Justifying, Characterizing and Indicating Sustainability

JUSTIFYING, CHARACTERIZING AND INDICATING SUSTAINABILITY

Book Series Sustainability, Economics, and Natural Resources Editor-in-Chief Shashi Kant, Faculty [email protected]

of

Forestry,

University

of

Toronto,

Canada;

Editorial Board Geir B. Asheim, Department of Economics, University of Oslo, Norway; [email protected] R. Albert Berry, Munk Centre for International Studies, University of Toronto, Canada; [email protected] Graciela Chichilnisky, Department of Economics, Columbia University, New York, USA; [email protected] David Colander, Department of Economics, Middlebury College, Vermont, USA; [email protected] M. Ali Khan, Department of Economics, Johns Hopkins University, Baltimore, USA; [email protected] Tapan Mitra, Department of Economics, Cornell University, Ithaca, USA; [email protected] About the Series An adequate economic theory of sustainability cannot be based on the neo-classical paradigm that is at the root of most sustainability issues. A new economic theory, rather than a new public policy based on the old theory, will be needed to guide humanity toward sustainability. The challenge to economists, with the help of other social scientists, is to build a new dominant economic paradigm–based on a more organic, holistic, and integrative approach. The book series Sustainability, Economics, and Natural Resources aims to integrate the concept of sustainability fully into economics and to provide a foundation for that new economic paradigm. The series is designed to reflect the multi- and interdisciplinary nature of the needed paradigm and will cover and integrate concepts from different streams such as agent-based modeling, behavioral economics, chaos theory, complexity theory, ecological economics, evolutionary economics, evolutionary game theory, institutional economics, post-Keynesian consumer theory, social choice theory, Smatrix theory, and quantum theory. The series will be a forum for new ideas and concepts concerning recent developments and unsolved problems in sustainability and the applications of these ideas to the policy scenario. Each volume in the series is self-contained. Together these volumes provide dramatic evidence of the range of approaches to sustainability issues found in economic thinking. The editors welcome proposals for new books in the series. Interested authors can contact the Editor-in-Chief, Shashi Kant, or the Publisher, Fabio de Castro for further details at [email protected].

Justifying, Characterizing and Indicating Sustainability GEIR B. ASHEIM Department of Economics, University of Oslo, Norway

123

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978–1–4020–6199–8 ISBN 978–1–4020–6200–1 Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springeronline.com

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Cover Figure: A feasible sustainable stream is superimposed on the discounted utilitarian optimum in the Dasgupta-Heal-Solow model of capital accumulation and resource depletion.

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CONTENTS

Preface Foreword Contributors

vii x xii

1. Economic Analysis of Sustainability

1

PART I. JUSTIFYING SUSTAINABILITY 2. Intergenerational Ethics Under Resource Constraints

19

3. Justifying Sustainability Geir B. Asheim, Wolfgang Buchholz, and Bertil Tungodden

33

4. Resolving Distributional Conflicts Between Generations Geir B. Asheim and Bertil Tungodden

53

5. The Malleability of Undiscounted Utilitarianism as a Criterion of Intergenerational Justice Geir B. Asheim and Wolfgang Buchholz 6. Rawlsian Intergenerational Justice as a Markov-perfect Equilibrium in a Resource Technology 7. Unjust Intergenerational Allocations

63

83 103

PART II. CHARACTERIZING SUSTAINABILITY 8. The Hartwick Rule: Myths and Facts Geir B. Asheim, Wolfgang Buchholz and Cees Withagen

125

9. Hartwick’s Rule in Open Economies

147

10. Capital Gains and ‘Net National Product’ in Open Economies

155

11. Characterizing Sustainability: The Converse of Hartwick’s Rule Cees Withagen and Geir B. Asheim

171

12. On the Sustainable Program in Solow’s Model Cees Withagen, Geir B. Asheim, and Wolfgang Buchholz

179

13. Maximin, Discounting, and Separating Hyperplanes Cees Withagen, Geir B. Asheim, and Wolfgang Buchholz

191

v

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CONTENTS

PART III. INDICATING SUSTAINABILITY 14. Green National Accounting for Welfare and Sustainability: A Taxonomy of Assumptions and Results

197

15. Net National Product as an Indicator of Sustainability

215

16. Adjusting Green NNP to Measure Sustainability

225

17. Does NNP Growth Indicate Welfare Improvement? Geir B. Asheim and Martin L. Weitzman

241

18. A General Approach to Welfare Measurement through National Income Accounting Geir B. Asheim and Wolfgang Buchholz

249

19. Green National Accounting with a Changing Population

271

Index

291

PREFACE

During the last two decades, I have subjected the concept of sustainable development to economic analysis. To a great extent this work has been done in co-operation with my co-authors Wolfgang Buchholz, Bertil Tungodden, Martin Weitzman and Cees Withagen, and it has lead to a series of journal articles. This book presents the results of this research program. The original articles are reproduced. However, I have updated information about references and corrected a few mistakes (mostly typographical). STRUCTURE OF THE BOOK This book consists of 19 chapters. Chapter 1 is new, written as a guide to the book and its content. It also gives an up-to-date survey of relevant literature and its relation to the later chapters. Chapters 2–19 are reproductions of published articles. The articles are organized into three parts. Part I, which comprises Chaps. 2–7, is concerned with the normative question of how to justify sustainability. Part II, consisting of Chaps. 8–13, considers how sustainable development can be characterized. Finally, in Part III, Chaps. 14–19 are devoted to the problem of indicating sustainability. Within each part, the initial chapter – i.e., Chap. 2 for Part I, Chap. 8 for Part II and Chap. 14 for Part III – is an overview article that functions as a survey for the later chapters in the corresponding part. NOTES ON THE HISTORY AND ORIGIN OF THE RESEARCH PROGRAM My interest in sustainability and intergenerational justice was spurred years before I published in 1986 the first of the articles that are included in this book. I was intrigued by the following problem posed in the context of the so-called Dasgupta–Heal–Solow model of capital accumulation and resource depletion: Even under assumptions that ensure that non-decreasing streams of consumption are feasible, discounted utilitarianism forces well-being towards a zero consumption level eventually, independently of how small the positive discount rate is. It was my opinion at the time – and it still is – that this casts serious doubts on the desirability of using discounted utilitarianism as a criterion for intergenerational justice. But if not discounted utilitarianism, what criterion should be used? Two books contributed in a significant way to the formation of my thoughts about how to trade off the opposing interests of different generations in the presence of resource constraints. First, Partha Dasgupta and Geoffrey Heal’s book Economic Theory and Exhaustible Resources from 1979 not only presented the problem discussed in the previous paragraph, but contained a wide-ranging resource economic analysis, both from a positive and normative point of view. Second, Talbot Page’s book vii

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Conservation and Economic Efficiency from 1977 provided a thought-provoking discussion of how to manage natural and environmental resources in a manner that ensures both economic efficiency and intergenerational justice. My two first papers on alternative normative criteria for intergenerational justice, which were written in the latter part of the 1980s and are included in this book as Chaps. 6 and 7, apply these criteria to the Dasgupta–Heal–Solow model. The newer contributions that seek to provide normative justifications for sustainability (i.e., Chaps. 2–5) have also used the Dasgupta–Heal–Solow model as an important testing ground. In addition, two short journal articles were very influential for my work. Dixit, Hammond and Hoel’s article “On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion”, which I first read shortly after its publication in Review of Economic Studies in 1980, contributed to my research in two ways. First, it spurred my interest in Hartwick’s rule, describing resource management along streams with constant well-being. Hartwick’s rule is the integrating concept for the different chapters of Part II, which addresses the problem of how to characterize sustainability. Second, it demonstrated to me the usefulness of the notion of competitive paths, which I have used in many of the contributions that are included here. Martin Weitzman’s article “On the welfare significance of national product in a dynamic economy”, published in Quarterly Journal of Economics in 1976, introduced me to the topic treated in the third and final part of this book: How to indicate welfare improvement and sustainability through national accounting. ACKNOWLEDGEMENTS I am very grateful for the insights that I have obtained through the co-operation with my co-authors, Wolfgang Buchholz, Bertil Tungodden, Martin Weitzman and Cees Withagen. Half of the articles that are reproduced in this book have been co-authored by one or more of them. The research program has benefitted greatly from visits to Stanford University (1985–1986, sponsored by Peter Hammond, and again 2001–2002, then invited by Kenneth Arrow and Lawrence Goulder), Harvard University (2000, sponsored by Martin Weitzman), and CES, University of Munich (1994 and 1997, invited by Hans-Werner Sinn). It was a privilege to take part in lectures, seminars, and discussions during these visits, and the hospitality of these institutions are gratefully acknowledged. In particular, the interest, advice, and support of Ken Arrow and Larry Goulder during my participation in their research initiative on the Environment, the Economy and Sustainable Welfare during my last visit to Stanford were of great value. Seven of the 18 articles included in this book were written as part of this research initiative. Thanks are due also to the organizers of the Ulvön Conference on Environmental Economics – in particular, Bengt Kriström – for creating an enjoyable meeting where many productive discussions on the topics of this book have taken place. In addition to the help and input from the scholars mentioned above, I am also grateful to comments from and discussions with, among others, Thomas Aronsson,

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Sir Partha Dasgupta, John Hartwick, Geoffrey Heal, Michael Hoel, Aanund Hylland, Karl-Gustav Löfgren, Karl-Göran Mäler, Tapan Mitra, Jack Pezzey, Debraj Ray and Atle Seierstad. The challenges that have been offered by journal editors, associate editors and referees are also greatly appreciated. Finally, I am grateful to Shashi Kant for having taken the initiative to this book, to Springer for giving me the opportunity to publish it, to Esther Verdries and Fabio de Castro for editorial assistance, to Blackwell, Elsevier, the Rocky Mountain Mathematics Consortium, Springer and the Swiss Society of Economics and Statistics for permission to reproduce published articles, to CESifo in Munich, the Research Council of Norway (including several Ruhrgas grants) and the William and Flora Hewlett Foundation for financial support, and to my own institution, the Department of Economics at the University of Oslo, for providing me with working conditions that have made this project possible. Geir B. Asheim January 2007

