common knowledge implies that every generation knows that every following generation willfollow Hartwick's rule; it also implies that every generation knows that every followinggeneration knows that every following generation will follow Hartwick's rule; and so on. Ifthe assumption of common knowledge does not hold, however, the resulting price path maybe inefficient. Consider the following example of the continuous time model used in section3.2.1. Generation 0 follows Hartwick's rule. However, generation 0 believes that generation1 will be selfish and not follow Hartwick's rule but instead consume all of the capital stockand all of the remaining resource (generation 2 would not be able to buy any of the resourcefrom generation 1 since it does not inherit a capital stock which it could use to purchase theresource). This belief of generation 0 would change the anticipated demand for the resourcefrom generation 1, and hence change the resource price in the present.Figure 1.7 depicts such an example (for So = 5, keo = 1, a = 0.5, z = 0.5, and= 0.25) and compares it to a situation of common knowledge about all generations followingHartwick's rule. If the first generation expected repudiation by the second, the resource pricewould be lower, and generation 1 and 2's welfare higher than under common knowledge.All generations after time 2 would suffer from welfare of minus infinity. Note that, ifgeneration 0 expects generation 1 to repudiate, however, generation 1 does not repudiate butfollows Hartwick's rule, the resource price would jump at time 1. Since generation 0 wouldhave over-consumed, based on its faulty assumption that generation 1 will repudiate, it wouldbe impossible for following generations to achieve the constant welfare level of the commonknowledge scenario even if all generations actually followed Hartwick's rule.The discussion has shown that a constant welfare path can, theoretically, be achievedunder private resource ownership through government investments following Hartwick's rulealone. However, the requirement of common knowledge underlines the fragility of marketprices as indicators of scarcity, and hence as indicators of the required compensatoryinvestment, in an intergenerational context. It follows that for Hartwick's rule to work in a53Following or Repudiating Hartwick's Ruleu l0.80.676111••■ •=1■ 11115.00.44 F0.230 •_^_2-0.2rammal..• wi.etms= 11•1011=11.10.1 VEIDMPIE.01111 -----4.4 0A^I A o o1 2^3^4^5 3Time TimeAll Generations Follow HR Second Generation Repudiates HR^Al Generations Follow HR Second Generation Repudiates HR•••=111^ easseRepudiation Expected But^ Repudiation Expected ButSecond Generation Follows HR Second Generation Follows HRIn case of repudiation, welfare is minus infinity for all generations after t-2^In case of repudiation, the resource Is depleted at t-2.Figure 1.7 Following or Repudiating Hartwick's Ruleprivate ownership economy, it is necessary that all market participants base their expectationson common knowledge that Hartwick's rule will be followed. Hence, if a sustainabilityobjective was implemented based on Hartwick's rule, it would be important for thegovernment to not just quietly invest, but to explicitly state the sustainability policy andcommit to following it in perpetuity. This suggests a rationale for introducing sustainabilityas an explicit policy objective, possibly including legal obligations that would have at leastsome commitment value for future generations.544^ConclusionsThe work of resource economists has long focussed on intertemporally efficiencyresource use. This has resulted in policies emphasizing the establishment and securement ofresource property rights within the current generation. Institutional arrangements for intra-generational resource ownership have evolved. Yet, there are no institutional arrangementsfor the distribution of resource endowments across generations. The intergenerational generalequilibrium models in this chapter have demonstrated that if an early generation assumesownership of the resource stock, subsequent generations would face steeply declining levelsof welfare even if the resource is depleted efficiently. The models show how "... a [resourcedepletion] program can be intertemporally efficient and yet be perfectly ghastly." (Dasguptaand Heal 1979, p.25'7). Since these models abstract from uncertainty and technologicalprogress, they are too simplistic to predict doom for future generations. Yet, they clearlyshow that efficient resource markets alone do not ensure a reasonable welfare level for futuregenerations. Hence, striving for intertemporal efficiency in an intergenerational contextwould be incomplete without agreement on some basic principles of intergenerationalresource distribution.The concern about intergenerational welfare distribution arises because of systematicbiases in favour of earlier generations. Earlier generations are in a position to makeassumptions on resource ownership and impose them unilaterally and irreversibly on latergenerations. The models have shown that shifting resource endowments from futuregenerations to the present unambiguously increases the welfare of the current generation.Hence, an early generation that maximizes its own welfare would have an incentive toassume ownership of the full resource stock generating a systematic bias against the welfareof future generations.The concerns about intergenerational justice are aggravated by observations of currentdepletion practices. While, in the past, population and technology were at a level thatprevented an early generation from depletion at a level that could cause global, long-term55welfare effects, this has changed with the increase in human population and technologicaladvances that allow resource depletion at a truly global scale. Casual evidence of the currentpractice of resource extraction confirms the view that the current generation assumesownership of the full resource stock. Political decision making on resource depletion seemto include some consideration of depletion efficiency. However, little thought appears to begiven to the issue of intergenerational welfare distribution. Subsequently, depletion rates andresource prices would be based on assumptions about resource distribution that are biasedin favour of the current generation. Since, in general, resource prices depend on theassumption on which generations own the resource. there is no basis for the claim thatinterference with resource market prices would imply deviations from the "right price",unless the current assumptions about resource ownership are explicitly endorsed.The models in this chapter are sufficiently general to allow a broad interpretation ofnatural resources. Natural resources would not only include the typical non-renewableresources, such as fossil fuels and minerals, but also other types of natural capital, such asdepletion of the ecosphere's capacity to absorb human waste products. While the scarcity ofsource constraints (fuels or minerals) has often been overestimated in the past, the scarcityof sink constraints (absorption capacity for emissions etc.) has traditionally beenunderestimated. Applied to emissions, the models of this chapter would suggest that theimplementation of a mechanisms that ensures efficiency (for example, a Pigou tax or anintergenerational market for pollution rights) is insufficient for an optimal policy. An optimalpolicy would have to includes consideration of the endowment effect from assigning pollutionrights to different generations since all types of natural capital depletion can lead to negativeendowment effects for the welfare of future generations unless those are explicitlycompensated for the damage imposed on them (see also Bromley 1989 who discussed thisincome effect).Howarth and Norgaard (1990) suggest that the optimal, as opposed to merelyefficient, allocation of resources can be determined by maximizing a social welfare functionby choice of intergenerational resource endowments. This chapter has shown, in a model56with more than two generations, that the resulting resource endowments can easily beinefficient because of the lack of markets between non-adjacent generations. This would leadto a resource price rising at a rate below the discount rate. In these cases, future generationsare given ownership over a share of the resources for distributional reasons. However, forefficiency reasons, earlier generations should be able to use the resource and hence, need tobe able to buy the resource from later generations. Since a desired intergenerational welfaredistribution may not be achievable through efficient redistribution of resource endowments,explicit compensation mechanisms that could efficiently achieve the desired intergenerationalwelfare distribution have been discussed in the second part of this chapter.The models in section 3 have demonstrated how Hartwick's rule for compensatingfor resource depletion through capital investment can bring about constant welfare acrossgenerations. For Hartwick's rule to work in an economy with privately owned resources, itmust be common knowledge that all generations will follow Hartwick's rule. This underlinesthe relevance of explicitly stating or legislating sustainability as a policy objective.Alternatively, a compensation mechanism under public resource ownership is explored.Under the suggested mechanism, the resource would not be owned by any single generation.Instead, every generation would have the right to acquire the resource if it leaves behindcompensatory investment that allows the achieved utility level to be sustained by futuregenerations. Under this mechanism, the efficient resource price would have to beadministered.Price administration would be feasible in a simple world of certainty. In the realworld, however, resource price administration would open the door to a range of problemsof misuse and potential inefficiencies involving the incentives of individual decision makers.Hence, institutional arrangements for the implementation of the optimal depletion pathwithout administering the resource price would be desirable. In a separate paper (vonAmsberg 1992) I explore a market mechanism that would lead to the desired resourcedepletion path under full private ownership of the resource stock by the current generation.However, those who own and deplete a resource would have to insure themselves against the57claims of future generations on welfare equal to the welfare enjoyed by the depletinggeneration. A competitive insurance industry would offer such insurance to resource ownersand would effectively act as the agent of future generations on today's resource markets.Future generations would hold claims against the insurance companies on payment ofcompensation for resource depletion. The insurance companies would, thus, undertake thetask of investing the compensation for resource depletion. Such a regime would beparticularly suitable for efficient handling of the uncertainty inherent in future generations'welfare in the real world. This approach would imply a drastic change in the concept ofresource ownership from the right to use or dispose of a resource as one pleases to the rightto use a resource such that the benefits of such use benefit all generations equally.The most significant omission of this paper is uncertainty. In real life, there aredifferent types of uncertainty that render the implementation of any mechanism forintergenerational justice highly complex. The stock size of natural resources is uncertain. Inthe last decades, predictions about the quantity of reserves of non-renewable resources werefrequently updated, mostly upwards. On the other hand, only recently did we learn that theabsorptive capacity of the biosphere for many waste products is already exhausted. Ourincomplete knowledge of nature leads us to deplete natural capital in some instances withoutknowing about it (such as CFC or carbon dioxide emissions in the past). Technologicalchange introduces another dimension of uncertainty. Technology can work in eitherdirection: it can alleviate concerns about resource depletion if it leads to discovery ofsustainable substitutes, but it can also aggravate concerns about resource depletion if it leadsto new uses for natural resources. Other dimensions of uncertainty involve the size of futuregeneration and even their preferences. How do we know that future generations will enjoydriving cars as much as the current generation does? All these uncertainties are likely causesof non-marginal changes in the economy that would change relative prices. However, ifrelative prices change, compensatory investment calculated at previous price levels can easilybecome inadequate. Some of the complications resulting from uncertainty will be addressedin the following three chapters.a 8 (Kt: +pNRNe)z1 + 8CRN,N+1 —PN(1+ 8)KNC)V+1(I (KNe +pNRieg)(1—Z)58Appendix I-A: Derivation of the Equilibrium for the N-Generation ModelThe first generation i = 1 faces the maximization problem:= zLog[Kie.1] + (1 -z) Log[Riej ++ 8 (zLog[Kie,2] + (1 -z) Log[K2])max(37)s.t.a (Kle —1C1c,1)—K1ca+ (Rie - Ric -Rie,2 = 0P2 The first order conditions can be solved for the following demands:Kica(a K 13p2R1e)za(1 +8)8 (a Kie +I3P2Rie)Z1 + 8Rle3R1e,2Kie + Pp2R1e) (1 -z)13p2(1 +8)•5 (a Kie +13p2Rie)(1—z)P2(1 +8)(38)The last generation i = N faces the following maximization problem:maxUN = ZLOAKNc,N1+(1—Z)Log[Rigc,N] ++ 8 (zLog[KNe,N+i] + (1 -z)Log[RNc,N+0)^(39)s.t. ICAel -^ICNc.N+1 + pN RNe —RNc,N — RP!N+1 )^0aThe first order conditions can be solved for the following demands:c^(IC; +PNKT)ZKN,N ^1 + 8(Kiev +p NR) (1 -z)Re —N,N p N(1 + 8)(40)59The utility maximization problem of all intermediate generations, i = 2,...,N-1, is morecomplicated since they can trade twice, at times t = i and t = j +1. Their problem is solvedin two steps. At time i +1, they face the following maximization problem contingent on theirinvestments in the previous period, K1 and Rii:maxI 4 aj = zLog[Ki] + (1 -z)(41)s.t.^a Ki + 1 +^+ 1 ( = 0The solution to this problem is:Kicti +1 - ‘a^+^p Dz^Atitci+1(a K: + r3 pi+i Rii) (1 -z) (42)Pi +1The choices at t = i+1 are substituted into the following maximization problem at t =max^= zLog[Kicj (1 -z)Log[Ricj ++ 8 (zLog[K+i] + (I -z)Log[Ri+i])^(43)s.t.^ -Riej) = 0The first order conditions produce the Hotelling price path: pi.„, = (0/10) pi. This conditionhas to hold for an interior solution. Now, there are three first order conditions left for eachof the N-2 intermediate generations. In addition, there are N-1 market clearing conditions:13^Rie^R: V i =^ (44)and Ril is trivially Re1 - ItcLi. This are 3(N-2)+N-1 equations, which can be solved for 4(N-2) choice variables and one price. All other prices are then determined through the Hotellingrule. Demands and prices for all generations can be expressed with pi = ((31 a) p2 as:60c^(Kie +piRie)Z1+ 6c^a 8 (Kie +piRie)z1 + 8_ (Kie+ 8)D c^P 8 (Kie P iRie)(1-z)p,(1 +6)(45)(46)N K.e^ NCC" (1-2)E -2- E=1 cti-1^ :1=1 etj 1 N13i-1ZEIntermediate generations' demands could be directly derived for every generation separatelyif the Hotelling price path is assumed. However, prices and investments can only becalculated by solving the first order conditions for all intermediate generations simultaneouslywith the market clearing conditions since individual generations are indifferent betweeninvestment in R and K if the Hotelling price path holds.Appendix I-B: Derivation of the Three Generation Equilibrium with Ki2= 0In a three generation equilibrium with Ki2 = 0, demands of generation 1 and 3 are the sameas in the interior solution (Appendix I-A). Since generation 2 invests resources only, theoptimal choices at time t = 3 are:K2c,3 = 13p3Rz R2e,3^R(1 -z)^(47)The optimization problem of generation 2 is:maxx2"2,NAU2 = zLog[K2`,2]+ (1 -z) Log[R2ca.]+ 8 (z Log[K2:3] + (1-z) Log[R2`,3]) (48)s.t. K2e K2ea+p2(R2e - R2i - R2ca)The resulting demands with z = 0.5 and fi = 1 are:61 K; +p2R2e2(1+8) R2c,223K2e +p2R2ep22(1 +8)(49)The prices are:8 p3(K2e +p2R2e)2(1 +8)8 (K2e +p2R2e)p22 (1 +3)P2 — (1 + 8) a Kr + (1 +28)K;(1 + 8)Rie + R2eP38 K2e Rie + a 8 KA` + 2 8 K2e ke + (1 + 8) a Kie R3e + (1 +28) K2e ke(50)((1 + 8)a Kie +(1+28) K2e) K3eCHAPTER IIINEFFICIENCIES FROM INCOMPLETEINTERGENERATIONAL INSURANCE MARKETS1^IntroductionThe current generation, due to its population size and its technological capabilities,has an unprecedented ability to affect the welfare of future generations. Many effects on thewelfare of future generations occur through externalities, such as the long-term impacts ofpollution or changes to the production possibilities through technological progress. Themarket failures resulting from such technological externalities' are well recognized for intra-generational decision making and are directly applicable to intergenerational decision making,as well. This chapter does not analyze these intergenerational externalities. Instead it dealswith situations in which an earlier generation affects future generations' welfare by makinginvestments that reduce or increase the uncertainty faced by future generations throughmarket mechanisms.Examples of investments that change the uncertainty faced by future generationsthrough market transactions include all types of insurance investment. Protective measuresagainst natural or human-made disasters, such as building a dam against flooding, fall intothis category. Similarly, the preservation of biodiversity can be viewed as an investment thatreduces the uncertainties faced by future generations by providing options that pay off in bad1 In common usage, the term externalities refers to technological externalities only.Technological externalities are those where non-price variables in the production orutility function of one agent are chosen by another agent. In contrast, pecuniaryexternalities result from price changes in the economy, and, in general, do notconstitute a market failure (see Baumol and Oates 1988).6263states of the world, such as climatic changes or degeneration of important seeds. Often, suchinsurance investments have to be made across generational time. They involve investmentdecisions of one generation that pay off only to future generations. There are otherinvestments of the current generation that lead to specialization and increase the uncertaintyfaced by future generations. In both cases, the returns to investment are assumed to be soldto future generations through markets. They do not constitute a technological externality, inwhich case the market failure would be obvious.This chapter is motivated by the question whether competitive markets lead toefficient levels of intergenerational investment under risk. The analysis leads to a marketfailure that is based on the incompleteness of inter-generational insurance markets. Considerthe following stylized example to motivate the following analysis. The example consists ofa simple endowment economy with only one consumption good and two overlappinggenerations. Both generations are risk averse. Generation 1 lives at time 1 and 2, receivesan endowment of 10 units of the consumption good at time 1 and consumes only at time 2.Generation 2 lives and consumes at time 2 only. Generation 2 receives an endowment of 10at time 2. At time 2, a flood may strike with a 50 percent chance. The flood would destroyall of generation 2's endowment.At time 1, generation 1 could build a dam that would protect the endowment ofgeneration 2 from the uncertain flood. At time 2, it would be too late to build the dam.Generation 1 does not receive any direct benefits from the dam. However, it can sell the damto generation 2 at time 2, after information is received on whether the flood will strike butbefore the flood actually occurs. The cost of building the dam is 5 units of the consumptiongood. First, it will be shown that generation 1 would not build the dam under thesecircumstances. Then, it will be shown that the decision not to build the dam is inefficient.If generation 1 decided to build the dam, it would sell the dam to generation 2 sincegeneration 1 itself does not receive direct benefits from the dam. Assuming a competitivemarket for the dam, generation 2 would buy the dam at a price equalling its marginalbenefits. If the flood is known to occur, generation 2 would pay at most 10 for the dam. If64the flood is known not to occur, there is no benefit from the dam and generation 2 wouldnot be willing to pay anything for the dam. Generation 1 would have the alternative of notbuilding the dam and consuming 10, or building the dam and consuming 15 if the floodoccurs and 5 if it does not occur. Note that the expected value of generation l's consumptionis the same in both cases. Since generation 1 is risk averse, it prefers certain consumptionof 10 to risky consumption with the same expected value and would not build the dam.Now suppose generation 2 was able to commit itself to buying the dam at time 1(before it is born and before the uncertainty is resolved). At time 1, generation 2 could buythe dam at its cost, 5, and leave generation 1 indifferent as to whether the dam was built ornot. It would be up to generation 2 to decide whether the dam was built. If the dam wasbuilt, generation 2 would consume 5 with and without the flood. If the dam was not built,generation 2 would consume 0 if the flood occurs and 10 if the flood does not occur. Sincegeneration 2 is risk averse, it would prefer certain consumption of 5 over risky consumptionwith the same expected value and decide that the dam should be built. Hence, in thecompetitive solution the dam would not be built, but in the coordinated solution the damwould be built. Generation 1 is indifferent between the competitive solution (no dam is built)and the coordinated solution (the dam is built and generation 2 commits to buying ex-ante).Generation 2 strictly prefers the coordinated solution to the competitive outcome. Hence, theoutcome of the coordinated solution Pareto-dominates the competitive solution. Therefore,the dam should be built, and the competitive solution is inefficient (see Table 2.1 for asummary of the pay-offs).(G1/flood, Gl/no flood) -(G2/flood, G2/no flood)Competitive Solution(Generation 1 decides)Coordinated Solution(Generation 2 decides)Generation 1 does not build the dam.Generation 1 builds the dam.Table 2.1^Consumption in the Competitive and the Coordinated Solution65The reason for the inefficiency of the competitive solution is the incompleteness ofintergenerational insurance markets. Since generation 2 is born into resolved uncertainty,there is no market on which it could buy the dam ex-ante. However, generation 2 would liketo shift part of its wealth from the good state (no flood) to the bad state (flood). One wayto do this is paying for the dam in the good state as well. Therefore, generation 2 wouldbenefit from being able to commit itself to buying the dam ex-ante. However, there is no ex-post incentive to honour this hypothetical commitment if generation 2 is born into the goodstate (no flood). Since there are no markets for moving the wealth of generation 2 acrossstates, building the dam requires generation 1 to take on the risk of the dam being worthlesseven though there would be no aggregate uncertainty if the dam was built. Subsequently, arisk averse generation 1 has an inefficiently low incentive to build the dam in the competitivesolution.The general observation that the incompleteness of markets between non-adjacentgenerations can lead to inefficiencies has first been made by Samuelson (1958) in a modelunder certainty. Without specific reference to the intergenerational framework, it isrecognized that, with uncertainty, the incompleteness of insurance markets is a market failurethat can lead to inefficiencies (Hart 1975, and Grossman 1977). Wright (1987) suggests theproblem that future generations may wish to insure themselves against appearing in a badstate. More specifically, Howarth (1991) makes the observation that the incompleteness ofinsurance markets can lead to inefficiency in an overlapping-generations model withtechnological uncertainty. Neither of these papers analyzes the consequences of the marketincompleteness. The results of this chapter are related to previous research that demonstrateshow the arrival of information may reduce welfare by eliminating the possibility of mutuallybeneficial exchanges (see for example Akerlof 1970). Similar to models with informationasymmetries, an uninformed agent (generation 1 is uninformed at the time it makes itsinvestment decision) trades with an informed agent (generation 2). As a result, beneficialexchanges are foregone that would have been possible if both agents were uninformed at thetime of trade.66This chapter formally analyzes the market failure that arises due to incompleteintergenerational insurance markets. This chapter shows the conditions under which themarket failure will occur and describes the direction of its effects. The first section analyzesthe nature of the inefficiency arising from incomplete inter-generational insurance marketsin detail and derives specific results regarding over- or under-investment by the currentgeneration. Examples for over- and under-investment are provided. The second sectioncontains applications of the main result to insurance investments, such as protective measuresagainst global warming, the conservation of biodiversity, and, finally, the depletion of a non-renewable resource.A uniform approach for analyzing the market inefficiency is taken throughout thischapter. First, the competitive solution is determined by solving the maximization problemof two overlapping generations for the equilibrium with incomplete markets. Second, acoordinated solution is determined by making the hypothetical assumption that markets arecomplete and solving both generations' maximization problem. From work by Debreu(1959), Arrow (1964), and Radner (1972), we known that the market equilibrium based oncomplete markets is efficient. Hence, the coordinated solution is efficient. The effect of themarket incompleteness is then determined by comparing the choice variables (i.e., theamount of insurance investment) in the competitive and the coordinated solution.Throughout this chapter, it is assumed that the welfare of a generation can be sensiblydefined across different states into which this generation could be born. There is somephilosophical controversy about the identity of future generations (for a brief discussion seeHowarth 1991). However, this assumption is not critical for the main intuition underlyingthe analysis of this chapter. If it is asserted that future generations cannot be sensibly definedacross states into which they are born, the potential future generations born into differentstates of the world would have to be treated as distinct identities. This is the approachsuggested by Wright (1987). Then, the welfare distribution across future generations borninto different states becomes a distributional problem that would have to be approached bymeans of a social welfare function. With a social welfare function that has the form of an67expected utility function, the results of this chapter would still be fully applicable. Themaximization of a generation's utility function across different states would then beinterpreted as the maximization problem of a social planner.2^An Intergenerational Investment Model under RiskThere are two generations j (j = 1, 2). Generation 1 lives at times 1 and 2;generation 2 at time 2 only. There are two states of the world k (k = A, B). At time 1,there is uncertainty which of the two states, A or B, will occur at time 2. There is only onegood that both generations consume at time 2 only. Both generations have von Neumann-Morgenstern utility functions, II; =ir uj[cIA] + (1-7) u[c03] where cik denotes consumptionof generation j in state k. 7r is the probability that state A occurs. Both generations arestrictly risk averse (u,' > 0, u;" < 0). At time 1, generation 1 receives an endowment, el,and can invest this endowment in state contingent production of the consumption good. Theinvestment in state A is denoted iA; subsequently i8 = e, - iA• The output of production isfA[iA] if state A occurs and fB[eriA] if state B occurs (fk' > 0, f," 0). Generation 2 isborn into one of the two states and receives the endowment e,2A or e2B depending on whichstate occurs.In this model, there are no trade opportunities since there is only one good in eachstate. Trade across states is impossible since generation 2 is born into one of the two statesand, therefore, cannot trade with generation 1 at time 1. Generation 1 solves the followingmaximization problem:max U1 = ui[fAUAn + (1 - ir)ui[fB[ei-iAl]^(1)jALet iA* denote the investment in state A chosen in the competitive solution. Then, iA* isdetermined by the following first-order condition:68It u/L4 11A = (1-101411B fiB^ (2)where u'iA and u'iB denote marginal utilities with iA = iA*• This first-order conditiondetermines iA* uniquely since the LHS of (2) is strictly decreasing in iA* while the RHS isstrictly increasing. Generation 2 has no choices to make and consumes its endowments in therespective state (U2 = IT %Le/Al + (1-7r) u7fee2d).To calculate the efficient coordinated solution, it is assumed that markets are completeand generations can trade the consumption good across states. Let t denote the quantity ofconsumption in state B that generation 1 buys from generation 2 against payment of thequantity pt of the consumption good in state A. Hence, p is the price of consumption instate B in terms of consumption in state A. Utilities are the same as before, however, theyare now denoted with V to distinguish equilibrium utility values from the competitivesolution. Now the maximization problem of generation 1 is:max Vi^vi[fA[iA] -pt] + (1 --rt)vi [f B[e 1- i A] + t]^(3)jA,twith the first order conditions:Tc v/L4 f/A = (1 -n) v/iB flB(1-n)vlie ^ (4)TCwhere v '1A and v'IB denote marginal utilities with iA = iA**, and iA** denotes the investmentin state A chosen in the coordinated solution. The second generation maximizes:max V2 = TC v2[e2A+p t] + (1-70v2[e-t]^(5)with the first order condition:(6)Combining (4) and (6):v/IB^vim (7)v/L4^v/24gives the intuitive result that the ratio of marginal utilities between states must be equalizedacross generations in the coordinated solution. Given the assumptions on v and f, the firstorder conditions determine the unique solution.Proposition 2.1:^Assume that the competitive solution is an interior solution(generation 1 invests a strictly positive amount in both states), andthat in the competitive solution, generation 2 is relatively worse offin state A than generation 1, then the competitive solution impliesthat generation 1 under-invests in state A, compared to theefficient coordinated solution.Proof:With iA* as the investment level for the competitive solution for which (2) holds, andiA** as the efficient investment level for the coordinated solution for which (4) holds,proposition 2.1 can be written as:u /IA^u/2A . * *^. *> ■. IA < lAulis^ul2Bu /IA^u /2A . * *^. *< lA > lAu/18^1412814 /IA_ 14/2AU /18^U /2B(8)Now, the first part of (8) will be shown by contradiction. Hence, it is assumed that:70u/IA^u /2A^(9)u /2BCombining (2) and (4) results in:vim f/A[i; *]^u/IA f/A[i;]^ (10)vita f1B[1-1A**1^ultB f/B[1Suppose iA** = iA*. Then, (10) collapses to v '1A/v 91B = 1-1'1A/U'lB• From this and iA** = iA*follows t = 0 by comparing the definitions of U, and V,. From U2, V2, and t = 0 followsv92A/v'2B = 11'2A/1192B- From this and (7) follows U'2A/U '2B = 1-1'1A/11'1B which contradicts (9).Now suppose iA** > A. Then, (10) implies v' MA' 1B > 1A/U 113. From this and iA** >iA* follows t > 0 by comparing U, and V,. From U2, V2, and t > 0 follows v /v'2A '2B <le2A/U'2B. From this and (7) follows u'2A/u 2B > U'1A/U '18 which contradicts (9). Since iA**= iA* or iA** > iA* are inconsistent with (9), from (9) must follow iA** < iA*. The prooffor the second part of (8) is identical with all signs reversed.The third part of (8) is also shown by contradiction. Suppose u'2A/u928 = 11'1A/11'1B andiA** > iA*. With (10) this implies v'iAiv'iB > 11'1A/U'ig. From this and iA** > iA* followst > 0 by comparing U, and V,. From U2, V2, and t > 0 follows v'2A/V'2B < U'2A/1192B. Fromthis and (7) follows u '2A -/u '2B >^1B which contradicts u'2A/u'2B = u'iain'113* With jA**< iA*, the same contradiction can be shown by reversing all signs. Since iA** > iA* or iA**< iA* are inconsistent with u'2A/u' 2B =^from u' 2A/ tf 2B = lAke 1B must follow IA**= iA*. The coordinated solution- was constructed as an Arrow-Debreu equilibrium withcomplete markets. Therefore, iA** is the efficient level of investment (Arrow 1964, andDebreu 1959).111This general result can lead to inefficiently low or inefficiently high investment forthe bad state in the competitive solution depending on the nature of the uncertainty. Ifgeneration 1 faces higher risk than generation 2, there will be excessive investment for thebad state; if generation 2 faces higher risk than generation 1, there will be insufficientRisk of Generation 2, Relative to Generation 1u'213Figure 2.1 Relative Risk of Generation 200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1u'2A112A-u2B u2A/u'2B-01A/V1 B71investment for the bad state. Ahigher risk for generation 2means that, in terms of ratiosof marginal utilities, the badstate for generation 1 is worsefor generation 2. This situationis depicted graphically inFigure 2.1. The following twoexamples analyze two types ofuncertainty. First, endowmentuncertainty (exogenousuncertainty) for generation 2 isassumed. This leads to under-insurance by generation 1.Second, uncertainty about the return to investments rather than endowments is assumed. Inthe latter case, uncertainty is the outcome of the optimal investment decision by generation1 (endogenous uncertainty) and leads to over-diversification of generation l's investment inthe competitive solution.2.1 An Under-Insurance ExampleThe result that markets lead to under-insurance of risks that are higher for futuregenerations than for the current generation is illustrated by means of the following simpleexample. The first generation receives an endowment, el, before the uncertainty is resolvedand splits this endowment between investments in the consumption good in state A, iA, andinvestment in the consumption good in state B, i8 = eriA. Technology is linear, fk[ikl=ik,and identical for the two states A and B, which are equally probable Or = 0.5). Generation2 is born into either state A or B, facing endowment uncertainty (e2A/7 < e2B/(1-7)). Hence,state A is the bad state. Both generations have log-utility. In the competitive solution, no72trade can take place. Generation 2 consumes its endowment in the respective state.Generation 1 faces the following problem:max U1 = IC Log[iA]+(1 - n) Log[e A]jA= eA^1However, through a coordination mechanism, generation 1 could insure generation 2 againstthe endowment uncertainty within the limits of generation l's own endowment. In thecoordinated solution, trade would be allowed and generation 1 would solve the problem:max 111 = it Log[i A - pt] + (1 -n) Log[e - iA + t]^(12)Generation 2 would solve the problem:max V2 = n Log[e2A+ pt] + (1 -n) Log[e2B - t]^(13)The resulting equilibrium would be:p = 1t = e24 n + e2B 7t - e2A^ (14)* *IA^e^e2B + (n -1)e2AInsurance investment (investment for the bad state) in the coordinated solution, iA**, exceedsthat in the competitive solution, i,*, if e2A < 7/(1-7r)en, which is the case whenever A is,in fact, the bad state for generation 2. The numerical example in Figure 2.2 shows how thecoordinated solution can lead to a Pareto improvement over the competitive solution. As aresult of the linear technology, all gains from trade accrue to generation 2. Of course, theunder-insurance result is the outcome of the assumed endowment structure. Over-insurancewould occur if the first, and not the second generation received an uncertain endowment attime t=2. In that case, generation 2 could share some of generation l's risk in a coordinatedsolution, resulting in less insurance investment compared to the competitive solution.ClA 1.5• 1.5CiA -1.5c2A •. 2.0i■ 0.41U2-0.41Ul 0.41U2 I. 0.35iA - 1.5^ IA - 1.0is 1.5N^ 1B 2.0cis •• 1.5c2s - 1.0cis 1.5• - 1.573Competitive Solution^Coordinated SolutionEndowments: e^3, e^2, 0 2B au 1, Productivity: f AmI A, fB leFigure 2.2 Under-Insurance with Exogenous Risk2.2 An Over-Diversification ExampleA different outcome results from endogenous uncertainty, as illustrated by thefollowing example. Again, log-utility is assumed for both generations. While technology isstill linear, investment in state A now yields a higher rate of return than investments in stateB (f.A[iA] = a. iA fg [id = b. iB air > b.(1-7)). In this example, there is no endowmentuncertainty (e2 = e2A = e2B). In the competitive solution generation I would again investone-half of its endowment in each of the two states:max U1 = Log[aiA] +(l-Tr)Log[b(el-iA)]jA^(15). *• 1A = -rt eIn the competitive solution, consumption of generation 2 would be e2 in either state. In acoordinated solution, the risk of high return investment could be shared between bothclA 3.0cm - 3.0clA - 3.0csA - 2.0Ul 0.75U2 - 0.75Ul 0.75Ue -. 0.69iA - 1.5^ iA = 2.0is .... 1.5^ is = 1.0Cis ma 1.5c2s - 2.0cis - 1.5c2s - 1.574Competitive Solution^Coordinated SolutionEndowments: el -3,. ee -2,. 2B 2, Productivity: f A - 2 I A, le - leFigure 2.3 Over-Diversification with Endogenous Riskgenerations which would induce generation 1 to increase investment in the high-return stateA. The maximization problems for the coordinated solution are:max V1 = tr Log[a i A - pt] + (1 --n)Log[b(ei - i A) + t]Atand:max V2 = rc Log[e2+ pt] + (1 -TO Log[e2- t]rThe resulting equilibrium is:(16)(17)75t - e2(nal-nb-b)^ (18)aTC^1 - TC e2=^+ b aIn this example, iA** (high return investment in the coordinated solution) is higher than iA*(high return investment in the competitive solution) if a > b(1-7)/7 which is simply thatinvestment in state A yields higher returns. Hence, the competitive solution impliesinefficiently low risk taking by generation 1 or excessive diversification of investment. Anumerical example is provided in Figure 2.3 and shows that the coordinated solution canresults in a Pareto improvement over the competitive solution.2.3 The Robustness of the Market Failure from Market IncompletenessSo far, the market failure resulting from incomplete intergenerational insurancemarkets has been shown in an extremely simplistic model with a single consumption goodonly. To understand the real life relevance of this market failure, it would be desirable toanalyze the implications of relaxing some of the strict assumptions made before. The firstextension of the model presented in the preceding section would allow generation 1 toconsume at time 1 as well. With log-utility, generation 1 would spend a fixed proportion ofits endowment value on consumption at time 1. Since prices are fixed through constant ratesof return, they would not change with allowing consumption at time 1. Hence, consumptionat time 1 would be unaffected by opening markets for consumption at time 2, and allprevious results would remain unaffected. Similarly, letting generation 2 live for a secondperiod and allowing it to consume at time 3 would not alter any results.The introduction of additional goods may lead to the transmission of risks acrossgenerations through the prices for these other goods within each state. In order to test this76intuition and the robustness of the main result in a more general setting, this sectionintroduces a second consumption good. In the following models with more than onecommodity, the efficient coordinated solution is derived as a Radner equilibrium, in which- in addition to spot markets within each state - contingent markets for one commodity ineach state are opened before the uncertainty is resolved (Radner 1972). Under perfectforesight, such an equilibrium is efficient under conditions that are prevailing in the simpleexamples used here. Consider a model like the one used in the under-insurance example ofthe previous section. Generation 2 faces endowment uncertainty for good I. Now, there isa second good II, of which both generations receive a certain endowment. Within each state,both generations are free to trade goods I and II. For the competitive solution, bothgenerations maximize their utility:max U =1,4,CA,taPg[iA tAPA] ±gL°g[elff tA])(19)(1 -70(1.og[eii- iA - t Hp H] + gLog[e 111 + ti])and:max U2 =^71 (Log[e2L4 + t Ap A] + g Log[e2H- tA])+rA,t„^ (20)(1 -7c)(Log[e2H3 + t Bp B] + g Log[e2H - t IA)where subscripts 1 and 2 designate generation 1 and 2, respectively, subscripts I and IIdesignate good I and II, respectively, and subscripts A and B designate state A and B,respectively. pk is the price of good II in terms of good I in state k. The relative importanceof good II is expressed by the parameter g. For the coordinated solution, trade across statesis introduced (not shown here). The equilibrium investments in the competitive case, iA*, andin the coordinated case, iA**, are:0^2^4^6^8^10Importance of the second good, gCompetitive Solution Coordinated SolutionOM = INN = = II1.421. InvestmentInsurance investment, iA2.1777t((1 +g)eueiff +g euf(e2m+ezo)+eue2H)-(gemed(1 + g)ein+ ew^ (21)...lA = n(eil÷e2m+e2n3)-e21AFigure 2.4 Insurance Investment and a Second Good's ImportanceFrom equation (21) follows, with all endowments non-negative, that iA** > iA* if eliA <eni r/(1-r)which is that A is, in fact, the bad state. Hence, the under-insurance result holds78for the case of a second good that enters the utility function concavely, though the extent ofthe inefficiency may be reduced. In particular, the extent of under-insurance is reduced withincreasing the relative importance of good II, expressed by the parameter g. This effect isdepicted in Figure 2.4 for a numerical example (en = 3, ell/ = 2, enA = 1, eng = 2, am =2, 7 = 0.5). g = 0 would refer to the one good case shown in Figure 2.2. The rationale forthe decrease of the inefficiency with introduction of a second good is that generation 2 facesendowment uncertainty for good I, and would like to shift some of this uncertainty to goodII through trade within each state. Hence, in the good state, generation 2 sells relativelymore (or purchases relatively less) of good I to (from) generation 1 as compared to the badstate. As a result, good II is cheaper in the bad state. Generation 1 faces some of generation2's endowment uncertainty through the price mechanism within each state, and has anincreased incentive to invest for the bad state. Hence, the relative risk of both generationsis closer in the two-good model than in the one-good model. If generation 2, however, facedendowment uncertainty for good II as well, it would have less incentive to shift uncertaintyto good II, and the introduction of a second good would not reduce the extent of theinefficiency in the competitive solution.As an extreme case, consider a utility function in which the second good enterslinearly. Now the maximization problems for the competitive solution are:max U1 =LAtA 11it (L9g[iA tAPA1 +g1(eiff tA))+(22)(1 -7) (Log[eu - iA - tBp B] (÷•-,1 (e111 -I-max U2 =^it (LOg[e2t4 + tApA] +g2(e211-tA))+tAtB(1 -it)Pg[e2m tBP g2(e211-tB))and:(23)The first order conditions are:79utuAPA = ulim4 = g1WuBPB = 14/1UB = g1u/21ApA = u/2RA = g2u/2/BPB = 1412HB - g2and can be rearranged to:u/IIA^u/21A^ _14/1IB^U/2113With a quasi-linear utility function, generation 2 would try to shift all uncertainty togood II. As a result, trade between goods I and II within each state would be sufficient toequalize risk across generations in the competitive solution. Hence, with a quasi-linear utilityfunction, the competitive solution is efficient. While proposition 2.1 is not violated, it hasno relevance in this case since the condition that gives rise to the inefficiency cannot arisewith this specific utility function. A quasi-linear utility function may have no empiricalrelevance, however, it is useful to point out this extreme condition under which the marketfailure would be remedied. The conclusion from this discussion is that the under-insuranceeffect will be large if generations are relatively risk averse, if there are relatively few goodsover which risks can be spread, or if the risks are relatively large. This would focus real lifeconcerns about this market failure on risks that have a large aggregate effect, such aspossibly global warming.3^Applications of the Under-Insurance ResultIn this section, the principal ideas analyzed in an abstract manner in the previoussection, are put to work in models closer to real-life environmental problems. First, a modelof intergenerational insurance investment with a general production function and more thantwo states of the world is analyzed. Second, the example of insurance investment in buildinga dam that was presented in the introduction to this chapter is discussed in more detail.