FOREWORD

The concept of “sustainable development,” which is commonly termed “sustainability,” may seem new to some economists, and for others, may even be viewed as inappropriate for inclusion in economics literature. Some economists have even termed this concept as morally repugnant and logically redundant. I believe that these economists, who are not willing to accept the concept of sustainable development as an integral component of the economics profession, are still living in the twentieth century, and are not ready to move the economics profession into the twenty-first century. Such an approach toward the concept of sustainable development, and at times, to any new concept, is tragic to the profession as a whole, and will only contribute to enhancing the gap between economic theory and evidence from real life situations. Fortunately, there is a group of economists who is committed to incorporating the concept of sustainable development into the economics profession. Geir Asheim, who has devoted his last 20 years to subjecting the concept of sustainable development to economic analysis, is among the top economists of this group. Asheim’s work on sustainability is of the highest intellect, greatest precision, and uncompromising rigor which may limit its readership to the technically well-motivated few economists who are able to see the economics profession well-beyond conventional economic efficiency theory based on discounted utilitarianism. The concept of sustainable development is a complex aspect of human welfare, and there are no simple solutions to complex problems. Geir Asheim is one of those rare economists who have realized the complexity of sustainable development, and he has analyzed three critical dimensions of sustainable development – justification, characterization, and indication – keeping the complexity in perspective. This volume is a collection of his work on all these three dimensions, and I believe that readescrs are fortunate to have this collection which is a premier on the economics of sustainable development. The key characteristic of Asheim’s work is to analyze the economic implications of different assumptions and relaxing the assumptions used by other economists and himself in the analysis of sustainable development. This characteristic results in numerous economic findings which are of immense value not only to economists, but also to environmentalists, planners, policy makers, development experts, and resource managers. This volume includes a large number of such findings, and I would leave it to the readers to search for these findings throughout the pages of this volume. However, some findings discussed next are of general interest as well as of critical importance to sustainable development. First, criteria for intergenerational equity should not only be judged by the ethical conditions on which they build, but also by their consequences in specific environments. For example, discounted utilitarianism may have appealing consequences in some technological environments but may lead to consequences indefensible from an ethical point in other environments. At the same time, there exist social preferences

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over infinite utility streams that protect the interests of future generations, while retaining sensitivity for the interest of the present. Second, under a wide set of circumstances the Hartwick rule for capital accumulation and resource depletion characterizes an efficient path with constant utility by the value of net investment being zero. This result does not even depend on substitutability between man-made and natural capital. Third, along paths where utility is not constant, however, it is not generally true that a positive value of net investments – i.e., that accumulation of man-made capital exceeds depletion of natural capital – entails sustainability, or vice versa. Hence, the value of net investments is only an imperfect indicator of sustainability even if the vector of net investments takes into account the depletion of natural resources and degradation of environmental resources. Finally, the different analyses in this book demonstrate that stronger results require stronger assumptions, which not only impose harder informational requirements, but stronger assumptions may not be realistic in many real-life situations. Hence, the applicability of economic results based on stronger assumption becomes limited, and it demands pluralism in economic analysis of sustainable development. In conclusion, this volume presents a path-breaking work on economics of sustainable development. I hope, this volume will not only serve as the premier on this subject, but also motivate many more economists to move the economic profession toward the realities of the twenty-first century. Shashi Kant Editor-in-Chief

CONTRIBUTORS

Geir B. Asheim Department of Economics University of Oslo P.O. Box 1095 Blindern, NO-0317 Oslo, Norway [email protected]

Martin L. Weitzman Department of Economics Harvard University Littauer Center Cambridge, MA 02138, USA [email protected]

Wolfgang Buchholz Department of Economics University of Regensburg DE-93040 Regensburg Germany wolfgang.buchholz@wiwi. uni-regensburg.de

Cees Withagen Department of Economics Tilburg University P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands Department of Economics, Free University De Boelelaan 1105, NL-1081 HV Amsterdam The Netherlands [email protected]

Bertil Tungodden Department of Economics Norwegian School of Economics and Business Administration Helleveien 30, NO-5045 Bergen, Norway [email protected]

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CHAPTER 1 ECONOMIC ANALYSIS OF SUSTAINABILITY

Abstract. This chapter provides a guide to the book and its chapters. It also gives a selective survey of relevant literature and its relation to the included articles.

Twenty years ago the notion of “sustainable development” was introduced into the political agenda by the World Commission on Environment and Development through its report WCED (1987), also called the Brundtland Report. The Report does not give a precise definition of “sustainable development.” The quotation that is usually taken as a point of departure is the following: “Sustainable development is a development that meets the needs of the present without compromising the ability of future generations to meet their own needs” (WCED, 1987, p. 43). The Brundtland Report looks at sustainability both as a requirement for intragenerational justice and as a requirement for intergenerational justice. In the contributions included in this book I (and my co-authors) limit the discussion by considering sustainability to be a requirement for intergenerational justice. The included articles present models where an infinite number of generations follows in sequence, and where distributional issues within each generation are not explicitly considered. In all but one chapter, population is assumed to be constant. In such a context sustainability requires that we, as the current generation, not use more than our fair share of the resource base. More precisely, we should manage the resource base such that the well-being that we ensure ourselves can potentially be shared by all future generations. The notion “well-being” includes everything that influences the situation in which people live. Hence, it includes much more than material consumption. It is intended to capture the importance of health, culture, and nature. There are two important restrictions, though well-being does not include the welfare that people derive from their children’s consumption. Likewise, only nature’s instrumental value (i.e., recognized value to humans) is included in the well-being, not its intrinsic value (i.e., value in its own right regardless of human experience); i.e., an anthropocentric perspective is taken. The general rationale behind these restrictions is that there is an argument to be made in favor of distinguishing the concept of justice applied in a society from the forces that are instrumental in attaining it. In the present context this means that it may be desirable to separate the definition of sustainability from the forces that can motivate our generation to act in accordance with the requirement of sustainability. To approach a formal sustainability definition, I follow Pezzey (1997, p. 451) in saying that development is sustained if the stream of well-being is nondecreasing. Using this term we can then define the concept of sustainable development as follows:

1 Asheim, Justifying, Characterizing and Indicating Sustainability, 1–15 c 2007 Springer


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Definition 1: A generation’s management of the resource base at some point in time is sustainable if it constitutes the first part of a feasible sustained development. A stream of well-being develops in a sustainable manner if each generation’s management of the resource base is sustainable. The idea of defining sustainability in this way corresponds closely to what is usually meant by sustainability, which “basically gets at the issue of whether or not future generations will be at least as well off as the present generation” (Krautkraemer, 1998, p. 2091). This is not the place to present a survey of the abundance of sustainability definitions; it suffices to stay that the above definition is closely related to the definition of sustainability proposed by Pezzey (1997). Note that sustainability in the sense of Definition 1 does not preclude that a generation makes a large sacrifice to the benefit of future generations, so that its own well-being is lower than that of its predecessor. Hence, sustainable development is a wider concept than sustained development: While sustained development implies sustainable development, the converse implication does not hold. Economic theories of natural and environmental resources usually seek to answer the following question: How can an efficient management of natural and environmental resources be achieved? The objective is to get the real economy to imitate a perfect market economy through internalizing external effects and to promote economic efficiency through regulating the use of natural and environmental resources when such internalization is not feasible. Traditionally, many economists have held the view that, in a perfect market economy, posterity will be made better off due to accumulation of man-made capital (including accumulation of knowledge). To the extent that the depletion of natural resources and the degradation of environmental resources have been explicitly taken into account, these economists have claimed that, due to rising resource prices and technological progress, new reserves will be added to existing resources and substitutes to these resources will be made available. A classic reference for this point of view is Barnett and Morse (1963) (see also Nordhaus, 1974). However, in general, this view cannot be defended. At any time the present generation determines how the resource base is being managed. Given our technological capacities, it is possible to exploit the resource base to our own advantage – at the expense of the well-being of future generations. That economic efficiency does not necessarily lead to intergenerational fairness was forcefully argued by Talbot Page (1977) in his book Conservation and Economic Efficiency. He illustrated the issue by the following analogy: If someone suggested that the ocean fisheries in the Pacific should be regulated by giving full rights to the entire resource stock to Japan for one year, to the United States for the next, to Russia for the third year, and so forth, it would be natural to claim that the country that came first would exploit the resources to too large an extent. This skepticism would be especially great if the harvest methods were technologically advanced. Still, if we abstract from the fact that generations overlap, this is the way a perfect market economy (without market failure of any kind) allocates natural and environmental resources between the generations: Future generations’ well-being depends on the altruism that we extend to them as

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3

well as our limited capacity to exploit stocks of natural and environmental resources to our own advantage. This leads to the following conclusions:

r Generational conflicts will not necessarily be solved in a perfect market economy. Distributional problems arise because the present generation through its capital and resource management policy determines the endowment of future generations.

r A requirement for sustainability, if binding, is a requirement for a more fair intergenerational distribution. It is not a requirement for an efficient management of natural and environmental resources. Page (1977) contains another analogy – of a sailing ship – which nicely illustrates this distinction: Sustainability corresponds to setting the rudder according to the destination, while efficiency corresponds to balancing the sails according to the wind. The two issues are related: How the rudder is set influences how the sails will have to be balanced. However, if we care about where we will end up, it is not sufficient to concentrate all attention on how the sails are balanced. In the collection of articles included in this book, I (and my co-authors) subject sustainable development to economic analysis by posing three questions. (1)

Justifying sustainability. From a normative perspective, why is it desirable for our generation to contribute to the implementation of sustainable development? In the analogy of a sailing ship: Why does sustainability correspond to a right destination?

(2)

Characterizing sustainability. If sustainable development is implemented, what does it look like? How do we describe the situation if we are heading for the right destination?

(3)

Indicating sustainability. If we would like to implement sustainable development, how can we tell whether development is in fact sustainable? How to detect if we are off course?

This book consists of three parts corresponding to the three questions posed above; each part consisting of six articles. In the following three sections of the introduction I give a guide to each part. I also provide an up-to-date but selective survey of relevant literature and its relation to the included articles. 1. JUSTIFYING SUSTAINABILITY Generations have conflicting interests in the management of the resource base: If the current generation increases its own well-being by exploiting natural resources and degrading the quality of the natural environment, without making sufficient compensating investments in man-made capital, then the interests of future generations are undermined.

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Can social choice theory guide us in the normative question on how such an intergenerational conflict should be resolved if all generations gathered behind a veil of ignorance (Vickrey, 1945; Harsanyi, 1953; Rawls, 1971), not knowing in what sequence they would be appear? What kind of criterion for intergenerational justice would we recommend if we did not know to what generation we belonged and considered intergenerational distribution from an anonymous perspective? Would we in such a hypothetical situation argue for sustainable development, thus giving this concept a normative foundation? When seeking a normative foundation for sustainability, one must impose some self-evident condition (or axiom) ruling out present behavior that leads to inequitable consequences for future generations. A commonly suggested equity condition is Weak Anonymity (WA). This condition ensures equal treatment of all generations by requiring that a permutation of the well-being of two (or a finite number of) generations not change the social evaluation of the stream. In the intergenerational context WA implies that it is not justifiable to discriminate against a generation only because it appears at a later stage on the time axis. However, equal treatment can be achieved by treating all generations equally poorly. Therefore, one must also impose some condition that leads to efficiency, thus ruling out such wasteful policies. A commonly suggested efficiency condition is Strong Pareto (SP). This condition ensures sensitivity to interests of any one generation by requiring that a stream of well-being is socially preferred to another if it at least one generation is better off and no generation is worse off. When applied to a reflexive and transitive binary relation, the combination of WA and SP is often referred to as the Suppes–Sen grading principle (Suppes, 1966; Sen, 1970). In Chap. 3 (which reproduces Asheim et al., 2001), Wolfgang Buchholz, Bertil Tungodden and I show that the Suppes–Sen grading principle rules out any unsustainable development if the technology is productive. The intuition is simple: Suppose one generation’s well-being is higher than the next. If the excess well-being of the first generation were shifted to the second, then by WA the resulting stream would be equally good in social evaluation. However, since the technology is productive, such an investment of the excess well-being of the first generation will actually have a positive net return, meaning that the second generation will end up with a higher well-being than what the first generation had in the original situation. Hence, by SP, there exists a feasible transfer from one generation to the next that leads to a socially preferred stream whenever the first generation has a higher well-being than the second. This means that only efficient and nondecreasing streams of well-being can be admissible in social evaluation under WA and SP. Such streams correspond to sustained – and hence, sustainable – development. Even after ruling out streams that are not efficient and nondecreasing, the following problem remains: How to resolve intergenerational conflicts that go beyond the sustainability question? In Chap. 4 (which reproduces Asheim and Tungodden, 2004), Bertil Tungodden and I address this problem by imposing further conditions on the social evaluations of well-being streams. We first impose (two versions of)

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a preference continuity axiom that establishes a link to the standard finite setting of distributive justice, by transforming the comparison of any two infinite streams to an infinite number of comparisons of streams each containing a finite number of generations. We may then supplement WA and SP with well-known equity conditions from the traditional literature on distributive justice. One such condition is Hammond Equity, giving absolute priority to generation i if generation i is worse off than generation j, in comparisons between two streams where all generations but i and j have the same well-being. When added to our other conditions, this results in lexicographic maximin (or “leximin”) extended by an overtaking procedure to an infinite number of generations. Leximin gives absolute priority to the worst-off generation, but takes in a lexicographic manner into account the well-being of better-off generations to resolve ties between alternatives. Under this criterion of intergenerational justice an efficient and completely egalitarian stream is the unique optimum if such a stream exists. Another condition applied in the traditional literature is 2-Generational Unit Comparability, which also considers comparisons between two streams where all generations but i and j have the same well-being. It requires that there be a cardinal scale of instantaneous utility, along which well-being is measured, so that social evaluation not change if there is a constant addition to the well-being of generation i in both alternatives and a (possibly different) constant addition to the well-being of generation j in both alternatives. When included, this leads to undiscounted utilitarianism extended by an overtaking procedure to an infinite number of generations. Undiscounted utilitarianism maximizes the sum of well-being and entails that any admissible stream is efficient and increasing in productive technologies.1 Hence, both leximin and undiscounted utilitarianism satisfy WA as a condition for equal treatment, while a commonly used criterion like discounted utilitarianism does not. It has been suggested (see Asheim and Buchholz, 2003, p. 407, for references) that the application of undiscounted utilitarianism imposes unacceptable demands on the present generations, by requiring it to implement a very high saving rate in productive economies. This in turn is used as an ethical justification for discounting, as it protects the present generation from the excessive saving that seems to be implied when future well-being is not discounted. In Chap. 5 (which reproduces Asheim and Buchholz, 2003), Wolfgang Buchholz and I question this justification of discounting by showing that undiscounted utilitarianism has sufficient malleability within important classes of technologies. Consider any efficient and nondecreasing stream of well-being. Then there exists some cardinal scale of instantaneous utility, along which well-being is measured, so that this stream is the unique optimum according to undiscounted utilitarianism when this utility scale is used.2 However, if we seek to resolve the intergenerational conflicts that go beyond the sustainability question, then this may not be an entirely attractive conclusion: Even if we add to the Suppes–Sen principle conditions sufficient to characterize some version of undiscounted utilitarianism, we need not limit the kinds of sustainable development that may be deemed as an acceptable social choice.

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The so-called Dasgupta–Heal–Solow (DHS) model of capital accumulation and resource depletion Dasgupta and Heal (1974, 1979); Solow (1974) is a useful testing ground for different criteria for intergenerational justice:

r Even under assumptions that ensure that nondecreasing streams are feasible, discounted utilitarianism forces well-being toward a zero consumption level eventually, independently of how small the positive discount rate is. The reason is that the capital productivity of the economy approaches zero as an increasing stock of reproducible capital substitutes for dwindling resource input. Hence, discounted utilitarianism seems not to respect the interests of the future generations in this model.

r Undiscounted utilitarianism, on the other hand, leads to unbounded growth if the function that maps well-being into instantaneous utility is strictly increasing (cf. Note 2), leading to unacceptable inequality: Why should we save for the benefit of descendants infinitely better off than ourselves?

r Finally, leximin leads to a constant consumption stream, implying that wellbeing at any point at time is at the maximal level compatible with sustainable development. Such a stream may not be attractive: (a) If the economy is poor at the outset (i.e., has a small stock of man-made capital), it becomes locked into poverty. The productive resources of the economy are managed in a sustainable way, but development is not created. (b) If generations actually care for their children, why should they not be allowed to save on their behalf? In Chap. 6 (which reproduces Asheim, 1988), I analyze the DHS model and ask whether there are criteria for intergenerational justice which (1) protect the distant generations against the grave consequences of discounting, (2) do not lead to unacceptable inequalities, and (3) do not waste the possibilities for development. Answers to this question can be provided by assuming that each generation’s dynamic welfare – according to its subjective preferences – is a convex combination of its own well-being (measured along some cardinal scale of instantaneous utility) and the dynamic welfare of its children. The term “subjective preferences” is meant to capture “selfish” altruism, which motivates a generation to contribute to the welfare of its children because it leads to increased welfare for the contributor. Note that the subjective preferences are nonpaternalistic (in the terminology of Ray, 1987) since each generation respects the subjective preferences of its children, and thereby, takes into account the well-being of all future generations. If each generation manages the resource base in order to maximize its dynamic welfare, then the resulting stream corresponds to a discounted utilitarian optimum. Hence, in the DHS model, this leads to unacceptable outcomes for future generations. However, if the criterion of intergenerational justice maximizes the dynamic welfare of the generation that according to its subjective preferences is worst off, then – as I show in Chap. 6 – this leads to streams where well-being grows initially

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when the economy is highly productive, followed by an eventual phase with constant well-being, thereby protecting the distant generations against the grave consequences of discounting. Thus, the possibilities for development are not wasted without leading to unbounded inequalities, while at the same time the economy’s resource base is managed in a sustainable manner. This criterion, which is due to Calvo (1978), includes undiscounted utilitarianism and leximin as special cases since (1) if each generation on the basis of its subjective preferences does not discount the welfare of its children, then the criterion of intergenerational justice is of no importance and we return to undiscounted utilitarianism, while on the other hand (2) if every generation discounts the welfare of its children heavily, then the criterion leads to a completely egalitarian stream and no development occurs. In Chap. 7 (which reproduces Asheim, 1991), I first give a justification of sustainability, as an alternative to the one presented in Chap. 3.3 I postulate that one stream is better than another if it has both less inequality (in terms of Lorenz domination) and a larger sum of well-being (measured along some cardinal scale of instantaneous utility). This amounts to a seemingly weak ethical restriction: A feasible development is excluded if there exists another feasible development that increases the total sum to be shared between the generations, and simultaneously, shares it in a more egalitarian way. I show that a stream of well-being must be nondecreasing in order to be an admissible social choice under this ethical restriction. I then consider the consequences in the DHS model of each generation managing the resource base in order to maximize its dynamic welfare, subject to the constraint that well-being is nondecreasing. This turns out to produce the same kinds of streams as in Chap. 6, where well-being grows initially when the economy is highly productive, followed by an eventual phase with constant well-being. Chapters 3–7 are introduced by Chap. 2 which reproduces Asheim (2005). 2. CHARACTERIZING SUSTAINABILITY Human economic activity leads to the depletion of natural resources and the degradation of environmental resources. Sustainable development requires that man-made capital (both real and human) be accumulated in order to make up for the decreased availability of natural capital. Hartwick’s (1977) rule is a well-known characterization result for sustainable development of a certain kind, namely development where well-being is held constant, meaning that at all times well-being is held at the maximum level consistent with sustainability. The rule assumes constant population and a stationary technology and gives the following characterization: In a perfect market economy, well-being is held constant indefinitely if the depletion of natural capital at any time corresponds in market value to the accumulation of man-made capital, i.e., the market value of net investments is equal to zero. Note that Hartwick’s rule does not entail that the total value of the capital stocks is constant along a path where well-being is held constant. This would be the case under the assumption of a constant interest rate. However, constant well-being and a constant interest rate may be inconsistent in the sense that they cannot both be

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realized. This is indeed the case in the DHS model of capital accumulation and resource depletion, where the decreasing capital productivity (as an increasing stock of reproducible capital substitutes for dwindling resource input) along the egalitarian stream corresponds to a decreasing interest rate. If the interest rate decreases, then the capital gains will be positive. In this case, constant well-being corresponds to an increasing total value of the capital stocks. In Chap. 10 (which reproduces Asheim, 1996), I explore the relation between capital gains and the interest rate along an egalitarian path. Hartwick’s rule holds both for an open (national) and a closed (global) economy. However, particular problems arise when applying Hartwick’s rule for sustainability in an open economy: The technology must then include the gains from trade (see Svensson, 1986). This means that the assumption of a stationary technology would necessitate that the relative international prices are constant. However, from Hotelling’s (1931) rule it follows that from a resource-rich economy’s point of view, the terms-of-trade facing future generations will be more favorable than the one facing the present generation. This implies that a part of the capital gains on the unexploited stocks of natural resources can be consumed at the present time without conflicting with the sustainability requirement, thus lowering the required compensating investments. Hence, sustainability for all countries require resource-rich economies to invest less that their own resource rents, and resource-poor economies to invest more. In Chap. 9 (which reproduces Asheim, 1986), I derive an analog to Hartwick’s rule for open economies and apply it to the DHS model. This analysis is further developed in Chap. 10. In Chap. 9, I also show how the combination of capital accumulation, resource depletion, and a decreasing interest rate in the context of the DHS model have consequences for how sustainable income is divided between three classes of people: workers, capitalists, and resource owners. Resource owners use an increasing resource price to offset their diminishing stocks and achieve constant consumption while investing nothing. In contrast, the capitalists do all the investing. The reason is the following: In order to achieve constant consumption they must accumulate capital, so that the greater capital stock compensates for the decreasing rate of return, as measured by the interest rate. Assume constant population and a stationary technology within a general multisector growth model, which may include natural and environmental resources. Hartwick’s rule states that if, along an efficient stream, the value of net investments (with resource depletion included as negative components) is zero at each point in time, then well-being is constant. This rule was established for a very general class of models in an elegant and important piece of work by Dixit et al. (1980). Dixit et al. also attempted to show the converse result: If well-being remains constant at the maximum sustainable level, then the value of net investments is zero at each point in time. However, they succeeded to do so only under additional (and perhaps not very attractive) assumptions. In Chap. 11 (which reproduces Withagen and Asheim, 1998), Cees Withagen and I give a direct and comprehensive proof of the converse of Harwick’s rule in a general setting without having to make any additional assumptions.4