(24)(25)80Third, the conservation of biodiversity is analyzed as a problem that combines both genericsituations that can lead to inefficiency: investment in insurance and specialization ininvestments that are more productive. Forth, the model is applied to the intergenerationaldepletion of a non-renewable resource. The purpose of that discussion is to analyze whetherthe incompleteness of markets impacts on the resource depletion path.3.1 A Model of Insurance InvestmentThe following model of intergenerational insurance investment extends the originalmodel to cases with more than two states and a generic type of insurance that is morerealistic than the original model in which the first generation could only make investmentsthat would directly offset the second generation's endowment uncertainty. As before,consider two overlapping generations 1 and 2. Both generations consume a singleconsumption good at time 2. However, generation 1 also lives at time 1 and receives itsendowment, el, at time 1. Generation 2 is threatened with uncertain damage that reduces itsendowment, e2, by damage, dk, depending on the state k into which generation 2 is born.Generation 1 can invest in insurance, i, at the cost c[i] (dc/di > 0, d2c/di2 > 0). Also eli 0 must be fulfilled. Generation 1 has no direct benefit from the insurance, however, itcan sell insurance at price Pk to generation 2 at time 2 in state k. Insurance decreases thedamage dk. Damage is always positive (dk > 0). For all k, dk[i] is positive and decreasingin i (ddk/di < 0, also assume d2dk/di2 > 0). Assume that any states A and B can be orderedsuch that A < B adA > dB (state A is "worse "than state B), then it is assumed to be thenature of insurance, i, that A < B <=> ddA/di < ddB/di for all i. This means, in a good stateof the world damage is small and benefits from insurance are small as well. The net marginalcosts of insurance (marginal cost minus marginal benefit) are lower in bad states than ingood states. An example of damage and net marginal costs in different states as a functionof insurance, i, is shown in Figure 2.5.Both generations have von-Neumann-Morgenstern utility functions and are strictlyrisk-averse. Utility for both generations is defined across states (U, = Ek TkUj(Cik), j = 1, 2;2.521.50.5I^I^■1,,I,,L■1,.1■,1.1,1.■1■.10Insurance investment, i_-^...0 /tX-AIIr pmInsurance Investment,(a) Damage and Insurance (b) Net Marginal Cost of InsuranceDamage, dk^ Net Marginal Cost, gk10.50-0.5-1-1.5281Figure 2.5 Damage and Net Marginal Cost of Insurance Investment> 0, u"i < 0). Generations have perfect foresight and behave competitively. First, thecompetitive solution will be determined. Second a coordinated solution for hypotheticallycomplete ex-ante markets will be determined and compared to the former.3.1.1 The Competitive SolutionTrade between generations is possible only at time 2 after uncertainty is resolved.Hence pk is the ex post price at which generation 1 sells insurance to generation 2 in statek. Generation 2 maximizes utility within each state:max u2 [e2 - djik] - pkresulting in the first order conditions:ddi.=^v kdik(26)(27)Generation 1 maximizes its utility by choosing the level of investment in insurance, i, givenperfect foresight of prices in different states k:82max U1 = E ku -di] + p ki]^ (28)resulting in the first order condition:E nkuik(Pk- (-14) =^ (29)With ik = i V k, and substituting (27) into (29), the equilibrium insurance investment, i*,can be determined from:(ddkul + (mil)E^E 7CkU1k lik[i] = 0di^di (30)where g[i] is the net marginal cost of insurance (marginal cost minus marginal benefit) instate k with:ddk[il dc[i] g k[i] di^diAll gk in (30) are strictly increasing in i (d2c/di2 > 0 and &clic/di' > 0). Also, all Tku'ik arepositive. Hence, (30) determines i* uniquely.3.1.2 The Coordinated SolutionAlternatively, consider that ex-ante markets were complete, and generation 2 couldtrade with generation 1 at time 1. Now, markets for insurance, ik, in different states of theworld are open at time 1 with prices qk in terms of the consumption good in state A. Marketsare also open for transfers in state contingent consumption goods, tk, from generation 1 togeneration 2 with prices qd,. Hence, the numeraire good is the consumption good in state A.Utilities are the same as above. However, to distinguish utilities between both equilibria,they are here denoted with v instead of u. Now, generation 1 maximizes its utility by choiceof investment in insurance, i, and state contingent transfers, tk:(31)83max V1 = icAv1{el -^+ E gki+E qtktd E k1'1 Lle1 - c[i] - tk j^(32)k=A^k=B^k=BThe resulting first order conditions are:dc7t AVIA .L.d qk = E dtnkvik qtkIt a VIAGeneration 2 maximizes its ex-ante utility by choice of insurance acquired in each state andstate contingent transfers:max V2 = 7C AV 2{e 2—dk[ik] -r Cr- ki k— E qtktkl^E TC kV 2[ e2 --dk[ik] +tki^(34)ik,tk^ k=A^k=B^k=BThe resulting first order conditions are:Tc v /k 2k^kqk^ V kcAll2A di k(35)qdc^It k V2k^V^kIrA V2AFor all markets to clear, ik = i for all k. With (33) and (35) relative marginal utilities areequalized across generations:k Vlk^k V2k qtk^ V k^ (36)nA 114^nAV2A(33)V kCombining (33) and (35), the equilibrium insurance investment, i**, can be determinedfrom:84^E nkvik ddk[i ^+ dc[i**] - E rckviik gkU*1 = 0di^di(37)Again, (37) determines i** uniquely since all gk are strictly increasing in i and Tkv'ik arepositive for all k.3.1.3 Comparison Between Both EquilibriaNow, the relationship between equilibrium insurance with incomplete markets, i*, andwith hypothetical complete markets, i**, can be analyzed by comparing equations (30) and(37). This leads toProposition 2.2:^The incompleteness of intergenerational insurance markets leadsto inefficiently low insurance investment by generation 1 (i**> i*).Proof: From (28), um depends on the state k only through pk• From (27), pk = -ddk/di.Insurance was defined by A < B dd,/di < ddB/di, hence A < B u- IA > ulB and withu" < 0, A < B <=> u'IA < u'18. This means given any choice of i with i > 0, generation1 is worse off in good states of the world than in bad states. This is because damage doesnot affect generation 1 directly. However, generation 1 benefits from damage by being ableto sell insurance to generation 2 at a higher price. As a result, marginal utility of generation1 is higher in good states of the world. Now, consider the hypothetical complete marketarrangement. From (36) follows that both generations' ordering of utility over states mustbe the same in equilibrium. Since aggregate wealth is larger in good states of the world(damage is always positive), both generations' utility must be higher in good states. Hence,A < B <=>v > v'm.85Equation (30) equalizes the weighted sum of the functions gk[i] with zero. Forillustration, the shape of the g[i] functions is shown in Figure 2.5 (b). Since the weightsIrku'lk are positive for all k and A < B <4 gA[i] < gB[i], some gk[il must be positive whileothers must be negative and there must be a state H such that gk[il 0 for k < H andgk[il > 0 for k > H. Now the equilibrium equations (30) and (37) are normalized by therespective marginal utilities in state H:ulk—Tck gkul =0(38)Vlk— rck gk[i**] = 0k V1HNow, for every k > H (gk[il > 0), u' lichr 1H > 1 and v'lkiv'1H < 1; also for every k <H (gk[il < 0), u'ik/u'lli < 1 and V' ik/V' 1H > 1. Hence, replacing all weights on positivegk by a smaller number and all weights on negative gk with a larger number, it must hold:VikL T k gkri <V1H(39)Since all gk are increasing in i, from (38) and (39) follows i* < i**. Since i** is the uniquevalue for i in a Pareto efficient coordinated solution, the competitive level of insuranceinvestment, i*, is inefficiently low. •3.2 Building a Dam to Protect Future GenerationsAn other application to a hypothetical real life situation is provided to illustrate therelevance of the foregoing results. This is a somewhat more complex version of the dam-building case presented in the introduction, which allows continuous levels of insuranceinvestments. At time 1, generation 1 can invest in building a dam that will protect generation2 from a possible flood. The probability of a flood striking at time 2 is 1-7 (state B). If the86flood strikes, the damage to generation 2 is Max[d4,0] where i is the hight of the dam builtby generation 1. The cost of building the dam is linear in its hight, c[i] = c.i. If the flooddoes not strike (state A), there are no benefits from the dam. First consider the case withoutex ante markets (competitive solution). Both generations have log-utility over the singleconsumption good with endowments el and el. Generation 1 's problem is:max Ul = IC Log[e + pAi c i] + (1 - Tc) Log[e + pBi - ci]^(40)The first order condition of generation l's maximization problem for an interior solution is:(c - p 2) (7c - 1)^(p A - c) 7C --ci+ipB^-ci +ipAGeneration 2's problem is:max U2 = it Log[e2-paiA]+ (1 - it) Log[e2 - p B - d + i]iAlBThe maximization of generation 2 simply fixes prices at marginal benefits (PA = 0, pB = 1).With these prices, (41) can be solved for i. Taking corner solutions into account, generation1 would build a dam with hight i*:(41)(42)i* = Min[Afe (c -1 +1 -C2 -c2^ 0 ,d (43) Consider a numerical example with ir = 0.5, d = 0.5, c = 0.5, el = e2 = 1 (seeFigure 2.6). Then, the dam hight under the competitive solution is zero. In the competitivesolution, utilities are U1 = 0 and U2 = -0.347Consider next the coordinated solution with hypothetically complete markets at time1. At time 1, there is trade in four goods: consumption in state A and in state B and damhight in state A and in state B. Consumption in state A is chosen as the numeraire. Transfersof the consumption good between both states are denoted t. p, is the payment of consumptiongood in state A for one unit of consumption in state B. pi, and p8 are the respective pricesDam HeightDam Height, n0.6 ^^I^ l iii 0.3^0.36^0.42^0.48^0.54^0.6Dam Cost, cCompetitive Solution Coordinated Solution87Figure 2.6 Dam Hight as a Function of Costsper unit of the dam in terms of consumption in state A. Now, the problem of the firstgenerations is:max V1 = Log[ei- ci + (p A + p B)i -pit] + (1 - It) Log[e - ci +t]^(44)with first order conditions:88^c(1 - it)^(PA +/28 -0" + -oci - t - e1 el+ '(PA +pB- c) - petn -1^Pen +  -oel - ci +t el + i(pA +pB- c) -petThe second generation's problem is:max V2 = it Log[e2-pAiA- p BiB + pet] + (1 - lc) Log[e2 - d + i B - t]A.ilPtThe first order conditions are (with i A = 114 = :^7C - 1^Ptn -oe2-d+i-t e2-i(pA+pB)+ptt1- 7C PBn - o^(47)e2 -d+i-t+ i(pA+pB) - pet - e2PA It -ot(pA+pd-ptt-e2From the first order conditions, the dam hight in the coordinated solution, i**, can bedetermined as:[i** = Min MaxThe other equilibrium values are:(c -1 +n)(ei^+ cdn,01,di^(48)C 2 - C(1-c - ic)ei +(c -c2 -cn)e2+c2dit(45)(46)(49)Pt - c -1For 7 1= 0.5, d = 0.5, el = e2 = 1, the dam would be built at full hight (i = 0.5) and theresulting utility values would be U1 = 0, U, = -0.288, which implies a Pareto improvement89compared to the competitive solution. Figure 2.6 compares the dam hight in the competitivesolution with the coordinated solution for a range of dam cost, c. Note that at c = 0.5, themarginal cost of the dam equals its expected marginal benefits. At this value, the dam wouldnot be built under market arrangements but would be built with full hight under an efficientcoordination mechanism. Also, from equations (43) and (48) follows that i**^i* if c <1 and d^e2. These two conditions simply require that the cost of the dam be less than itsbenefit in the bad state and damage to generation 2 not exceed its endowment.3.3^Conservation of BiodiversityNext consider another simple application that shows the trade-off between diversity(insurance investment) and specialization (economies of scale). This model relates thefindings of the general model to the loss of species occurring as a result of productiveactivities of the current generation. Again, generation 1 produces at time 1 and consumes attime 2; generation 2 is born at time 2, and produces and consumes at time 2. There is a landarea of size L that is cultivated by the first generation at time 1 and by the second generationat time 2. This area is the habitat of two grain species, rice and wheat. Both grains areperfect substitutes in consumption. One unit of land yields one unit of grain whether it iscultivated with rice or wheat. At time 1, there is no uncertainty about yields. For everyspecies that is cultivated or preserved without cultivation, there is a setup-cost s L/2,expressed in land area that has to be set aside, i.e., for seed development or research. If aspecies is not cultivated, or preserved on an area of size s, it becomes extinct. Withoutconsidering generation 2, generation 1 has no benefits from diversity. It would grow onlyone of the two species and let the other one become extinct.At time 2, generation 2 is born into one of two states. For example due to a pest ora change in climate, wheat does not grow in state A, and rice does not grow in state B. Theseeds for the species that was cultivated by generation 1 are passed on to generation 2 forfree. However, if generation 1 cultivated only the crop that does not grow in the state intowhich generation 2 is born, generation 2 would be willing to buy the seeds for the other0.5-1.50^0.1^0.2^0.3^0.4Cost of conservation, sConservation of One Species Only Conservation of Both SpeciesSpecies Conservation - Competitive SolutionUI0.50-0.5-190Figure 2.7 The Conservation of Species and Welfare of Generation 1species from generation 1 at any price below L. Anticipating this at time 1, generation 1compares its welfare, U1, between conserving one and two species, considering the cost ofpreserving the second species, s. Both states are equally probable; both generations have log-utility and consume grain only. Let g denote the number of species preserved (g E {1,2}).Then, generation 1 faces the problem:max U1 = 0.5 Log[L - g + 0.5 Log[g (L - s)]^(50)ge{1,2)With L = 1, generation 1 would conserve the second species in the competitive solution onlyif s < 1/3 as shown in Figure 2.7. However, if ex-ante contracting was allowed, generation912 would commit itself to paying up to L in both states for the conservation of the secondspecies in order to avoid starvation in the bad state.To see the Pareto improvement resulting from inter-generational coordination,consider the following numerical example. With L=1, s=1/3+ E and 7r =0.5, generation 1will grow only one species and receive utility 111 = Log[L-s] = -0.405. Generation 2 wouldstarve in the bad state and hence receive utility U2 = 0.5.Log[L]+0.5.Log[0] = - OD . Withan agreed price for the conservation of the second species between 1/3 and 1 in both states,both generations could be made better off. Assume a price for conservation of 0.4. Now, U1= Log[L-2s + 0.4] = -0.310 and U2 = Log[L-0.4] = -0.511, which implies a Paretoimprovement over the competitive solution. Hence, the incompleteness of inter-generationalmarkets would have led to the inefficient extinction of one of the two species in thecompetitive solution. The extinction of a species is an irreversible decision that is associatedwith the loss of an option value (see Arrow and Fisher 1974 and Pindyck 1991). This optionvalue is implicit in the above maximization problem of generation 1 and is lower underincomplete markets than under an efficient coordination mechanism.3.4 Consumption of a Non-Renewable ResourceIn this application, the implications of incomplete inter-generational insurance marketsfor the depletion of a non-renewable resource are analyzed for the case of uncertainty aboutthe resource stock. Consider the simplest possible model with two generations, 1 and 2, twostates, A and B, and a single consumption good, which is the non-renewable resource. Thereis no production in this economy. Generation 1 lives and consumes at times 1 and 2;generation 2 lives and consumes at time 2 only. The endowments of both generations dependon the state that is revealed at time 2. These endowments are denoted elA, eIB, e2A and elB.At time 1, generation 1 may consume up to the minimum of eiA and em (en min[e1A,e1B]).The competitive solution does not include any trade opportunities since within each statethere is only one good, and trade across states is impossible since generation 2 is born into92one of the two states. Hence, generation 2 simply consumes its endowment in the respectivestate. Generation 1 chooses its consumption at time 1 by maximizing its utility, U1:°MX U1 = Ull[C11] + 1r 1 i 12[e m - c ii] + (1 -7t)u de 1- c11]^(51)C H(with u' > 0 and u" < 0) resulting in the first order condition:^u/ll = rcu/iA+(l-n)u/113^ (52)Clearly, the competitive solution is inefficient if the resulting risks of both generations differ,that is if:^ * ^um[eL4- c ]^u2A[e 2A]1 1(53)u ta [ e 113 -C1*1]^u5[e28]where c*ii is the utility maximizing consumption of the resource at time 1 in the competitivesolution.To find the efficient solution, hypothetical ex-ante trade across states is introduced.Both generations now maximize their respective utilities by choice of trade across states. Lett denote the amount of the resource sold (bought) by generation 1 (generation 2) in state Bagainst payment of resources in state A. Let p denote the price of the resource in state B interms of the resource in state A. Utilities in the coordinated solution are denoted by v. Thenthe maximization problems are:max V1 = v ii[c ill 4. It vi2,[e 1,4 -C11 -pi + (1 --7c) v de 1,9 - c11 + t]cipt^ (54)max V2 = n v2[e2A +pt] + (1 -n)v2[e2B-t]tAfter p is eliminated from the resulting three first order conditions, the coordinated solutionis determined by the following two equations:93v/t4^v/2Avi2B^(55)v/11 = rc v/IA + (1 - 7c) v/IBNow, two questions can be analyzed. First, given any choice of c11, how does the sharingof resources at time 2 change between the competitive and the coordinated solution; andsecond, how does c11 compare between both solutions. The first question is answered by thefirst equation in (55). If the competitive solution leads to different risks across generations,(53), trade would equalize these risks. Hence, if generation 2 faces larger endowment riskthan generation 1, generation 1 would take on some of this risk in the coordinated solution,and vice versa.The second question of the effect of trade on cl, is more complex and addressesinefficiencies in the aggregate resource depletion path that may arise due to the marketincompleteness. Recall that, in the insurance model in section 2.3, it was possible forgeneration 1 to simultaneously increase insurance investment and increase consumption attime 1. Therefore, unambiguous under-insurance results could be obtained. In the resourcedepletion case, generation 1 cannot invest for one state only. It can only postpone resourceconsumption (reduce c11), which increases its wealth in both states equally. Reducing c11 hasan insurance effect since leaving more of the resource behind reduces aggregate uncertaintyat time 2. However, since trade at time 2 makes generation 1 better off, there is anadditional income effect that would tend to increase ci, if consumption at time 1 is a normalgood. Since the insurance and income effect are working in opposite directions, the aggregateimpact of opening trade on c11 is ambiguous. Because of the importance of understanding theimpact of incomplete markets on the resource depletion path, the aggregate effect will beexplored in more detail in this section.For the following analysis, 7 = 0.5 is assumed to simplify notation. A is consideredthe bad state (elA < eIB). Also, in the competitive solution, generation 2 faces higheruncertainty than generation 1 (u'i,/u'IB < u'2A/1.1'2,3). These assumptions would reflect a94realistic setting in which the current generations faces some endowment uncertainty aboutthe resource stock, and future generations face higher uncertainty in the same direction.Hence, if trade between both generations across states was allowed, generation 1 would sellresource in the bad state, A, for resource in the good state, B. First, there is the price, orinsurance, effect of allowing trade across states on cu. Let p be the price of the resource instate B in terms of the resource in state A. Then, the relative price of c11 is the sum of theresource prices in both states, divided by the value of generation l's endowment,(1 +p)/(em+p.em). Trade leads to a price, p, that is lower than the price implicit in the no-trade solution. With eiA < eiB, the relative price of consumption at time 1 increases with adecrease in p. Hence, trade increases the relative price of consumption at time 1, and theprice, or insurance, effect of allowing trade would lead to a reduction of cu. Second,however, trade leads to higher welfare. The resulting income effect would tend to increasecll. The net effect of opening markets on c11 depends on the curvature of the utility function.The illustration in Figure 2.8 shows generation I 's indifference map for resourceconsumption at time 2 with the optimal resource consumption in states A and B, c*IA andc*IB, for the competitive solution assumed to be at point E (c*IA = 1, c*iB = 2). Theimplicit price at this no-trade equilibrium is determined by the ratio of marginal utilities(shown by the price line p = 0.5). If markets across states were opened, an explicit pricewould be formed below p = 0.5 (since generation 2's resource endowment is assumed to bemore risky than that of generation 1). Generation 1 would trade from point E along the newprice line to the new utility maximizing consumption point, which is the tangent of the newprice line through E with an indifference curve. Two such points are shown as F and G. Theoffer curve shown in Figure 2.8 is the line connecting the utility maximizing consumptionchoices of generation 1 for all possible equilibrium prices. The impact of trade on cii isdetermined by the change of expected marginal utility of resource consumption at time 2 thatoccurs because of allowing trade across states (compare equations (52) and (55)). If, for agiven c11, trade leads to a rise in expected marginal utility from consumption at time 2, cllwill decline, and vice versa. Hence the expected marginal utility at the competitive solution(point E) needs to be compared to the trade equilibrium (on the offer curve). Note that the95Indiff (U-1.0) Indiff (U-t1) Indiff (U-1.2) P-0.5Figure 2.8 Offer Curve96trade equilibrium is not necessarily the coordinated solution since c11 may change in thecompetitive solution. The result of the analysis is summarized inProposition 3.3: If generation 1 has increasing or constant absolute risk aversion,and generation 2 faces higher endowment uncertainty thangeneration 1, incompleteness of intergenerational insurancemarkets leads to inefficiently low resource consumption at time 1.If generation 1 has decreasing absolute risk aversion, resourceconsumption at time 1 can be inefficiently high.Proof: Let c*, and c*18 denote consumption in the competitive solution and CIA and cmconsumption choices if trade across generation was allowed, given choice of consumptionat time 1 in the competitive solution, c*n. Then, c, < c*„ < c*18 < cm. Also CIA =c*,-p.t and ci8 = c*18-Ft. Hence:cm - cm(56)C1B- C1BFirst consider increasing or constant absolute risk aversion of generation 1:dull(c)With ciA < cm, we can write:ui(c) )0^V c(57)dcuil(cm)-ull(c1B)^U 1(C IA) ul(ciA) (58)ul(cm) ukC1B)^U 1B) U l(C1B)Utility maximization by generation 1 implies:97ul(cm) _ 1^(59)ulfrid^PUsing this and (56):u "(cm)^ciB -ci*B^ s ^ .... u "(c )(c - c )^u (60)iiu (cm)^cm* -cm^IA IA IA lifrld(C1B - c1B)With u" <0 and CIA < c*iA c< *1B < CIB:u 1(cm* )-ui(cm) > u "(cm) (cm* -cm)u /I(c1,9) (c1B-ciB**) > u '(cIB) -14 1(c1B)Combining (60) and (61), therefore:ul(cm* )+ul(ci*B) > ul(cm) + ul(c idSince trade leads to a lower expected marginal utility from consumption at time 2, theefficient coordinated solution implies higher consumption at time 1 than the competitivesolution: c11** > cil*, where c11** is the (efficient) resource consumption at time 1 in thecoordinated solution.Next consider decreasing absolute risk aversion. In addition, an assumption is madethat the decrease in absolute risk aversion is large compared to the price change broughtabout by trade. This assumption is:_ u^"(c,`,4)^u/(c,')ukc) > ui(c)^ (63)u"0,)^P_^ukc,j,)(61)(62)Note that the RHS of (63) is the implicit price of the resource in state B in the competitiveno-trade solution divided by p, the price with trade. Since the price is lower with trade, the98RHS is greater than one. The LHS is greater than one because of decreasing absolute riskaversion. However, the additional assumption is made that the LHS is greater than the RHS.Rearranging (63), and using (56):u "(ci*,,)^- c,*„ *^*^ / *u (cm)(cm - cu) < u I (cL)(cul ciB)With u" <0 and cm < c*m < els <u i(cm* )-u/(cm) < u "(cm* )(cm -cm)u ll(43)(cii3 -c i*B) < u V18) - u V i*B)Combing (64) and (65), therefore:u VIA* ) u /(Ci*B) < u L4) uSince trade leads to a higher expected marginal utility from consumption at time 2, theefficient coordinated solution implies lower consumption at time 1 than the competitivesolution: c11** < c11*. There is inefficient over-consumption in the competitive solution. •Now consider examples for both cases (over- and under-consumption of resources attime 1). In order to analyze the change in expected marginal utility between the no-tradeequilibrium, E, and another point on the offer curve, the iso-expected marginal utility(IEMU) curve is introduced. This is a curve connecting all points in the consumption spaceof generation 1 at time 2 that result in the same expected marginal utility as point E. Theposition of the IEMU curve depends on whether the utility function shows decreasing,constant, or increasing absolute risk aversion. In the simplest case of constant absolute riskaversion (CARA: d(-u' '/u')/dc = 0, i.e., exponential utility), the IEMU curve lies exactlyon the indifference curve through E since any change in utility is accompanied by aproportional change in marginal utility. Since the offer curve is tangent to the indifferencecurve in E and lies above the indifference curve anywhere else (voluntary trade must makeu"(c ^cm* -cm(64)(65)(66)'•-•.. .--^... ,.- ..^ .., s'... ........''...- . .,. ............^ ...'•:-°-....^1111111111^1^1^I^1^,^Ii^1^i^II^I^1^1^i^1^1^1^I^1^I^1^1^1^I^1^17)-T111111111111111111 ^2.5 -_•,,21.51.••1111..1'1'11111111199generation 1 better oft), trade must results in a point above the IEMU curve. This implieslower IEMU from consumption at time 2 and, hence, higher c11 than in the competitivesolution.Quadratic Utility (IARA)Cb1^1.5 2CaOffer Curve Indifference Curve IEMU Curve Certainty (Ca-Cb)iaraFigure 2.9 Resource Depletion with Quadratic UtilityAs an example of increasing absolute risk aversion (IARA: d(-u' '/u')/dc > 0),quadratic utility is shown in Figure 2.9. Moving along an indifference curve away from thecertainty line, expected marginal utility increases with IARA utility; hence, the IEMU curveis flatter than the indifference curve when CB > cA and steeper when CB < cA. Here, to theleft of the competitive solution, E, in Figure 2.9, the IEMU curve must lie below theindifference curve and the offer curve above the indifference curve. Hence, whenever100generation 2 faces higher endowment risk than generation 1, generation 1 will inefficientlyunder-consume the resource at time 1. As can be seen from Figure 2.9, the impact of tradeon c11 is ambiguous if generation 1 faces higher endowment uncertainty than generation 2(to the right of E).Log Utility (DARA)Cb3.53_ \2.5•••••••••2.••••••••••••••••••••••.••••• ••••• ••••• ••••••••••••• •••••1.5 •••••••.••••• ••••••• ••••••1 .••••• .•••• ..•••• • ••• • :.•••;^------^•_0.5 111111111111111 111111111111111111'111111111 111111 111111111111111)11111111^1.5^2CaOffer Curve Indifference Curve IEMU Curve Certainty (Ca-Cb)daraFigure 2.10 Resource Depletion with Log-UtilityDecreasing absolute risk aversion (DARA) is intuitively more appealing.2 Anexample of DARA (log-utility) is shown in Figure 2.10. With DARA utility, the IEMU2 Arrow (1970) shows that decreasing absolute risk aversion implies that the risky assetis a normal good, i.e., the demand for risky assets increases with an increase in anindividual's wealth.Competitive Solution^Coordinated SolutionCii - 0.764CIA - 3.2362.000U1=1.606U: - 1.386Cie 5.236Ceas '■ 8.000Cii - 0.576CIA - 2.808.• 2.616U1 1.630U: -1415ois 6.950C29 6.475Endowments: lA = 4,e ID -6, e -2, e -8101curve is steeper than the indifference curve when CB > CA and flatter when CB < CA. Sincethe offer curve is tangent to the indifference curve in E, the offer curve must lie beneath theIEMU curve at least for some range to the left of E. Hence, in this area, incompleteness ofmarkets leads to excessive resource consumption at time 1.Figure 2.11 Over-Consumption of a Non-renewable ResourceA numerical example will illustrate the over-consumption problem resulting fromhigher endowment risk of the second generation under log-utility. Assume utility functions:33 Consumption at time 1 enters the utility function linearly in order to obtain a simpleexplicit solution. This functional form is not required for obtaining the over-consumption result.102=^+ 0.5 Log[c m] + 0.5 Log[c^(67)U2 = 0.5 Log[c2A] ÷a5L4g[c2B]and endowments eiA = 4, ell3 = 6, e2A = 2, e.213 = 8. In the competitive solution c*11 =0.764. This results in u'iA/u' ig = 1.62 < u '2A/u $2B = 2. Since the risk faced by generation2 is greater, this implies over-consumption. Consumption in the coordinated solution wouldonly be c**11 = 0.576. The over-consumption in the competitive solution and the Paretoimprovement through the coordinated solution are confirmed by the equilibrium values forthis example shown in Figure 2.11. The intuition for the dependence of the over-consumption result on decreasing risk aversion is that generation 1 assumes a risk by notconsuming the resource at time 1 since the value of the resource at time 2 is uncertain. Sincetrade increases generation 1 's wealth (in utility terms), decreasing risk aversion leads to ahigher willingness to take on the risk of postponing consumption, and hence lower resourceconsumption at time 1 in the trade case (coordinated solution).4^ConclusionsThis chapter has shown that the incompleteness of intergenerational insurance marketsconstitutes a market failure that leads to inefficient intergenerational investment decisionsunder risk. Early generations over-diversify if they face risks that are larger than those ofthe following generation and could, therefore, be shared across generations. On the otherhand, if risks are increasing from generation to generation, the current generation wouldunder-insure against those risks. Furthermore, a generation with decreasing absolute riskaversion would, in many cases, over-consume a natural resource if the stock uncertainty islarger for future generations. The main message of this chapter is caution against the beliefthat markets work efficiently to transmit across generations the right signals for economicdecisions under risk. The larger the social risks involved, the more reason is there to believethat markets do not bring about efficient decision making by the current generation. Thedirection of the inefficiency, however, depends on the nature of the risk assumed. Therefore,103the policy implications that can be drawn from this chapter depend on the empiricalassessment of the risks that the current and future generations are facing.For many types of natural capital depletion (i.e., resource depletion, reduction ofbiodiversity, large-scale emissions), short-term effects are minor compared to possible, butuncertain, future effects. Hence, natural capital depletion would generate risks that fit thepattern of those risks which would be inefficiently under-insured by market forces.Therefore, the effect of under-insurance of future risks and over-consumption of resourcesappears to be empirically more relevant for environmental problems than the over-diversification result. Many global environmental issues involve large-scale and long-termuncertainties that would lead to under-insurance. The uncertain costs of global warming,ozone layer depletion and land degradation would be expected to increase with time and,hence, fall into the category of risks that would be under-insured through competitivemarkets. With respect to global warming, note that a reduction in the release of greenhousegases itself does not fall into this category since such an insurance policy impacts on futuregenerations through an intergenerational externality and not through markets. Greenhousegas emissions are inefficiently high due to the external costs such emissions create. However,this chapter shows that there is not only an inefficiently high incentive to release greenhousegases but also an inefficiently low incentive to undertake protective measures against theconsequences of the greenhouse effect, even if such insurance could be sold to futuregenerations.Another application with policy relevance is the preservation of biodiversity. Throughthe required preservation of habitat, it is costly to conserve a species. If there are no or onlysmall expected benefits from a species during the lifetime of the current generation but largeruncertain benefits to future generations, current generations would have an inefficiently lowincentive to make the investments required to conserve biodiversity. Similarly, long-terminvestment in back-stop technologies for the provision of energy (i.e. solar energytechnologies) would be inefficiently low. Many other examples would involve irreversible1 04activities (such as land conversion) that generate uncertain opportunity costs that are expectedto arise only after the lifetime of the current generation.The results of this chapter suggest that in all these situations, an intergenerationalcoordination mechanism could lead to gains for all generations. Such intergenerationalcoordination mechanism would link the relative risks of generations, and would require latergenerations to compensate earlier generations for insurance investments even if the bad statedoes not occur and insurance is worthless, ex-post. A possible coordination mechanism couldinvolve debt contracts that create obligations for future generations to pay for insuranceinvestments of earlier generations even if the investment turns out to be worthless in theparticular state that is realized. A sustainability constraint on the activities of the currentgeneration, as an alternative coordination mechanism, will be discussed in chapter IV. Asustainability constraint would commit the current generation to ensure consumptionpossibilities of future generations equal to the current generation's consumption. Asustainability constraint would require higher provisions for future generations if they areborn into a bad state of the world, and vice versa, and, thus, link relative risks acrossgenerations.CHAPTER HIENVIRONMENTAL DECISION MAKINGUNDER AMBIGUITY AND IGNORANCE1^IntroductionThe standard model for decision making under uncertainty is the expected utilitymodel, either with objective probabilities (von Neumann and Morgenstern 1947) or extendedto subjective probabilities (Savage 1954) over states of the world. In expected utility theory,uncertainty is modeled as risk. Risk is defined as a situation of uncertainty with a singleprobability distribution over a well defined set of states of the world. Hence, both approachesto expected utility require that all possible states of the world and their consequences underany of the alternative actions be known. Also, expected utility theory requires that additiveprobabilities can be assigned to all states. Ambiguity is a situation in which it is impossibleto assign a single probability distribution to all states of the world. Ignorance is a situationsin which it is impossible to describe all states of the world completely. This chapter ismotivated by the belief that, by abstracting from ambiguity and ignorance, subjectiveexpected utility theory (SEU) assumes away critical aspects of many environmental decisionmaking problems. It is argued that, since SEU is not applicable to situations of ambiguityand ignorance, it has significant limitations as a normative model for environmental decisionmaking.Environmental decision making involves choices about the depletion of natural capital,i.e., choices about resource depletion, emissions, land use, and other issues. Many of thesechoices involve large uncertainties that are not well captured by modeling them as risk.Ambiguity and ignorance are of relevance in many environmental problems, such as globalwarming, ozone layer depletion and the reduction of biodiversity, which are large in scale105106and could potentially lead to significant changes in social welfare. Ignorance and ambiguityare important in environmental decision making because of our incomplete knowledge of thefunctioning of natural systems, the long lag between the time a decision is taken and the timeall impacts will be felt, and the fact that many environmental effects are external to theagents making the decisions, which reduces their economic incentives for research into theconsequences of their activities.In addition to the conceptual concerns about using SEU for environmental decisionmaking, there are practical reasons for the desire to find a more suitable model for decisionmaking under uncertainty. Deficiencies in the practical evaluation of uncertain costs andbenefits in the context of environmental decision making are widespread. The treatment ofuncertain environmental costs and benefits in the cost-benefit analyses of several recentWorld Bank projects is examined in von Amsberg (1993). This examination shows asurprising lack of sophistication in the analysis of uncertainty. In fact, uncertainenvironmental costs or benefits are often excluded from the analysis completely. Inparticular, the costs of resource depletion, land use, habitat destruction and emissions arefrequently ignored, sometimes with the explicit justification of excessive uncertainty aboutthese costs. For example, none of the analyzed energy projects attempts to quantify thepossible damages resulting from global warming and carbon dioxide emissions. Suchtreatment of uncertainty can clearly lead to a systematic bias in the evaluation of projects atthe cost of excessive environmental destruction. One reason for these deficiencies can befound in the absence of widely accepted models that allow the representation of ambiguity,or vague uncertainties. The present chapter is motivated by these deficiencies and theimportance of providing environmental policy makers with more applicable tools for theevaluation of uncertain environmental costs and benefits.The focus of this chapter is the review of several models of decision making underuncertainty and the proposal of a model that is more suitable for normative environmentaldecision making than SEU. The focus of the chapter is not the development of a new theory.Rather, the purpose is to analyze existing models and combine elements of those models in107order to provide a decision theory that is of practical use for environmental decision making.Section 2 will introduce the formal notation of uncertainty and discuss the requirements fora useful normative model for environmental decision making. Section 3 will review andanalyze several existing decision theories. It begins with an analysis of the conceptual andpractical problems of using SEU and reviews several relevant alternative models of decisionmaking under ambiguity and ignorance. Section 4 suggests a decision making model basedon a combination of the Dempster-Shafer belief-function theory and Choquet expected utilityas the most suitable approach from a policy perspective. The proposed theory is illustratedby means of an example. The implications of using this model for environmental decisionmaking are discussed. In order to focus the discussion on the problems of uncertainty inenvironmental decision making, this chapter abstracts from other problems of evaluatingenvironmental costs and, such as the need to shadow price external effects, to take intoaccount intergenerational equity effects and to choose the appropriate social discount rate forfuture costs and benefits.2^Environmental Decision Making: Models and RealityMany environmental decision making problems are highly complex. Therefore, inorder to analyze a problem with limited cognitive capacities, we have to build a model ofthe decision making situation, which we then analyze. A model is an abstraction from realityand is supposed to reduce the analysis to those aspects of reality that are relevant for a givenproblem. Of course, what aspects of reality are relevant for a specific problem depends onthe nature of that problem. When a new class of problems arises it is useful to examinewhether the models used for other classes of problems are still the most useful abstractionof reality for the given purpose. Subjective expected utility theory (SEU) is so widely usedas a tool for decision making under uncertainty that it is often forgotten that the elements ofthis theory are only an abstraction of reality and not reality itself. The analysis in this chapteris driven by concerns that SEU may not be a very useful model for large-scale environmentaldecision making. The present section lays the groundwork for analyzing this question. First,common elements and notations for decision theories will be introduced. Second, some108important aspects of the reality of environmental decision making problems are discussed.Third, a list of requirements for a good model of environmental decision making issuggested.2.1 Building Blocks of Decision TheoryIn this sub-section, some building blocks of theories for decision making underuncertainty are introduced, together with some notation to be used throughout the chapter.The discussed elements are models of beliefs, acts, and values, as well as the basis forderiving normative statements from decision models. Beliefs represent the assessment ofthose aspects of the world that cannot be influenced by the decision maker. Beliefs aremodeled by defining a list of possible states of the world and expressing the likelihood ofoccurrence of these states. There is a set of states of the world, 0 = {col,...