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Formally, the main result of Chap. 11 states that if a constant well-being stream maximizes the sum of discounted well-being for some path of discount factors supporting the well-being stream, then the value of net investments is equal to zero at all times. In a critique of our work, Cairns and Yang (2000) argue that, in the context of streams where well-being is held at the maximum level consistent with sustainability, discounting “is contrived and inconsistent with the motivation of sustainability analysis.” They thus suggest that in the DHS model – which is the basic model in which Hartwick’s rule for sustainability was originally derived – falls outside the realm for the main result in Chap. 11. Chapter 12 (which is a corrected version of Withagen et al., 2003a)5 was originally written as a response to the critique presented by Cairns and Yang (2000). In this paper, Cees Withagen, Wolfgang Buchholz, and I supplement the analysis of Chap. 11 by showing that any stream in the DHS model having the property that the well-being of the worst-off generation is maximized satisfies the following: It has constant consumption and maximizes the sum of discounted consumption for some path of supporting discount factors. This means that the premise of our general result on the converse of Hartwick’s rule is satisfied in the case of the DHS model. The view presented by Cairns and Yang (2000) is based on a misunderstanding that stems from confounding discounted utilitarianism as a primary ethical objective with having supporting discount factors in a model where intergenerational equity is the objective. In Chap. 13 (which reproduces Withagen et al., 2003b), Cees Withagen, Wolfgang Buchholz, and I clarify our approach further and relate it to the concept of separating hyperplanes. Thus, we connect the characterization of sustainability with the basic results of modern microeconomic theory, as originating with Arrow (1951), Debreu (1951), and Arrow and Debreu (1951) in a general setting, and with Malinvaud (1953) in the setting of dynamic infinite-horizon economies. Chapters 9–13 are introduced by Chap. 8 which reproduces Asheim et al. (2003). In this chapter, Wolfgang Buchholz, Cees Withagen, and I shed light on Hartwick’s rule, provide semantic clarifications, and investigates the implications and relevance of this rule. 3. INDICATING SUSTAINABILITY The question which is posed in part III of this book – “how can we tell whether development is in fact sustainable” – is often associated with a question posed by Hicks (1946) in his book Value and Capital: What is the maximum that a population of an economy can consume in a given period and still be as well off at the end of the period as it was in the beginning? (cf. Hicks, 1946, p. 172). In an economy with constant population and a stationary technology, this question can easily be answered if there is only one aggregate capital good: Well-being does not exceed the sustainable level if and only if the stock of the aggregate capital good is not reduced. It is, however, a complicated task to answer this question in an economy with heterogeneous capital. The reason is that if human economic activity depletes the stocks of natural capital, then it is necessary to determine how much accumulation

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of man-made capital is required to make up for the depletion. How can relative prices be found that “correctly” value the different kinds of capital? It is a natural point of departure to investigate whether market prices – under the assumption of a perfect market economy with constant population and a stationary technology – can be used to determine the “correct” relative price between natural and manmade capital: Does it hold that well-being does not exceed the maximum sustainable level if the market value of net investments is nonnegative, i.e., if the accumulation of man-made capital at least compensates in market value for the depletion of natural capital? Is a non-negative market value of net investments sufficient for sustainability? By Hartwick’s (1977) rule, well-being is constant and at its maximum sustainable level if the market value of net investments is equal to zero at all future times. In the context of a competitive economy, Hartwick’s rule states that an intertemporal competitive equilibrium leads to a completely egalitarian path if, at all times, the value of depleted natural capital measured at competitive prices equals the reinvestment in man-made capital. However, Hartwick’s rule does not entail that a competitive economy that for the moment measured at competitive prices reinvests depleted natural capital in man-made capital manages its stocks for natural and manmade capital in a sustainable manner. For it is conceivable that such reinvestment is achieved because the competitive prices of natural capital are low. This in turn can be caused by the economy not being managed in a sustainable manner: If future generations are poorer than we are, they will be unable to “bid” highly through the intertemporal competitive equilibrium for the depletable natural capital we manage, leading to low prices of such capital today. Hence, although Hartwick’s rule implies that the economy follows an efficient and egalitarian path if the market value of net investments is equal to zero at all times, one cannot conclude that if the market value of net investments at some time is equal to zero, then the well-being at that time is sustainable. I show this formally in Chap. 15 (which reproduces Asheim, 1994) by means of a counterexample in the context of the DHS model. Thus, since the value of net investments is not a perfect indicator of sustainability even if the vector of net investments takes into account the depletion of natural resources and the degradation of environmental resources, it is likewise not possible to indicate sustainability by comparing the value of consumption with comprehensive (or “green”) net national product (where net national product is defined as the sum of the value of consumption and the value of such a comprehensive vector of net investments). The reason why this does not hold is that the relative price of man-made capital in terms of natural capital in an intertemporal competitive equilibrium depends on the entire future equilibrium path. Or in the words of Pezzey and Toman (2002) in their discussion of my 1994 article: “. . . changing an economy from unsustainability to sustainability changes all its prices. Sustainability prices and sustainability itself are thus related in a circular fashion: Without sustainability prices, we cannot know whether the economy is currently sustainable; but without knowing whether the economy is currently sustainable, currently observed prices tell us nothing definite about sustainability. In particular, net national product . . . equals the

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maximum sustainable level of consumption only if an economy is already at a constant consumption path.”

These insights entail that only approximate indicators of sustainability can be derived from observable market prices, even in an economy implementing an intertemporal competitive equilibrium without imperfections of any kind. One approximate indicator of sustainability takes the following quotes from Hicks (1946) as its point of departure. After considering the possibilities of changing interest rates and changing prices, Hicks (1946) writes on p. 174: Income No. 3 must be defined as the maximum amount of money which the individual can spend this week, and still be able to spend the same amount in real terms in each ensuing week.

Then, on p. 184: The standard stream corresponding to Income No. 3 is constant in real terms. . . . We ask . . . how much he would be receiving if he were getting a standard stream of the same present value as his actual expected receipts. This amount is his income.

In an economy where well-being depends on a single consumption good, this concept of income can be defined as the constant level of consumption with the same present value as the actual future stream of consumption. In Chap. 16 (which reproduces Asheim, 1997), I show how comprehensive net national product must then be adjusted for anticipated capital gains and interest rate effects in order to measure this kind of Hicksian income. Unfortunately, this analysis is hard to generalize to the empirically relevant case of multiple consumption goods. However, the Hicksian question – what is the maximum that a population of an economy can consume in a given period and still be as well off at the end of the period as it was in the beginning – can be interpreted in an alternative manner. One important alternative is to associate “as well off ” with the level of dynamic welfare, e.g., of the kind discussed in Sect. 1 of this chapter, entailing that dynamic welfare corresponds to a discounted utilitarian welfare function. In this case it follows from Weitzman (1976) (by combining his (10) with the equation prior to his (14)) that dynamic welfare is improving if and only if the value of net investments is positive. Although Weitzman (1976) appears to be the first to show formally that the value of net investments has such welfare significance, his seminal paper emphasizes the following related result: Increasing real net national product indicates welfare improvement if the constant discounting of a utilitarian welfare function applies to a single consumption good, or to a linearly homogenous aggregate of a vector of consumption goods. Weitzman’s result is remarkable – as it means that changes in the stock of forward looking welfare can be picked up by changes in the flow of the value of current net product – but strong assumptions are invoked. Fortunately, the assumptions can be weakened as long as we stay within the setting where national accounting is comprehensive so that the vector of net investments takes into account the depletion of natural resources and the degradation of environmental resources. The result that welfare improvement is indicated by (1) a positive value of net investments and (2) an increasing real net national product holds:

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r Even if the constant discounting of a utilitarian welfare function does not apply to a linearly homogenous aggregate of a vector of consumption goods. This is shown by Martin Weitzman and me in Chap. 17 (which reproduces Asheim and Weitzman, 2001) under the provision that net national product is deflated by a consumer price index.6

r Even if dynamic welfare does not correspond to a discounted utilitarian welfare function. This is shown by Wolfgang Buchholz and me in Chap. 18 (which reproduces Asheim and Buchholz, 2004) under the provision that dynamic welfare is maximized. Chapter 18 also applies the DHS model of capital accumulation and resource depletion to illustrate how a positive value of net investments and an increasing real net national product can be used to indicate welfare improvement when the economy has other objectives than discounted utilitarianism, e.g., objectives which incorporates a concern for sustainability. In practical applications, a host of different problems makes it hard to use the indicators above for determining whether dynamic welfare is increasing. The assumption that technology is stationary means that technological progress is captured by capital components that measure accumulated knowledge. The change in these components must be included in the vector of net investments and valued. How restrictive this assumption is, relates closely to a second problem, namely that not all investment flows can be valued given the available price information. This applies not only to accumulated knowledge, but also to stocks of natural and environmental capital. A third problem is related to the fact that our capital and resource management does not have deterministic consequences. Finally, if we allow for growth in population, then – even if we assume that an aggregate capital good exists – it is inappropriate to require that the per capita capital stock be nondecreasing if the rate of population growth varies over time. If, e.g., the present generation is half as large as all future generations (i.e., population is constant beginning with the next generation), then sustainability does not require that the present generation accumulates the stock of the aggregate capital stock to a size twice as large as the one it inherited. Because this would leave the current generation at a level of per capita well-being which is lower than that of later generations. In Chap. 19 (which reproduces Asheim, 2004), I discuss the meaning of sustainability and welfare improvement under population growth and present a general analysis of how to indicate welfare improvement with a changing population. The analysis of Chaps. 17–19 shows that a positive value of net investments (appropriately adjusted if technology is not stationary or population is growing) indicates welfare improvement. Returning to the discussion of Chap. 15, we recall that a non-negative value of net investments is not a sufficient condition for sustainability, entailing that increasing dynamic welfare does not ensure that well-being is below the sustainable level. However, we have not addressed the question whether it is a necessary condition: Does a decreasing dynamic welfare imply that well-being exceeds the sustainable level? Pezzey (2004) demonstrates that this is indeed the

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case under discounted utilitarianism, thereby showing that the welfare analysis of Chaps. 17 and 19 can be used as a one-sided sustainability test. Chapters 15–19 are introduced by Chap. 14 which reproduces Asheim (2003). In this chapter, I emphasize the role that different assumptions play for the various results in welfare and sustainability accounting. Acknowledgments: I thank Jack Pezzey and Bertil Tungodden for helpful comments. NOTES 1 Two recent contribution similar to ours, but not imposing our preference continuity axiom, have

appeared: Bossert et al. (2006) show the consequences of adding Hammond Equity (and this condition only) to WA and SP, while Basu and Mitra (2007) show the consequences of adding (only) a unit comparability condition similar to the one that we use. 2 This result is subject a qualification of mathematical nature: We allow the function that maps wellbeing into instantaneous utility to be constant beyond some level of well-being, while ensuring that SP is satisfied by means of a lexicographic construction (see Asheim and Buchholz, 2003, Appendix A, for details). 3 The justification of sustainability provided in Chap. 7 requires that well-being is measured along a cardinal scale of instantaneous utility and that both levels of and differences in such instantaneous utility are comparable between different generations. For the justification of sustainability provided in Chap. 3, on the other hand, it is sufficient that well-being is measured along an ordinal scale, and that levels of well-being are comparable between different generations. 4 The necessity of Hartwick’s rule has subsequently been addressed by Mitra (2002) and Buchholz et al. (2005). 5 The main claim of the original version published as Withagen et al. (2003a) remains valid, but its proof included incorrect components. By assuming that resource input is important in production (Mitra, 1978, p. 121), meaning that its production elasticity is uniformly bounded away from zero, we are able to present an amended version as Chap. 12. 6 See Sefton and Weale (2006) for an analysis which demonstrates the importance of consumer prices indices in the context of national accounting.