(4}. Each stateof the world, co, is a complete description of all relevant aspects of the world. Sometimesit is convenient to consider a partition, f, of O. A partition is a set of disjoint subsets of il,that together exhaust a The elements of a partition are called events. If it is said that anevent occurred, this means that information is received that the true state is one of the statescontained in this element of the partition. There is a measure p on 0 that reflects beliefs inthe likelihood of the states in O. Those aspects of the world that can be influenced by thedecision maker are modeled as a set of acts, A = {a1,...,am}, which describe the objects ofchoice. The acts, a, map the states, co, into a description of outcomes, a(w). Finally, thedecision maker holds values about different outcomes of the world and their likelihood ofoccurrence. These values, or preferences, held by the decision maker are modeled by a valuefunction, u(a(co),p(00), that maps acts and measures into real numbers. This value functioncan be used to scale the acts, a. If values, or preferences, meet some basic consistencyconditions, which are assumed here, they can be expressed as a real-valued, continuousutility function (see Varian 1978, p. 113).Consider the example of a decision about the level of emissions of a toxic chemical.The acts, or objects of choice, would be the possible levels of emissions. The states of the109world would be complete descriptions of the world that determine the possible impacts ofdifferent emission levels, i.e. health impacts, damages to buildings, climatic changes, andso on. The function a(w) describes the impacts of emission level a if state co is the true stateof the world. Finally, the value function, u(a(co),p(c0)), would evaluate the physical impactsof different emission levels, and assign to them a level of utility or welfare, which can beused to scale the different emission levels a. The decision maker would chose the level ofemissions, a, for which u(a(co),p()) is highest.The most commonly modeled type of uncertainty is risk. The term risk impliesassumptions in addition to the structure discussed so far. First, the decision maker isassumed to know all states of the world (know all possible complete descriptions of theworld). Second, p is assumed to be a single probability measure, Ira, over the set of states,0, that reflects all relevant information about beliefs in the true state. This means, r(0) mustfulfil the axioms of probability (0 ..2r 1; r(0) = 1; r(A)+ r(B) = r(A UB) V A,B Cwith A 11 B = 0). Concerns about the insufficiency of modeling uncertainty as risk goback at least to Knight (1921) and Keynes (1921). Knight introduced the distinction betweenrisk, when probabilities are known, and uncertainty, when they are not known. Keynesdiscussed the distinction between implications and weight of evidence. He considered theimplications of evidence as leading to a probability judgement and the weight of evidenceleading to a particular degree of confidence in this probability judgement. The distinctionbetween the implications of evidence and the weight of evidence could be related, forexample, to an expert's probability assessment versus the presumed reliability of the expertin maldng this assessment.Extensions of risk can be captured by the terms "ambiguity" and "ignorance".Ambiguity describes a situation in which the decision maker knows all states of the worldbut p does not need to be a single probability measure; i.e., p can be a non-additive measureor a set of additive probabilities measures. Ignorance describes a situations in which thedecision maker does not know some or all relevant states of the world. Consider the set ofwell defined states, 0, as a subset of all states, 0* (SI c (2*). Then 0*\ = {w*1,...,(.0*L}110is the set of states that are not well defined and will be called scenarios. A situation ofignorance is characterized by 0*\ 0. A scenario may consist of an incompletedescription of the world or a state about which no information at all is available. With a real-valued utility function it is implicitly assumed that every state of the world results in a welldefined level of utility expressed in a real number. Even if a state is not completely defined,utility resulting from it is defined. Information about a scenario is assumed to be expressedas the interval of minimum and maximum utility values from this scenario. These utilityvalues are denoted, urnif,(a(w*)) and u„(a(cost)). If no information at all is available abouta scenario, this interval would include the lowest and highest utility values possible, i.e.minus and plus infinity. Finally, complete ignorance describes a situation in which neithera set of well known states, 0, nor a measure, p, exist. Under complete ignorance, each act,a, is associated only with minimum and maximum utility, u(a) and umax(a).The concern of this chapter is how to make good environmental decisions (normativeor prescriptive) rather than how to describe real life environmental decision making (positiveor descriptive). Hence, the question arises how the discussed elements of decision modelscan be combined into a normative theory. The utility of individuals as revealed throughpreferences is generally accepted as the source of value and the basis for normativejudgements in economics and related disciplines (see Sen 1982a). Moreover, preferences areoften assumed to be stable, exogenous to individual choices, and known with adequateprecision (see March 1981). While there are reasons to question these assumptions and theiruse for constructing a normative theory of choice (see Elster 1979, Rhodes 1985, and Sen1987), these concerns are outside the scope of the discussions in this chapter.Individual rationality is defined as behaviour that maximizes individual utility. Ifindividuals acted rationally, their observed choices would maximize their utility. If theirobserved choices maximize their utility, a model of individual choice that performs welldescriptively, will also lead to maximization of utility in sufficiently similar problems andwould, therefore, be able to serve as a normative theory. Hence, the use of a descriptivemodel of choice as a normative theory requires that, first, individuals act rationally and,111second, the situation to which a model is applied normatively is sufficiently similar tosituations for which the model performs well descriptively. The development of decisiontheories from axioms of rational choice can be viewed as an attempt to expand the scope ofa normative theory, particularly, for situations in which one of these two conditions are notmet. For example, if Savage's axioms of rational choice were accepted as a normativefoundation, then SEU would follow and could be used as a normative theory. The problemremains that some axioms with intuitive appeal, such as von Neumann-Morgenstern'sindependence axiom or Savage's sure thing principle, turn out to be systematically andconsciously violated (see for example MacCrimmon and Larsson 1979). This wouldundermine these axioms' validity as foundations for a normative theory if individualbehaviour leading to these violations is nevertheless considered rational.2.2 Some Aspects of the Reality of Environmental Decision MakingThis section attempts to describe several important aspects of the reality ofenvironmental decision making which will lead to the requirements for a good model ofenvironmental decision making discussed in the following section. Environmental decisionmaking involves choices about the depletion or restoration of natural capital. Three importantexamples of such choices are the disposal of waste products, the depletion of naturalresources, and the conversion of wildlands. The disposal of waste products includesemissions into the air or water as well as waste disposal on land. Issues arising fromdecisions about the disposal of waste products include the threat of global climate changesthrough the greenhouse effect and the depletion of the ozone layer through emissions ofchlorofluorocarbons (CFCs). Decisions about the depletion of natural resources involve theconsumption of fossil fuels and minerals, reductions in the stocks of forestry and fisheries,as well as the depletion of top soil resources through unsustainable agricultural practices andresulting erosion. Decisions about the conversion of wildland involve possible climatechanges, the destruction of the habitat of species and subsequent loss of biodiversity, suchas it is occurring at a rapid rate in many forests areas all over the world.112There are two main classes of environmental problems that decision makers arefacing. One class of environmental problems involves a decision on whether to allow aneconomic activity, such as a project, that generates certain benefits to the entity undertakingthe project and uncertain external costs through the depletion of natural capital. For example,the construction of a power plant or a hydroelectric dam generates (relatively) certainbenefits, such as the value of the electricity produced. However, the costs of depletingnatural capital (emitting carbon dioxide, or flooding a rainforest area) are highly uncertainand often occurring as externalities in the distant future. Similarly, the economic benefitsfrom logging a forest or using CFCs are relatively certain while the costs associated withemissions or wildland conversion are highly uncertain. The other class of environmentalproblems requires a decision on whether to undertake a project that restores natural capitalor reduces its depletion. In this class of problems, costs and benefits are a mirror image ofthose in the first class of problems. Here, costs are certain (i.e., the costs of pollutionabatement equipment, or the costs of a reforestation project), and the benefits, which are theavoided costs of natural capital depletion, are highly uncertain. Of course, a practicalsituation may involve uncertainties in costs as well as benefits; however, the describedgeneric structure appears to be typical of many problems in which the environmental costsand benefits are external, pervasive and often occurring at a distant time and are, therefore,more uncertain than internal and immediate costs or benefits.Environmental decision making is often characterized by a high degree of ignoranceabout the costs of natural capital depletion or the benefits of its restoration. There are twomain reasons for this ignorance about relevant states of the world. The first reason is thelimitation of the cognitive capacity of humans compared to the complexity of naturalsystems. Even if we understood the functioning of all natural system, limited cognitivecapabilities would still prevent us from imagining and considering all relevant states of theworld. Beyond limited processing capacities of individuals, social information processing isdeficient as well. For example, inadequate communication between scientists and decisionmakers would further reduce the completeness of state descriptions considered in a decisionsituation.113The second reason for ignorance is the combination of incomplete knowledge ofnatural systems with our capacity to undertake innovative activities whose outcome dependson the functioning of natural systems. Thousands of years of history of the natural sciencescan be interpreted as a learning process about the functioning of natural systems. At everystep, humans have obtained new information about the functioning of natural systems thatwas not available before. Obviously, this learning process is ongoing. So far, there is noindication that the natural sciences are approaching anything like a complete understandingof all natural systems. The continuing process of discovering means that states which arepossible to consider after a discovery is made cannot be considered before the discovery.Even though we do not fully understand natural systems, we know how the environmentresponds to particular activities that were repeated many times in the past. Today, however,the extent and new types of natural capital depletion often introduces unprecedented changes.For these unprecedented forms of natural capital depletion, we have no historic experiencefrom which we could form precise expectations about their outcomes. As a result, we areignorant about possible outcomes to a large degree.There are a wide range of environmental decision making problems for whichignorance is relevant. Examples of the past show how the impacts of environmentaldestruction could not have been considered because they were not yet known. It took manydecades of using and emitting CFCs until the possible destructive effects on the ozone layerwere discovered in the early 1970s. Regulators, faced with the emergence of CFCs, couldnot have considered the depletion of the ozone layer before this effect was first discussed inscientific circles. For a long time, carbon dioxide emissions were considered harmless, untilscientists directed attention toward the possibility of global warming through the greenhouseeffect. The introduction of a new drug or a new chemical is an innovative activity that canlead to relevance of unknown states of the world despite extensive testing, as theThalidomide case demonstrates. There are many more examples for past decisions that weremade not in consideration of these risks but in ignorance about possible states of the world.Similarly, we are ignorant about possible outcomes of unprecedented activities currentlyundertaken. The debate about possible health impacts of electromagnetic fields would be an114example. The uncertainty is clearly not well described as risk. Possible interactions betweenhuman health and different types of electromagnetic fields are not yet known. Therefore,possible states of the world cannot be fully described.In many of the mentioned examples, it is impossible to define all relevant states ofthe world. However, we know that we are ignorant. In fact, in some instances, we may notbe surprised about the occurrence of a scenario that we could not have described beforehand.Often, it is possible to describe a scenario, or a class of similar states that, individually, arenot well defined. For the introduction of a new drug, one possible scenario would includethe discovery of birth defects resulting from the use of the drug by pregnant women. Anotherscenario would include unexpected side-effects or interference with other drugs. For thedestruction of habitat and the reduction of biodiversity, an important scenario would includeall states in which an extinct species could have been used later for the development of a newdrug, even for a disease that is not yet known, or could have provided important geneticmaterial for crop research. For atmospheric emissions, one scenario would include all stateswith unexpected climatic consequences. In each of these cases, it would be impossible todescribe all individual states. However, for a scenario, there may even be historic data thatshows, for example, the relative frequencies of pharmaceutically useful discoveries from aspecies.Since many environmental problems involve innovative activities, ambiguity is ofgreat importance. Probability assessments for innovative activities would be considered lessreliable than those for activities for which there is considerable experience, or even statisticaldata. In the latter case, experts would tend to agree on a probability assessment for aparticular event's occurrence. In the former case, conflicting probability assessments ofexperts would be likely. Consider real-life situations in which all possible outcomes of anact can be well described but no reliable, or only conflicting, information regarding theirprobability is available. The consequences of a large scale accident, such as an oil spill, achemical leak, or an accident in a nuclear reactor, may be well understood (the states are115well defined) but due to limited experience it is not possible to assign reliable probabilitiesto the possible states of the world.Another important aspect of many environmental decision problems is that they arevery different from decision situations in which the behaviour of individuals can be observed.An obvious difference from typical individual decision making is the scale of someenvironmental problems that can involve decisions about future living conditions on the entireplanet. Also, many environmental decisions have a very long time-horizon due to the inertiaof many natural systems. For example, CFCs are estimated to remain in the atmosphere foran average of 65 to 120 years after their emission (see Deutscher Bundestag 1989, p.158).It is said that deforestation undertaken by the Romans in the Mediterranean countries hassignificant negative effects on the climate in North-Africa, even today. Due to their shortlife-span, individuals would not consider such long-term aspects of their decision making.Similarly, most of the uncertain costs in environmental problems arise as externalities thatare excluded from consideration in individual choice. Finally, many environmental problemsare different from typical individual choices, especially those that have been studied in detail,in that they are highly complex and involve knowledge from a large number of differentdisciplines.2.3 Requirements for a Theory of Environmental Decision MakingIn the previous section, some important aspects of real-life environmental decisionproblems were discussed. They included a generic structure in which there is moreuncertainty about the environmental costs than about the benefits of an investment, theprevalence of ignorance and ambiguity, and a significant difference between environmentaldecision making and typically observed individual choices. A theory for environmentaldecision making should be based on a model that reflects and does not abstract from thoseparticularly important aspects of real life environmental decision problems. In this section,several requirements for a good normative theory of environmental decision making underuncertainty are suggested. The requirements for a suitable normative model of environmental116decision making that will be discussed in this section can be summarized by tworequirements for the modeling of beliefs and three additional requirements for the modelingof values. A suitable model of beliefs should, first, allow the representation of ambiguity andignorance, and, second, allow the construction of beliefs from independent pieces ofevidence. This includes the requirement for easy elicitation and aggregation of beliefs fromexperts. A suitable model of values should, first, be consistent with observed behaviour andbe derived from an intuitively appealing set of axioms. Second, the model should be able toclearly separate decisions that are based on a theory with strong normative foundations fromthose based on more speculative aspects of the theory. Third, the choice rule implicit in themodel of values should be robust to possible misspecifications of the model.The discussion in the previous section has emphasized the importance of ambiguityand ignorance in environmental decision making. Therefore, a good decision model forenvironmental problems should allow the representation of beliefs with ambiguity orignorance. Consider different reliabilities of probability assessments as an example ofambiguity. In environmental decision making, there is often incomplete knowledge of someof the relevant natural processes. On the other hand, some aspects of a problem may be verywell understood. Similarly, an analysis may entail very different levels of detail and alikelihood judgement based on a detailed analysis with in-depth understanding of theproblems involved should be distinguished from likelihood judgements based on sketchyunderstanding and "back of the envelope" type analysis. The question whether the reliabilityof probabilities does or should affect a decision will be discussed later. However, there isno reason to a-priori give away the information contained in knowledge of differentreliabilities of probability assessments. In case of ignorance, there are states that are not welldefined. Hence, a model that does not allow for the representation of ignorance, would forcea decision maker to either ignore a scenarios, or make arbitrary assumptions about a scenariothat would complete the description of the world. An important reason for explicitconsideration of ambiguity and ignorance lies in the asymmetry of many environmentalproblems. As discussed above, controlled human-made systems and internal effects are, ingeneral, better understood than uncontrolled natural systems and external effects. Therefore,117there is more ambiguity and ignorance about environmental costs than internal benefits.Hence, the use of a model that suppresses ambiguity and ignorance can introduce asystematic bias in the decision analysis that depends on the evaluation of ambiguity andignorance.Many environmental problems have great complexity compared to the cognitivecapacities of humans. For some complex issues, no single individual may posses all relevantinformation that is available in different domains of knowledge. Because of the complexityof the issues, a wholistical assessment of beliefs may be rather unreliable. For example, thewillingness of individuals to purchase insurance against an environmental threat could beused to infer beliefs about the likelihood of that event. If individuals have to make decisionsthat depend on their assessment of environmental uncertainties, they are unlikely to use allavailable evidence in determining their beliefs; yet, the aggregation of different bodies ofknowledge can generate new conclusions. In a complex situation, individual behaviour islikely to be influence by published opinions or existing decision making models. Therefore,individual betting rates (i.e. the amount of insurance acquired by individuals) would not bean independent guide to a good decision making model. Similarly, the elicitation of wholisticbeliefs from experts would reduce the transparency of the decision making process and openthe door for biases created, for example, by individual incentives of experts.The difficulties in assessing wholistic beliefs suggests that a good theory forenvironmental decision making should focus on the construction of beliefs from individualpieces of evidence (see Shafer 1981). The construction of beliefs from evidence requires thedecision analyst to elicit knowledge or evidence from experts. The representation of beliefsthat is required by the decision making model would influence the way information is elicitedfrom the expert. Information obtained from the expert, in turn, may be biased through theway it is elicited. The more the experts are forced to express beliefs in a form that isunnatural to their thinking, the more it would be expected that framing biases influenceelicited beliefs. Thus, a good model for environmental decision making should accommodatethe natural way of belief representation in the relevant domains of experts and allow the118incorporation of vague beliefs. The latter requirement is, of course, related to the importanceof representing ambiguity and ignorance.The complexity of many environmental problems leads to the need to break down aproblem into different components and assess beliefs over the uncertainty in sub-problems.Therefore, a good decision model would accommodate the aggregation of beliefs that wereobtained from different bodies of evidence in order to assess the uncertainty in sub-problems.Even within a given sub-problem, the belief assessments of experts may diverge. In simpleproblems of risk, it is likely that experts would agree in their belief assessment. Forexample, little disagreement would be expected if different experts analyzed a standardlottery machine and had to express their beliefs about drawing a particular number. On theother hand, the complexity of environmental problems will lead different experts to come todifferent assessments of belief based on the same body of evidence. Therefore, a gooddecision model would also allow the aggregation of different beliefs based on the same bodyof evidence. This aggregation of divergent beliefs is particularly important since manyenvironmental problems involve social and not individual choice due to the prominent roleof externalities.The second part of a decision model is the value function, or choice rule, that scalesacts based on the assessment of the consequences of different states and acts and the beliefsover the states and scenarios. In contrast to beliefs, which have some impersonalinterpretation (a person holding a particular belief would think that another person shouldhold the same belief based on the same body of knowledge), values are an expression ofattitudes and preferences that are personal (a person would not think that another person musthave the same attitude toward risk or uncertainty, even if that person was holding the samebeliefs). Since the objective of environmental decision making is to increase utility ofindividuals, and choices of rational individuals would reveal the preferences from whichutility can be derived, a choice rule should be consistent with observed behaviour toward riskand uncertainty. In order to be able to translate attitudes toward risk and uncertainty that1 19were observed in one situation to a different situation, a choice rule should follow fromintuitively appealing axioms that are consistent with observed behaviour.In the previous section, the conditions were discussed under which a descriptivetheory of choice can serve as a good prescriptive theory. Three reasons will be given herewhy the normative foundation for environmental decisions making is rather weak. First, onewould be comfortable with applying a decision model that performs descriptively well in aparticular domain of problems to a normative decision situation which is sufficiently similar.Due to the scale, the long-term implications, and the externalities, most environmentaldecision situations, however, are not even remotely similar to observable decision makingby individuals. Since there are no observable choices, preferences of individuals over suchproblems are not revealed. There is no necessity for a model that describes preferences wellin one domain to also describe them well in a very different domain. Second, applying anormative theory to a domain in which little empirical observations of behaviour areavailable may be acceptable if the theory is based on intuitively appealing axioms that areconsistent with behaviour in other domains. However, as discussed in a later section, specificaxioms of choice underlying SEU, as the most widely accepted decision making theory, aresystematically violated in situations of ambiguity. Therefore, there is little basis for applyingthese axioms to environmental decision problems that are characterized by a high degree ofambiguity. Third, as discussed before, the complexity of environmental problems is largecompared to human cognitive capacity. Therefore, rational individual behaviour is less likelythan in simpler problems. Therefore, even if individual behaviour could be observed inenvironmental problems, it would not give a strong foundation for normative decisionmaking.The weak foundation for normative decision making, particularly in the presence ofambiguity and ignorance, makes the explicit and transparent evaluation of ambiguity andignorance desirable. Despite many drawbacks, the most widely accepted theory for decisionmaking under risk is expected utility theory. While many other approaches have beensuggested, there exists no such widely accepted theory of decision making under ambiguity120and ignorance. The difficulties with inferring preferences outside the domain of typicalindividual choices makes it unlikely that a generally acceptable normative model forenvironmental decision making under uncertainty is easily developed. Therefore, a goodenvironmental decision theory should separate the treatment of risk from the treatment ofambiguity and ignorance. This would make it possible to distinguish conclusions that can bederived on the basis of expected utility, independent of the normative treatment of ambiguityand surprise, from those conclusions that rely on additional assumptions about the treatmentof ambiguity and surprise. This would allow the isolation of conclusions from an extensionof expected utility theory from potentially more controversial conclusions. Because of itsgeneral appeal, expected utility should be the special case of a more general theory fordecision maldng under uncertainty.The final requirement for a good theory of environmental decision making is basedon the recognition that the specification of uncertainty under ambiguity and ignorance ismore likely to be faulty than under risk. The discussion so far has shown that designing agood model for environmental decision problems is no trivial task. The misspecification ofa model can lead to significant, possibly irreversible consequences, in environmental decisionmaking. This suggests that the choice rule should be robust to specification errors. Thisrequirement of robustness is not to be confused with risk aversion. In many situations, adecision maker will consciously take on some risk. Even if this risk taking leads to anegative outcome, the decision maker would still consider the original decision as correctsince it took the risk explicitly into account. On the other hand, a negative outcome that wasnot considered by the decision maker because of a misspecification of the model would leadto regret. The decision maker would consider the original decision incorrect. The choice rulein a good decision model should, therefore, avoid regret about misspecification throughrobustness.1213^Approaches to Decision Making under UncertaintyThis section reviews several models for decision making under uncertainty and theirsuitability for environmental decision problems. After a brief description of the theories,some discussion of their aptness according to the criteria developed in section 2.3 isprovided. This review is incomplete and focusses on those theories that are either widelyaccepted or appear to have particular potential for the analysis of environmental problems.First, subjective expected utility theory is discussed and criticised as the most widely usedand accepted theory of decision making under uncertainty. Second, theories of decisionmaking under ambiguity are analyzed. These models can be separated into those modifyingthe utility from ambiguous states, those based on a set of probability distributions on thestates, and those based on upper and lower probabilities or belief assessments through somenon-additive measures. Third, some theories of decision making under ignorance arediscussed. This review excludes a large number of models that were developed to addressthe weaknesses of SEU to describe individual behaviour under standard risk, as for exampleshown in the Allais paradox. Many interesting and important theories fall into this category(i.e., prospect theory by Kahneman and Tversky 1979, or anticipated utility theory byQuiggin 1982). However, they are excluded from this review in order to focus the analysison the deficiencies of SEU with respect to the treatment of uncertainty that is not risk.3.1^Subjective Expected Utility TheoryThe most commonly used model for decision making under risk is subjective expectedutility theory (SEU). SEU assumes a measure, 7, over states, co, that is a (additive)probability distribution and a utility function of the form:122U a Eit. u(a()) (1)In order to strengthen the theoretical foundation of this model, axioms of choice have beendeveloped from which the representation of preferences in this particular functional form canbe derived. Two of these axiomatizations were developed by von Neumann and Morgenstern(for objective probabilities) and Savage (for subjective probabilities). SEU is applicable tosituations of risk only and, thus, requires the highest degree of specification of uncertainty.SEU is the model underlying most of the literature on environmental evaluation underuncertainty including evaluation techniques in cost-benefit analysis. In this section, I willargue that the inability of SEU to represent ambiguity and ignorance is a significantlimitation for many environmental decision situations and is likely to bias the evaluation.SEU assumes that all states of the world are well defined and that a uniqueprobability distribution over these states exists. Walley (1991, p.3) calls this the "Bayesiandogma of precision". He discusses this deficiency of SEU in depth and provides referencesfor proponents and critics of SEU. In SEU, neither ambiguity nor ignorance can berepresented. All uncertainty has to be expressed in the form of risk. Two examples shallillustrate the problem of ignoring ambiguity and ignorance. First, consider two differenttennis matches (for this example see Gdrdenfors and Sahlin 1982). The decision maker hasdetailed knowledge of the skills, experiences and strategies of the players in match A and,after careful deliberation, assigns probabilities of one half to each player winning the match.On the other hand, the decision maker has never even heard the names of the players ofmatch B and no information about them is available. In this situation, the decision maker isassumed to also assign probability one half to each player winning the match. SEU wouldbe incapable of distinguishing these two situations with the same utility outcomes and thesame probability assessment but sharply different levels of reliability of these probabilities.This means, SEU would not allow different treatment of these two matches, i.e. differentbetting rates.123Second, consider a situation with two states of the world, in which no informationis available about the probability of those states. For example, a ball is to be drawn from anurn which contains green and red balls in an unknown number and proportion. The decisionmaker is asked to assess the probability of drawing a red ball. For an analysis based onSEU, a "Bayesian non-informative prior distribution" of the balls in the urn would have tobe assumed. Based on the principle of insufficient reason, one sensible prior probabilitydistribution might be one half for each of the two states. However, Arrow and Hurwicz(1972, p.2) remark:A state of nature is a complete description of the world. But how we describethe world is a matter of language, not of fact. Any description can be madefiner by introducing more elements to be described; hence, any state of naturecan be expressed as a union of more elementary states of nature.The state green could be divided by more detailed description into the states dark green andlight green. Would this consideration change the prior probability assessment for green andred to two-thirds and one-third, respectively? Then, what happens if the state red was dividedinto more states by more detailed description? What about dividing the state dark green evenfurther? Clearly a non-informative prior must be based on some well defined set ofelementary state of nature. Since the partition of states is a matter of description and not offact, using a non-informative prior makes a decision arbitrarily dependent on the descriptionof states.These concerns apply not only to the assumed uniform distribution over two statesbut to all non-informative prior distributions. In the urn example, one could seek anotherbasis for assigning a non-informative prior distribution. One could assume a uniform priordistribution across the spectrum of colours. This would make probability assignmentsarbitrarily dependent on the measurement of colours. Measuring the light frequency ofcolours would result in a different distribution than measuring heat absorption or hue.Alternatively, probability could be distributed uniformly over the range of outcomes in terms124of utility. Then, the probability assessments for drawing different colours would depend onan individual's utility function, which also does not appear appealing. The simple conclusionis that a precise probability distribution cannot express lack of precision. If it was possibleto define elementary, equi-probable states of the world, there would be no need to expressimprecision. However, in real-life decision making, there is no pre-defined set of elementarystates, and, therefore, a specific prior distribution has to be assumed (by definition ofambiguity it cannot be obtained from information about the problem) and introducesarbitrariness in the decision making process.SEU assumes the existence of a set of well defined states of the world. If SEU is usedin a situation of ignorance, problems similar to those discussed for ambiguity arise. In thepresence of ignorance, there are states that cannot be fully described and cannot beassociated with a single utility outcome under every act. In order to provide the informationrequired by SEU, some assumption would have to be made about an average utility valuefor a scenario under a given act. Again, SEU would force the decision maker to introducean arbitrary assumption. (The assumption is arbitrary since, by definition of ignorance, it isnot based on available information.)The inability of SEU to represent ambiguity and ignorance would restrict theelicitation of beliefs from experts. The requirement of a precise probability distributionsuppresses all available information that cannot be expressed in a single probabilitydistribution. Also, the elicitation of ambiguous beliefs would be prone to framing biases.Beliefs that are forced into single probability distribution over a fixed list of states may bemore biased by uncontrollable aspects of perception or description of these states comparedto elicitation that leaves more room for expressing vague beliefs. The arbitrary assumptionsthat experts or decision makers have to make to obtain precision could lead to systematicbiases, for example, induced by a more detailed description of particular possibilities, orsimply an implicitly higher weight on well understood states. While these specific biases arehypothetical, the requirement of precision requires implicit and uncontrollable assumptionsthat would tend to generate some bias.Balls^30^60Act^Red(R)^White(W) Yellow(Y)f^$1000^$0^$0g $0^$1000^$0h $1000^$0^$1000i^$0^$1000^$1000with Savage's axioms:f > g implies p(R) > p(W);h < i implies p(R) < p(W).Table 3.1 The Ellsberg Paradox125By assuming a probabilitydistribution over a well-defined set ofstates, SEU assumes away thedifference between choice problemswith probability assessments ofdifferent reliability. A normativetheory that is oblivious toward thepossibility of uncertainty aboutprobabilities could be justified if therewas empirical evidence that thereliability of probability assessments does not matter. To the contrary, there is considerableempirical evidence that ambiguity does matter in individual choice (see for exampleMacCrimmon 1968, MacCrimmon and Larsson 1970, and the review in Camerer and Weber1992), most of which is related to the Ellsberg paradox (Ellsberg 1961). One version of theEllsberg Paradox is shown in Table 3.1. An urn contains 30 red balls and 60 white or yellowballs. Individuals are offered a choice between two bets whose payoffs depend on the colourof one ball that will be drawn from the urn. Most individuals prefer bet f to bet g (whichimplies with Savage's axioms p(R) > p(W)) and prefer bet h to bet i (which implies p(R)< p(W)), where p(R) and p(W) denote the assessed probabilities of drawing a red or whiteball, respectively. Hence, individuals are unable to assign consistent probabilities to theevents R and W. Here, individuals show systematic preference for unambiguous events, thatis they demonstrate ambiguity aversion. A large part of the literature on decision makingunder ambiguity deals with solutions to the Ellsberg paradox.The contradicting probability assessments in the Ellsberg paradox follow fromSavage's "sure-thing principle". A few of the approaches that were taken to weaken orreplace Savage's sure thing principle are summarized in section 3.2. Defenders of SEUsometimes point out that, if individuals held beliefs that do not conform to the axioms ofprobability, it would be possible to construct bets that would be accepted by these individualsand would generate a sure loss (this is the so called "Dutch-book argument"). The Dutch-126book argument, however, is based on the assumption that an individual would always accepteither one or the other side of any bet. The argument does not hold, if an individual wouldnot be willing to accept either side of an ambiguous bet. The latter behaviour, would appearquite reasonable and invalidates the Dutch-book argument (see Gdrdenfors and Sahlin 1982).Following from the empirical analysis of choice under ambiguity, SEU does not welldescribe preferences in situations of ambiguity. Since individual preferences are consideredthe basis for normative decision making, the Ellsberg paradox and related evidence wouldundermine the justification for using SEU as a normative model in situations involvingambiguity. It was suggested above that the environmental costs of many activities are highlyambiguous while the benefits are less ambiguous. If ambiguity aversion was part ofindividual preferences, modeling environmental choices with SEU would then systematicallybias decisions against environmental conservation. The deficiencies of SEU, particularly asa normative theory, and the absence of a generally accepted alternative normative model,underline the importance of making transparent the mechanisms used for deciding amongambiguous alternatives.3.2 Decision Making under AmbiguityTheories of decision making under ambiguity are mainly motivated by attempts torationalize the Ellsberg paradox and find a descriptive model that is consistent with theempirical evidence regarding individual choice under ambiguity. Camerer and Weber (1992)have recently surveyed this literature. Following Camerer and Weber, the models proposedin the literature on decision making under ambiguity can be roughly divided into three majorapproaches. The first approach is based on modifying the utility obtained from ambiguousversus unambiguous states. The second approach retains the assumption of state independentutility and a probability measure over the set of states. However, instead of a singleprobability distribution, there is a set of probability distributions, which are related througheither a second-order probability distribution or some measure of reliability. The thirdapproach is based on expressing beliefs about the likelihood of states by a measure that does127not fulfil the axioms of probability, such as probability intervals or non-additive probabilities.This section reviews a few of the decision models and discusses their suitability as normativemodels for environmental decision making.Before entering the discussion of ambiguity models, it is important to note that inmodels which are not based on a single second-order probability measure (or a singleprobability measure of some higher order), some familiar statistical statements becomemeaningless. Without a single probability measure, there is no longer an expected value ofthe utility resulting from beliefs. Hence, if one questions the existence of a single probabilitymeasure, one would have to question the meaning of a statement that, for example, comparesthe expected cost of global warming with the costs of mitigating measures. Also statementsabout ambiguity aversion or ignorance aversion become less meaningful. With a singlesecond order probability measure, ambiguity aversion would be shown in preference for abet with a precise probability 7 over a bet with risky probability with expected value T. Thisstatement becomes meaningless if an expected value for 7 is not defined. It would bepossible to speak of pessimism or optimism about ambiguity only if all decision weight isput on the lowest or highest outcome. No other decision weights would warrant a label suchas ambiguity-averse or ambiguity-seeking.Some models for decision making under ambiguity are based on a modification ofutility from ambiguous events rather than modification of the probability weights on theseevents (see Sarin and Winkler 1992 for an example). For such a theory to be useful, somestructure would have to be imposed on the modifications of utility due to ambiguity. WhileSarin and Wakker impose such a structure for a simple Ellsberg-type situation, there is noobvious route for extending their approach to more complex situations of ambiguity as itwould prevail in many environmental problems. Without a structure that would allow theconstruction of ambiguous beliefs from evidence, this approach would be of limited valuefor environmental decision making. Moreover, the modification of utilities for therepresentation of ambiguity blurs the distinction between the representation of beliefs andvalues. This distinction, however, appears to be useful in order to increase the transparency128of the decision making process in face of the weak normative foundation for decision makingunder ambiguity.The simplest ambiguity models based on modification of the representation of beliefsare those with a single second-order probability distribution. This amounts to a descriptionof a well-defined two stage lottery. To reflect ambiguity aversion, some of these models usenon-linear weights on probabilities. While such models are able to explain empiricalanomalies of second-order risk aversion, their normative applicability is questionable. Thesame real life situation could be described by different decompositions of risk which wouldthen result in different ranking of acts. For example, a simple lottery could also be describedas a two-stage lottery: a first lottery that results in a particular position of the balls throughmixing; and a second, biased, lottery, in which the winner is determined by drawing a ballgiven the position of balls determined in the first stage. Thus, different descriptions of thesame lottery could lead to different rankings of acts. A single second-order probabilitydistribution avoids the real problem of ambiguity. It is applicable only to situations in whichuncertainty is completely specified, albeit somewhat more complex than under risk.Therefore, a model with unique higher order probabilities does not help address thefundamental question how to decide in the absence of a complete specification of risk.An example of a decision theory based on a set of probability distributions withoutassuming the existence of a second order probability distribution is Gdrdenfors and Sahlin(1982). In their model, every probability distribution in the set of possible distributions isassociated with a level of "epistemic reliability". In a first step, probability measures withan unsatisfactory level of epistemic reliability are eliminated. In a second step, every act isevaluated with the one probability measure that leads to the lowest expected utility amongall probability measures that remain after the elimination process in step one. The act withthe largest minimal expected utility is chosen. This model clearly represents the distinctionbetween implications and weight of evidence. It allows representation of incompleteknowledge by introducing several distinct probability measures. Yet, evidence would needto be represented in form of probability measures, which, following the discussion of SEU,129is an unnatural way to represent incomplete knowledge. Without a clear interpretation of thesatisfactory level of epistemic reliability, the elimination rule introduces a somewhat arbitraryelement in the decision malcing process. The aggregation of different beliefs is given througha maximin rule on admitted probability measures. This does not allow a distinction betweendifferent degrees of reliability of admitted probability measures or a combination of theinformation contained in different measures.There is a large number of decision theories that are based on some notion of upperand lower probabilities or probability intervals with different interpretations. The essentialconstruct to represent beliefs in these models is the lower probability, n(o), of a well definedstate, co, or a set of states. The simplest type of lower and upper probabilities are minimumand maximum probabilities. In a lottery example, the minimum probability for drawing a ballof a particular colour could be determined from an incomplete specification of the lottery.That is, observing that one out of three balls in an urn is red (without knowing the colourof the others) allows assigning a minimum probability of one third and a maximumprobability of one to the state red. In a more complex real life situation, lower probabilitywould be some lower level of subjective belief in the occurrence of a particular state.Like proper probabilities (that fulfil the axioms of probability), lower probabilitiesare real numbers between 0 and 1. However, they do not need to be additive. Hence it ispossible that n(A)+n(B) < n(A UB) and A n B = 0. In most models it is assumed thatlower probabilities are monotonic. This means n(A)+n(B) 5_ n(A UB) is required. Lowerprobabilities can be used to express situations of ambiguity. Suppose it is only known thatan urn contains red (R) and green (G) balls. Then ambiguous beliefs could be expressed by,for example, n({il}) = 0, n({G}) = 0 but n({R,G}) = 1. Let 2' denote the set of all subsetsof the set of all states, a The lower probability values of all members of 20 would expressall beliefs in a given situation. Ambiguity would be present if n(A)+n(B) < n(A UB) andA n B = 0, where A and B are subsets of O. While all information is contained in the lowerprobabilities of 2', one can easily define an upper probability, p(A), from the lowerprobability of the complement of A:Figure 3.1 Upper and Lower ProbabilitiesUpper (p) and Lower (n) Probabilities130p(A) = 1:n(;) V A c CI^ (2)(A = LI \A)Equation (2) reflects the intuition that the probability of an event cannot be greater than oneminus the lower probability of its complement.For every subset of fZ,A, there is a lower and anupper probability. Figure 3.1shows all possible probabilityintervals as points on a twodimensional graph. On thediagonal line, upper and lowerprobabilities are equal (n(A) =p(A)). This implies that A isan unambiguous event(n(A)+n((hA) = 1). Thewhite area, n(A) < p(A),describes ambiguity with the extreme of vacuous probabilities (represented by n(A) = 0 andp(A) = 1). Beliefs in the shaded area, n(A) > p(A), would not be permitted if lowerprobabilities are assumed to be monotonic. Otherwise, they would represent contradictionsin the beliefs. The case of contradictions is not further analyzed. However, in most of thediscussed theories, contradictions could be treated analogously to ambiguity with quiteintuitive results.Kyburg (1983) develops a decision theory based on a probability interval for everystate. Expected utility for every act is calculated as an interval from the probability intervalsover states. The lower expected utility of an act is determined by using that (additive)probability measure out of the family of probability measures falling within lower and upperprobabilities that yields the lowest expected utility. The upper expected utility is calculated131correspondingly by using that additive probability measure that yields the highest utility.Kyburg also defines upper probability of an event as one minus lower probability of thecomplement event. Kyburg's decision rule suggests that all acts, a, are rejected for whichthere is an alternative act, aj, for which the lower expected utility exceeds the upper expectedutility of ai. This rule would order acts only partially in case of acts with overlappingexpected utility intervals. Kyburg's model has been criticised since it does not provide anintuitive way for updating beliefs to reflect new evidence (see Gdrdenfors and Sahlin 1988,p. 243).Larsson (1977) shows that a non-additive measure, called P-measure, can be derivedfrom a modification of Savage's axioms. This P-measure can be interpreted as a lowerprobability and fulfils the requirements listed for a lower probability. Walley (1992)discusses the importance of lower and upper probabilities at length and introduces variousmathematical techniques for updating and conditioning "imprecise probabilities". Walley'supper and lower probabilities are based on a behavioural interpretation as betting rates. Thelower probability of an event A, n(A), would be the supremum buying price for a lottery thatpays one if event A occurs and zero otherwise. The upper probability of an event A, p(A),would be the infimum selling price for the same lottery. For an ambiguous lottery, n(A) <p(A) could be interpreted as ambiguity aversion.Lower probabilities and non-additive probabilities provide a more flexible frameworkfor representing beliefs than an additive probability measure. Ambiguity can be expressedby explicitly assigning probability weight not to a single state but to a set of states.Ambiguity is clearly isolated as the difference between lower and upper probability of astate. This framework appears to be better suited for elicitation of beliefs from experts.Models for decision making under ambiguity do not, in general, allow the representation ofignorance. A suggestion for incorporating ignorance, however, will be presented in section4. The models discussed so far do not provide an intuitive way for updating or combiningbeliefs. The Dempster-Shafer belief function theory is able to fill this gap and will bediscussed in section 4. Their theory is particularly appealing for environmental decision132making since it has a strong evidential rather than behavioural interpretation. The modelingof values in the approaches discussed so far does not have a strong normative foundation,partly because most of these models were designed primarily to address empirical anomaliesof SEU under ambiguity.In the areas of artificial intelligence and fuzzy set theory, some concepts are used thatcan be related to upper and lower probabilities. However, the primary objective of thesetheories is not a conceptual treatment of ambiguity as defined in this chapter. These theoriesare, therefore, only mentioned briefly. In possibility theory, upper and lower limits onbeliefs are referred to as necessity and possibility. Possibility is understood as an ordinal andnot a cardinal concept (see Dubois, 1988). Hence, possibility theory is concerned with theinterpretation of statements such as "A is at least as necessary as B". Fuzzy decision theory,on the other hand, is extending the Bayesian decision making framework to fuzzyprobabilities and utilities (see Zadeh, 1978, and Freeling, 1984). Acts are linked to sets ofprecise probabilities and utilities through membership functions. The theory amounts to anextensive sensitivity test of Bayesian decision making. Furthermore, it provides the tools toincorporate the ambiguity of real life statements such as "A is quite probable" into asystematic decision making framework. The focus of these theories is on translatingstatements of vague every-day language into an automatic decision support system rather thanon conceptually dealing with the treatment of ambiguity. Walley (1991, p.266) remarks, "Iffuzzy sets have a useful role to play, it is in modeling the ambiguity of ordinary language... Fuzzy decision analysis ... does not appear to add anything useful to sensitivity analysis.On the contrary, [it] may obscure the decision problem, by adding second-order structurewhich is difficult to assess and whose meaning is unclear."3.3 Decision Making Under IgnoranceThere are few models that allow the explicit representation of ignorance. However,some very simple models of decision making under complete ignorance were proposed andare discussed here. In addition, a very different approach to decision making under ignorance133developed by Shackle is briefly presented. SEU can be contrasted with models for decisionmaking under complete ignorance, as the opposite extreme in terms of informationrequirements. Under complete ignorance, the only information assumed to be available is theminimum and maximum utility that can result from each act, a, denoted u(a) and un(a).A decision criterion can be based on a linear combination of these two utility values:U(a) = a umin(a) + (1 - a)u (a), Osasl (3)This is the so-called Hurwicz criterion (Hurwicz 1951). The Hurwicz criterion has twoextreme cases. With a =1, it is the maximin criterion (the act with the highest minimumutility is chosen). With a=0, it is the maximax criterion (the act with the highest maximumutility is chosen).Other proposals have been made including minimizing maximum regret. Underminimax regret the alternative would be chosen under which the highest possible regret (thedifference between utility obtained and utility that could have been obtained from the act withthe best outcome) is smallest. However, this criterion requires information about thealignment of states (or the correlation of outcomes across acts). Another criterion, sometimesreferred to as the Bayes-Laplace criterion, applies the principle of insufficient reason. Ifufith,(a) and urn(a) is the only information available, these two utility values would beconsidered two elementary states which are assigned equal probability. Hence, this criterionwould rank acts by comparing the arithmetic average of u,„„,(a) and u,„.),(a). In the contextof using a non-informative prior distribution, it was discussed that unfin(a) and um(a) are notwell defined elementary states of nature but merely a description of the possible range ofutility outcomes. This concern applies to the latter criterion.It is difficult to imagine a decision situation under complete ignorance. Even withinnovative activities, we would have some expectation of the outcomes and some historicalparallels to draw upon. For example, the release of a new chemical in the environment mayclearly result in surprises. However, past experience with similar chemicals or contemplationof scenarios of possible outcomes could provide more information than only a utility range.134A decision model based on complete ignorance would discard all information other than thepossible utility range. It does not appear reasonable to suggest that vague or unreliableinformation should not enter the decision making process at all. Therefore, decision theoriesfor complete ignorance do not have direct policy relevance. However, these theories areuseful thinking devices for contemplating an extreme hypothetical situation, and can be usedas building blocks for a more complex model for situations of partial ignorance.An interesting theory of decision making under ignorance was introduced by Shackle(1949, 1969). This theory allows a richer structure than complete ignorance while explicitlyretaining the possibility of states that are not well defined. Shackle fundamentally rejects theusefulness of the probability concept to decision situations without historical precedent. Hesuggests a theory for decision making under ignorance that assumes the existence of apotential surprise function (ranging from zero to one) over an incomplete subset of all statesof the world. He also assumes an attractiveness function (expressing what most attracts thedecision maker's attention rather than referring to attraction as the decision maker's utility)over the space of utility outcomes and potential surprise. Attractiveness increases withincreasing distance of the outcome of a state from the current utility level and decreases withincreasing potential surprise. Those points of the potential surprise function that yield thehighest attractiveness on the gain and loss side are designated as focus gain and focus loss,respectively. Acts are ranked by a decision function on focus loss and focus gain (twoutility/potential surprise pairs). Perrings (1989) applies Shackle's theory to determining thesize of an environmental bond that someone, who undertakes an innovative activity, shouldhave to post in order to generate the desirable incentives for research into the outcome ofsuch activities. Arrow and Hurwicz (1972) discuss how Shackle (1949) would interpret totalignorance as zero potential surprise of all states. Decision making under complete ignorancewould then depend on the maximum and minimum payoff only regardless of the definitionof elementary states of nature.Shackle's decision theory addresses fundamentally important issues that are ignoredby SEU. It accommodates the possibility of surprise and allows the expression of beliefs in135the occurrence of a state through potential surprise which imposes less structure than aprobability distribution. However, there would be many obstacles to using Shackle's theoryfor environmental decision making. There is no obvious way for constructing beliefs fromevidence, integrating evidence, or combining divergent beliefs. Moreover, the model requirescomplex behavioural assumptions (the existence of a potential surprise function, anattractiveness function and a decision function) that would be difficult to test and appearsomewhat ad-hoc. The implicit assumption that outcomes other than focus-gain and focus-loss are irrelevant is difficult to reconcile with intuition. This theory is fundamentallydifferent from and cannot be viewed as an extension of SEU to situations of ambiguity orignorance. At this stage, it would not be attractive to replace SEU with a theory that leavesso many questions unanswered.4^A Model for Environmental Decision Making under UncertaintyThis section proposes a decision theory that is judged to be more suitable fornormative environmental decision making than SEU, based on the requirements developedin section 2.3. The two principal elements combined in this theory are the belief function-theory by Dempster-Shafer and Choquet expected utility theory. Belief function theoryprovides a natural way to express beliefs that can account for ambiguity as well as ignorance.Choquet expected utility and its axiomatization provided by Sarin and Wakker (1992) providea decision theory for non-additive probabilities that can be used on beliefs expressed by aDempster-Shafer belief function. Combined, these two theories provide an extension of SEUto problems of ambiguity and ignorance. The components of the suggested theory fordecision making under ambiguity and ignorance are not novel. The contribution of thissection lies in some extensions of the Dempster-Shafer belief function theory and thedemonstration how this theory can be combined with Choquet expected utility to a usefulmodel for practical environmental decision making.This section begins with a characterization of the representation of beliefs with theDempster-Shafer belief function theory including Dempster's rule for combining evidence.136Some extensions of belief-function theory are suggested. These extensions would allow therepresentation of ignorance in a belief function. Also Dempster's rule is generalized for theintegration of conditional evidence and the aggregation of different beliefs based onassessment of the same evidence. Then, the decision theory based on Choquet expectedutility is presented. A lottery example illustrates the use of the suggested theory and appliesit to the value of additional information about the lottery. The implications of relaxingassumptions about individual choice under ambiguity are analyzed. This latter modificationleads to a model of choice under ambiguity that may not be able to order all acts completelybut avoids a normative assumption about decision making under ignorance and ambiguity.Rather, the ultimate treatment of ambiguity is isolated in a behavioural parameter on whicha real-life decision may, or in some instances may not, depend. Finally, the implications ofthe suggested theory for environmental decision making are discussed.4.1 Representation of BeliefsDempster (1967) and Shafer (1976) have developed a mathematical theory of evidencethat is based on belief functions. A belief function can be interpreted as a lower probability.With an additive probability measure, a total probability mass of one is divided amongindividual states. With a belief function, the total probability mass of one is divided amongindividual states but also sets of states (including possibly the set of all states). TheDempster-Shafer theory combines lower probabilities with intuitively appealing rules for thecombination of independent evidence and the incorporation of statistical inference. Theirtheory emphasizes the construction of beliefs from evidence rather than the behaviouralinterpretation of beliefs as betting rates. The following is a very brief summary of thoseelements of Shafer's (1976) theory of evidence that are used for the proposed theory forenvironmental decision making.Beliefs over the occurrence of states are expressed by a belief function, Bel. Let fldenote the set of all states of the world and 20 denote the set of all subsets of a The samebeliefs represented in a belief function can also be expressed in basic probability137assignments, and belief functions are easiest explained by introducing basic probabilityassignments, first. The basic probability assignment, m, is a function mapping the set 20 intoreal numbers between zero and one with m(0) = 0 and EA corn(A) = 1. This means, a totalprobability mass of one is distributed among the subsets of 0 which may be singletons, setsof several states or 0 itself. A subset of 0, A, is called a focal element of the belief functionif m(A) >0. The union of all focal elements is called the core of the belief function. Thebelief function, Bel, is a non-additive and monotonic measure that maps the set 20 into realnumbers between zero and one with Bel(0) = 0, Bel((1) = 1 and Bel(A) = EscAm(B)- Theupper probability, P*(A), is defined by P*(A) = 1-Bel(Ä). Bel(A) can be interpreted as theminimum and P*(A) as the maximum probability mass that can be assigned to all membersof A.A belief function generalizes subjective probabilities and allows the representation ofambiguity and probabilities with different reliabilities. Consider a situation with only twostates of the world, A and B. In a situation of risk, all of the belief function's focal elementsare singletons, that is m(A) = 7 and m(B) = 1-7. In a situation of risk, the belief functionis an additive probability measure (or a "Bayesian belief function"). A situation of ambiguitycan be represented by including non-singletons in the core of a belief function. For example,m(A) = T and m(A,B) = 1-7 would represent ambiguous beliefs. Different reliability ofprobabilities can be represented by making different basic probability assignments toindividual states and their union. Recall the discussed example of a tennis match. Beliefsover the outcome of a match, where equal chances are assessed based on very detailedinformation, may be Bayesian: m(A) = 0.5 and m(B) = 0.5. For another match where equalchances are assessed on very sketchy information, beliefs might be represented by m(A) =0.2, m(B) = 0.2 and m(A,B) = 0.6. Finally, if equal chances are assessed purely on theprinciple of insufficient reason, beliefs would be represented by m(A,B) = 1.Shafer discusses several tools for the construction of belief functions from evidence.Only some of these tools are presented here. Many important details of the Dempster-Shaferbelief function theory are not reported here and can be found in Shafer (1976). A basic idea138of belief function theory is that evidence in real life situations is often too complex to beassessed wholistically. Therefore, Shafer suggests breaking down the evidence into simple,intuitively independent pieces, representing these pieces by a belief function, and combiningthe belief functions in a systematic way. Two belief functions, Bell and Be12, can becombined using Dempster's (1967) rule. If the intersection of the cores of two belieffunctions, Bell and Be12, is non-empty, that is if:E m1(A1)m2(B) < 1 (4)the combined belief function, Bel, is expressed by the following basic probabilityassignments, m(A):m(A) -E m1(A1)m2(B1)A,n11/=AV A c 0, A e^(5)1 - E mi d m2(B j)ijAinBj=0If the intersection of the cores of Bell and Be12 is empty (the extreme case of irreconcilableevidence), the combined belief function will be vacuous, m(0)=1. The belief functiondetermined by (5) is called the orthogonal sum of Bel, and Be12, denoted Bel = Bell El) Be12.More than two belief functions can be combined using Dempster's rule pairwise, i.e. (BellED Be12) e Be13. The orthogonal sum is transitive. Dempster's rule leads to a more focussedbelief function (higher basic probability assignments on subsets with a smaller number ofstates) if evidence is conforming. It leads to a less focussed belief function if evidence iscontradictory.Shafer also provides a rule for the representation of beliefs from statistical inference.Suppose the set of states, 0, consist of the possible values of a statistical parameter, 0.Shafer suggests the following equation for calculating beliefs over the parameter 0:139max q0(x)Bel(A) = 1 -P: (it) = 1 °EA^ V A ce^(6)max q e(x)Oeewhere q9(x) is the likelihood of the observation, x, if the true value of the parameter was 0.This rule is a generalization Bayes' rule for updating a Bayesian belief functions. Severalindependent observations, xi, can be combined using Dempster's rule.The rule (6) shows that the theory of belief functions can provide stronger conclusionsthan a theory based on minimum and maximum probabilities. Consider the example of anurn with two balls of unknown colour. If one red ball was drawn from an urn, a minimumprobability of one half could be assigned to the event red. However, this procedure wouldnot exploit the full information contained in the observation. Drawing one red ball allowsstronger statistical inference since it was more likely that a red ball was drawn from apopulation of two red balls (R,R) than from a population of one red and one other ball(R,0). This information is used with equation (6) by refining the set of states from adescription of outcomes (red or other) to a description of both balls in the urn. Theparameter 0 (0E 0, 0 = {0,1/2,1)) would represent the share of red coloured balls in theurn. With (6), the observation of a red ball would lead to a belief function m((R,R)) = 1/2,m((R,0);(R,R)) = 1/2. This belief function would then be combined with the (Bayesian)belief function, allocating probabilities for drawing red given the composition of the urn. Theresult is m((R,R)) = 1/2, m(R,0);(R,R)) = 1/4, m((R,0);(R,R)) = 1/4 (the colour whichis actually drawn is underlined). This leads to a basic probability assignment for drawing ared ball of 3/4 and for drawing a red or another ball of 1/4. Since beliefs do not coincidewith minimum probabilities, it is possible that the true probability turns out to be outside theinterval of belief and upper probability. In the above example, the belief in drawing a redball is 3/4, and the upper probability is one. However, it may turn out that the urn containsonly 1 red ball (the true probability is 1/2).140Three extensions of the Dempster-Shafer belief function theory are proposed here forthe incorporation of conditional evidence, the aggregation of conflicting interpretations of thesame evidence, and the representation of ignorance. Dempster's rule combines independentevidence symmetrically. However, evidence combined by Dempster's rule needs to be basedon belief functions on compatible sets of states. In particular, it is required that the sets ofstates of two belief functions have a common partition. However, Dempster's rule is easilyextended to the case where one belief function is combined with another on a subset of theset of states of the first belief function. This is the important case of integrating conditionalevidence. Note that such conditional beliefs were already used in the urn example above (thebelief that if the urn contains one red and one other ball, the probability of drawing a redball is one half). The set of states of the latter belief function is a subset of the set of statesof the former belief function. Suppose Bel, is a belief function over the set of states fl, andBel2 is a belief function over the set of states if, with NI' C O. Then:m1(A)1 -^E^m1(i9m2(B;)(Akruip(Ak\,)=0E^miodm,(B)k,1(Akrut)u(Akvp)=A1^m,(A).2(B)ij(AknI3i)U(AkVir) =m(A) =ifAnit =0(7)ifAnT*0V A c 0, A * 0This equation updates those beliefs in Bell that are held over sets of states that include statesover which beliefs are held in Be12 (states that are members of 4f) in analogy to (5). Beliefsheld over states that are not members of if are only normalized to account for contradictionsin Ir. (7) is a generalization of (5) and collapses to (5) if if14 1Dempster's rule is designed to combine independent evidence. It cannot be used tocombine beliefs resulting from different interpretations of the same evidence. Underambiguity and ignorance, it is perfectly reasonable that two experts with access to the samebody of evidence would hold different beliefs. It would be inappropriate to combine thesetwo beliefs by Dempster's rule. To see this consider a situation in which two experts holdthe same beliefs based on the same evidence. The combination of these two belief functionsdoes not add information to either one of them. However, use of Dempster's rule would leadto an inappropriately focussed belief function. Belief function theory, however, would offera very convenient way of combining different beliefs resulting from the same evidence. Let"Bell is included by Bel211, denoted by Bel, c Be12, be defined as:Beli(A) Bel(A)^VAcQ^ (8)Note that this definition also implies:Bel i(A) Be12(;4-)^P * (A) s P 2 * (A)^(9)Hence, if Bell is included by Be12, the interval of beliefs and upper probabilities, Bel, andP*1, is included in the interval Bel2 and P*2:c Be12^Be12(A) s Bel i(A) P 1* (A) s P(A) V A e^(10)Hence Bell is more focussed and less ambiguous than Be12. If different experts interpretevidence differently, resulting in different belief functions, Bel„ this is an additional sourceof ambiguity that can be represented in a combined belief function, Be!, which would be themost focussed belief function that still includes all beliefs, Bet, or:Bel(A) = minmi(B)^V A eL B cAFinally, ignorance is not explicitly considered in the Dempster-Shafer theory.However, it can be represented in a belief function as well. Ignorance was defined as asituation in which there are some states w*E1/*\SZ which are not fully described. These states142were called scenarios. The maximum and minimum utilities that could be obtained inscenario co* under act a were denoted umax(a(c)*)) and u(a(0`)). Hence, to incorporateignorance, beliefs would be associated with the range of utility values possible under ascenario. This can be achieved through representing a scenario by two "pseudo-states" thatgenerate maximum and minimum utility under each act. The belief in the occurrence of ascenario can be represented by belief in the occurrence of these two pseudo-states. Thepseudo-states associated with a scenario, w*, are denoted by co* = {ces,co*s} with u(a(ces))= uni(a(ce)) and u(a(ces)) = umin(a(co*)) for all a. Now, the two pseudo-states thatconstitute a scenario can be treated like any other states. However, basic probabilityassignments can only be made to the couple of associated pseudo-states. Hence, m({ce"})= 0 and m({(.0*s}) = 0 would be required. Basic probability assignments can be made to acouple of associated pseudo-states (representing an individual scenario) as well as to sets ofscenarios and other states. This allows a unified treatment of ignorance and ambiguity anduses all the information that is available about scenarios. Because of this unified treatmentof ambiguity and ignorance, I will generally only use the term ambiguity in the remainderof this section.4.2 Decision TheoryDempster and Shafer do not explicitly consider their belief function theory as adecision theory. However, belief function theory can be suitably combined with a decisionrule that is an extension of SEU to non-additive probabilities. Sarin and Wakker (1992)provide an axiomatization for calculating expected utility with non-additive probabilitiesbased on an earlier paper by Choquet (1953-54). This theory is called "Choquet expectedutility" (CEU). CEU is a generalization of SEU to non-additive probabilities. Implicitly,CEU assigns the probability weight that is not assigned to a single state but a set of statesto the individual state out of this set that leads to the lowest utility under an act. Thisambiguity-pessimism built into CEU is derived from an axiom of cumulative dominance thatreplaces the sure thing principle in the Savage axioms. The axiomatization has intuitive143appeal, and the results of this theory are consistent with empirical evidence, such as theEllsberg paradox.Since the sum of non-additive probabilities over all states can be less than one,expected utility cannot be calculated directly with non-additive probabilities (see also Larsson1977, p.147). However, Choquet (1953-54) has introduced an approach that makes itpossible to calculate the expected utility from a non-additive measure such as the belieffunction introduced above. To calculate CEU of an act, a, with finite states, co, states aresorted by their utility outcome: u(a(oh)) 5_ ... u(a(coN)). Then:CEU(a) = u(a(6 )1)) + E (u(a(wn)) - u(a(c n_i))13e1.(1._1)^(12)n=2where Be1,(„_1) is defined as the belief in all states that, under act a, generate utilityu(a(con_i)) or higher:Bel(I1) = Bel({G) ell I u(a(6)))14(a(con_i)))) = E m(A)^(13)A c1:1,CEU calculates an expected value from non-additive probabilities by forming the expectationbased on the difference between cumulative probabilities of better-than utility sets. Sincep(0) = 0 and p(0) = 1, the sum of the differences of cumulative probabilities is always oneeven if p is non-additive, which means that an expected value can be calculated. If p is anadditive probability measure (or Bel is a Bayesian belief function), CEU is equal to SEU.An appealing feature of CEU is that several axiomatizations for CEU with non-additive probabilities are available. After previous axiomatizations of CEU by Gilboa (1987)and Schmeidler (1989), a more intuitive axiomatization has been provided recently by Sarinand Waldcer (1992). In their axioms, Savage's sure thing principle (which is violated by theEllsberg paradox) is replaced with an axiom of cumulative dominance. Cumulativedominance requires "... that if receiving consequence a or a superior consequence isconsidered more likely for an act f than for an act g, for every a, then the act f is preferred144to the act g." (Sarin and Wakker 1992, p.1256). In addition, Sarin and Wakker require asufficiently rich set of unambiguous acts.The cumulative dominance axiom, and hence CEU, implies that basic probabilityassignments for a set of states are allocated to the member of this set that yields the lowestutility outcome for the act to be evaluated. To see this, consider the (additive) probabilitymeasure N(11) with pa(co) defined as the sum of the basic probability assignments over allsubsets of II whose member that generates the lowest utility under act a is co:Pa() =E^m (A)(Acfl IA and u(n(co))su(a(coi)) V (al EA)Then:CEU(a) = u(a(0) 1)) +E[(u(a(6).))-u(a(co)) E Pa(i)1 =n=2^ i=n-1= E pa(wdu(a(w„))n=1which is SEU with probability measure p„(). Note that, by definition of pa(i1), every act,a, is evaluated with a different probability measure, MO)• This shows that the informationcontained in a belief function cannot be reduced to the information in a single probabilitymeasure. As a result of the cumulative dominance axiom, the evaluation function assignsprobability weights on non-singleton sets of states to the state that yields the lowest utilityunder the act to be evaluated. This "pessimistic" treatment of ambiguity is discussed in alater section in more detail.4.3 An Illustrative ExamplesThe example for illustrating the proposed theory is based on drawing balls from anurn. Even though such an example cannot illustrate the complexities of a real life problem,it demonstrates the operation of the proposed theory in as simple a manner as possible. One(14)(15)145ball is to be drawn from the urn. Utility is assumed to be 1 for drawing a green ball (G) and2 for a red ball (R). The urn contains two balls. Each of the two balls may be either red orgreen. Further information about the composition of the urn and how it was obtained is notavailable. This set-up can be regarded as a two-stage lottery. The first stage is not welldefined and determines the composition of the urn. The second stage is well specified witha ball being drawn from an urn with the composition determined in the first stage.The combined lottery has four relevant states (describing the composition of the urnand, if the urn contains balls of different colours, the colour of the ball drawn, shown asunderlined): = {(G,G);(G,R);(G,R);(R,R)}. There are two sources of beliefs about thislottery. One source of belief is the assessment that the probability of drawing a particularcoloured ball equals the proportion of balls of this colour in the urn. This belief can beexpressed in a conditional Bayesian belief function, Bel,. In the example, the core of thisbelief function is mi((C,R) I (G,R)) = 0.5 and m1((G,R)1(G,R)) = 0.5. The second sourceof belief is knowledge that will be acquired about the composition of the balls in the urn.This belief can be expressed in a belief function Bel2 over 2' where Y is the set of possiblecompositions of the urn; Y = {(G,G);(G,R);(R,R)}. Without any additional knowledge aboutthe composition of the urn, Bel2 is vacuous, and the core of Bel2 is triviallym2((G,G),(G,R);(R,R)) = 1. Combining Bell and Be12 with the rule in (7) leads to a newbelief function, Be13, over 20 with the core m3((G,G);(C,R);(R,R)) = 0.5 andm3((G,G);(G,R);(R,R)) = 0.5.Panels (a) through (d) in Figure 3.2 show how the evaluation of this lottery evolveswith increasing information acquired about the colour of the balls in the urn. Panel (a) showsthe evaluation of the lottery based on Be13. From Be13, the CEU for this lottery can becalculated. The line on top of the light shaded area shows the belief that the realized levelof utility will be at least U*. Hence in Panel (a), the light shaded area equals CEU of Be13.Bel3 represents the information that the urn contains two balls that may be red.or green (seePanel (a)). No state yields utility less than one. Therefore, Bel(U = Be13(0) = 1 if1. The belief that utility is greater than one equals the belief in occurrence of the set146(c) Twice One Red Ball DrawnCEU(Bel)-1.875(d) One Red and One Green Ball DrawnCEU(Bel)-1.5Figure 3.2 Evaluation under Ambiguity: A Lottery Example147of states that yields utility greater than one: Bel(U. U*) = Be13((G,R);(R,R)) = 0 if U*> 1.Hence CEU equals one. To show the degree of ambiguity, it is instructive to compare beliefswith upper probabilities. The line on top of the light and dark shaded areas togetherrepresents the upper probability that the realized level of utility will be at least U*. Upperprobabilities, P*, are calculated by subtracting the belief in the complement set from one.For 1, trivially P*(U = 1-Be13(0) = 1. With U*> 1, P*(U = 1-Bel3((G,G);(Q,R)) = 1. The dark shaded area shows the prevailing ambiguity.Subset X E 2' P2*(Y \ X) Be1200 P2*(X) M2(X)^1{(G,G)} 1 0 0 0{(G,R)} 1 0 1/2 0{(R,R)} 1/2 1/2 1 1/2{(G,G);(G,R)} 1 0 1/2 0{(G,G);(R,R)} 1/2 1/2 1 0{(G,R);(R,R)} 0 1 1 1/2{(G,G);(G,R);(11,R))0 1 1 0able 3.