REFERENCES Arrow, K.J. (1951), An extension of the basic theorems of welfare economics, in J. Neyman, (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley and Los Angeles Arrow, K.J. and Debreu, G. (1954), Existence of an equilibrium for a competitive equilibrium, Econometrica 22, 265–290 Asheim, G.B. (1986), Hartwick’s rule in open economies, Canadian Journal of Economics 19, 395–402 [Erratum 20, (1987) 177] (Chap. 9 of the present volume) Asheim, G.B. (1988), Rawlsian intergenerational justice as a Markov-perfect equilibrium in a resource technology, Review of Economic Studies 55, 469–483 (Chap. 6 of the present volume) Asheim, G.B. (1991), Unjust intergenerational allocations, Journal of Economic Theory 54, 350–371 (Chap. 7 of the present volume) Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Asheim, G.B. (1996), Capital gains and ‘net national product’ in open economies, Journal of Public Economics 59, 419–434. (Chap. 10 of the present volume)

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Asheim, G.B. (1997), Adjusting Green NNP to measure sustainability, Scandinavian Journal of Economics 99, 355–370 (Chap. 16 of the present volume) Asheim, G.B. (2003), Green national accounting for welfare and sustainability: A taxonomy of assumptions and results, Scottish Journal of Political Economy 50, 113–130. (Chap. 14 of the present volume) Asheim, G.B. (2004), Green national accounting with a changing population, Economic Theory 23, 601–619 (Chap. 19 of the present volume) Asheim, G.B. (2005), Intergenerational ethics under resource constraints, Swiss Journal of Economics and Statistics 141, 313–330 (Chap. 2 of the present volume) Asheim, G.B. and Buchholz, W. (2003), The malleability of undiscounted utilitarianism as a criterion of intergenerational justice, Economica 70, 405–422 (Chap. 5 of the present volume) Asheim, G.B. and Buchholz, W. (2004), A general approach to welfare measurement through national income accounting, Scandinavian Journal of Economics 106, 361–384. (Chap. 18 of the present volume) Asheim, G.B. and Tungodden, B. (2004), Resolving distributional conflicts between generations, Economic Theory 24, 221–230 (Chap. 4 of the present volume) Asheim, G.B. and Weitzman, M. (2001), Does NNP growth indicate welfare improvement?, Economics Letters 73, 233–239. (Chap. 17 of the present volume) Asheim, G.B., Buchholz, W. and Tungodden, B. (2001), Justifying sustainability, Journal of Environmental Economics and Management 41, 252–268 (Chap. 3 of the present volume) Asheim, G.B., Buchholz, W. and Withagen, C. (2003), The Hartwick rule: Myths and facts, Environmental and Resource Economics 25, 129–150 (Chap. 8 of the present volume) Barnett, H.J. and Morse, C. (1963), Scarcity and Growth: The Economics of Natural Resource Availability. John Hopkins University Press, Baltimore, MD Basu, K. and Mitra, T. (2007), Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization, Journal of Economic Theory 133, 350–373 Bossert, W., Sprumont, Y. and Suzumura, K. (2006), Ordering infinite utility streams, Journal of Economic Theory, forthcoming, doi:10.1016/j.jet.2006.03.005 Buchholz, W., Dasgupta, S. and Mitra, T. (2005), Intertemporal equity and Hartwick’s rule in an exhaustible resource model, Scandinavian Journal of Economics 107, 547–561 Cairns, R.D. and Yang, Z. (2000). The converse of Hartwick’s rule and uniqueness of the sustainable path, Natural Resource Modeling 13, 493–502 Calvo, G. (1978), Some notes on time consistency and Rawls’ maximin criterion, Review of Economic Studies 45, 97–102 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Debreu, G. (1951), The coefficient of resource utilization, Econometrica 19, 273–292 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Harsanyi, J.C. (1953), Cardinal utility in welfare economics and in the theory of risk-taking, Journal of Political Economy 61, 434–435 Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Hicks, J. (1946), Value and Capital, 2nd edition. Oxford University Press, Oxford Hotelling, H. (1931), The economics of exhaustible resources, Journal of Political Economy 39, 137–175 Krautkraemer, J. (1998), Nonrenewable resource scarcity, Journal of Economic Literature 36, 2065–2107 Malinvaud, E. (1953). Capital accumulation and efficient allocation of resources, Econometrica 21, 233–268 Mitra, T. (1978), Efficient growth with exhaustible resources in a neoclassical model, Journal of Economic Theory 17, 114–129 Mitra, T. (2002), Intertemporal equity and efficient allocation of resources, Journal or Economic Theory 107, 356–376 Nordhaus, W.D. (1974), Resources as a constraint on growth, American Economic Review 64 (Papers and Proceedings), 22–26 Page, T. (1977), Conservation and Economic Effiency. John Hopkins University Press, Baltimore, MD

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Pezzey, J. (1997), Sustainability constraints versus “optimality” versus intertemporal concern, and axioms versus data, Land Economics 73, 448–466 Pezzey, J. (2004), One-sided sustainability tests with amenities, and changes in technology, trade and population, Journal of Environment Economics and Management 48, 613–631 Pezzey, J. and Toman, M. (2002), The economics of sustainability: A review of journal articles, in J. Pezzey and M. Toman, (eds.), The Economics of Sustainability. Ashgate, Aldershot, UK Rawls, J. (1971), A Theory of Justice. Harvard University Press, Cambridge, MA Ray, D. (1987), Nonpaternalistic intergenerational altruism, Journal of Economic Theory 41, 112–132 Sefton, J.A. and Weale, M.R. (2006), The concept of income in a general equilibrium, Review of Economic Studies 73, 219–249 Sen, A.K. (1970), Collective Choice and Social Welfare. Oliver and Boyd, Edinburgh Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Suppes, P. (1966), Some formal models of grading principles, Synthese 6, 284A–306 Svensson, L.E.O. (1986), Cormment on R.M. Solow, Scandinavian Journal of Economics 88, 153–155 Vickrey, W. (1945), Measuring marginal utility by reactions to risk, Econometrica 13, 319–333 WCED (The World Commission on Environment and Development) (1987), Our Common Future. Oxford University Press, Oxford, UK Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162 Withagen, C. and Asheim, G.B. (1998), Characterizing sustainability: The converse of Hartwick’s rule, Journal of Economic Dynamics and Control 23, 159–165 (Chap. 11 of the present volume) Withagen, C., Asheim, G.B. and Buchholz, W. (2003a), On the sustainable program in Solow’s model, Natural Resource Modeling 16, 219–231 (Chap. 12 of the present volume) Withagen, C., Asheim, G.B. and Buchholz, W. (2003b), Maximin, discounting, and separating hyperplanes, Natural Resource Modeling 16, 213–217 (Chap. 13 of the present volume)

Part I JUSTIFYING SUSTAINABILITY

CHAPTER 2 INTERGENERATIONAL ETHICS UNDER RESOURCE CONSTRAINTS

Abstract. When evaluating long-term policies, economists usually suggest to maximize the sum of discounted utilities. On the one hand, discounted utilitarianism was given a solid axiomatic foundation by Koopmans (Econometrica, 1960). On the other hand, this criterion has questionable implications when applied to economic models with resource constraints. This raises the question: What ethical conditions for intergenerational distribution should and can be imposed? I use my discussion of such conditions to illuminate the conflict between equity and efficiency that has been a central theme in the axiomatic literature on intergenerational justice. Moreover, a guide to some recent contributions is presented.

1. INTRODUCTION How should we treat future generations? From a normative point of view, what are the present generation’s obligations toward the future? What ethical criterion for intergenerational justice should be adopted if one seeks to treat different generations in an impartial manner? These questions can be approached and answered in at least two ways: 1.

Through an axiomatic analysis one can investigate on what fundamental ethical conditions various criteria for intergenerational justice are based, and then proceed to evaluate the normative appeal of these conditions.

2.

By considering different kinds of technological environments (e.g., growth models without or with the restrictions imposed by natural resource constraints), one can explore the consequences of various criteria for intergenerational justice, and compare the properties of the intergenerational utility streams that are generated.

It is consistent with Rawls’ (1971) reflective equilibrium to do both: criteria for intergenerational justice should not only be judged by the ethical conditions on which they build, but also by their consequences in specific environments. In particular, we may question the appropriateness of a criterion for intergenerational justice if it produces unacceptable outcomes in relevant technological environments. This view has been supported by many scholars, including Atkinson, who writes

Originally published in Swiss Journal of Economics and Statistics 141 (2005), 313–330. Reproduced with permission from Swiss Society of Economics and Statistics.

19 Asheim, Justifying, Characterizing and Indicating Sustainability, 19–32 c 2007 Springer


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GEIR B. ASHEIM “. . . the relation between economics and ethical principles is not linear but rather iterative. Examination of the implications of moral principles in particular models may lead to their revision. By applying ethical criteria to concrete economic models, we learn about their consequences, and this may change our views about their attractiveness” (Atkinson, 2001, p. 206)

and Dasgupta and Heal, who conclude “. . . it is legitimate to revise or criticize ethical norms in the light of their implications” (Dasgupta and Heal, 1979, p. 311).

When evaluating long-term policies, economists usually suggest to maximize the sum of discounted utilities. On the one hand, such discounted utilitarianism has been given a solid axiomatic foundation by Koopmans (1960). On the other hand, this criterion has questionable ethical implications when applied to economic models with resource constraints, as demonstrated by Dasgupta and Heal (1974) in the so-called Dasgupta–Heal–Solow (DHS) model of capital accumulation and resource depletion Dasgupta and Heal (1974, 1979); Solow (1974). Hence, the DHS model points to the importance of testing criteria by their consequences in specific technological environments. This paper illustrates the concept of reflective equilibrium by first (in Sect. 2) considering the consequences of a class of utilitarian criteria (with and without discounting) in the DHS model (with and without resource constraints). Then (in Sect. 3) various ethical conditions for intergenerational preferences are presented, including a set similar to the one that Koopmans (1960) uses to axiomatize discounted utilitarianism. This section brings forward a central theme of the axiomatic literature on intergenerational justice, namely that there is a basic conflict between conditions that ensure efficiency on the one hand, and conditions that ensure equity, on the other hand. In particular, discounted utilitarianism, while leading to Paretoefficient optimal streams, does not satisfy conditions that ensure equity in the sense of equal and impartial treatment of different generations. The conflict between efficiency and equity arises when one insists on intergenerational preferences that can be represented by a social welfare function (cf. Basu and Mitra, 2003), they are not incompatible by themselves. Building on Asheim et al. (2001), I indicate in Sect 4 how conditions of efficiency and equity can be used to justify the notion of sustainable development. Moreover, I present a guide to some recent contributions, which explore different ways out of the ethical dilemma posed by the impossibility of combining conditions ensuring both efficiency and equity with a social welfare function that represents the intergenerational preferences. Some concluding remarks are offered in Sect. 5. 2. THE ETHICAL SIGNIFICANCE OF RESOURCE CONSTRAINTS Consider the class of utilitarian criteria:
∞ max u(C(t))e−ρt dt , 0