^Belief Function after Observing One Red BallIn a second step it becomes known that one red ball was obtained in a random drawfrom the urn (see Panel (b)). From this statistical evidence a new belief function, Belo overcan be calculated from (6). As discussed before, the core of Bel4 would be m4((R,R)) =0.5 and m4((R,R);(G,R)) = 0.5. The values for Bel,' are shown in Table 3.2. Combined with148Bell, we obtain Be15 with the core m5((R,R)) = 0.5, m5((R,R);(G,R)) = 0.25 andm5((R,R);(G,R)) = 0.25. Now Bel(U.U*) = Be13(0) = 1 if U*5.1 and Bel(U .U*) =Be15((G,R);(R,R)) = 0.75 if U*> 1. Hence, CEU equals 1.75.In a third step, new information is received that in a second, independent, randomdraw from the same urn, a red ball is drawn again (see Panel (c)). Since both observationsare independent and both observations are represented by the belief function Belt, thecombined belief function, Be16, can be obtained from Dempster's rule: Be16 = Bel4EDBe14.Combine Be16 with Bell to obtain Bel,. The core of Bel, is m7((R,R)) = 0.75,m,((R,R);(G,R)) = 0.125 and m7((R,R);(G,R)) = 0.125. Now Bel(U .U*) = Bel7(0) =1 if U*__ 1 and Bel(U.U*) = Bel7((G,R);(R,R)) = 0.875 if U*> 1. Hence, CEU equals1.875. It can be easily verified that upper probabilities are not affected by the evidenceobtained in steps (b) and (c).In a final step, the draw of a green ball is reported. This observation leads tocomplete knowledge of the composition of the urn, (G,R), expressed in Be18 with the corem8((G,R)) = 1. Hence, the resulting belief function, Bel9=Be119Be18, is Bayesian with thecore In9((Q,R)) = 0.5 and m9((G,R)) = 0.5 with CEU = 1.5 (see Panel (d)). In this case,all ambiguity is resolved and CEU = SEU.Table 3.3 summarizes the evolution of this lottery's evaluation. Also in this table,evaluation with CEU is compared with SEU using two different Bayesian prior distributions.The example illustrates the problems discussed above concerning selection of a Bayesian non-informative prior distribution. The question is whether two indistinguishable states, (G,R)and (R,G), should be treated as one or two elementary states of nature. If they are treatedas distinct elementary states, a uniform prior distribution would assign probabilities of 0.25to the states (G,G), (R,G), (G,R), and (R,R), respectively (shown as SEU(1)). If they aretreated as one state, a uniform prior would assign probability 0.333 to the states (G,G),(G,R) and (R,R), respectively (shown as SEU(2)). The evaluation in Table 3.3 is obtained149from a posterior probability distribution, tk(Y), updated from the respective priordistribution, 4)(Y), with Bayes' rule:Stage Belief Function CEU SEU(1) SEU(2)(a) NoKnowledgeBe13 = BelIEBBel2:m3((G,G);(Q,R);(R,R)) = 0.5 andm3((G,G);(G,R);(R,R)) = 1.500 1.500()) One RedBall DrawnBel5 = Bel3eBel4:m5((R,R)) = 0.5,m5((R,R);(_Q,R)) = 0.25 andm5((R,R);(G,R)) = 1.750 1.833(c) Twice OneRed Ball DrawnBel, =Bel5EB Bela:m,((R,R)) = 0.75,m,((R,R);(G,R)) = 0.125 andm,((R,R);(G,R)) = 1.833 1.900(d) One Redand One GreenBall DrawnBel9=Be179Be18:m9({Q,R)) = 0.5 andm9((G,R)) = 1.500 1.500a e j.^Summary of Lottery Evaluation41(v I x) -  Ax I v) 4(u) E Ax I 040) (16)where x is the observation (i.e. one or two red balls drawn) and f(x I v) is the likelihoodfunction for drawing a ball of specified colour from an urn with a share u of balls of thiscolour. Note that depending on the prior distribution used, SEU can be larger or smaller than150CEU. Of course, if the belief implicit in using one of these priors was genuinely held, thiscould be expressed in an additional belief function. If such a Bayesian belief function wascombined with the belief function obtained from statistical inference, SEU and CEU wouldbe identical at all stages of this lottery. The point of this example, though, is to show howthe lottery can be evaluated without belief in a prior distribution.In a practical decision making situation, learning more about a problem may be anadditional alternative to accepting or rejecting a lottery. For making a choice between thesethree acts, it is necessary to determine the value of information obtained from learning. Thevalue of information can be calculated for the above example. Consider a situation in whichone red ball has been drawn from the urn at random (the situation in Panel (b)). Now, it isassumed that the decision maker faces three alternatives: buying the lottery at a price of 1.6(in utility), rejecting the lottery (and receiving 0 utility), or learning more about the lotteryby drawing one more ball at random and deciding afterwards whether to accept or reject thelottery. We are interested in the value of the third alternative which is the difference betweenthe value of the lottery with and without the information obtained from learning.Under the Bayesian approach, calculation of the value of information isstraightforward, but it depends on the assumed prior distribution. Using the prior of SEU(1),without the learning option the lottery should be accepted, and its value would be 0.15(=1.75-1.6). The value of the lottery with the learning option depends on the expectationabout the outcome of the second draw from the urn and the action chosen as the result oflearning. The probability of drawing a red ball would, at this stage, be assessed as 0.75. Ifa red ball is drawn, the lottery's value would be 0.233 ( =1.833-1.6; see Table 3.3). If agreen ball is drawn (probability 0.25), the lottery would be rejected since its value would benegative (1.5-1.6), and the outcome under this alternative would be zero. Hence the valueof the lottery with learning option is 0.175 ( = The value of information is 0.025(=0.175-0.15). Similar calculations lead to a value of information of 0.017 for the priordistribution used in SEU(2).151The value of information can be calculated analogously for CEU. The value of thelottery without the learning option is 0.15 (=1.75-1.6). If a red ball is drawn, CEU of thelottery would be 0.275 (=1.875-1.6; from Table 3.3). If a green ball was drawn the lotterywould be rejected (outcome zero). Of course, the expectations about the outcomes of learningare not expressed in probabilities but in a belief function (m(R) = 0.75, m(R,G)=0.25; seePanel (b)). CEU can be calculated by replacing the colour of the ball drawn with the utilityoutcome, and would be 0.20625 (=0.75-0.275). The resulting value of learning is 0.05625(=0.20625-0.15). The results are summarized in Table 3.4. The fact that the value oflearning is significantly higher for CEU than for SEU can be intuitively explained by theabsence of a prior distribution in the calculation of CEU. Since CEU is based on less priorinformation, new information has a larger effect on the assessment of beliefs, and hence ahigher value.(a) Value ifred balldrawn(b) Value ifgreen balldrawn(c) Valuewithlearning(d) Valuewithoutlearning(e) Value ofinformation[(c)-(d)]CEU 0.275 0 0.20625 0.15 0.05625SEU(1) 0.233 0 0.175 0.15 0.025SEU(2) 0.3 0 0.25 0.233 0.017Table 3.4^The Value of Information4.4 An Extension with Incomplete OrderingAs a result of the cumulative dominance axiom, CEU implies ignorance toward thepossibility of superior consequences unless there is actual belief in the occurrence of superiorconsequences. To see this, consider the first stage of the lottery described in section 4.3 (see152Panel (a) in Figure 3.2). At this stage, beliefs are vacuous and CEU =1. Compare thislottery to a certain outcome of U=1 which would also be evaluated with CEU =1. Hence,the proposed theory implies indifference between obtaining U=1 for sure and a lottery withminimum utility 1 and the possibility of a superior consequence, i.e. U=2, if no strictlypositive basic probability is assigned to consequences with utility strictly greater than 1. Thisdemonstrates the pessimism about ambiguity built into CEU. The defense of this theory isthat if the lottery was preferred to the sure outcome, this shows that there is some strictlypositive belief in the occurrence of a superior consequence. For example, the mereknowledge of the existence of red balls in the universe may lead the decision maker to assignsome belief to a red ball being contained in the urn. Then the lottery would be preferred tothe sure consequence.The weak foundation for a normative model of choice under ambiguity was discussedbefore in the context of SEU. CEU may be considered to have a somewhat strongernormative foundation than SEU since it is not as clearly inconsistent with empirical evidenceshown in connection with the Ellsberg paradox. Also, CEU is consistent with a cautious "no-regret" approach to ambiguity. While the use of CEU appears to be preferable to the use ofSEU, and other reviewed theories, it would, nevertheless, be an overstatement to claim thatthe intuitive appeal of the cumulative dominance axiom alone provides a very solidfoundation for the normative use of CEU. Therefore, this section seeks to explore whetherany meaningful statements about choice under ambiguity can be made without an assumptionabout the treatment of ambiguity, that is without accepting the cumulative dominance axiom.Ambiguity in a given problem is reflected by the existence of many (additive)probability distributions, 70), that are included by the belief function, Bel(0) (see thedefinition in (8)). If the same ranking between two acts could be obtained by using SEU withany i((I) that is included in Bel((), it could be said that a ranking between these two actsis independent of the normative treatment of ambiguity. Put differently, if max(SEU(aori((I)))< min(SEU(bor1((1))) for all 70) that are included by Be1(0), it can be concluded that actb can be chosen over act a without having to resort to any assumptions about the normative153treatment of ambiguity. This approach, of course, does not guarantee a complete orderingof acts and is strongly related to Kyburg's (1983) theory, which was introduced before.With the cumulative dominance axiom, the value function allocates the basicprobability assignment of a set of states to the state with the lowest utility in this set. From(15) follows that:n/(0)cBel(Q)min^SEU(a,Tc i(C))) = CEU(a)^(17)where c denotes 'included by' as defined in (8). Similarly, max(SEU(aor1((1))) is equal toCEU*(a), which denotes Choquet expected utility calculated from the upper probabilitiesrather than beliefs. Analogous to (12), CEU* is defined as:CEU * (a) = u(a(u) 1)) + E (u(a(u n)) - u(a(6)n-i))11:(C)n-1)^(18)n=2where 13.*(0„_1) is the upper probability of all states that, under act a, generate utilityu(a(con_1)) or higher:P(C -1) = 1 - Bel ({u) ea I u (a (G))) < u(a(con_1))))^(19)where states are again assumed to be sorted by their utility outcome: u(a(0.0) .....u(a(coN)).CEU*(a) would reflect the most optimistic attitude toward ambiguity. (CEU* could also beobtained by reversing the cumulative dominance axiom. With the reversed axiom, the valuefunction would allocate the basic probability assignment of a set of states to the state withthe highest utility in this set. This assumption would lead to ignorance toward downsidepossibilities.)Let V(a) be a value function for the act a that does not depend on assumptions aboutthe treatment of ambiguity. Then, V(a) can be bounded by CEU(a) (based on an allocationof all basic probability assignments to the state with the lowest utility) and CEU*(a) (basedon an allocation to the state with highest utility):154CEU(a) s V(a) s CEU * (a) (20)If all ambiguity is resolved (the core of the belief function consists of singletons only),CEU(a) = V(a) = CEU*(a). If there is ambiguity, CEU(a) < CEU*(a). Then V(a) isbounded by an interval and does not necessarily order all states completely. However, if V(a)> V(b), or CEU(a) > CEU*(b), a can be chosen over b without the need to resort toassumptions about the treatment of ambiguity. If the intervals of two acts, V(a) and V(b),overlap, the two acts cannot be ordered unless assumptions are made about the attitudetoward ambiguity, as expressed, for example, in the cumulative dominance axiom. In theexample of Figure 3.2, CEU is represented by the light shaded areas in Panels (a) to (d).CEU* would be represented by the light and dark shaded area together (CEU*(a) = 2,CEU*(b) = 2, CEU*(c) = 2, CEU*(d) = 1.5). The advantage of this extension is that itcan isolate decisions that can be made without assumptions about attitude toward ambiguity,based on the interval V(a) alone, from those decisions that require additional assumptions,i.e. the cumulative dominance axiom. This approach appears to be useful to make transparentthe treatment of ambiguity in a given decision problem.Analogous to the Hurwicz criterion for complete ignorance, one could propose adecision theory in which V takes the form of a linear combination of CEU and CEU* thatwould allow complete ordering of acts:V(a) = (1- a) CEU(a) + a CEU * (a) with a E [0,1] (21)where a is behavioural parameter of optimism about ambiguity. While this may be a routefor future research, currently there is neither an axiomatic foundation for such arepresentation nor empirical evidence that would support choice of a specific a. Also, itshould be emphasized again that in a model of decision making under ambiguity, probabilityweights are assigned to sets of states. Since there is no assumption about a well defined setof elementary states, there is no basis for assigning probability weights to any of the statesin this set, except for the states with minimum and maximum utility within the set. Sinceexpected utility within a set of states is not defined, ambiguity aversion and ambiguity155seeking are not defined. Hence, a specific a cannot be used to express a specific degree ofambiguity aversion.4.5 Implications for Environmental Decision MakingIn the previous sections, I have proposed a combination of the Dempster-Shafer belieffunction theory and Choquet expected utility based on Sarin and Wakker's axiomatizationas a normative theory for environmental decision making. This theory was selected based onthe requirements for a suitable decision model that were discussed in section 2.2. Let usexamine the proposed theory in light of these criteria. Ambiguity can be expressed by basicprobability assignments to sets of more than one state. Similarly, the degree of confidencein a probability assessment can be expressed by combining a belief function representing theevidence with a belief function on the reliability of the evidence obtained. Ignorance can beincorporated by basic probability assignments to pseudo-states for the maximum andminimum utility from a scenario. The Dempster-Shafer belief function theory is explicitlydesigned to represent beliefs obtained from evidence rather than inferred from behaviour.As a generalization of an additive probability measure, a belief function gives experts abroader choice for expressing their beliefs in the presence of ambiguity. Dempster's ruleprovides for the elegant aggregation of beliefs obtained from independent evidence. I haveshown how beliefs of different experts based on the same evidence can be pooled. Theproposed theory is a generalization of SEU for situations of ambiguity and ignorance. Incontrast to SEU, the theory is consistent with empirical evidence of individual choice underambiguity. Sarin and Waldcer provide intuitively appealing axioms for CEU as the choicerule for non-additive probabilities. Even if the normative conclusions obtained from thecumulative dominance axiom are not accepted, partial ordering of acts is still possible basedon SEU and all probability distributions that are included by the belief function. Theextension with partial ordering shows which decisions can be made without assumptionsabout attitudes toward ambiguity. The use of CEU leads to more robustness towardspecification errors since CEU tends to favour postponing an ambiguous act in favour ofadditional learning about the problem.156For decision making under risk it is well recognized that the contexts of an individualdecision needs to be considered. This means that possible gains are treated as additions toand losses as subtractions from current wealth. As a result, a risk averse individual is averseto risks in losses as well as gains. Similarly with risk aversion, the certainty equivalent ofa cost is higher than its expected value, and the certainty equivalent of a benefit is lower thanits expected value. The context may include risks that depend not on the current decisionalone. Then, it is necessary to analyze the correlation between the risks resulting from aspecific choice and other risks. Wilson (1982) shows how to consider risks in projects withreturns that are correlated with social risks. Even though an act, considered in isolation, isrisky, it may reduce social risk if it is negatively correlated. Then, with risk aversion, thisrisky act would be evaluated more favourably than a certain act.Now, the same considerations apply to the evaluation of an ambiguous act. CEUimplies pessimism about ambiguity, Hence, ambiguous costs would weigh higher, andambiguous benefits would weigh lower than costs or benefits without ambiguity. A projectwith ambiguity that is correlated with social ambiguity would be relatively more attractiveif the correlation was negative and less attractive if the correlation was positive. Note thateven in the presence of ambiguity, it may be possible to make a statement about correlationwith already existing ambiguity if an act is chosen that offsets or increases the physicalcondition that gives rise to the ambiguity. We can now recall the introductory discussion inwhich it was argued that the unknown consequences of natural capital depletion are animportant source of ignorance and ambiguity in environmental decision problems. Considera decision on whether to reduce the depletion of natural capital, i.e., to limit logging or toundertake a reforestation project. Such an activity that, viewed in isolation, yields ambiguousgains and might, therefore, be evaluated at a low CEU, would in fact reduce or insureagainst an ambiguity that is already present. For example, a reforestation project may beassociated with ambiguous gains since there is ambiguity and ignorance about the long-termbenefits from a forest. However, a reforestation project that is undertaken to offsetdeforestation, in fact reduces the ambiguity about future well-being since it helps maintainthe biological status quo that is associated with less ambiguity than a situation of large-scale157deforestation. Hence, taking into account this context, the gains from this reforestationproject would be evaluated more highly under CEU.From this discussion follows that, in typical environmental decision problems, thereare two stylized acts to choose from. The first act implies the depletion of natural capital(i.e., the decision not to restrict emissions, not to undertake a reforestation project, or to goahead with a resource depletion project) and will be called the unsustainable act, au. Thesecond act implies the conservation of natural capital or compensation for natural capitaldepletion (i.e., the decision to restrict emissions, undertake a reforestation project, or notto undertake a resource depletion project without compensation) and will be called thesustainable act, as. In this stylized form, the decision problem involves a choice between theunsustainable act, which is ambiguous, and the sustainable act, which is unambiguous. Anunambiguous act is defined as an act whose outcome does not depend on ambiguous states.Therefore, CEU and SEU of an unambiguous act would be equal.Now, it can be shown that the use of SEU in a situation of ambiguity leads to asystematic bias in favour of the unsustainable act. Since the sustainable act is assumed to beunambiguous, CEU(as) = SEU(as). It is reasonable to assume that the probabilitydistribution, 7((I), used for calculating SEU of the unsustainable act, would be included bythe Belief function that would be used for calculating CEU. It was shown before that:min SEU(a, rci(0)) = CEU(a)711(0) c Bel(CI)(22)Therefore, CEU(au) ._ SEU(a,) if 7 c Bel. Unless CEU(au) = SEU(a,), SEU over-valuesthe unsustainable act compared to CEU. Since CEU and SEU value the sustainable actequally, SEU leads to a systematic bias in favour of the unsustainable act, as compared toCEU, which was judged to have a stronger normative basis for decision making underambiguity than SEU. CEU would choose the sustainable act whenever SEU does. However,there are also situations in which CEU would choose the sustainable act while SEU wouldchoose the unsustainable act. CEU puts the "burden of proof", or the burden to reduce the158ambiguity, on the unsustainable act, which leads to the ambiguity. In the absence of anyknowledge about the unsustainable act (a vacuous belief function over the possible utilityrange), use of CEU would lead to choosing the sustainable act as a default.It was discussed before that even though normative use of CEU is preferable to SEU,the foundation for normative use of CEU is not fully satisfactory. Nevertheless, I suggestthat a decision rule based in CEU be adopted based on the following reasoning. If thenormative basis for CEU, obtained from the cumulative dominance axiom, is not accepted,and the value function intervals for a, and au overlap (see (20)), no normative statement couldbe made about choice under ambiguity. If a decision between a, and au needs to be madenevertheless, the critical question becomes which of the two acts needs a normativejustification to be chosen. In this case, I would argue that using CEU (instead of CEU*) andfavouring the sustainable act reflects an intuitively appealing no-regret or robustness strategytoward ambiguity. Recall the example in section 4.3, which demonstrated how the value ofinformation would be considerably higher under CEU compared to the use of a Bayesianprior distribution. This suggests that under CEU, the learning option, which implies delayingthe ambiguous activity, would often be the optimal choice. Hence, the unsustainable actwould require a normative justification. Then, CEU should be used for choosing betweenau and a„ even if its normative basis is not fully accepted. It should be added that there is,clearly, no symmetry between CEU and CEU*. Neither empirical evidence nor intuitionwould suggest the optimistic attitude about ambiguity implied in CEU*. CEU* wasintroduced only to define a sensible (but extreme) upper bound on the value function.The proper definition of alternative acts is of crucial importance in any decisionmaking problem. Above, the sustainable act was compared with the unsustainable act. Thisreflects a typical situation of deciding about whether to implement a project or go ahead withan activity. In many real life situations there is a third option: learning more about theuncertainty before deciding whether or not to undertake the activity in question (see alsoPindyck 1991 for a discussion of the option value of delaying an irreversible investmentdecision). It was suggested that this third alternative would be chosen more often under159CEU. Of course, information has value only if actions are delayed until the information isobtained. It is worthless to study ambiguity while the act is undertaken that generates theambiguity (i.e. while the forest is cut, or the emissions are made). This point has policyrelevance, for example, in the context of the discussion about emissions of greenhouse gasesand CFCs. Often, the argument is made that more information is required before emissionreduction should be imposed. However, if current emissions are the cause of ambiguousdamages, the relevant alternative acts are continued emissions, emission reduction, oremission reduction with learning. The suggested alternative of continuing emissions andlearning simultaneously is misleading since information will have no value if it does not leadto additional choices (i.e. if the damage has been caused by the time information will bereceived). In the real-life policy debate, it often appears that the burden of proof lies onthose who want to change a current course of action. However, delaying a decision for thepurpose of further learning should not mean continuing the current activity but delaying theactivity that is cause of the ambiguity.5^ConclusionsThis chapter was motivated by the desire for a normative decision theory forenvironmental policies that accommodates the existence of ambiguity and ignorance. Itconcludes that SEU has severe limitations as a model for decision making under ambiguityand ignorance. Out of a number of alternative decision theories, a combination of theDempster-Shafer belief-function theory and Choquet expected utility is suggested as the mostsuitable model for the purpose at hand. The pessimistic attitude toward ambiguity that ismanifest in CEU is the result of a cumulative dominance axiom. If this axiom is consideredunacceptable, an incomplete ordering of acts is still possible that is independent of attitudestoward ambiguity. It is shown how the suggested theory addresses many requirements fora suitable model including the representation of ambiguity and ignorance, and theconstruction and aggregation of beliefs from evidence.160An important conclusion from the discussions in this chapter is that, while CEU ispreferable to SEU, there is no strong normative foundation for any model of decisionsmaldng under ambiguity and ignorance. Therefore, it would be most important in anydecision model to make the treatment of ambiguity and ignorance explicit rather than buryit in assumptions, as it occurs in SEU. The proposed theory allows this transparency bysetting bounds on the value function that hold regardless of the assumed attitudes towardambiguity. Thereby, those decisions that require assumptions about the treatment ofambiguity can be isolated from those that do not. As a direction for further research, it mayprove useful to empirically analyze a decision model with a behavioural parameter ofoptimism about ambiguity.Hopefully, the suggested theory offers a methodology that will help overcome thedeficiencies in the practical evaluation of uncertain environmental costs and benefits. Thesepractical deficiencies are often manifest in ignoring environmental effects altogether. If CEU,and hence the cumulative dominance axiom, is accepted as a normative basis, the impliedpessimism about ambiguity would tend to increase the cost or benefits assigned to uncertainenvironmental damage or improvements. Even if the axiom was not accepted, this chaptershows that there would not be a normative basis for accepting an activity that might beconsidered acceptable under a decision theory that suppresses ambiguity, such as SEU.Under either condition, the suggested theory would increase the attractiveness of sustainablealternatives to natural capital depletion, such as pollution abatement or compensatinginvestments, since the costs of ambiguous environmental damages would weight relativelyhigher compared to the risky, but often not or less ambiguous, costs of environmentalsustainability.While the proposed theory overcomes some of the shortcomings of SEU, itsusefulness remains to be analyzed in practical situations involving a complex choice problem.At least three problems with the suggested theory deserve further attention. The first problemis that there remains some doubt about the descriptive and prescriptive power of thecumulative dominance axiom. If this axiom is not invoked, and no other assumption about161behaviour under ambiguity is introduced, ordering of acts may be incomplete. At this stage,the only suggestion made is to use the behavioural parameter of optimism about ambiguity,a, as a parameter for sensitivity analysis on the given problem. The sensitivity of a decisionwith respect to the parameter of optimism about ambiguity, a, would have to be explicitlydiscussed which makes the treatment of ambiguity and ignorance transparent in the decisionmaking process. The weak normative basis for any decision that depends on choice of aspecific a, would justify a no-regret approach of not undertaking any activities that generateambiguity and would be undesirable with at least some a value.The second problem relates to information that is available but not used in calculatingCEU. Evidence about a problem situation is represented in the belief function over allsubsets of the set of states. However, the calculation of CEU is not based on the beliefsabout all subsets. CEU only recognizes the states resulting in the highest and lowest utilityin a subset of states. This means that in the calculation of CEU, a possible difference inbeliefs over the set of states {col, 6)2, ()3} and another set {0.)1, (03} is ignored ifu(a(coi)) > u(a(6.)2) > u(a(w3). With the cumulative dominance axiom for CEU one has to acceptthat information such as the difference between beliefs in the two sets above is irrelevant tothe problem. Similarly, with Savage's axioms for SEU one accepts that the informationrelating to a probability's reliability is irrelevant to a given problem.The third concern about the suggested theory relates to the requirement of specifyingbeliefs over subsets of states and upper and lower bounds for the utility obtained fromscenarios. Just as difficult as it is to define an additive probability measure for SEU, it maybe difficult to define beliefs and upper probabilities of particular states, in some situation.However, a belief function appears to be a more natural way to express evidence andrepresent incomplete knowledge of the decision problems. Moreover, eliciting a probabilityrange for a state would be less prone to framing effects and other biases that result from thedescription of the problem (i.e. the presentation of elementary states or the scale on whichoutcomes are presented (physical impacts versus dollar versus utility). Also, it is difficult toimagine an approach for considering scenarios unless some bounds can be determined on162their utility outcomes. Again, in some decision problems this may still be too rigid astructure. More often, however, it appears that there are natural upper and lower bounds onthe impact that a specific act can have, determined by either geographic scale of the act orthe number of people that can possibly be effected or the like. The major advantage ofputting probability weight for scenarios on the upper and lower bounds only lies in avoidingthe arbitrariness arising from assigning weights to intermediate utility values.CHAPTER IVA SUSTAINABILITY CONSTRAINT:REQUIREMENTS, SPECIFICATIONS, AND APPLICATIONS1^IntroductionThe purpose of this chapter is to consider the conclusions from chapters I-III andprovide a constructive perspective on policies designed to remedy the deficiencies ofconventional economic analysis of natural capital depletion. Specifically, this chapterattempts to show how the imposition of a sustainability constraint on the economic activitiesof the current generation would address the concerns about the deficiencies of market forcesto bring about efficient and intergenerationally equitable use of natural capital. Whilesustainability is an increasingly popular policy objective, there is little agreement on theprecise meaning of this concept. Therefore, an additional objective of this chapter is toprovide an operational definition of sustainability that can be applied to practical policymaking, such as the decision about a resource project. This chapter will show how economicanalysis tools, such as cost benefit analysis, that were developed for an almost empty world,can be adjusted in order to be useful for analysis in a full world. The chapter presents amethod for evaluation and comparison of projects or policies that involve depletion of naturalcapital, for example, in the form of depletion of a non-renewable resource, over-exploitationof a renewable resource or exhaustion of the biosphere's waste absorbtion capacity.Often, detailed formal analysis is required in order to better understand a specificproblem. In other cases, a broad and simultaneous view of many issues is required to gainunderstanding. While the first three chapters have followed the former approach, this chapterfollows the latter. In this chapter, I try to assess the aggregate impact of many differenteffects: those analyzed in chapters I-III as well as others that are not analyzed in detail in this163164thesis. This chapter attempts to synthesize the discussion of these effects and to explore thelimitations and implications of formal economic modeling of natural capital depletion. As aresult, this chapter relies less on formal analysis than on summaries and intuitive argumentsthat seem more appropriate for the material at hand.In section 2, I will summarize the deficiencies of conventional analysis of naturalcapital depletion as it follows from the discussions in chapters I, II, and III. It will be shownthat, due to various market failures, conventional economic evaluation is likelyunderestimating the costs of natural capital depletion. It will be argued, based on principlesof intergenerational justice, that adequate compensation of future generations for thedepletion of natural capital is required. The discussion in section 3 will show how asustainability constraint would be able to deal with the deficiencies of conventional analysis.Subsequently, a practical approach for the integration of a sustainability constraint withproject analysis is suggested. The "sustainable supply rule", proposed in section 4, wouldallow derivation of a sustainable price for the depletion of natural capital which should beused as the shadow price for natural capital depletion in project analysis. Section 4 alsodiscusses applications of the sustainable supply rule. Section 5 presents a stylized case studiesthat applies the sustainable supply rule to an oil development project.2 The Need for a Sustainability ConstraintThis section discusses a variety of reasons why the evaluation of natural capitaldepletion with conventional methods is deficient. In the first part of this section, I willabstract from issues of intergenerational welfare distribution. I will discuss the marketfailures that lead to deviations of the market price from the economic value of natural capitaland the systematic biases against remedying these market failures. Also, the inherentincompleteness of shadow prices for natural capital due to the uncertainty about long-termenvironmental costs will be demonstrated. The second part of this section will address thespecial concerns arising from the intergenerational problem with respect to depletion of165natural capital. Both parts of this section together demonstrate the need for a sustainabilityconstraint on economic activities which would have to be reflected in economic evaluation.2.1 Inefficient Market and Shadow Prices of Natural CapitalThis sub-section will demonstrate why externalities and the use of a private discountrate that is higher than the social discount rate will lead to inefficient market prices of naturalcapital. These market failures suggest the need for government intervention. However, inorder to design efficient environmental policies, it is necessary to shadow price naturalcapital. Therefore, the practical deficiencies in shadow pricing natural capital will bediscussed. Subsequently, suggestions will be made for a better approach to shadow pricingunder uncertainty based on the decision theory proposed in chapter III.2.1.1 Lack of Reliable Market Prices for Natural CapitalAn investor in human-made capital would expect that property rights for theinvestment are established and that markets exist on which the capital and outputs of theinvestment can be exchanged, in order to be able to reap the benefits of the investment.Natural capital, on the other hand, exists without an investor to ensure the existence ofproperty rights and markets for such capital. Hence, in contrast to human-made capital,property rights as the prerequisite for the existence of markets do not exist for many typesof natural capital. Since markets do not exist traditionally, they would have to be created.Creating markets for natural capital, however, is often prohibitively costly since many typesof natural capital provide public goods, and their depletion is associated with negativeexternalities. Examples include forestry, fisheries, public rangelands, ground-water resourcesand the waste absorption capacity of air, land, and water. The use of these resourcesgenerates external costs that are not considered in individual decision making. Even whereproperty rights can be created, they are often less secure than those for other assets. Forexample, governments in developing countries may lack the institutional strength to restrictaccess to natural resources, resulting in inefficient over-usage and under-pricing of the resource.166When the depletion of natural capital is caused by production, and the costs ofdepletion are not included in the product's price, an environmental externality exists.Externalities can be remedied by, for example, Pigou taxes or the introduction of trade withemission certificates (see Baumol and Oates 1988). However, these instruments are rarelyused. Public decision makers seem to have few incentives to implement policies for theinternalization of external effects, since they are more exposed to the concentrated influenceof internal beneficiaries (the polluter or user of natural capital) than to the dispersed voicesof damaged individuals. For a single polluter, lobbying politicians is much cheaper than fora large number of damaged individuals. who would suffer from the costs of organizing andcoordinating their collective lobbying efforts (see Mueller 1989, pp 235-238, for a reviewof the pertinent literature on lobbying and government regulation). Facing intense lobbyingby the polluters, politicians can use the prevailing uncertainty and the significant informationrequirement for determining external costs as a justification for delaying efficient policies.It would be insufficient to remedy recognized market failures only. Even if Pigoutaxes were implemented for all known and proven external costs, the long lag from the timeenvironmental damage is caused to the discovery of an environmental problem and topolitical recognition and action, would lead to systematic underestimation of external costs.Static consideration of policies to internalize externalities neglects dynamic incentives for thegeneration of external costs. For utility maximizing individuals, profit maximizingcorporations and social welfare maximizing local or national governments, there is asystematic incentive to generate internal benefits not only by productive activities but alsoby shifting costs to external entities. Costs of depletion of natural capital can be shifted tooutside individuals, corporations, regions, or countries. This continuous and pervasiveincentive to invent new ways of depleting natural capital causes governments to notoriouslylag behind in charging a price for the use of natural capital or securing property rights.Together, these factors would explain some of the apparent deficiencies in theimplementation of efficient environmental policies and the lack of reliable market prices fornatural capital.167The depletion of some types of natural capital, notably some non-renewableresources, is not directly linked to negative externalities. Following conventional resourceeconomic analysis, in a world without extraction costs and with certainty or completemarkets, maximizing social benefits from depletion of a non-renewable resource wouldrequire depletion at a rate such that the price of the resource rises at the marginal rate ofreturn to investment in the economy (see Hotelling 1931). Similarly, the price of a renewableresource would have to change with the rate of return minus the natural rate of growth ofthe resource. The initial price for a non-renewable resource (or a renewable resource witha growth rate below the interest rate for all stock levels) would be set such that the resourceis depleted exactly at the time when the rising price reaches the cost of a backstoptechnology, or demand is reduced to zero. The inter-temporal resource allocation, resultingfrom the Hotelling rule, would be efficient since rents derived from the resource could beinvested at the rate of interest such that no other resource depletion path would Pareto-dominate the outcome. In a world with perfect and complete markets for natural capital, thisefficient result would be brought about by the market mechanism.Perfect and complete markets, however, are unlikely to exist. Chapter II has shownthat complete markets are, in fact, impossible due to the intergenerational structure of theproblem. Even if perfect markets for a natural resource existed and no obvious externalityprevailed, the market price would underestimate the social opportunity cost of resourcedepletion if the private discount rate of resource owners exceeds the social discount rate. Theopportunity cost of depletion (the user cost) is the discounted future