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where u is a utility function that specifies the well-being derived from non-negative consumption C, and ρ ≥ 0 is the utility discount rate. By letting u have constant elasticity of marginal utility, i.e., u(C) = C 1−η /(1 − η), and assuming η > 1, the utility integral may converge, even if the discount rate ρ equals zero (this corresponds to how Ramsey, 1928, applies classical utilitarianism to infinite utility streams). We may interpret η as a parameter of inequality aversion. Consider testing criteria in this class in the DHS model of capital accumulation and resource depletion: Q = K α Rβ = C + I , where Q denotes non-negative production, K non-negative capital, and R nonnegative resource input, and where I := K˙ and α ≥ β > 0, α + β < 1. The initial resource stock S is finite, implying that the integral of resource inputs is ∞ constrained to be finite: 0 Rdt ≤ S. The DHS model, in the version presented above, is pessimistic by not allowing for technological progress. It is also pessimistic by not including renewable natural resources, only extraction of a nonrenewable resource is combined with capital to produce output. On the other hand, the model is optimistic with respect to the possibilities for substitution between capital and resource when the capital stock increases and resource input diminishes.1 In the version presented above, it is also optimistic by not assuming that capital depreciates. I do not claim that this model describes accurately available production possibilities in the real world. As the subsequent analysis will show, however, it is well-suited to illustrate how a small variation in the parameters of the model may lead to very different consequences when combined with criteria for intergenerational justice. In order to evaluate the ethical significance of resource constraints, one can now consider two different technological environments:

r The DHS model without resource constraints: β = 0 r The DHS model with resource constraints: β > 0 and for each of these, apply two utilitarian criteria for the evaluation of intergenerational utility streams:

r Classical utilitarianism, entailing equal treatment of generations: ρ = 0 r Discounted utilitarianism, entailing unequal treatment of generations: ρ > 0 This gives four cases in which utilitarian criteria can be tested in the DHS model, as illustrated in Table 2.1.

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Table 2.1. Testing Utilitarian Criteria in the DHS Model

Classical utilitarianism Discounted utilitarianism

Without resource constraints  ∞ 1 1−η dt max 0 1−η C

With resource constraints  ∞ 1 1−η max 0 1−η C dt

s.t. C = K α − K˙  ∞ 1 1−η −ρt max 0 1−η C e dt

s.t. C = K α R β − K˙  ∞ 1 1−η −ρt max 0 1−η C e dt

s.t. C = K α − K˙

s.t. C = K α R β − K˙

2.1. Classical Utilitarianism Without Resource Constraints The solution to the problem


∞

max 0

1−η 1 dt 1−η C

,

subject to C = K α − K˙ , is given by: K˙ = s K α

and C = (1 − s)K α ,

where s is a constant savings rate satisfying s = 1/η < α. If 1/α < η < ∞, then a positive and constant fraction s = 1/η of output is invested, while the remaining part is consumed. In this case, consumption grows beyond all bonds since, with a positive and constant savings rate, capital and output grow beyond all bonds. One may argue that classical utilitarianism thus leads to unacceptable inequalities in this model. If, one the other hand, η = ∞, then this limit of classical utilitarianism corresponds to the (Rawlsian) maximin criterion, entailing that the savings rate is zero. This leads to constant consumption. If the economy is poor to begin with, in the sense of having a small initial capital stock, then the maximin criterion perpetuates poverty. In either case, classical utilitarian (including the limiting case of maximin) can be criticized, for leading to unacceptable inequalities or for perpetuating poverty.

2.2. Discounted Utilitarianism Without Resource Constraints The solution to the problem


∞

max 0

1−η −ρt 1 e dt 1−η C

,

INTERGENERATIONAL ETHICS UNDER RESOURCE CONSTRAINTS

23

subject to C = K α − K˙ , satisfies that   1 α 1−α K → ρ

  α α 1−α and C → ρ

as t → ∞. This means that an ethically appealing consumption stream that avoids unacceptable inequalities and does not perpetuate poverty can be implemented as an optimal stream by choosing ρ and η appropriately. In particular, with moderate discounting so that consumption is an increasing function of time, the discount rate ρ determines the upper bound on consumption, while the parameter of inequality aversion η determines the pace at which consumption converges to this upper bound. Hence, without resource constraints one may claim that discounted utilitarianism “works better” than classical undiscounted utilitarianism. If this technological environment corresponds to the mental model that most economists have, then it may also serve to explain their endorsement of discounted utilitarianism.

2.3. Classical Utilitarianism With Resource Constraints The solution to the problem


∞

max 0

subject to C = K α R β − K˙ and

∞ 0

K˙ = s K α R β

1−η 1 dt 1−η C

,

Rdt ≤ S, is given by: and C = (1 − s)K α R β ,

where s is a constant savings rate satisfying s = β + (1 − β)/η < α. If (1 − β)/(α − β) < η < ∞, then a positive and constant fraction s = β + (1 − β)/η of output is invested, while the remaining part is consumed. In this case, consumption grows beyond all bonds and one may argue that classical utilitarianism also with resource constraints leads to unacceptable inequalities. One the other hand, η = ∞, corresponding to the (Rawlsian) maximin criterion, entails that the savings rate compensates only for resource extraction. By in this way reinvesting resource rents (cf. Harwick’s rule; Hartwick, 1977), such savings behavior leads to constant consumption. If the economy is poor to begin with, in the sense of having a small initial capital stock, then the maximin criterion perpetuate poverty, as pointed out by Solow (1974) in the context of this model. In either case, the conclusions are exactly the same as in the model without resource constraints (except that the savings rate has to compensate for resource extraction): classical utilitarianism (including the limiting case of maximin) can be criticized, for leading to unacceptable inequalities or for perpetuating poverty.2

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2.4. Discounted Utilitarianism With Resource Constraints The solution to the problem


∞

max 0

subject to C = K α R β − K˙ and

∞ 0

1−η −ρt 1 e dt 1−η C

,

Rdt ≤ S, entails that

K → 0 and C → 0 as t → ∞, even though unbounded consumption growth is feasible. This conclusion holds independently of how small the positive utility discount rate is. The reason is the following: Along a stream where consumption is bounded away from zero, net capital productivity must approach zero as a result of capital accumulation and resource depletion. Such a stream cannot be optimal with a positive and constant discount rate. Hence, a positive utility discount rate ρ in the presence of resource constraints, by forcing consumption to (eventually) approach zero, leads to unacceptable inequalities and undermines the livelihood of future generations. These four cases illustrate the importance of testing the consequences of ethical criteria in different environments. Even though discounted utilitarianism may compare favorably to classical utilitarianism in the model without resource constraints, by avoiding unacceptable inequalities and perpetual poverty, the conclusion is reversed when resource constraints are included. In the DHS model with resource constraints, discounted utilitarianism – for any positive utility discount rate – leads to consequences that undermine the livelihood of future generations and seems to be indefensible from an ethical point of view. Introducing positive discounting does not improve upon the utilitarian criterion in the case with resource constraints. 3. SOCIAL PREFERENCES OVER INTERGENERATIONAL UTILITY STREAMS The demonstration of Sect. 2.4 shows both the importance and deficiency of testing criteria for intergenerational justice by their consequences in a class of technological environments. It is an important message that we should not extrapolate from an observation that a criterion leads appealing consequences in some environments, because this may not generalize to other environments. It is not sufficient since such testing does not give us an understanding of the underlying ethical principles on which various criteria for intergenerational justice build. This motivates an investigation of fundamental ethical conditions that can be imposed on social preferences over infinite utility streams. Koopmans’s (1960) axiomatic analysis of discounted utilitarianism is a seminal investigation of the fundamental ethical conditions on which a criteria for intergenerational justice builds. Following in his tradition, this section applies social choice theory to problems of intergenerational distribution, and considers the relationship

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25

between important ethical conditions, on the one hand, and well-known criteria for intergenerational justice, on the other hand. While for growth theoretic analysis it is convenient to use continuous time, ethical conditions are most often formulated in a discrete time setting, with an infinite but countable number of generations. Let the instantaneous well-being of generation t be represented by utility u t that can take on values in a set Y ⊆ R, where R is the set of real numbers.3 Let X = Y ∞ be the set of all infinite utility streams, where ∞ = |N| and N denotes the set of natural numbers. Denote by 1 u = (u 1 , u 2 , . . . ) an element of X , and denote by 1 uT = (u 1 , u 2 , . . . , u T ) and T +1 u = (u T +1 , u T +2 , . . . ) the T -head and T -tail of the utility stream 1 u, respectively. Assume that there are social preferences R over the utility streams in X , where for any 1 u, 1 v ∈ X , 1 u R 1 v entails that 1 u is deemed socially at least as good as 1 v. Denote by I and P the symmetric and asymmetric parts of R; i.e., 1 u I 1 v entails that 1 u is deemed socially indifferent to 1 v and 1 u P 1 v entails that 1 u is deemed socially preferable to 1 v. There are different types of ethical conditions that can be imposed on intergenerational preferences. Firstly, equity conditions ensure, e.g., that different generations are treated in an equal and impartial manner. Secondly, efficiency conditions ensure that optimal utility streams are nonwasteful. Thirdly, independence conditions postulate that what happens in different periods are treated independently. Lastly, so-called technical conditions impose consistency and continuity requirements on the social preferences. The most common equity condition is Weak Anonymity (WA). This condition ensures equal treatment of all generations by requiring that any finite permutation of utilities should not change the social evaluation of the stream. In the intergenerational context condition WA implies that it is not justifiable to discriminate against a generation only because it appears at a later stage on the time axis. Condition WA (Weak Anonymity) For any 1 u, 1 v ∈ X , 1 u I 1 v if 1 v is derived from through a finite permutation of utilities.

1u

The most common efficiency condition is Strong Pareto (SP). This condition ensures that the social preferences are sensitive to utility increases by any one generation by requiring that a utility stream is socially preferred to another if at least one generation is better off and no generation is worse off. Condition SP (Strong Pareto) For any 1 u, 1 v ∈ X , 1 u P 1 v if u t ≥ vt for all t and u s > u s for some s. Following Koopmans (1960), independence conditions come in two forms. Firstly, by Independent future (IF), preference between streams that differ only from the second period on is the same as if the present time (period 1) was actually at period 2; i.e., as if generations {1, 2, . . . } would have taken the place of generations {2, 3, . . . }. Condition IF (Independent Future) For any 1 u, 1 v ∈ X with u 1 = v1 , 1 u R 1 v if and only if 2 u R 2 v.

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Table 2.2. Important Ethical Conditions for Intergenerational Preferences

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.    & transitive   Indep. future   IP IF  Indep. present  

Table 2.3. Maximin Satisfies . . .