benefit foregone becauseof depletion. Hence, the user cost depends on the discount rate used by the decision maker.If the decision maker uses a discount rate higher than the social discount rate, the resourcewould be depleted too fast and prices would be less than the social opportunity cost. Anextensive body of literature deals with the complex question whether private and socialdiscount rates do or do not coincide (for an overview of the issues see the introduction inLind et al. 1982). The main arguments for a difference between the two rates are based ondifferences between social and individual risk, taxes, and capital market imperfections. Here,I will highlight only some of the concerns suggesting that the decision maker's discount rate168is likely to exceed the social discount rate. The first group of arguments suggests that thediscount rate of profit maximizing resource owners is above the social discount rate. Thesecond group of arguments suggests that individual decision makers apply a discount ratehigher than the discount rate of a profit maximizing resource owner.First, the discount rate of profit maximizing resource owners will be excessive ifprivate marginal rates of return on investment exceed social rates of return. In the realworld, the social return on investment would be below private returns because of theexistence of external costs that are not internalized for the reasons discussed above. Ingeneral, investment in natural capital seems to be more associated with external benefits(such as a forest that provides external benefits in the form of recreational value, climaticstabilization, soil stabilization and habitat for wildlife) while industrial investment producesprimarily external costs (such as pollution and waste products of an industrial plant).Incomplete internalization of external effects means that market forces would equalize privatebut not social rates of return on investment. With declining marginal rates of return, thiswould lead to excessive industrial investment and insufficient investment in natural capitalcompared to the social optimum. The private discount rate would be well above the socialrate of return on investment. Note that this argument is based on the existence of negativeexternalities from general investment in the economy, not necessarily from depletion of thenatural resource.Second, market interests rates are based on the decisions of short-living individualswho exhibit a positive rate of pure time preference or impatience (the rate at which futureutility is discounted). However, for the consideration of the social discount rate, it would beinappropriate to extrapolate from pure time preference over the short lifetime of anindividual to intergenerational time. While a moderate variation of well-being during anindividual's lifetime would be rationally accepted, the same rate of pure time preferenceapplied to very long time periods would imply acceptance of extreme hardship during latertimes for moderately increased well-being earlier on. The extrapolation of pure time169preference to periods that exceed human lifetime seems inappropriate. Social time preferencewould, therefore, be lower than individual pure time preference.Third, private discount rates are based on the consideration of risks that are absentfrom a social point of view. Private discount rates would be excessive due to the risk ofappropriation and the instability of property rights for natural resources. Private owners ofa resource provide for the risk of restrictive government regulation by increasing extractionand, implicitly, using a higher discount rate. This implicit discount rate would be above thesocial discount rate since resource appropriation, would merely imply a change of ownershipand not represent a social risk that requires discounting. In addition, market interest ratesreflect the individual risk of death and the resulting uncertainty of future consumption. Thisleads to higher discounting of future consumption. On the other hand, social discountingshould only reflect the corresponding, but much lower, risk of extinction. However, the riskof extinction of the human species is, at least in part, endogenously determined by theactivities and the discount rate used by the present generation. Discounting for anendogenous risk of extinction could justify the present generation's deliberate decision to bethe final generation and deplete all natural capital. Assured extinction after the presentgeneration would justify a zero weight on future consumption (equivalent to an infinitediscount rate). This would deny future generations their right to existence and stronglyviolate our ethical intuition. Social discounting on the basis of endogenous risk of extinctionis, therefore, unacceptable.Finally, capital market imperfections can lead to resource owners under financialdistress depleting a resource excessively, implying a discount rate above the social rate. Acompany under financial distress, facing risk of bankruptcy, may disregard the opportunitycosts of depletion in order to avoid the costs of bankruptcy and reorganization. Similarly,a country under a foreign exchange constraint may choose to increase depletion and exportof natural resources at a depressed price. This would explain the increased, and from aglobal perspective inefficient, liquidation of natural capital in several highly indebteddeveloping countries in recent years.170The second group of arguments why decision makers would apply a discount ratehigher than the social discount rate is based on evidence of resource owners' behaviour thatis inconsistent with profit maximization. When resources are owned by governments,extraction licenses are often granted through tender and bidding procedures that do not takeinter-temporal welfare maximization into account and encourage excessive extraction.Behaviour of decision makers that is inconsistent with profit maximization can be explainedby the principal-agent problem, or the conflict of interest, between the owner of the resource(a country or a company) and the individual decision maker (a manager or politician).Individual incentives may lead to explicit manipulation of decisions, however, more likelyis a subtle but systematic bias in the decision making of government, companies and lendingagencies.If politicians use GNP figures, or other related income measures that omit accountingfor natural capital depletion (see Ahmad et al. 1989) as signals of their performance to theelectorate, they would have an incentive to maximize GNP rather than, unobservable, socialwelfare and, hence, encourage excessive resource depletion. Similarly, the manager of acompany who knows more than the shareholders about the value of the natural resourceowned by the company would have an incentive to underestimate future opportunity costsin order to boost present profits and his own corresponding compensation. Also, managersof lending agencies that decide about the funding of resource projects often face incentivesthat are biased toward achieving short-term benefits due to the difficulties of any system ofpersonal long-term accountability. The incentives of project managers may explain why thecost-benefit studies of extraction projects often do not include a user cost. For example, theWorld Bank frequently includes the full resource rent as a benefit in the calculation ofeconomic rates of return for extraction projects (see von Amsberg 1993). Hence, under theconditions that, first, decision makers within an organization maximize not benefits to theorganization but personal utility, second, their tenure is shorter than the time span overwhich their decisions have effects and, third, information is incomplete, there would be asystematic incentive to undertake activities that generate benefits in the present and costs inthe future even if discounted future costs outweigh present benefits. Since information171imperfections can be especially severe for future opportunity costs which are often lessvisible than financial costs, which are reflected in financial statements, opportunity costs maybe hidden from the principal (the shareholders or the electorate) relatively easily. If projecteconomists, managers and politicians ignore or underestimate user costs, the supply ofextracted natural resources will be higher then in the efficient market equilibrium. Inaggregate, this effect is likely to be non-marginal and would depresses market prices fornatural resources.In summary, markets for natural capital often do not exist due to the lack of propertyrights. Policy makers seem to have little incentive to implement efficient environmentalpolicies to internalize externalities. Even if policy makers desire implementation of efficientpolicies, it is difficult to establish markets for natural capital due to the pubic good natureof many natural resources and dynamic incentives to use natural capital in increasinglypervasive and evasive ways. Even when markets for natural capital exist, market prices arelikely underestimating economic values since decision makers often have individualincentives to ignore opportunity costs and use an implicit discount rate above the appropriatesocial discount rate.2.1.2 Difficulties with Shadow Pricing Natural CapitalFor many environmental policies, such as setting environmental standards or taxes,or evaluating a project that depletes natural capital, an estimate of the economic cost ofnatural capital depletion is required. Due to the discussed distortions of market prices, thedepletion of natural capital needs to be shadow priced. The correct shadow price is theopportunity cost of natural capital depletion, or the value of natural capital in its next bestalternative, present or future use. Determining this shadow price is a complex task since forglobal resources (such as oceans, atmosphere), non-tradable resources (such as forests) andeven tradable resources evaluated from a global perspective (maximizing social welfare ona global and not a national level), the best of all possible alternative uses, now or at any timein the future, has to be determined in order to be able to shadow price the resource. If the172depletion of a tradable resource is evaluated from a national point of view, the expectedinternational price of the resource, after depletion, has to be estimated.The opportunity cost of depleting a stock of a non-renewable resource is generallyrecognized as the user cost which is the discounted value of future benefits foregone throughcurrent depletion (see Hotelling 1931). The applicability of the opportunity cost concept toother types of natural capital depletion is often neglected (see also Akerlof 1981). Anydepletion of natural capital that excludes or diminishes the potential benefits from identicalor alternative activities, now or in the future, implies a social opportunity cost that shouldbe reflected in the shadow price. Consider, for example, an emission generating factory thatdue to low levels of emission does not cause any environmental damage. If a second factorywas economically feasible but would lead to environmental damage due to the aggregateemissions of both factories, the first factory would have caused a social opportunity costwhich, for efficiency reasons, should be allocated to that factory as a user charge for a non-marketed production factor. This non-marketed production factor is natural capital in theform of the limited absorption capacity of the natural environment. As this example withdiscrete investment opportunities shows, a social cost for using the environment can occureven in the absence of environmental damage. This broader view of the opportunity costconcept highlights the pervasiveness of external costs of economic activities in industrialeconomies and demonstrates that many "free goods" are in fact not free but valuable naturalcapital.The problems with shadow pricing natural capital depletion can best be illustrated byexamining shadow pricing practice in cost-benefit analysis. For that purpose, I havepreviously analyzed the economic evaluation of several World Bank projects that depletenatural capital (von Amsberg 1993). Many World Bank economists have made importantcontributions to the natural resource economics literature, and the World Bank is arecognized leader in the development and application of cost-benefit analysis techniques. Itis striking, therefore, that in most World Bank project evaluations natural capital is treated173as a free good. Thus, the depletion of natural capital is not treated as a cost. This deficiencyis most noticeable for the evaluation of natural resource depletion, land use and emissions.In most projects that involve depletion of a natural resource, such as mining or oil/gasprojects, no user cost or depletion premium is included in the calculation of economic projectcosts. For example, a gas extraction project would be evaluated as if the project actuallyproduced natural gas while, in fact, it is merely depleting a natural gas reservoir. In somecases, a depletion premium is calculated, however, the opportunity cost of depletion isdiscounted at a rate higher than the assumed increase in real prices of the in-situ resource.This approach leads to a negligible depletion premium and is inconsistent with efficientmarkets, which would lead to the user cost of a specific in-situ resource rising at a rate equalto the opportunity cost of capital.In many World Bank project reports, the opportunity cost of land is assumed to bezero, often with the justification that there is no alternative commercial use for the land. Forprojects that not only use but also degrade land (as is the case with many agricultural andmining projects), this assumption is clearly questionable. Most land serves many functionsbeyond their obvious commercial use, including use for various subsistence activities, andrecreation. Moreover, in an increasingly crowded world, many remaining wilderness areasare important for preserving biological diversity.Also, the costs of atmospheric emissions are usually not quantified in the economicanalysis of projects even though possible damage from emissions is often acknowledged inthe reports. Carbon dioxide emissions are not even mentioned in most of the analyzedreports. Two justifications for not including damage costs from emissions are often given,but should be rejected. First, for almost all pollutants there is considerable uncertainty aboutthe damages arising from emissions. As a result, a quantification and evaluation of damageis often not attempted. Omitting damage costs from the analysis implies the assumption ofzero damage costs. Assuming zero costs because of uncertainty is an obviously undesirablepractice. Second, many projects include some pollution abatement measures. Even if the174remaining emissions do not exceed international emission standards, damages arising fromthese emissions need to be considered. If emission standards were set efficiently, marginalabatement costs would be equalized with marginal damage costs. Hence, damage arises evenin the presence of an efficient pollution standard. While the adherence to pollution standardsis commendable, it does not alleviate the requirement to evaluate damage arising fromemissions within given standards. Both, abatement costs and damage costs arising fromremaining emissions need to be included in a complete economic analysis.All the examples of deficient practices have in common that shadow pricing of naturalcapital is seriously impaired by the fact that the services rendered by natural capital arepervasive and difficult to evaluate completely. The opportunity cost of clear-cutting a forestincludes all benefits that could have been derived from the standing forest. These includerecreational benefits, climatic stabilization, return on sustainable forestry and others. Whilethe number of effects to be considered and the information requirements may beoverwhelming, more problems arise because of limited knowledge about the functioning ofthe biosphere. The costs of releasing CFCs into the atmosphere could not have beencorrectly evaluated twenty years ago because the destructive effects of CFCs on the ozonelayer were not yet known. Moreover, there is great uncertainty about cultural andtechnological development of humanity; hence, the economic value of resources in the distantfuture is almost impossible to estimate. Too many unexpected events and potentialdiscoveries lie ahead to make sensible point estimates of resource values hundreds of yearsahead. As a result, practical evaluation of the opportunity costs of natural capital depletionis often incomplete since only those effects that are already known and understood areincluded. Our knowledge that there are potential, yet unknown, costs of depletion is notreflected. Thus, conventional shadow pricing of natural capital depletion implies a systematicunderestimation of depletion costs. The benefits from depletion, on the other hand, areevaluated more or less completely since they are usually captured by the market for outputproducts, such as steal, pulp, electricity, or others.175Shadow pricing techniques that are more sophisticated than those used in practice areavailable. For the evaluation of risks, the option value concept has been developed (seeCichetti and Freeman 1971). Irreversibility can conceptually be included by adding a quasi-option value (see Arrow and Fisher 1974)) However, sophisticated evaluation techniquesare rarely used in practice due to the high information requirements and the involved effortthat does not seem to be justified in light of general data imprecision. Moreover, all thesetechniques have conceptual limitations since they are based on subjective expected utilitytheory and, subsequently, do not appropriately deal with situations of ambiguity andignorance (see chapter III) even though ambiguity and ignorance are most important for thecosts of natural capital depletion.2.1.3 A Better Default Value for Natural CapitalConsider the extreme case of complete ignorance about the cost resulting fromdepletion of natural capital. I will call the cost assigned to depletion under completeignorance the 'default value'. The current practice of evaluation is based on positiveenumeration of those costs that are already known; hence, the default value of natural capitalis zero. Already, a good Bayesian decision maker would reject current practice. The neglectof natural capital in economic evaluation strongly conflicts with our understanding that theearth's natural capital is a proven system resulting from millions of years of evolution thatprovides life support for humans. This argument is based on the view of natural capital notas an arbitrary accumulation of natural resources but as a complicated web of interactionsthat has emerged from evolutionary competition. Therefore, there is a good chance that anyinterference with this system causes a cost to humans that a Bayesian decision maker wouldtake into account even if the precise mechanism by which this cost is generated is notcompletely understood. Past experiences of unexpected negative consequences of interference' The environmental economics literature uses the term 'option value' for what isusually referred to as a risk premium. A 'quasi-option value' in environmental economicswould be called an option value in financial economics.176with natural systems (ozone layer depletion, greenhouse effect etc.) would suggest that acautious approach with respect to depletion of natural capital is in our best interest. Such acautious approach would reflect the expectation of unpredictable costs arising from thedepletion of natural capital.The problems of analyzing costs under ambiguity and ignorance have been discussedin chapter III. According to that discussion, a belief function would be a more appropriaterepresentation of uncertainty than the subjective probability distribution a Bayesian decisionmaker would use. The proposed decision theory, based on Choquet expected utility (CEU),would lead to a more cautious approach in the evaluation of natural capital depletion. Undercomplete ignorance, the cost assigned to depletion would be the maximum of the range ofpossible costs. This highest possible cost would be the cost of offsetting the damage(providing a sustainable substitute or implementing measures that mitigate any damage tonatural capital). Hence, the default value of natural capital would be the cost of offsettingany damage (I will refer to this as the cost of a sustainable substitute from here on). Thesuggestion for evaluating natural capital at the cost of a renewable substitute goes back toJohn Ise (1925). Even if the normative basis for CEU is not accepted, the discussion inChapter III shows that, in face of complete ignorance, there are no normative grounds foraccepting a decision that is based on a lower value of natural capital depletion. The resultingapproach reflects the view that we do not fully understand the functioning of the naturalsystems; therefore, natural capital in its original form deserves considerable benefit of doubtfor its existence.Implementation of a more cautious approach to natural capital depletion would notimply that all natural capital must be left untouched. Rather, the pessimistic attitude towardambiguity, implicit in CEU, leads to a reversal in the burden of proof. Current practiceimplies that natural capital should be depleted unless it can be shown that depletion makesus worse off. A cautious approach would require that natural capital be left intact unless itcan be shown that depletion makes us better off. The difference can be significant in a worldof large uncertainties. Life on earth has emerged from co-evolutionary adaptation to the177current composition of the atmosphere. Hence, changing the composition of the atmosphereis likely to disrupt the functioning of the biosphere in fundamental ways. Therefore, it wouldbe acceptable to change the composition of the atmosphere (for example through excessivecarbon dioxide emissions) only if we understood the functioning of the biosphere sufficientlywell to be able to show confidently that a change in the composition of the atmosphere wouldmake us better off. On the other hand, current practice would suggest to pollute theatmosphere until we find out that the damage, which results from this change, outweighs thebenefits. We are currently learning about the functioning of the biosphere by trial and error.However, due to the increased human population and our ability to amplify our impacts onthe environment through technology, the stakes of this trial and error have grown so largethat it is now generating unacceptable uncertainty. Moreover, many forms of natural capitaldepletion, and thus our possible errors, are irreversible. This increases the risks and providesadditional support for a cautious approach and a reversal in the burden of proof.While CEU suggests that the default value for resource depletion would be the costof a sustainable substitute, we can, step by step, reduce this value as our knowledge aboutthe functioning of the world increases, and we conclude that it is to our benefit to depletenatural capital. The theory presented in chapter III would be used to evaluate the depletioncost at every stage in this learning process. This is in contrast to current practice that wouldbegin with a default value of zero for a presumably free good. As unexpected damage arises,or the resource becomes scarce, the cost of natural capital depletion would be increased stepby step. Once uncertainty is resolved, the resulting value assigned to natural capital is thesame under either approach. The difference is that during the learning process, under thecautious approach, the true value of natural capital depletion is approached from a highervalue. The cautious approach leads to a reversal of the default assumption of natural capitalas a free good to the assumption that every unsustainable activity must bear the cost of itsconversion to a sustainable activity.Some examples can illustrate this change in the default assumption. For the evaluationof an oil field's depletion, the cautious approach would imply initial pricing of the energy178resource at the cost of a sustainable substitute, such as solar energy. This shadow pricewould be reduced if it can be shown with reasonable confidence that depletion does not causeunexpected environmental damage, that there are no alternative future high-value uses foroil, and that depletion makes us better off. A renewable resource, such as a forest would bepriced according to the sustainable yield at a desired stock level. Cutting trees beyond thesustainable yield would be acceptable only if it can be shown that all costs arising fromreduction of the forest are outweighed by the benefits. If a sustainable substitute for naturalcapital does not exist, the default value for depletion would be the maximum possibledamage. The extinction of a species would carry an infinite cost unless its future value canbe bounded from above. This cautious approach can help avoid large costs that arise if analternative high-value use is discovered after a resource is depleted. For example, sustainableyield forestry would have reduced the loss of a large numbers of Pacific Yew trees that wereonly recently discovered to contain the ingredient for a potent cancer fighting drug ('Taxol').If the future value of a non-renewable resource is known with some confidence, depletionwould be acceptable at a price that provides for compensation of the opportunity costimposed on the future.2.2^The Intergenerational ProblemThe depletion of natural capital has impacts that reach beyond the time span of onegeneration. Chapter II has shown inefficiencies arising from natural capital depletion underuncertainty in an intergenerational context. Other intergenerational market failures will bediscussed in this section. Chapter I has shown problems of intergenerational welfaredistribution that result from natural capital depletion. Both types of intergenerationalproblems are examined in this section, leading to the definition of requirements forovercoming the intergenerational inefficiencies and the distributional concerns.1792.2.1 Intergenerational Market FailuresIt was already discussed in an intra-generational framework, that there is inefficientlyhigh depletion of those types of natural capital that have the attributes of a public good. Inmany cases, the costs of natural capital depletion will be realized only years or decadesahead. In an intergenerational context, the externality problem is aggravated by the fact thatthe damaged future generations are not represented in the political decision making processesthat could remedy the externality problem. Hence, individuals of the current generation donot only have an excessive incentive to deplete natural capital but also do not have acollective incentive to curtail depletion to efficient levels since the costs of depletion areimposed on future generations as an externality. While it is fairly obvious that pollution andsome other types of natural capital depletion generate externalities, there are other, lessobvious, types of intergenerational externalities that lead to concerns about the efficiency ofmarkets across generations. An example would be investments that yield long-term returnsthat are inherited by future generations, such as large public projects (a dam or a subwaysystem) or economy-wide learning-by-doing effects (see Arrow 1962). In these examples,early generations would not undertake efficient investments because the returns are realizedonly to future generations. This diffusion of benefits to future generations leads to a marketfailure similar to those failures that are well recognized for research and developmentinvestments in the intra-generational context (see Tirole 1988, pp. 389-419, chapter 10).Chapter II has analyzed another type of intergenerational market failure.Intergenerational insurance markets are necessarily incomplete since no generation canparticipate in trade before it is born. Therefore, markets do not bring about efficientintergenerational allocation of the risks that are arising from natural capital depletion.Chapter II has shown that there is an excessive incentive to deplete natural capital (i.e.,reduce the diversity of species) if the resulting risk is primarily born by future generations.Similarly, there is an insufficiently low incentive to invest in sustainable substitutes thatreduce future risks from the depletion of natural capital. On the other hand, incentives to180invest into activities with risks that could be diversified across generations are inefficientlylow.In these examples of intergenerational market failures, a coordination mechanismbetween generations could make all generations better off. Such a coordination mechanismwould lead to the first generation taking certain actions, such as investing in building a dam,in return for the second generation honouring an implicit intergenerational contract (it cannotbe an explicit contract since the second generation is not yet born at the time this contractwould have to be closed), such as paying for the dam. While such a contract would makeall generations better off, the second generation does not have an ex-post incentive to honourit. Alternatively, an implicit intergenerational contract may allow the first generation toimpose an external cost on the second generation if the former compensates the latter for thiscost once the second generation is born. In this example, the first generation would not havean incentive to honour the contract. In both cases, legal and institutional barriers, such asdebt arrangements could prevent generations from repudiating the contract. Note that manysocial security systems operate on a pay-as-you-go basis. Their functioning is obviouslybased on a similar implicit intergenerational contract. The two elements of the suggestedintergenerational contract would be a mechanism to compensate another generation forimposed external effects (costs or benefits) and a mechanism to equalize risks acrossgenerations (equalize the ratio of marginal utility between different states of the world).2.2.2 The Problem of Intergenerational Welfare DistributionThe analysis in chapter I has shown that problems of intergenerational welfaredistribution lie at the heart of the problem of natural capital depletion. This section does notattempt to deal comprehensively with the complex philosophical questions involved inintergenerational choice. Only some important concerns of relevance to the evaluation ofnatural capital depletion will be discussed here (see Berry 1983 for a broader discussion).In chapter I, the public good nature of caring for future generations was discussed. Here itis assumed that, in their individual choices, generations do not consider the impacts of their181decision making on the welfare of future generations. This section summarizes some resultsfrom chapter I and suggests an intuitive requirement for a rule that would lead to equitableintergenerational welfare distribution. The imposition of such a rule can be seen as oneapproach to collective action in order to overcome the public good problem of caring forfuture generations.Practical economic analysis usually emphasizes the efficiency aspects and disregardsthe possible distributional impacts of activities that deplete natural capital. Projects, forexample, are commonly evaluated according to the potential Pareto criterion (also referredto as the Kaldor or compensation criterion). This criterion states that an activity should beundertaken if those who gain from the activity could compensate those who lose and still bebetter off. Since hardly any project is conceivable that does not lead to losses for at leastsome individuals, projects are justified on efficiency grounds even though they only achievea potential Pareto improvement and not an actual Pareto improvement. The use of thepotential Pareto criterion is quite justifiable in cases where no individuals suffer large lossesfrom a project, where effective income redistribution mechanisms are in place and where nosystematic bias exists against any group. Under these circumstances it can be expected thatcompensation can be implicitly achieved by the sum of all projects which in aggregate wouldprovide net benefits to every individual.The depletion of natural capital often imposes costs on future generations. In thesecases, the conditions that would justify use of the potential Pareto criterion are unlikely tobe met. Large losses to specific generations may be possible because long-term discountingdiminishes the impact of even catastrophic events in the future (i.e. a nuclear accident).There is no reason to assume that the income distribution between present and futuregenerations is explicitly endorsed. Furthermore, the decision not to compensate losers canbe reversed within the same generation. In intergenerational choice, however, gainers willhave died by the time losers live. Hence, the decision to not actually compensate isirreversible. Also, the selection of projects by the current generation would be systematicallybiased in favour of projects benefitting the current and burdening future generations. For182these reasons it was suggested by Page (1983) to exclude intergenerational problems fromevaluation with the potential Pareto criterion. Concentration of economic analysis onefficiency may be justified for intra-generational choice because of the existence of effectivemechanisms to bring about the socially desired income distribution through governmenttransfer payments. In intergenerational choice, on the other hand, there is a fundamentalasymmetry between present and future generations since distribution of resources isdetermined by the present generation alone. Future generations have no influence on currentdecisions. A pure efficiency criterion could, therefore, not prevent any single generationfrom consuming all resources leaving future generations uncompensated. Hence,ntergenerational justice needs to be explicitly considered.The analysis in chapter I, building on work by Howarth and Norgaard (1990), showsin a general-equilibrium model with overlapping generations that for every possibledistribution of resource ownership across generations, there is a different efficient depletionpath. Market prices would be correct indicators of economic value only if the implicitassumptions about inter-generational resource distribution were considered sociallyacceptable. However, an increase in the share of the resource owned by the first generationunambiguously increases welfare of the first and reduces welfare of the second generation.Hence, a self-interested first generation would assume ownership of the full resource stock,leading to gloomy welfare levels for future generations even if the resource is depletedefficiently. The channelling problem described in chapter I aggravates this income effect.There is an additional income effect against the welfare of future generations arising fromthe depletion of non-marketed types of natural capital (negative externalities). A systematicincentive for the current generation to generate current benefits at the expense of futurecosts, leads to additional concerns about future generations' welfare even if the actions ofthe first generation are efficient.Preferences about the desired intergenerational welfare distribution can be expressedin an intergenerational social welfare function that reflects the desired ethical position. Oneconceptual approach for choosing a social welfare function has been suggested by Rawls183(1971). The conflicting interests of different generations would be assessed by thosegenerations behind a "veil of ignorance", meaning that individuals would have to decide ona social welfare function without knowing which generation they are going to be part of.Rawls suggests (specifically in an intra-generational context, though) that a maximin welfarefunction would be chosen under these circumstances. Yet, it is difficult to imagine thatconsensus on a social welfare function could be achieved, in particular since Arrow (1951)has shown that it is impossible to construct a social welfare function based on individualpreferences that meets some basic intuitive requirements. If agreement could be found on asocial welfare function, the optimal distribution of resources to different generations couldbe derived from it. However, a social welfare function is hardly operational in anintergenerational context. On top of uncertainty about the impacts of earlier generations'decisions on the consumption opportunities of later generations, there is uncertainty aboutthe preferences of future generations. With the assumption that preferences of futuregenerations are like those of the present generation, the present generation would undulyrestrict future choices. Other assumptions about future generations' preferences would becompletely arbitrary. More operational than a social welfare function would be Page's (1983)suggestion to require equal opportunities for every generation rather than a specified levelof welfare.It is fairly obvious that the assumptions made by a selfish current generation aboutresource distribution would be unacceptable and future generations would need to beexplicitly compensated for the depletion of natural capital. However, since consensus on asocial welfare function appears illusive, and the allocation of resource property rights basedon a social welfare function would not be operational, the intergenerational welfare problemwill instead be approached by attempting to define rules that are intuitively appealing basedon Page's notion of equal opportunities for all generations. The details of applying theserules would depend on specific circumstances and will be discussed later. First, everygeneration can use natural capital without depleting it. This would include harvesting thesustainable yield from renewable resources and using the environment as a sink for wasteswithin the natural capacity for regeneration. Second, compensation of all future generations184would be required for any depletion of natural capital since depletion excludes the use ofnatural capital by any future generation. The precise nature and quantity of suchcompensation will be the topic of much of the remainder of this chapter. However, twointuitively appealing requirements for adequate compensation can be established that wouldlet all generations accept the compensation as a substitute for the depleted natural capital.First, the benefits derived from depletion of natural capital should be shared equally amongthe depleting and all future generations. Second, a later generations must receivecompensation at least equal to the benefits it would receive if this later generation itself, andnot earlier generations, depleted the resource and compensated all following generations.This latter condition is meant to restrict wasteful uses of a resource that would be acceptableunder the condition of equal rent sharing alone. I will call compensation of future generationsthat meets these requirements adequate compensation.Under these rules for intergenerational resource distribution, the endowment of everygeneration would include the sustainable yield of the earth's natural capital plus the benefitsfrom depletion of natural capital if adequate compensation is made to future generations.This requirement of explicit compensation for any unsustainable use of natural capital wouldimply a major change in our cultural and legal understanding of ownership of land and othernatural resources. This is consistent with the conclusion that not all natural capital can beowned by the current generation for reasons of intergenerational justice. Owning naturalcapital would only include the right to harvest its sustainable yield while leaving its capitalvalue intact. For example, owning a mine would imply the right to exploit the resource onlyif adequate compensation of future generations was provided. Land and resource ownershipwould become comparable to trusteeship rather than to ownership as understood for human-made capital.3^The Appropriate Sustainability ConstraintsMost normative models of economic policies are based on the objective of socialwelfare. Since social welfare is neither observable nor directly measurable, all economic185policies to maximize social welfare are based on some implicit or explicit model of socialwelfare. Such modeling leads to two problems. First, as discussed there is no consensus ona normative basis for either an intergenerational social welfare function, in particular underuncertainty, or decision making under ignorance and ambiguity, which is pertinent for theproblems discussed here. Second, the involved problems are of such complexity that acomplete model would exceed individual and institutional processing capacities. As a result,a normative basis for decision making has to be assumed, and the model requires significantabstractions from reality. The discussion in section 2 has shown how these two steps leadto a significant biases against the welfare of future generation, in favour of depletion ofnatural capital.Since there is currently no basis for a consensus on either the normative assumptionson which decisions should be based, or the nature of the abstractions that are necessary toobtain an operational model, the decisions that are made should be robust to different modelsof social welfare. The requirement of robustness means that our decisions should performrelatively well for a variety of reasonable models of the world, at the cost of maximumperformance if the one model selected by the decision maker turns out to be the right one.Therefore, maximizing social welfare subject to a constraint that guards against mistakes thatcan result from incorrect specification of the model may be preferable to maximizing a wellspecified social welfare model. The introduction of such a constraint is the route suggestedto alleviate the deficiencies discussed before, and will be discussed throughout the remainderof this chapter. The imposition of a constraint itself is not based on maximization, but itfollows from the recognition of the limits of formal modeling in a complex world. It reflectsa strategy of reducing possible regret about decision making based on maximizing anincorrectly specified model. Of course, if one day one could agree on a