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.    & transitive   Indep. future   IP IF  Indep. present   

Secondly, by Independent Present (IP), preference between streams that differ only in the first two periods does not depend on the continuation of the stream; i.e., the trade-off between the well-being of the first two generation is not influenced by the utility level of later generations. Condition IP (Independent Present) For any 1 u2 , 1 v2 , ∈ Y 2 and 3 w, 3 x ∈ X , (1 u2 , 3 w) R (1 v2 , 3 w) if and only if (1 u2 , 3 x) R (1 v2 , 3 x). Lastly, the so-called technical conditions include that the social preferences are complete and transitive, and continuous (relative to the sup norm topology). The ethical conditions as listed above are summarized in Table 2.2. Consider now different kinds of criteria for intergenerational justice. The (Rawlsian) maximin criterion entails that 1u

R 1 v if and only if

inft≥1 u t ≥ inft≥1 vt .

Hence, only the worst-off generation matters! This criterion is often associated with Rawls (1971) and it was applied in the intergenerational setting by Solow (1974). It is straightforward to demonstrate that maximin is a complete, transitive, and continuous criterion which satisfies condition WA, but not the efficiency and independence conditions, as summarized in Table 2.3. What if the worst-off generation matters most, but the second worst-off generation is used to resolve ties, and so on? This corresponds to a criterion which, in the literature, has been referred to as lexicographic maximin, or leximin for short. When applying leximin to the evaluation of intergenerational utility streams, it satisfies conditions WA and SP, as well as the independence conditions. However, leximin is

INTERGENERATIONAL ETHICS UNDER RESOURCE CONSTRAINTS

27

Table 2.4. Leximin and Classical Utilitarianism Satisfy . . .

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.    & transitive   Indep. future   IP IF  Indep. present  

Table 2.5. If Only the Infinite Future Matters, then . . .

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.    & transitive   Indep. future   IP IF  Indep. present  

neither complete nor continuous, as summarized in Table 2.4. The same conclusion holds for classical utilitarianism, as suggested by Ramsey (1928) and adapted to infinite utility streams by Atsumi (1965) and von Weizsäcker (1965):4 T T ut ≥ vt . 1 u R 1 v if and only if ∃ Tˆ s.t. ∀T ≥ Tˆ , t=1

t=1

Another possibility is to have that only the infinite future matters by, e.g., letting 1u

R 1v

if and only if

lim inft→∞ u t ≥ lim inft→∞ vt .

This criterion satisfies all the conditions, except for condition SP, as summarized in Table 2.5. This is not an attractive ethical criterion, as it is insensitive for the wellbeing of generations 1, 2, 3, . . . , T , independently of how large the finite number T is chosen. In particular, it constitutes a dictatorship of the future, in the terminology of Chichilnisky (1996). Finally, discounted utilitarianism means that ∞ ∞ δ t−1 u t ≥ δ t−1 vt , 1 u R 1 v if and only if t=1

t=1

where δ is the factor between 0 and 1 with which future utilities are discounted. This criterion satisfies all the conditions, except for condition WA, as summarized in Table 2.6. In fact, discounted utilitarianism is the only criterion that satisfies conditions SP, IF, and IP, as well as completeness, transitivity, and continuity. This characterization of discounted utilitarianism is very similar to the one given by Koopmans (1960). Since discounted utilitarianism discriminates future generations by discounting their utility, it does not treat generations equally, as required by

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Table 2.6. Discounted Utilitarianism Satisfies . . .

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.    & transitive   Indep. future   IP IF  Indep. present  

Table 2.7. No Social Preferences Satisfy . . .

WA Weak Anonymity SP Strong Pareto

 Continuity   Compl.  & transitive    Indep. future   IP IF  Indep. present  

condition WA, and it constitutes a dictatorship of the present, in the terminology of Chichilnisky (1996). When going through the above mentioned ethical criteria for intergenerational utility streams, it is noteworthy that none of them satisfies all the conditions that have been introduced. In fact, no criterion can, as a consequence of result reported by Diamond (1965) (and which he attributes to Yaari): There exist no complete, transitive, and continuous (in the sup norm topology) social preferences satisfying conditions WA and SP, as summarized in Table 2.7. This points to a fundamental conflict between equity and efficiency. The technical conditions that Diamond (1965) requires entail that the intergenerational preferences can be numerically represented by a social welfare function. His result has subsequently been strengthened by Basu and Mitra (2003), who show that the assumption of numerical representability comes in conflict with conditions WA and SP, even without imposing continuity. 4. SOLUTIONS TO AN ETHICAL DILEMMA As reported in Section 3, it is impossible to combine the usual ethical conditions ensuring both efficiency and equity with a social welfare function that represents the intergenerational preferences. This constitutes an ethical dilemma. There are essentially two ways out of this dilemma: 1.

Stick with conditions WA and SP, but do not insist on representability.

2.

Stick with representability, but modify equity and efficiency conditions.

INTERGENERATIONAL ETHICS UNDER RESOURCE CONSTRAINTS

29

In relation to the first of these alternatives, it is important to note that conditions WA and SP are not incompatible by themselves, and it is of interest to explore the consequences of imposing these conditions. It is not too difficult to show that conditions WA and SP correspond to the Suppes–Sen grading principle Suppes (1966); Sen (1970), whereby one utility stream is preferred to another if a finite permutation of the former Pareto-dominates the latter. In Asheim et al. (2001) we show that the Suppes–Sen grading principle (and thus, conditions WA and SP) has far-reaching implications in technological environments that satisfy the following productivity condition: A set of feasible utility streams satisfies “productivity” if, for any feasible 1 u with u t > u t+1 for some t, the utility stream (u 1 , . . . , u t−1 , u t+1 , u t , u t+2 , . . . ) is feasible and inefficient. Hence, if a utility stream is not nondecreasing – i.e., there exists some t such that u t > u t+1 – then it is possible to save the additional utility of generation t for the benefit of generation t + 1 such that t + 1’s gain is larger than t’s sacrifice. By condition WA the new stream would have been socially indifferent to the old one even if the additional utility of t were transferred to t + 1 without any net productivity (since this would have amounted to a permutation of the utilities of generations t and t + 1). By condition SP it follows that the new stream is (strictly) preferred since t + 1’s gain is larger than t’s sacrifice. This argument means that only nondecreasing utility streams are undominated by social preferences satisfying conditions WA and SP in technological environments satisfying our productivity condition. Since any nondecreasing utility stream is sustainable – according to any of the most common definitions of the notion of sustainable development – conditions WA and SP thus justify sustainability. The DHS model (even with resource constraints; i.e., β > 0) is productive in the above sense. Hence, if the social preferences satisfy conditions WA and SP, then only efficient and nondecreasing utility streams are undominated. However, as I have already discussed in Sect. 2, discounted utilitarianism leads to optimal utility streams that are not nondecreasing, although efficient and nondecreasing utility streams exist (due to the assumption that α > β). This entails that, in the DHS model with resource constraints, there are utility streams acceptable under social preferences satisfying conditions WA and SP, but such streams are not optimal under discounted utilitarianism. In contrast, efficient and nondecreasing utility streams may be optimal under discounted utilitarianism without resource constraints. Consequently, in such a technological environment, optimal utility streams under discounted utilitarianism need not be dominated under social preferences satisfying WA and SP. This may serve as formal support for the ethical intuition that discounted utilitarianism fails in a fundamental manner in the DHS model with resource constraints, although leading to attractive consequences in the same model without resource constraints. Even though conditions WA and SP combined with productivity may justify sustainability, there exists the further problem concerning how to resolve distributional conflicts between generations that go beyond the sustainability question.

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However, as shown by Basu and Mitra (2007), Asheim and Tungodden (2004a), and Bossert et al. (2006), it is possible to resolve such conflicts by introducing additional conditions, leading to characterizations of leximin and classical utilitarianism. On the one hand, such intergenerational preferences are appealing as they include the usual equity and efficiency conditions. On the other hand, they are both insensitive toward the information provided by either interpersonal level comparability or interpersonal unit comparability. Leximin makes no use of interpersonal unit comparability (even if utilities are unit comparable), while classical utilitarianism makes no use of interpersonal level comparability (even if utilities are level comparable). Hence, it is of interest to explore a middle ground between utilitarian criteria and egalitarian criteria like maximin and leximin, with the aim of developing criteria for intergenerational justice that make nontrivial used of both level and unit comparability. This serves as a motivation for the analysis presented in Asheim and Tungodden (2004b). There we follow the other ways out of the ethical dilemma posed in the beginning of this section, by sticking with representability, while modifying equity and efficiency conditions. Within a framework that resembles the one used by Koopmans (1960), we show that there exist representable social preferences that assign priority to the infinite number of future generations if the present is better off than the future, but trade off the interests of present and future generations in the reverse case. We thereby axiomatize social preferences that lead to ethically appealing outcomes in the DHS model with resource constraints, by allowing for development and thus preventing perpetual poverty, without leading to unacceptable inequalities.

5. CONCLUDING REMARKS In this paper I have illustrated Rawls’ (1971) reflective equilibrium in the context of the social evaluation of infinite utility streams: criteria for intergenerational justice should not only be judged by the ethical conditions on which they build, but also by their consequences in specific environments. I have shown how a discounted utilitarianism, while promoting appealing consequences in some technological environments, may lead to consequences indefensible from an ethical point in other environments. I have also illuminated the conflict between equity and efficiency conditions, which have been a central theme in the axiomatic literature on intergenerational justice, and which represents an ethical dilemma in the social evaluation of infinite utility streams. I have argued that there are ways out of this ethical dilemma. In particular, the existence of such a dilemma does not constitute a definitive case for discounted utilitarianism, which in the terminology of Chichilnisky (1996) represents a dictatorship of the present. Rather, as shown in recent contributions, there exist social preferences over infinite utility streams that protect the interests of future generations, while retaining sensitivity for the interests of the present.

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Acknowledgments: The paper builds on lectures given at Waseda University in Tokyo, and the Congress of the Swiss Society of Economics and Statistics in Zurich, March 2005. I thank Bertil Tungodden and participants at these events, in particular Lucas Bretschger and Koichi Suga, for helpful comments. NOTES 1 If the elasticity of input substitution is below unity, then positive and nondecreasing consumption is not

feasible with resource constraints. If the elasticity is above unity, then resource inputs are not essential. Hence, the Cobb-Douglas case – with unit elasticity of input substitution – is interesting, as it makes the resource essential, without ruling out sustainable development. 2 For a general analysis of classical utilitarianism and maximin in the DHS model, see Asheim et al. (2007). This analysis considers also population growth. 3 A more general formulation is, as used by Koopmans (1960), to assume that the well-being of generation t depends on a n-dimensional consumption vector Ct that takes on values in a connected set. However, by representing the well-being of generation t by a scalar u t , one can focus on intergenerational issues. In doing so, I follow, e.g., Diamond (1965), Chichilnisky (1996), Asheim and Tungodden (2004b), Hari et al. (2005), and Bossert et al. (2006). 4 Why do not leximin and discounted utilitarianism satisfy continuity? To see this, let v > u > w 1 > w 2 > w 3 > · · · > x where w n → x as n → ∞, and consider the utility streams 1 un = (u, w n , w n , w n , . . . ), n ∈ N, and 1 v = (v, x, x, x, . . . ). Both leximin and discounted utilitarianism imply that, for each n ∈ N, 1 un P 1 v, while by condition SP 1 v is socially preferred to the utility stream that 1 un approaches as n → ∞, namely (u, x, x, x, . . . ). This contradicts that leximin and discounted utilitarianism are continuous in the sup norm topology.