well specified modelof the world, welfare would be higher without the imposition of such a constraint, whichwould then no longer be appropriate.In fact, some agreement appears to be emerging that a sustainability constraint oncurrent economic activities is the preferable approach to addressing the intergenerational186problem (see Marlcandya and Pearce 1988, p.42). A sustainability constraint would offset thediscussed systematic biases of decision makers against considering opportunity costs andagainst actually compensating future generations. A sustainability constraint would becomepart of the framework within which economic analysis takes place. The constraint wouldexclude economic activities that impede on basic rights of future generations, just like basicindividual rights are often considered immune to economic analysis. Within a sustainabilityconstraint, standard economic analysis based on efficiency would be fully applicable.In this section, I propose a sustainability constraint on current economic activities thataddresses the deficiencies of markets and conventional shadow pricing practices to bringabout an efficient and intergenerationally equitable use of natural capital. Three mainrequirements followed from the discussion in section 2. First, due to the inherentincompleteness of conventional evaluation and the lacking normative basis for decisionmaking under ignorance and ambiguity, the default assumption of natural capital as a freegood should be reversed, and natural capital depletion should be evaluated at the cost of asustainable substitute unless it can be shown that the economic costs of depletion are less.Second, to alleviate intergenerational market failures, a mechanism is required that wouldcompensate a generation for external effects imposed by another generation, and equalizerisks across generations. Third, concerns about intergenerational welfare distribution suggestcompensation of future generations for the depletion of natural capital and equal sharing ofthe benefits derived from depletion. In this section, a sustainability constraint on economicactivities will be discussed that would address these three requirements.The purpose of a sustainability constraint is to restrict current economic activities byrequiring that the current generation leave assets (human-made capital as well as naturalcapital) behind such that a specified variable, such as consumption or utility, can bemaintained at its current value in perpetuity. Various definitions of sustainability have beenproposed. Pezzey (1989) discusses a variety of different sustainability definitions. Thesedefinitions differ mainly in the variables to which a sustainability constraint is applied. Thecontroversy between proponents of different sustainability constraints is based on different187assumptions about the nature of substitutability between different types of assets. Daly andCobb (1989) discuss the distinction between strong and weak sustainability. A weaksustainability constraint requires the sum of the values of natural and human-made capitalstocks to be kept non-declining. Compensation for natural capital depletion would beinvestment in any other form of capital of equal value. On the other hand, a strongsustainability constraint would require non-declining stocks of human-made and naturalcapital, separately. Compensation for the depletion of natural capital would have to be madethrough investment in natural capital. This section suggests a systematic approach to theclassification of different sustainability constraints and discusses the appropriateness ofdifferent constraints.3.1^Strong and Weak SustainabilityThe difference between strong and weak sustainability can be explained with thediagram in Figure 4.1. The stream of human welfare, depicted at the top of the diagram, isthe ultimate concern of the economist with an inherently anthropocentric point of view. Theweakest sustainability constraint would apply at this highest level of aggregation. It wouldrequire that all human-made and natural capital together be kept intact such that it cansupport a non-declining stream of welfare in perpetuity. The concept of human welfare iseminently detached from the physical realities of natural capital. Welfare is the result of avariety of consumption streams that are combined in a welfare or utility function. Tworepresentative consumption streams that contribute to welfare are shown at level 2 inFigure 4.1. Questioning the substitutability between different consumption streams, the nextstronger sustainability constraint would apply to the different consumption streamsseparately. This sustainability constraint would require that groups of capital that generatedifferent consumption streams be kept intact, such that they can support consumption streamsthat are all separately non-declining. Similarly, consumption streams are created by aproduction process that combines various commodities. Level 3 would representsustainability of the supply streams for production. Finally, the strongest sustainabilityconstraint (level 4) would apply at the level of extraction of physical assets. This strongestLevels of SustainabilityLevel 1:sustainability of welfare1111Level 3:sustainability of supplies(energy, water, capital)Level 4: lite 0:0sustainability of extraction(oil, gas, iron ore)^ILevel 2:sustainability of consumption(heat, transportation, food)Substitution betweenresourcesSubstitution betweenconsumption streamsSubstitution betweenproduction inputsTypes of capital188sustainability constraints would require that the stock of specific assets be left intact as to beable to provide non-declining extraction streams.Figure 4.1 Levels of SustainabilityThe four discussed levels of sustainability are representative steps along thecontinuum between the abstract concept of human welfare and the physical concept ofindividual resource stocks or assets. They represent a decreasing level of abstraction and anincreasing proximity to physical realities. Clearly, a sustainability constraint imposed at ahigher level of aggregation and abstraction restricts the choices of the living generation lessthan a sustainability constraint imposed at the level of individual resources. The strongconstraint requires to leave capital intact such that a large number of extraction streams can189be maintained non-declining separately. The weak constraint, on the other hand, allows fora variety of substitution possibilities and only requires maintaining the welfare stream to benon-declining. The four representative sustainability constraints (non-declining welfare, non-declining consumption streams, non-declining input supply streams and non-decliningextraction streams) and their applicability are discussed in more detail in the followingparagraphs.The weakest sustainability constraint (level 1 in Figure 4.1) requires that everygeneration limits its activities such that a non-declining level of utility can be obtained by allfuture generations. Under certainty, every generation would deplete resources such that thisconstraint holds with equality. The maximin welfare path would be the result. Hartwick(1977) and Dixit et.al . (1980) have shown in models with one infinitely living generation thatthe maximin welfare path would be obtained if net investment in the economy was zero atall times, meaning that all competitive profits from resource depletion are invested inreproducible capital (this is Hartwick's rule). Hence, to meet the sustainability constraint,every generation would be required to invest all resource profits and leave this investmentbehind as compensation for future generations. In an intergenerational world, the price pathwould have to be administered such that the resource is depleted efficiently under suchconstraint (see chapter I). There would be no restriction on the type of compensatinginvestment.Under certainty, the weakest sustainability constraint would be the appropriate onesince future economic prices of all goods and all forms of capital could be predicted. If thestock of all capital, aggregated at real future prices is non-declining, utility would be non-declining and a weak sustainability constraint should suffice. However, the uncertaintiesinvolved in long-term forecast of economic prices are numerous. Uncertainty about theeconomic value of goods in the future can result from demand or supply uncertainty.Demand uncertainties can result from changes in income through differing incomeelasticities. Unexpected development of technological substitutes or additional uses for goodswill change demand. Also, preferences of individuals may change, particularly in the long-190term. Supply uncertainty can be caused by changes in technology, changes in the quantityof known natural resource deposits or changes in knowledge about natural systemcharacteristics, such as the regeneration capacity of the atmosphere. Considering non-quantifiable uncertainty such as the possibility of drastic technological advances, culturalchanges or catastrophic events, it is clear that ambiguity and ignorance prevail about futureprices.In the absence of reasonable estimates of future economic prices, the weakestsustainability constraint would have to be based on aggregation at present prices. However,using present prices for the valuation of future capital stocks is inappropriate since prices aremeasures of marginal rates of substitution while depletion of natural capital at the currentscale clearly implies non-marginal changes in the economy. Such non-marginal changes willlead to changes in relative prices, not only of non-renewable resources. Investment at presentprices is, therefore, unlikely to provide adequate compensation. For example, compensationof future generations for the depletion of oil reserves through investment in a highwaysystem and gasoline-powered vehicles at their present economic values would clearly beinadequate. Gasoline-powered vehicles and oil are complements and their future values arelikely to be negatively correlated. In case oil turns out to be very valuable in the future, ahigh compensation for earlier depletion would be required, however, the vehicles or thehighway system, intended to serve as compensation, would be worthless since there is notenough oil to use them. Alternatively, if oil proves to be not as scarce as expected, theearlier generation would have compensated excessively and restricted their own consumptionunduly.Under the multidimensional uncertainties of real life, general investment ascompensation would lead to a high variance of the welfare of future generations. All thediscussed uncertainties can affect future rates of substitution at the level of a futuregeneration's welfare function and, hence, the welfare effects of compensatory investment.Either a high risk about the adequacy of compensation would have to be accepted, orcompensating investment would have to be made well in excess of Hartwick's rule to ensure19 1adequacy of the compensation with the desired likelihood. Since the concept of welfare isnon-operational a weak sustainability constraint is non-operational under real lifeuncertainties. It appears impossible to determine the expected welfare of a remote generationresulting from current depletion of natural capital depletion and compensation throughgeneral investment. This weak sustainability constraint, therefore, does not satisfy therequirements of intergenerational justice under real life conditions.The uncertainty about substitutability at the level of a future generation's welfarefunction can be alleviated by imposing sustainability constraints separately on consumptionstreams that generate welfare (level 2 in Figure 4.1). These consumption streams are theservices derived from depletion of natural capital, such as heat, transportation, nutrition, andothers. Hence, consumption from a non-sustainable source would only be acceptable if theconsumption stream can be obtained at-the same real cost in perpetuity. Hence, the stocksof groups of capital providing certain services such as energy, water, waste absorption orclimatic stability, would have to be kept intact. A switch from one to another type of capitalwithin such a group would be admissible. The consumption streams are the output of aproduction process that combines natural capital with other forms of capital. The nature ofthe required compensation would depend on the specific production function. Only specificinvestments that leads to sustainability of the consumption stream would be acceptable ascompensation. Specific investment could, for example, increase the efficiency of naturalcapital use. Sustainability of consumption streams is more operational than sustainability ofwelfare since the production function for services is more likely to be known with someconfidence than all relative prices in the future. Sustainability of consumption streams isstronger than sustainability of welfare since it requires specific rather than generalcompensatory investment, and the former is included in the latter. Uncertainty aboutadequate compensation would be reduced but not eliminated since technological changes mayunexpectedly alter production functions. Therefore, sustainability of consumption streamswould be the appropriate constraint if there was sufficient confidence in understanding andpredicting the production function but insufficient knowledge of future relative prices.192The next stronger sustainability constraint would apply to resource supplies (level 3in Figure 4.1). This constraint would allow the use of natural capital only if the supplystreams entering production are made sustainable. This constraint would apply separately tothe supply of energy, water, the atmosphere's capacity to absorb emissions etc.. Thisconstraint applies to the inputs to production and is stronger than the one applying toconsumption streams since it does not allow for substitution at the level of the productionfunction. Sustainability of outputs can be achieved through sustainability of inputs but alsothrough substitution between inputs. Compensation under sustainability of resource supplieswould have to be made through sustainable substitutes, such as renewable energy sources fornon-renewable ones. Hence, each stock of capital providing certain basic inputs to productionsuch as energy, water, waste absorption or climatic stability, would have to remain intact,however, a switch from one to another type of capital within any such a group would beadmissible. Sustainability of resource supplies further restricts the choices of earliergenerations, however, it is also more operational and further reduces uncertainty aboutadequate compensation since the uncertainty about the production function is avoided.However, uncertainty remains about potential alternative high-value uses of individual naturalresources. This sustainability constraint would be applicable if there is considerableuncertainty about technological change or the nature of the production function but sufficientconfidence that no alternative high-value uses of the depleted resource will be discovered inthe future.Finally, the strongest sustainability constraint would separately require sustainabilityof extraction from all different types of natural capital (level 4 in Figure 4.1). Onlysustainable yields of renewable resources could be harvested and no non-renewable resourcescould be used. The only admissible substitution would be restoration of equal natural capitalat another location. This sustainability constraint would eliminate the uncertainties, exceptfor uncertainty about stock size, harvest and sustainable yields, and would be veryoperational. However, it would drastically restrict choices and ignore the existence ofsubstitutability between different resources and, to some degree, between natural resources,technology and human-made capital. Since such a strong constraint seems to exclude many193activities that would unambiguously benefit all generations, it would be too strong for mostinstances unless there is a significant possibility for not yet known alternative high-value usesfor a resource. The potential future value of genetic material and diversity would require thisstrongest sustainability for biodiversity which would, for example, preclude the possibilityto compensate for the extinction of species.A sustainability constraint at levels 2 or 3 involves compensation through functionalsubstitutes that provide the same services as the resources they are compensating for. Forexample, different sources of end-use energy would be functional substitutes. Compensationfor the depletion of non-renewable energy resources would have to be investment insustainable energy sources rather than in energy intensive production facilities. The quantityof such compensating investment has to ensure that the total capital for provision of energyremains intact. This means, the capital stock retains the ability to generate the presentlyconsumed amount of energy in perpetuity. By restricting compensation to functionalsubstitutes, the problem of estimating future prices would be alleviated, since comparisoncan be made in physical units of energy service.The stronger sustainability constraints reflect the intuition that future generations canbe more adequately compensated through functional substitutes and provide a built-ininsurance for the future costs of current economic activities. Since the prices of functionalsubstitutes are positively correlated, the value of the compensation is high whenever thevalue of the depleted resource is high and vice versa. A stronger sustainability constraint,therefore, ensures adequate compensation with a higher degree of robustness. As a result,compensation through functional substitutes is a cheaper way for current generations toprovide adequate compensation. The overly optimistic assumption that technological progresswill always increase society's opportunities is unrealistic and should be avoided. Hence,evaluation of substitution opportunities should be based on currently available technology orforeseeable technological development. This reflects prudent behaviour in view of theexperience that technology and increasing knowledge can lead to information thatsignificantly reduces the opportunities of future generations (see Daly and Cobb 1989, p.198).194For most situations, the foregoing discussion would suggest a sustainability constraintbased on groups of functionally substituting types of capital. However, the choice of aspecific sustainability constraints in a particular situation involves considerable judgementand should depend on the degree of risk aversion, the nature and degree of uncertainty aboutthe adequacy of intergenerational compensation and the availability of substitutes. The searchfor the appropriate sustainability constraint in a given situation involves the trade-off betweenthe risk of inadequate compensation and expected welfare gains for all generations. Thestronger the sustainability constraint, the more the choices of the current generation areconstrained by it. A stronger constraint reflects a more cautious approach involving lessinterference with the natural support system at the cost of reduced welfare. Under completecertainty, the weakest sustainability constraint would suffice. The less knowledge is availableon the nature of substitutability, the stronger the sustainability constraint should be. If nosubstitute is available at the level of the selected sustainability constraint, the depletion ofnatural capital would not be permitted. Since intergenerational compensation is a surrogatefor voluntary trade between distant generations, between which direct trade is not possible,the selected sustainability constraint should leave early generations sufficiently confident thatlater generations would have agreed with the substitution undertaken and the compensationprovided in the given situation.3.2^The Geographic Scale of SustainabilityThe chosen sustainability constraint can be implemented at various geographic scales.Non-declining capital stocks can be required on a global, national, regional or local level.For example, a carbon dioxide emission constraint could be imposed on a global scale.Alternatively such constraint could be imposed on countries separately. Even stronger, everycity or every project could be required to meet a carbon dioxide emission constraint. Similarto the choice of the level of sustainability, the choice of the geographic scale depends on thetrade-off between gains from substitution and the risk of inadequate compensation. Hence,the geographic scale should depends on the substitutability between natural capital at differentlocations. The atmosphere is a global resource, and carbon dioxide emission reductions at195any place are perfect substitutes. For global public resources (atmosphere, oceans) and fortradable resources, such as oil which has a high value compared to transport costs, thesustainability constraint should be implemented on a global scale. Otherwise, the market orshadow prices of natural capital depletion would differ across countries, and the potentialgain from equalizing the marginal cost of reducing natural capital depletion across countrieswould not be realized.The desirable use of a sustainability constraint on a global scale for global andtradable resources is hampered by the absence of effective mechanisms for internationalpolitical coordination. Therefore, in the absence of a global sustainability constraint,individual countries could impose a national or local sustainability constraint even for globaland tradable forms of natural capital. In this case, compensatory investment would not bedetermined on the global scale but depend on every individual countries' production orconsumption. While a global constraint should be the ultimate objective, international lendingagencies could play an important role in supporting the application of a national sustainabilityconstraints even for global public resources. Similarly, if there was not already asustainability constraint implemented at the national level, it could be imposed at a local oreven project level. This approach would lead to the "bottom up" implementation ofsustainability without the long delays inherent in national or even global decision making.There are examples of natural capital that cannot be easily substituted by naturalcapital at another location. Where natural capital is local in nature, a sustainability constraintshould naturally be applied at the local level. The services rendered by a watershed are notreadily tradable. Hence, such natural capital has to be maintained intact at the local level andthe destruction of one watershed cannot be compensated by restoration of a watershed inanother country. Also, fresh-water on one continent is not a good substitute for fresh-wateron another continent. Since transportation costs are usually prohibitive, sustainable watersupply should be required at the regional level.1964^Project and Policy Evaluation Under a Sustainability ConstraintThis section applies the sustainability constraint suggested in the previous section tothe evaluation of projects or policies which increase or decrease the depletion of naturalcapital. In this section, a "sustainable supply rule" for shadow pricing natural capitaldepletion is suggested The sustainable supply rule reflects the sustainability constraint andthe requirements of evaluation under uncertainty, as well as intergenerational equity andefficiency, on which the sustainability constraint was based. There is no inherent reason whyan individual project needs to be sustainable. It is perfectly acceptable to undertake a projectthat generates positive net benefits over the finite lifetime of a project if all costs areproperly included in the analysis and if no costs are imposed on future generations.However, the shadow prices used for the depletion of natural capital should reflect theappropriate sustainability constraint in order to overcome the inherent limitations ofconventional evaluation discussed at length in section 2. If a sustainability constraint wasimposed and compensation was made at the national or global level, market prices for naturalcapital would already reflect this sustainability constraint and no further adjustments wouldneed to be made for project evaluation. However, if a sustainability constraint is not yetimplemented at a higher level (currently this is likely to be the case), shadow prices thatreflect the appropriate sustainability constraint need to be calculated and used at the projectlevel. Also, compensation needs to be made for natural capital depletion caused by theproject.4.1^The Sustainable Supply RuleThis section puts the theoretical concepts discussed so far into practice. It proposesan operational sustainable supply rule for shadow pricing natural capital depletion anddetermining the appropriate intergenerational compensation for depletion of natural capital.The sustainable supply rule is an application of the theoretical requirements discussed before.The sustainable supply rule combines elements of Pearce's (Barbier, Maricandya and Pearce1988, and Pearce 1990) proposal to impose a sustainability constraint on a portfolio of197projects, Page's (1977) suggestion to tax resources such that their real price would remainconstant over time and El Serafy's (1989) approach for adjusting national income accountsto reflect resource liquidation. Through the requirement of sustainability for groups offunctional substitutes, however, the sustainable supply rule is more operational than, i.e.,the Pearce approach.The sustainable supply rule allows the depletion of natural capital only if the depletednatural capital is replaced by compensating investment that functionally substitutes for thedepleted capital. Natural capital depletion would be evaluated at the sustainable price of theservices derived from it. This sustainable price is the cost at which the services from thedepleted natural capital can infinitely beprovided through a sustainable substitute if for everyunit of depleted natural capital the sustainable price is invested in production of thesubstitute. If a functional substitute for the depleted natural capital is not available and thereis considerable uncertainty about its future value, natural capital of the same type needs tobe restored or depletion of natural capital would not be allowed (the shadow price would beinfinite). If there is sufficient confidence that the future value of natural capital can beestimated, depletion would be acceptable if compensation is made for future generationssufficient to offset the foregone benefits they could have derived from natural capital.The sustainable supply rule is applicable for a sustainability constraint at anygeographic scale. At the level of national policy making, the rule could be used to determinean adequate resource depletion tax. Whenever the sustainable supply rule is not applicablein a given situation, the appropriate alternative approach for evaluation should be derivedfrom the three theoretical requirements derived for adequate evaluation. In the followingdiscussion, the derivation of the sustainable supply rule is explained for the simplest possiblecase geared toward application at the project level: the depletion of a non-renewable resourcethat has a perfect sustainable substitute. Following El Serafy's (1989) approach, the benefitsfrom depletion of natural capital need to be divided into an income and a compensationcomponent. The compensation component would be allocated as a cost to resource depletion.It would be determined such that when the compensation component is invested in productionSustainable Supply CurvePrice/CostCsPsQuantity per Period, 0Cost of Sustainable Substitute Marginal Benefits Sustainable Supply41■1■ 0M.Mla■■ • OW1 •=11=Ii MIIMI. 198of a sustainable substitute for the non-renewable resource, it would, after exhaustion of thenon-renewable resource, lead to an infinite benefit stream from consumption of thesustainable substitute equal to the income component. Benefits from the non-renewableresource would be shared with all following generations through the constant incomecomponent which is generated through compensatory investment.Figure 4.2 The Sustainable Supply CurveThe demand curve for a service that can be provided by a non-renewable resourceor its sustainable substitute is shown in Figure 4.2. For simplicity, it is assumed that the non-renewable resource can be extracted at zero cost. The current unit cost of producing thesustainable substitute is assumed to be C. This is shown through the horizontal supply curve199at C. The line ABCD represents the demand (or marginal benefit) curve. At the price Ps,no sustainable substitutes would be offered and demand would have to be satisfied with thenon-renewable resource. Market demand at price Ps would equal the distance EK and totalbenefits from consumption of the resource would be the area under the demand curve up tothe quantity used, ACKE. This rent consists of two components: the owner of the resourcewould receive revenues equalling FCKE; the consumers would receive the (non-monetary)consumer surplus ACF. The sustainable supply rule requires that the revenues FCKE be thecompensation component to be invested and the consumer rent ACF be the incomecomponent accruing to the present generation for consumption. The point C on the demandcurve is determined such that the income component can be sustained for all generations.Compensating investment into the sustainable substitute would, in time, reduce theunit cost of the sustainable substitute from its current level Cs. This cost reduction occursbecause compensating investment into improvements of technology actually reducesproduction costs or because compensating investment is used as a free addition to the capitalstock for production of the substitute. Now, the sustainable price, P„ which also definespoint C on the demand curve, is determined such that the cost of producing the sustainablesubstitute is reduced to Ps exactly at the time when the non-renewable resource is depleted.At that time, production of the sustainable substitute will begin. Hence, this procedureensures that a sustainable price is chosen at which the same quantity of the service from theresource will be provided in perpetuity; first from the non-renewable resource and later fromthe sustainable substitute. The term sustainable price refers to the fact that this price doesnot change with depletion of the non-renewable resource.In the simplest case, the non-renewable resource is homogenous with zero extractioncosts and supply of the sustainable substitute is infinitely elastic at C. For the quantity ofthe resource extracted per period, Q, the sustainable price, Ps, would have to be set such thatPs-Q (the area FCICE in Figure 4.2) invested in every period until depletion of the non-renewable resource, would generate a return of (C5-PQ (area GHCF) in every period afterdepletion. Here, r is the (continuous time) rate of return on investment in sustainable200substitutes, S is the total stock of the non-renewable resource and S/Q is, therefore, thelifetime of the resource. The return to the compensating investment reduces the cost of thesustainable substitute which, in turn, leads to a constant sustainable price. The quantity Qof the resource will be available at the price P., before depletion from the non-renewableresource and thereafter from the sustainable substitute.The compensation component invested throughout the lifetime of the non-renewableresource must be sufficient to yield an infinite subsidy stream that reduces the cost of thesustainable substitute to the sustainable price for an equal consumption quantity per period.This is the case if the present value of the finite stream of the compensation componentequals the infinite stream of the subsidy after resource exhaustion which occurs after S/Qperiods. The present value of the compensation component, K, from time 0 to S/Q is:Q P^ s\K = f e'^dt - ^ I‘1 - e r0The present value of the subsidy stream, T, from time S/Q until infinity is:T = f e Q(Cs-Ps) = e1 Q(Cs-Ps)equating T and K and dividing both sides by Q/r:-r—\^-r—Ps (1 — e^= e^Cs-PsSolved for P„ this results in the following equation for the sustainable price:-r-Ps = e Q(1)(2)(3)(4)Equation (4) is based on highly simplistic assumptions and not directly applicable inmost instances. However, the assumptions of zero extraction costs for the non-renewable201resource and perfectly elastic supply of the sustainable substitute can be relaxed at the costof complicating equation (4). If there is a constant extraction cost, E, per unit of the non-renewable resource, the compensation component would be:K - Q(P-E) (1 - e :2)-r-Similarly, Ps is calculated by equating K and T, and solving for Ps:-r-Ps = E + e Q (Cs-E)where Ps is total price including extraction cost. The appropriate user cost is Ps-E. Similarequations for sustainable supply can be calculated for rising extraction costs or a situationin which a stock of compensating investment already exists. Also, a similar expression canbe found if no perfect sustainable substitute exists. If the service which is to be suppliedsustainably is produced from natural capital and another reproducible input, the expressionfor the sustainable price can be derived from the production function by optimizing the inputsover time such that a constant stream of the service is obtained at least cost.If the extraction rate of the non-renewable resource, Q, was increased, thecompensation component would have to increase as well, since the reduced lifetime of theresource would leave less time for returns of the compensating investment to compound. Thesustainable price would, therefore, be higher if the resource depletion rate was higher andvice versa. Hence, equation (4) describes a positive functional relation between quantity ofthe service from the resource consumed per period and the sustainable price. The typicalshape of the resulting sustainable supply curve is shown in Figure 4.2. The term sustainablesupply curve reflects the steady nature of this supply curve as opposed to an imaginarysupply curve with Hotelling-depletion which would shift upward to reflect opportunity costswhich would be rising with the scarcity of the non-renewable resource. The sustainable pricecan be read from this curve with knowledge of annual depletion of the resource. If theresource was supplied according to the sustainable supply curve, point C would be the(5)(6)1^I^I^II^I^I^I^I^II^I^1^I^II^I^I^I^I^IIResource Price Path ComparisonPriceDepletionCsPs.0"00'TimeNote!ling Price Sustainable Price202market equilibrium. The sustainable price would be the shadow price to be used for thevaluation of depletion in a resource extraction project or for the valuation of the output froma project producing a substitute for the resource. Under this rule, actual investment of thecompensation component (sustainable price times quantity of the resource depleted) wouldbe required for the project to be acceptable.Figure 4.3 Resource Price Path ComparisonFigure 4.3 compares the price paths resulting from use of the sustainable supply ruleand conventional depletion according to the Hotelling rule. With depletion according to thesustainable supply rule, the price would remain constant at level P. with depletion occurringsomewhere along the horizontal line. Optimal depletion in perfect markets without a203sustainability constraint would follow the Hotelling rule. C, would be the cost of a back-stoptechnology. Therefore, the price of the resource would rise at the rate of interest with theinitial price set such that depletion occurs exactly when the price reaches C, as shown inFigure 4.3. In the initial years, a larger amount of the resource would be available at a pricelower compared to the sustainable supply rule. In later years, a lesser quantity would beavailable at a higher price. With conventional depletion, the total rent accruing in the earlyyears would be close to the area ADE in Figure 4.2. This rent would fall gradually to ABJEin the last period before depletion. For all periods after depletion, total rent would be onlyABG. With the sustainable supply rule, rents would be ACF in all periods. The timing ofresource exhaustion under the different regimes would depend on the relation betweendiscount rate and the rate of return on the specific compensating investment.Use of the sustainable supply rule is clearly preferable to the use of the currentlymost applied rule, "ignore the costs of natural capital depletion". However, the justificationof the sustainable supply rule does not lie in formal social welfare maximization and it maybe possible to show that the sustainable supply rule is inferior to a conventional efficiencyapproach (the Hotelling price path) IF the conditions for first-best decision making are met.These conditions would include certainty or alternatively complete forward markets,internalization of all external costs, profit/social-welfare maximizing decision makers and theexistence of effective wealth transfer mechanisms between generations. The defense of thesustainable supply rule rests primarily on the assessment that these assumptions are highlyunrealistic. In particular, there are no explicit institutional arrangements in existence thatwould guarantee the desired level of welfare for future generations. The conventionalapproach (depletion according to the Hotelling path) would be preferable only if there wasa situation where market failures and systematic biases were removed, uncertainty aboutsubstitutability and inter-dependencies in complex natural systems was sufficiently resolved,and intergenerational welfare transfer mechanisms were in place.2044.2^Application of the Sustainable Supply RuleIn this subsection, practical applications of the sustainable supply rule will bediscussed and illustrated by means of several examples. First, applications to non-renewableand renewable natural resources, land use, and emissions will be discussed. Then,complications arising from unsustainable use of project outputs, international trade, anddifferent types of compensating investment will be assessed.The common practice of ignoring the user cost in evaluating natural resourceextraction projects is clearly unacceptable. The sustainable supply rule would be used tocalculate the appropriate depletion premium under a sustainability constraint. Consider theexample of an agricultural project based on ground-water depletion. A sustainable supplycurve for water can be found by estimating the cost of compensating investment insustainable water supply, for example a solar-energy powered water desalination plant. Usingthe present unit cost of producing desalinized seawater, C„ the sustainable price for ground-water, Ps, would have to be determined such that if an amount equal to Ps times the depletionrate was invested in desalination technology, then this investment would be able to providedesalinized seawater of the same quantity at the cost Ps for every year after depletion ofground water. The sustainable price of ground water would be calculated by using equation(4) with the current depletion rate Q, total stock left in the ground, S, and rate of return oncompensating investment, r. To evaluate a project that uses ground water, the sustainableprice of water, Ps, would be subtracted from project benefits as a unit cost for ground-waterdepletion.For shadow pricing the depletion of a non-renewable energy resource, such as oil,Cs would be the cost of producing a sustainable substitute, such as a unit of solar energy.Compensating investment might be most profitable in research and development in order toincrease the efficiency of photovoltaic energy generation. The size of the requiredcompensation component would be determined such that solar energy would becomeavailable at the sustainable price of one energy unit from oil in the same quantities after205depletion of the oil. Today, oil depletion would be evaluated at the sustainable price. Theroyalty or extraction tax levied from extracting companies should be set according toequation (6) for the sustainable price with non-zero extraction costs. Alternatively,compensating investment could be made in technologies to increase energy use efficiency.An increase in energy use efficiency would decrease the cost of sustainable energy supplyin the future accordingly and, thereby, lead to a lower sustainable price of energy servicefrom non-renewable resources.The sustainable supply rule can be applied to renewable resources as well. If arenewable resource is harvested sustainably, no special problems arise since depletion ofnatural capital does not occur. Therefore, the appropriate price is the marginal social benefitof the resource at the level of the sustainable yield. For the depletion of a renewableresource, the sustainable supply rule can be applied in analogy to non-renewable resources.Compensatory investment for depletion has to ensure that the same quantity can be extractedinfinitely at the same price. It would be difficult to imagine a functional substitute forforests. Therefore, compensating investment for harvesting a forest above its sustainableyield would be investment into enlarging the forest area such that the harvested amountwould become the sustainable yield of the enlarged forest area. For example, if a forest isharvested at a rate x, leading to depletion after y years, then compensation would take theform of acquiring additional land and investing in reforestation such that this new forest hasa sustainable yield of x after y years. The sustainable price of forest depletion would haveto cover the costs of expanding the forest area accordingly.The common assumption of a zero opportunity cost of land would, in most instances,be unacceptable under a sustainability constraint. The lack of observable commercial useswould be insufficient for the assumption of zero opportunity cost of land, especially if aproject does not only use but also degrade the land While the ecological value of wildlandswould differ considerably, assigning a value of zero would be justified only in extremecircumstances such as sustainable use of desert lands. In all other instances, a default valueabove zero would be appropriate. To determine the land value in the absence of apparent206commercial uses or a reliable market price, the replacement cost should be considered. Thiswould be the cost of rehabilitating or restoring similar lands that are already degraded. If aproject takes land out of the existing stock of productive land or wilderness areas, it shouldcarry the cost of maintaining the stock intact by rehabilitating or restoring a degraded areaof equal size. For example, a mining project would have to carry the cost of landrehabilitation not at the time of mine closing but at the time of mine opening. Due to theeffect of discounting, this change would significantly increase economic rehabilitation costs.The common practice of ignoring the costs of uncertain damage resulting fromemissions is equally unacceptable under a sustainability constraint. The expected damagecosts for most pollutants would be strictly greater than zero and an explicit evaluation ofexpected damages is preferable even under significant uncertainty. In the absence of anyknowledge about the damage costs, the costs of a sustainable substitute (the cost of completepollution abatement or the cost of compensatory reduction of equivalent emissions elsewhere)should be used as default value for the unknown damage costs. This default value is reducedif it can be shown that damage costs are less in a specific instance.There are no functional substitutes available for the services provided by theatmosphere. The composition of the atmosphere constitutes natural capital which is depletedby increasing the carbon dioxide concentration. Since, there is no known substitute for theatmosphere in its natural composition, compensatory investment would have to maintain thistype of natural capital intact. Hence, a new project that would increase the atmosphericcarbon dioxide concentration would have to bear the costs of investment, for example inreforestation, that would absorb the same amount of carbon dioxide that is discharged by theinitial project. Alternatively, investment could be undertaken in increased energy efficiencythat would lead to the same offsetting effect on net carbon emissions. The compensatinginvestment has to be undertaken and the costs are allocated to the initial project. The initialproject should only be undertaken if it yields a positive return after subtraction of the costof the compensating investment.207So far, resource depletion, land use and emissions resulting directly from the projectwere considered. Some projects produce output that is put to unsustainable use. If the marketprice of the output does not reflect a sustainability constraint on that output's use, outputprices need to be adjusted. For example, even if the coal reserves of the earth wereconsidered infinite and there was no depletion premium associated with a coal miningproject, the current use of coal is unsustainable due to the carbon dioxide emissions resultingfrom coal combustion that contribute to the greenhouse effect. Unless a sustainabilityconstraint is already imposed on the use of coal, and this is reflected in the market price, theoutput price used in evaluating the mining project needs to be adjusted for unsustainable useeven though the carbon dioxide emissions are occurring outside of the project.Environmental problems such as carbon dioxide emissions, loss of biodiversity anddeforestation reach beyond country borders. Even national environmental degradation maylead to international externalities through resulting migration and political tension. However,project evaluation is usually done from a national point of view, maximizing net socialbenefits accruing in one individual country. In the presence of international externalities,national welfare maximization leads to inefficiencies. Since several environmental issues canonly be dealt with at a global scale, analysis under a sustainability constraint must take aglobal perspective. This means that even international prices need to be adjusted if they donot reflect the appropriate sustainability constraint. Hence, even if coal is exported, theoutput price needs to be adjusted for the cost of unsustainable coal use resulting in carbondioxide emissions. This global view may counter the individual interests of the nationalgovernment that is implementing the project. Therefore, transfer payments between nationswould likely be required to achieve sustainability for traded resources and internationalexternalities. Concessionary lending through institutions, such as the Global EnvironmentFacility, administered jointly by the World Bank, UNDP, and UNEP, would be appropriateto facilitate the required transfer payments.Finally, the nature of compensating investment and the calculation of the returns onsuch investment requires further discussion. In principle, any form of compensating208investment is acceptable if it leads to provision of a sustainable supply of the substitute atthe sustainable price. The simple case, for which equation (4) was derived, refers to asituation in which facilities for the sustainable production of the substitute already exist.Then, compensating investment does not need to provide the sustainable substitute itself. Incases where technological improvement in the existing sustainable substitute is unlikely,compensating investment should be made in the form of general sustainable investment inthe economy. Proceeds of this investment would then be used to subsidize the production ofthe sustainable substitute in the existing facilities after the non-renewable resource isdepleted. In the more likely case that production facilities for sustainable substitutes do notyet exist, compensating investment would be made in such production facilities for asustainable substitute. Only after such investment is sufficient to produce the requiredquantity of the substitute, further investments should be made into general sustainableproduction to subsidize the cost of the sustainable substitute. In either case, r, would be thereal rate of return on the actual compensating investment undertaken.In many cases, the most effective investment for the reduction of the cost of asustainable substitute would be into research and development (R&D). While the return onR&D is difficult to estimate, the problem is alleviated in this case since the relevant rate ris the rate of return on R&D measured at the fixed output price C. Therefore, uncertaintyprevails only about the success of the R&D program in physical and not in economic terms.Measurement of return on R&D in terms of output price Cs explains why through suchcompensating investment worthwhile R&D into sustainable substitutes might be undertakenthat would not be undertaken under a pure market arrangement. Due to the public goodnature of information and knowledge, there are many reasons to believe that market forcesalone will not bring about the efficient level of R&D activities. Since patent protection israrely complete, private investors could normally not reap the full return on their R&Dexpenditures. Competitors would imitate the innovation or circumvent the patent which, inturn, would lead to a drop of output prices below Cs (see Tirole 1988, pp. 389-419, chapter10).209Use of the sustainable supply rule may lead to the rejection of projects that depletenatural capital and that were considered desirable without a sustainability constraint.However, other projects are restoring natural capital and would be more desirable under asustainability constraint. Imagine a country that uses forests or ground water resourcesunsustainably. A proposed conservation project that would increase water use efficiency orrestore forests, would under conventional analysis be evaluated by comparing a with-projectscenario and a without-project scenario. To justify the project, the pervasive benefits fromreforestation would have to be enumerated and evaluated. Because of the pervasive natureof benefits generated by natural capital, this would lead to a systematic underestimation ofthe economic benefits from the project. Under a sustainability constraint, the defaultassumption would be reversed and the appropriate comparison would be between a with-project scenario and a scenario with the next best project that would achieve sustainability.A reforestation project would be compared with a project that provides a sustainablesubstitute for fuelwood, such as energy use efficiency increases or a solar energy project,and should be implemented if it was the least cost alternative for achieving sustainability.This approach would be reflective of the view that the analysis of an individual project doesnot need to confirm that sustainability is desirable. Rather, the analysis should ensure thatthe least-cost option for achieving sustainability is pursued.5^A Stylized Case Study: An Oil Development ProjectThe stylized case study in this section is presented to illustrate the application of thesustainable supply rule with a real life example. This presentation is based on case studiesof several World Bank projects (von Amsberg 1993). However, many simplifyingassumptions have been introduced in order to focus attention on methodology rather than ontechnical details of the underlying project. The case study shows that reasonable shadowprices can be calculated with very moderate effort. While care has been taken to usereasonable cost estimates and make reasonable assumptions, there is certainly ample roomfor improving the estimates and calculations. The basic information underlying the casestudy, such as prices, exchange rates and cost estimates, is based on the Staff Appraisal210Report of the project. The analyzed project consists of the commercial development of anoff-shore oil field that is jointly owned by the national petroleum company and the domesticsubsidiary of a multinational corporation. The recoverable reserves of the oil field areestimated at 330 million barrels and would be extracted over the 21-year lifetime of theproject. Peak production will be reached in year two with production declining after yearfour. The project was appraised by the World Bank with an economic rate of return of 51%.This analysis did not include a user cost or depletion premium for the extracted oil. Hence,the economic rate of return reflects the full resource rent without considering the opportunitycosts of depletion. The price projections for oil that were used for the calculation of theeconomic rate of return are flat for the lifetime of the project (around $22/bbl in constant1990-$).5.1 The IssuesTwo main issues arise with respect to the sustainability of hydrocarbon extraction andits evaluation. First, the extraction of a non-renewable resource is unsustainable due to thelimited reserves. Second, the use of hydrocarbons as energy source is unsustainable since,at current consumption levels, its combustion leads to accumulation of carbon dioxide in theatmosphere. Hence, a sustainability constraint needs to reflect these two types of naturalcapital depletion.Oil from other oil fields in the country is obviously an almost perfect substitute foroil from this project. Since oil is primarily used as an energy commodity, substitution of oilwith another storable form of energy would also be acceptable under a sustainabilityconstraint. Other hydrocarbons, such as gas, evaluated at their energy content, would bealmost perfect substitutes for oil. However, these substitutes are finite as well. They canstretch the lifetime of the non-renewable resource but they cannot substitute for it inperpetuity. Beyond that, compensation would have to be based on truly renewable energysources.211Substitution would also be acceptable at the level of production of energy servicesfrom primary energy and capital. Within limits, capital can substitute for primary energythrough increased efficiency in energy use. First, at any given level of technologicaldevelopment, there is some substitutability between capital and energy, i.e. through increasedinsulation against heat loss. Second, investments in research and development can lead totechnological progress that would allow production of more energy service from the sameamount of capital and primary energy. Clearly, there are limits to the substitution of energyposed by thermodynamic constraints. Lighting a room requires some minimum amount ofenergy regardless of the efficiency of the bulb; transport requires some minimum amount ofenergy regardless of the efficiency of the vehicle. However, in many instances, physicallimits have not yet been exploited, leaving room for significant further substitution. Theestimation of substitution between capital and energy poses problems because of the manydifferent uses energy is put to and the difficulties inherent in anticipating technologicalprogress. However, reasonable estimates are available that can serve as the basis forsensitivity analysis.It is assumed that compensatory investment for current depletion would have toprovide not only for a constant consumption stream but for an increasing consumption streamat a constant price. Since the size of future populations is, at least partially, dependent ondecisions made at present, it is reasonable to assume that the current generation should bearat least some responsibility for satisfying the demands of an increasing human population.Also, taking the large disparities in energy consumption between countries into account, itcan be argued that rich nations whose consumption accounts for the largest share ofunsustainable energy resource depletion, should pay a price for depletion that provides theopportunity for poorer nations to increase their energy consumption at the same low costcurrently enjoyed by the high-energy consuming countries.The use of hydrocarbons leads to the release of carbon dioxide that contributes to thegreenhouse effect which is expected to lead to global warming. While the precise impactsof an increasing carbon dioxide concentration in the atmosphere on climate are still212controversial, it is now well established that burning of fossil fuels has already led to anincrease in the atmospheric carbon dioxide concentration. Annual carbon dioxide releases arecurrently estimated at 5.6 billion tons of carbon from fossil fuels and 0.6 billion tons fromchanges in land use (data from Deutscher Bundestag 1989). It is estimated that 3 to 4 billiontons of carbon accumulate in the atmosphere every year. Hence, carbon dioxide releaseswould have to be roughly cut in half to avoid further accumulation of carbon dioxide in theatmosphere. Even if a drastic reduction in carbon emissions is unrealistic in the short term,carbon releases would have to be reduced to the absorption capacity of the atmosphere,estimated at 2 to 3 billion tons of carbon per year, in the long-term.In summary, a sustainability constraint would be imposed on supply of the economywith energy services, under acceptable levels of carbon dioxide emissions. The sustainableprice for the depletion of oil should take into account substitution possibilities betweendifferent fossil fuels and between non-renewable and renewable energy sources. It shouldalso reflect possible increases in energy use efficiency and long-term limits on acceptablecarbon dioxide emissions. It should be based on a sustainable price for the country's oil andgas reserves; it should take the expected growth in energy demand into account; it shouldincorporate expected efficiency gains in energy use; and it should consider restrictions onacceptable carbon dioxide emissions.5.2 AssumptionsExtraction costs. The extraction costs for oil from this field are $4.4 per barrel($2.4/bbl development costs and $2/bbl recurrent costs).Oil reserves. The country has many other, yet undeveloped, oil reserves. Since otheroil reserves are perfect substitutes for the oil field to be depleted, those other oil reserves canbe depleted before energy supply has to be converted to solar hydrogen as the sustainablesubstitute. The oil reserves are estimated at about 22 billion barrels. At the current extraction213rate of 1.6 million barrels per day or 584 million barrels per year, these reserves would last37.7 years. Extraction costs are assumed to be constant at $4.4/bbl.Gas reserves. The country also has significant gas reserves, estimated at 150 trillioncubic feet (including undiscovered reserves). The gas reserves contain the energy equivalentof 27 billion barrels of oil (1.1 MI per cubic feet of natural gas). Since gas can beconsidered an almost perfect substitute for oil, a uniform sustainable price should becalculated for all gas and oil reserves instead of different sustainability premia for differentdeposits. Using the total oil and gas reserves equivalent to 49 billion barrels of oil andcurrent extraction of oil and gas equivalent to 612 million barrels per year, the lifetime ofoil and gas reserves together would be 80 years. Hence, renewable energy would not needto be produced until the year 81. Extraction costs for gas are assumed to equal those of oil.Consumption. In order to abstract from the complications of international trade, it isassumed that all gas and oil is consumed domestically. Since this assumption is obviouslywrong, the impact of considering oil exports are discussed at the end of this section. Demandfor energy services is expected to rise significantly over the next decades (see World Bank1992). Both population growth and increasing per capita energy consumption wouldcontribute to this increase that is assumed to be 3.7% per annum.The sustainable substitute. There are several renewable energy sources that are goodfunctional substitutes for oil. In this case study, hydrogen produced from solar energy is usedas a representative renewable energy source. Hydrogen can be used in much the same wayas natural gas. It can be stored and moved in tankers or through pipelines. Hydrogen can beproduced sustainably by electrolysis from water and photovoltaic electricity. Hence, theenergy source of solar hydrogen would be sustainable. The use of solar hydrogen issustainable since the combustion of hydrogen releases only water and does not contribute tothe build-up of carbon dioxide in the atmosphere. Solar hydrogen is used as a representativesustainable energy source because, in the long-run, it has the potential to become the least-cost substitute for hydrocarbons in many of their uses, and a variety of cost estimates are214available. Also, production of solar hydrogen seems to be feasible in this specific countrywhich has large semi-desert areas with low agricultural productivity and high solarinsolation. In reality, however, solar hydrogen is likely to be only one of many renewableenergy sources that would be one part of a sustainable future energy supply system.The future cost of producing solar hydrogen is estimated at $15 per GJ (see Ogdenand Williams 1989, p.39). The cost of producing solar hydrogen equivalent in energy contentto one barrel of oil is $90 (with 6 GJ per barrel of oil). While others estimate the prospectivecosts of hydrogen higher, "overall, long-term cost assumptions in the range $70-100/BOE[barrel of oil equivalent] for the backstop technologies in markets hitherto served by non-electric fuels seem justified" (Anderson and Bird 1992, p.16).Efficiency increases. Expected increases in energy use efficiency mean that eventhough the demand for energy services is assumed to rise at 3.8% per annum, primaryenergy extraction would not need to rise at the same rate. Substitution of capital for primaryenergy and investment in research and development can also satisfy part of the increasingdemand for energy services. The assumptions about the costs of increasing energy useefficiency are derived from Lovins' (1990) preliminary estimates of the full technicalpotential for energy savings. These cost estimates are based on already availabletechnologies. Figure 4.4 shows the cost curve, fitted to Lovins' data. The costs (in $ perbarrel of oil saved) express the present value of the capital investment required to save thespecified share of current primary energy input for the production of one unit of energyservice. The estimated cost curve isK = -3 -10.4Log10.85-S1[ 0.85 j(7)where K is the capital cost in $ per barrel of oil saved, and S is the share of energy saved.The assumption about the introduction of energy saving technologies are based onestimates that annual increases in energy efficiency of 2% are possible over several decades215Marginal Cost of Increases in Energy Use EfficiencyK50 —403020100 AssumedEfficiencyLimit: 85%-10 ^0^0.1^0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9^1Share of Current Energy Saved, SFigure 4.4 Costs of Increasing Energy Efficiency(Worldwatch Institute 1988, pp. 41-61, chapter 3). This estimate is based on average energyefficiency increases of 1.7% per annum during the 1973-83 period and the potential toincrease these achievements through increased investment.Hence, for the following calculations it is assumed that the annual 3.7% increase indemand for energy services is accommodated by 2% efficiency increase and 1.7% increasein primary energy production per year. This is consistent with a doubling of primary energyconsumption within the next forty years. With an increase in extraction of 1.7% per annum,the country's gas and oil reserves would be depleted after 50.7 years. It is assumed thatefficiency increases can be obtained until the time of depletion. Also, demand is assumed tolevel off after year 51.2030^2050Year216The cost for increased energy efficiency per unit of energy service, ct, is the integralof the cost function depicted in Figure 4.4 from zero to the achieved level of energy savings,which, as discussed above, is assumed to be a function of time:CI-0.02r1 -tI 3+ 10.4^0.85Losi  0.85- Sids =o(8)7.4 +7.4e -°.°2' +(10.4e 4"21-1.6)Log[-0.2 + 1.2C"21Carbon Dioxide Reduction Scenariobn BOE^ Carbon Dioxide Emissions, mio t C1.6- -100.0 Oil Gas Renewables • CO2 Emissionsgem.;EUImpFigure 4.5 The Carbon Dioxide Reduction Scenario217Year I^TotalinOil^Gasbbl oil equivalentmioRenew^I CO2Emiss1mb^t C1989 640 612 28 0 831990 651 612 39 0 841991 662 612 50 0 851992 673 612 61 0 861993 685 612 73 0 871994 696 612 84 0 881995 708 612 96 0 891996 720 612 108 0 901997 732 612 120 0 911998 745 612 133 0 921999 758 612 146 0 932000 770 600 151 19 922005 838 541 181 116 872010 912 482 210 220 822015 992 423 239 329 762020 1079 364 269 446 712025 1174 305 298 571 662030 1277 246 327 704 612035 1390 187 357 846 552040 1487 129 386 972 502045 1487 70 415 1002 452050 1487 11 445 1031 402055 1487 0 450 1037 392060 1487 0 450 1037 39tons of C per BOE: 0.1316 0.0863 0Carbon emissions are calculated from Deutscher Bundestag(1989), p.489: 0.29 kg CO, per Kwh from oil; 0.19 kg CO,per Kwh from gas; mass of CO, is 3.67 times the mass ofC.Table 4.1 Carbon Dioxide Reduction ScenarioCarbon dioxide emission constraint. In order to meet carbon dioxide emissionreduction targets, an alternative scenario is developed under which extraction of the country'soil and gas reserves is constrained by the requirement that carbon dioxide emissions fromall energy sources are gradually reduced to one half of 1990 levels from 2000 to 2050. Thisreflects an ambitious schedule for the reduction of carbon dioxide emissions, however, itfollows from the requirement to reduce emissions to the estimated absorption capacity. Underthis scenario, shown in Figure 4.5, oil would be depleted until 2050. The carbon dioxideemissions of gas are only 65% of the emissions of oil, compared on an energy content basis.Hence, gas extraction would be increased to the level at which total emissions from gas equal50% of total current carbon dioxide emissions from gas and oil. All remaining energydemand (as above rising at 1.7% per annum until year 50) would have to be met with solar218hydrogen. Hence, the annual extraction of gas would be constrained by the admissibleemissions of carbon dioxide rather than depletion of the resource. Production of the differentfuels and aggregate carbon dioxide emissions are shown in Table 4.1 and Figure 4.5. Underthis scenario, gas would be depleted after 85.7 years. Thereafter, it would be fully replacedby hydrogen.Discount rate. The sustainable prices derived in this case study are sensitive to theassumed rate of return on compensatory investment. The appropriate rate is the expected realrate of return on actual compensatory investment. The opportunity cost of capital used in theeconomic evaluation of most projects is between 10 and 12%. However, considering a long-term real rate of return on international capital markets in the range of 3 to 5%, it is unlikelythat a real rate of return of 10 to 12% can be realized on long-term investments of verysignificant amounts. Furthermore, as the worldwide integration of national financial marketsprogresses, it would be expected that the rate of return on investment would converge to therates obtained on international markets. Hence, a discount rate of 7% has been assumed forthe calculation of a sustainability premium, with additional sensitivity analysis for a discountrate of 5%.5.3 Project EvaluationIn order to separate the effects of the resource depletion constraint from the carbondioxide emission constraint, sustainable prices are calculated for these two constraintsseparately. First, consider the resource depletion constraint. Under the imposed sustainabilityconstraint, the price of energy service, not necessarily the price of primary energy, must besustainable. The sustainable price of energy service is obtained by equating the present valueof the cost of extraction (extraction rising at 1.7% per annum), Cx, the cost of providingsolar hydrogen after depletion, CH, and the assumed expenditures necessary to achieve the2% efficiency increases, CE, with the present value of revenues, R, received for energyservices (rising at 3.7% per annum) provided at the sustainable price:21950.7.1050.7C x^050.750.7CEf0e("3"612.106P sdt + f eo.o36.503612.10613 sclt50.7emon-6t612 .106.4.4dte °'°17°7612 .106.90dtei.°36-6`612 -106ctdt + f e 0.036 -50.7 612 106c507 dt50.7(9)The sustainable price for the energy service from one barrel of oil today, resulting fromusing (8) and solving Cx +CH+CE=R for P„ would be $6.5/bbl and the depletion premium$2.1/bbl. If depletion of fossil fuels was the only concern with respect to the sustainabilityof the project, $2.1/bbl would be the appropriate depletion premium to be used in evaluatingthe project.Next, a sustainability premium is calculated for the carbon dioxide constraintscenario. The sustainable price of energy services can be calculated for this scenario byequating the present value of revenues and costs under constrained extraction of fossil fuels.Revenues and costs of efficiency increases are the same as in equation (9). However, thecosts of extracting oil and gas and producing hydrogen change according to the depletionscenario (see Table 4.1). Under the carbon dioxide reduction scenario, the sustainable priceof one barrel of oil today would be $14.6/bbl (the sustainability premium, excludingextraction costs, would be $10.2/bb1). The sustainable price of $14.6/bbl reflects thedepletion of fossil fuel as well as the carbon dioxide constraint.A third scenario is calculated under which unlimited gas reserves are assumed.Hence, the extraction of gas would continue at the rate admissible under the carbon dioxideconstraint even after year 86. The resulting sustainable price is $14.5/bbl. Hence, thedepletion premium under the carbon dioxide constraint would be only $0.1/bbl (the depletionEconomic Rate of Return of the Project-20^I^I^I^i I^I, 0^2^4^6^8^10 12 14 16 18 20Sustainability Premium ($/bbl)-10Economic Rate of Return6040503020100(%)220Figure 4.6 Economic Rate of Return with Sustainability Premiumpremium is the difference between the sustainable price in a scenario with and withoutdepletion constraint, respectively). This reflects the restrictions imposed on the use of fossilfuels by a carbon dioxide constraint. Hence, the appropriate sustainability premium on thedepletion of the Oil field would be $10.1/bbl for unsustainable use of the atmosphere ascarbon dioxide sink plus $0.1/bbl for unsustainable extraction as depletion premium. In termsof shadow prices, the market price of oil output needs to be reduced by $10.1/bbl forunsustainable use while an economic cost of $0.1/bbl of input (oil extraction) should beincluded. The equivalent sustainability premium per ton of carbon emissions would be about$77. The sustainability premia for the three scenarios are summarized in Table 4.2.221EvaluationApproachYears untilDepletionSustainabilityPremium ($/bbl)(5% ROR)SustainabilityPremium ($/bbl)(7% ROR)DepletionConstraint Only50.7 5.8 2.1Carbon DioxideConstraint Onlyop 13.5 10.1Depletion andCarbon DioxideConstraint85.7 13.7 10.2Table 4.2^Summary of Depletion PremiaTable 4.3 shows the calculations for adjusting the project's net benefit stream for thesustainability premium. At a sustainability premium of $10.2/bbl, the economic rate of returnis reduced from 51.2% to 25.0% which is still above the assumed opportunity cost of capitalof 12%. Under the assumptions made, the project would still be acceptable in the base case.However, the sensitivity to oil market price changes would be very large and might lead torejection of the project. Now, a drop in oil prices by about $3.8/bbl would be sufficient toreduce the project's ERR to 12%. This could be considered unacceptable. Since the project'sadjusted ERR is quite sensitive to several of the assumptions made, the ERR is plotted as afunction of the depletion premium in Figure 4.6. For a sustainability premium of less than$14/bbl, the base case ERR would be above 12%. If the project was undertaken, actualcompensatory investment in the supply of renewable energy and the increase in energy useefficiency would have to be made. If all compensatory investment was made out of theproject's initial net present value (calculated without subtracting the opportunity costs ofdepletion) of $1.72 billion, $1.25 billion would have to be invested as compensation for222Year Net BenefitStream(mio $)OilProduction(mio bbl)SustainabilityPremium ($10.2)(mio $)AdjustedNet Benefits(mio S)1990 -88 0 0 -881991 -319 0 o -3191992 -263 o o -2631993 264 18.3 186.66 77.341994 614 36.5 372.3 241.71995 633 36.5 372.3 260.71996 657 36.5 372.3 284.71997 628 33.5 341.7 286.31998 587 30.1 307.02 279.981999 529 26.1 266.22 262.782000 457 21.7 221.34 235.662001 374 17.9 182.58 191.422002 304 14.7 149.94 154.062003 250 12.2 124.44 125.562004 204 10 102 1022005 163 8.2 83.64 79.362006 127 6.6 67.32 59.682007 104 5.3 54.06 49.942008 82 4.3 43.86 38.142009 61 3.3 33.66 27.342010 44 2.5 25.5 18.52011 31 1.9 19.38 11.622012 20 1.4 14.28 5.722013 9 0.9 9.18 -0.18Sum 328.4ERR 51.2% 25.0%NPV (12% OCC) 1718 1255 463Table 4.3^Adjusted Net Benefits with a Sustainability Premium of $10.2/bbldepletion of the oil field and the atmosphere's absorption capacity for carbon dioxides.5.4 CommentsTwo complications arise from the point of view of national welfare maximization,especially if the unrealistic assumption of domestic consumption is eliminated. The countryalone clearly has no incentive to curtail extraction of hydrocarbons according to a globalcarbon dioxide constraint. The decision to implement a global carbon dioxide constraint mustbe taken through international collective action; the sustainable pricing rule would merelybe the tool for the implementation of such collective action. On the other hand, the requiredcompensatory investment in sustainable substitutes points toward a sensible strategy on howthe country can mitigate the negative consequences that global carbon dioxide emission limits223would have for an oil exporting country. In particular, since the country is geographicallywell positioned for investment in solar hydrogen, such investment is suitable forcompensating for the loss in revenues that would result from restrictions on carbon dioxideemissions in oil importing countries.Also, if the oil was exported, the country would have little incentive to invest partsof the proceeds from the project in energy efficiency increases in other countries (presumablyin industrial countries in which most oil consumption takes place). However, if all countriesagreed on the use of a sustainability constraint, this apparent problem could be solved. Ifcompensatory investment was required for consumption or depletion of natural capital in allcountries, it would not matter in which country compensatory investment is undertaken sincethe market price would reflect the sustainability premium paid by the producing country.There would be a market price for "unsustainable oil" and another price for "sustainable oil"and the difference would be the sustainability premium. Hence, the country would obtain ahigher price for sustainable oil (for which compensatory investment has been undertaken)than for unsustainable oil, and compensatory investment would be undertaken in the countryin which it yields the highest return. As long as a sustainability constraint is not implementedglobally, it would be the role of international lending agencies to promote, and supportthrough concessionary lending, global thinking and the application of a sustainabilityconstraint even if it is not in the interest of narrowly defined national welfare maximization.6 ConclusionsThe practical evaluation of natural capital depletion in projects shows a surprisinglack of completeness and sophistication. Serious deficiencies prevail primarily with respectto the evaluation of land use, resource depletion and emissions from projects. The extensiveliterature on environmental evaluation and evaluation under uncertainty has obviously not yetfully penetrated practical project analysis. This neglect of environmental evaluation can beunderstood if it is considered that environmental impacts are often only considered a sideaspect of the economic evaluation of a project. Moreover, theoretical evaluation224methodologies often require data that is simply not available in the context of practicalproject evaluation in developing countries. On the other hand, neglect of environmentalevaluation leads to a systematic and other significant bias against the conservation of naturalcapital. This highlights the need for developing and applying simple rules and methodologiesfor environmental evaluation, such as the sustainable supply rule presented in this chapter.The case study in this chapter has shown that sensible estimates of shadow prices fornatural capital can be derived from a sustainability constraint as a rule for evaluation underuncertainty. However, in order to improve the quality of environmental evaluation in projectanalysis, a more systematic approach is needed for those cases where conventionalapproaches for environmental valuation fail because of the involved uncertainty orimplications for intergenerational justice. For example, generic shadow prices for variousemissions should be derived from a sustainability rule. Such shadow prices would be basedon physical flow models of individual pollutants. It cannot be expected that individual projectanalysts deal with the complexities of deriving all shadow prices for natural capital. It wouldbe infeasible for the analysts of every energy project to concern themselves separately withcomplex models of global climate changes in order to arrive at a shadow price for carbondioxide emissions. Therefore, it would be desirable to compile a "Sustainability PricingManual" for projects that would consist of practically applicable methodologies and estimatesfor important environmental shadow prices and could guide and improve the economicanalysis of projects with environmental impacts.The sustainable supply rule presented in this paper represents an application of thesustainability principle to the economic evaluation of natural capital depletion. It reflects amodel of the economic system as subsystem of the biosphere and is based on limitedsubstitutability between natural and human-made capital. This paper has established therequirements for appropriate shadow prices for natural capital depletion and the need foradequate intergenerational compensation. First, a shadow price below the price of asustainable substitute should only be used if it can be shown, with reasonable confidence,that the costs are not higher. Second, rents from depletion of natural capital would have to225be shared equally with all future generations. Third, sharing of rents should beaccommodated through investment in sustainable functional substitutes. Furthermore, in orderto be of practical use, the information requirements for the use of such a pricing rule shouldbe limited.By linking the shadow price of natural capital to the price of a sustainable substitute,the use of the sustainable supply rule would lead to a reversal of the default assumption thatnatural capital is a free good. Instead, the default assumption is that every unsustainableactivity must bear the cost of its conversion to a substituting sustainable activity. Thesustainable supply rule is suitable as a preemptive measure against the biases in evaluatingnatural capital. By using the sustainable supply rule, differences in relative scarcity ofdifferent resources would be appropriately reflected in the sustainable prices. The sustainablesupply rule alleviates the intergenerational justice problem since the benefits from naturalcapital depletion are shared equally with all future generations. This is achieved by chargingevery generation a sustainable price for the services derived from natural capital. Thissustainable price reflects the benefits all generations receive from depletion since it is belowthe initial cost of a sustainable substitute but, likely, above the current market price of thedepleting resource. The rule is based on a semi-strong sustainability constraint that requiressustainability for groups of functionally substitutable types of capital. The intuition of thisapproach rests on the assumption that there is far less uncertainty about functionalsubstitutability than about economic substitutability. The suggested rule is based oncompensation through investment in sustainable functional substitutes. Finally, a majoradvantage of the sustainable supply rule is that information requirements for determinationof shadow prices for natural capital depletion are moderate. The estimated lifetime of aresource, the cost of providing a sustainable substitute and the rate of return on specificcompensating investment suffice to calculate the sustainable supply curve. Even withoutknowledge of the demand curve, the sustainable price can be determined iteratively until themarket equilibrium lies on the sustainable supply curve. Furthermore, the rate of return ona specific compensating investment can likely be assessed more objectively and would be lesscontroversial than the appropriate social discount rate to be used in conventional analysis.226Work in the area of ecological economics is exploratory in nature. A more technicalanalysis of the involved problems is called for. While the general approach for a sustainablesupply rule is presented in this paper, more work has to follow in order to explorealternatives, resolve open theoretical questions and make the approach a workable one forpractical evaluation. Remaining theoretical questions concern the definition of sustainablesubstitutes and functional substitutability. Since there is uncertainty about functionalsubstitutability as well, the remaining risk of inadequate compensation needs to be addressed.An interesting extension would involve inquiry into the suitability of the proposed approachfor a general shift from taxation of labour to taxation of natural capital, which has beenproposed as an approach with intuitive appeal in a situation of high unemployment andenvironmental degradation. For this end, the overall impact of universal application of thesuggested sustainability constraint on the economy would have to be assessed. Finally, theissue of population growth leaves unresolved questions on the appropriateness of thesuggested distribution of resources between generations.More practical issues to be addressed include the institutional arrangements forimplementation of the intergenerational compensation schemes. Who undertakes andsupervises the undertaken investment and to whom do the returns to such investment accrue?In a related paper (von Amsberg 1992), I try to show that the allocation of intergenerationalproperty rights and creation of intergenerational markets can bring about the desiredcompensation through market mechanisms without huge government bureaucracies. Thesustainable supply rule needs to be modified for practical cases such as heterogenousresources, substitutes with inelastic supply and gradual phasing-in of the sustainablesubstitute. Similarly, transition toward a sustainable technology could occur in several steps.In more complicated examples, the sustainable supply rule remains applicable in principle,however, the appropriate equations need to be worked out to guide the practitioner in suchinstances.This chapter does not explicitly address the issues of intra-generational justice eventhough implementation of a sustainability constraint would have profound implications for227the allocation of resources within the living generation. Most compensating investment wouldhave to be undertaken by industrial countries with the highest amount of unsustainableconsumption. On the other hand, many opportunities for compensating investment with thehighest return would likely be in developing countries. This could motivate an increasedresource flow from industrial to developing countries. Intra-generational equity concernscould be addressed explicitly by modifying the sustainable supply rule such that sustainablesupply of a resource would be required not at the current level of consumption but at theincreasing level of consumption resulting from higher consumption in developing countries.Then, sustainability would ensure that consumption takes place at a level that cantheoretically be achieved in all countries and would resolve the dilemma that ecologicaldisaster would likely follow if all countries achieved the level of material consumption thatis currently enjoyed in the industrial countries. Hopefully, this chapter stimulates thinkingand contributes to the definition and implementation of a sustainable development path forindustrial as well as developing nations. 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