REFERENCES Asheim, G.B., Buchholz, W, Hartwick, J.M., Mitra, T., and Withagen, C. (2007), Constant savings rates and quasi-arithmetic population growth under exhaustible resource constraints, Journal of Environmental Economics and Management, 53, 213–229 Asheim, G.B., Buchholz, W., and Tungodden, B. (2001), Justifying sustainability, Journal of Environmental Economics and Management 41, 252–268 (Chap. 3 of the present volume) Asheim, G.B. and Tungodden, B. (2004a), Resolving distributional conflicts between generations, Economic Theory 24, 221–230 (Chap. 4 of the present volume) Asheim, G.B. and Tungodden, B. (2004b), Do Koopmans’ postulates lead to discounted utilitarianism? Discussion paper 32/04, Norwegian School of Economics and Business Administration Atkinson, A.B. (2001), The strange disappearance of welfare economics, Kyklos 54, 193–206 Atsumi, H. (1965), Neoclassical growth and the efficient program of capital accumulation, Review of Economic Studies 32, 127–136 Basu, K. and Mitra, T. (2003), Aggregating infinite utility streams with intergenerational equity: the impossibility of being Paretian, Econometrica 32, 1557–1563 Basu, K. and Mitra, T. (2007), Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization, Journal of Economic Theory, 133, 350–373 Bossert, W., Sprumont, Y., and Suzumura, K. (2006), Ordering infinite utility streams, Journal of Economic Theory, forthcoming, doi:10.1016/j.jet.2006.03.005 Chichilnisky, G. (1996), An axiomatic approach to sustainable development, Social Choice and Welfare 13, 231–257 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Diamond, P. (1965), The evaluation of infinite utility streams, Econometrica 33, 170–177

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Hari, C., Shinotsuka, T., Suzumura, K. and Xu, Y. (2005), On the possibility of continuous, Paretian, and egalitarian evaluation of infinite utility streams. Mimeo, Institute of Economic Research, Hitotsubashi University Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Koopmans, T.C. (1960), Stationary ordinal utility and impatience, Econometrica 28, 287–309 Ramsey, F.P. (1928), A mathematical theory of saving, Economic Journal 38, 543–559 Rawls, J. (1971), Theory of Justice. Harvard University Press, Harvard, MA Sen, A.K. (1970), Collective Choice and Social Welfare. Oliver and Boyd, Edinburgh Solow, R.M. (1974), Intergenerational equity and exhaustible resources. Review of Economic Studies (Symposium), 29–45 Suppes, P. (1966), Some formal models of grading principles, Synthese 6, 284–306 von Weizsäcker, C.C. (1965), Existence of optimal program of accumulation for an infinite time horizon, Review of Economic Studies 32, 85–104

CHAPTER 3 JUSTIFYING SUSTAINABILITY

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern NO-0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics, University of Regensburg DE-93040 Regensburg, Germany Email: [email protected]

BERTIL TUNGODDEN Department of Economics Norwegian School of Economics and Business Administration Helleveien 30, NO-5045 Bergen, Norway Email: [email protected]

Abstract. In the framework of ethical social choice theory, sustainability is justified by Efficiency and Equity as ethical axioms. These axioms correspond to the Suppes–Sen Grading principle. In technologies that are productive in a certain sense, the set of Suppes–Sen maximal utility paths is shown to equal the set of nondecreasing and efficient paths. Since any such path is sustainable, Efficiency and Equity can thus be used to deem any unsustainable path as ethically unacceptable. This finding is contrasted with results that seem to indicate that an infinite number of generations cannot be treated equally.

1. INTRODUCTION Motivated by a concern about environmental deterioration and natural resource depletion, sustainability is by now one of the key concepts in environmental discussion and, at least partly, in environmental policy. It was a major topic in the Originally published in Journal of Environmental Economics and Management 41 (2001), 252–268. Reproduced with permission from Elsevier.

33 Asheim, Justifying, Characterizing and Indicating Sustainability, 33–51 c 2007 Springer


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Brundtland-report WCED (1987) and it has become a main objective of many international organizations like the UN where, after the 1992 United Nations Earth Summit in Rio, it was put on the Agenda 21. Sustainability has also found its firm place as a “leitmotif ” in the programs of political parties and green political movements. The increasing importance of sustainability as a guideline of environmental policy is also reflected in environmental and resource economics; see, e.g., Pezzey (1997) where sustainable paths are confronted with standard optimal solutions as described in the traditional theory of economic growth. An ethical concern is at the heart of the interest in a sustainable environmental policy (cf. Toman et al., 1995, pp. 140–142, and Sandler, 1997, p. 64). In particular, sustainability implies that environmental and natural resources have to be shared with future generations. These resources are seen as a common heritage of mankind to which every generation should have the same right of access. Following Sidgwick (1907, p. 414), Pigou (1932, p. 25), and Ramsey (1928, p. 543) there is also a long tradition in economics for considering the unfavorable treatment of future generations as ethically unacceptable. Not much work has, however, been done on the relationship between an ethical postulate of equal treatment of all generations on the one hand and sustainability on the other. The present paper seeks to provide such a contribution by giving a justification for sustainability in the framework of ethical social choice theory. Our main result is that within a relevant class of technologies only sustainable behavior is ethically justifiable provided that the social preferences satisfy two focal normative axioms; equal treatment being one, efficiency being the other.1 There is a technical literature on intergenerational social preferences that contains rather negative results concerning the possibility of treating generations equally. This literature includes Koopmans (1960), Diamond (1965), Svensson (1980), Epstein (1986, 1987), and Lauwers (1997), and it essentially presents the finding that complete social preferences that treat an infinite number of generations equally need not admit optimal solutions. This negative conclusion appears not to have been much noticed by environmental and resource economists, Dasgupta and Heal (1979, pp. 277–281) being a remarkable exception. Still, it represents a challenge to everyone concerned with sustainability: Is the quest for the equal treatment of an infinite number of generations (implicitly) assumed in the recent literature on sustainability a vain one since earlier technical contributions have shown that such equal treatment need not be possible? This paper resolves this apparent conflict by looking directly at the possibility of having intergenerational preferences that are effective (in the sense of having a nonempty set of maximal paths) in a relevant class of technologies.2 The paper is organized as follows: After describing the proposed axiomatic basis for intergenerational ethical preferences in Sect. 2, we introduce the technological framework of the analysis and define the concept of sustainability in Sect. 3. Throughout the paper we apply two different productivity assumptions, Immediate productivity and Eventual productivity, and we show that both assumptions apply in many important classes of technologies. In Sect. 4 we develop a justification for sustainability by showing that an axiom of equal treatment (Equity) combined with the strong Pareto axiom (Efficiency) is sufficient to rule out nonsustainable

JUSTIFYING SUSTAINABILITY

35

intergenerational utility paths as maximal paths and thus as optimal solutions for any social preferences satisfying these axioms, as long as Immediate productivity holds. This result depends only on the assumption that the utility of any generation is (at least) ordinally measurable and level comparable to the utility levels of other generations. In this respect it strengthens earlier results (cf. Asheim, 1991), where the more demanding assumption of full cardinal unit comparability had to be assumed. Finally, we show that social preferences satisfying Efficiency and Equity are effective (i.e., yield a nonempty set of maximal paths) under the provision that the technology satisfies Eventual productivity. As demonstrated in Sect. 5, Eventual productivity is even sufficient to ensure the existence of complete social preferences obeying both Efficiency and Equity and resulting in a unique and sustainable solution to the problem of intergenerational justice. Thus, this paper yields results in a positive spirit as compared to much of the literature on intergenerational social preferences: An infinite number of generations can be treated equally, and such treatment justifies sustainability. We acknowledge that some may not subscribe to equal treatment as an ethical axiom in the intergenerational context (cf. Arrow, 1999). Here we do not directly address the normative issue of whether this axiom should be endorsed. Rather, we establish the feasibility of imposing such an axiom in the context of an infinite number of generations and investigate its implications for sustainability, one conclusion being that this axiom should not be identified with undiscounted utilitarianism. 2. ETHICAL PREFERENCES In deriving criteria for intergenerational distributive justice we adopt a purely consequentialistic approach which completely abstains from judging, e.g., the intentions and procedures lying behind each generation’s actions. Then the problem of giving an ethical basis for sustainability is reduced to making comparisons between feasible intergenerational distributions. There are many possible ways of solving conflicts of interests between generations in this framework. Here we will look for “a political conception of justice that we hope can gain the support of an overlapping consensus of reasonable . . . doctrines,” “a political conception the principles and values of which all citizens can endorse” (Rawls, 1993, p. 10). Formally, there is an infinite number of generations t = 1, 2,. . . . The utility level of generation t is given by u t which should be interpreted as the utility level of a representative member of this generation. Thus we do not discuss the issue of intragenerational distribution. We assume that u t measures the instantaneous wellbeing that generation t derives from its current situation. The term “instantaneous well-being” signifies that u t does not include altruism or envy toward other generations. We take instantaneous well-being as a starting point, because we consider it important to separate the definition and analysis of sustainability from the forces (e.g., altruism toward future generations) that can motivate generations to act in accordance with the requirement of sustainability; see also Rawls (1971, Sect. 22). Moreover, we assume that the utilities need not be more than ordinally measurable

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and level comparable.3 Hence, it is sufficient to require that the utility levels of each generation can be ranked on an ordinal scale and that these levels can be compared between generations. In order to compare different intergenerational utility paths, some binary relation R over paths 1 u = (u 1 , u 2 , . . . ) starting in period 1 is needed. Any such binary relation R is throughout assumed to be reflexive and transitive,4 but R may be complete or incomplete. If 1 v R 1 u and 1 u R 1 v hold simultaneously, then there is indifference between 1 v and 1 u, which is denoted by 1 v I 1 u. If, however, 1 v R 1 u but not 1 u R 1 v holds, then there is (strict) preference for 1 v over 1 u, which is, as usual, denoted by 1 v P 1 u. Thus I gives the symmetric and P the asymmetric part of the social preferences R. In this paper the social preferences R will be used to determine solutions that are ethically acceptable. Such an approach might be questioned (cf. Pezzey, 1997, pp. 450–460) since any norm stems from subjective value judgements that cannot be scientifically substantiated. Nevertheless, there may exist some basic norms whose ethical appeal seems rather uncontroversial and which can thus be used as axioms for characterizing ethical preferences. Anyone disagreeing with the conclusions that can be drawn from these ethical preferences will then have to argue against the basic norms. Such an axiomatic method makes an ethical debate about normative prescription more transparent by reducing it to an evaluation of the underlying axioms. The least controversial ethical axiom on R is that any social preferences must deem one utility path superior to another if at least one generation is better off and no generation is worse off. Efficiency (of R) Axiom. If 1 u = (u 1 , u 2 , . . . ) and 1 v = (v1 , v2 , . . . ) are two utility paths with vt ≥ u t for all t and vs > u s for some s, then 1 v P 1 u. We call this axiom Efficiency as it implies that any maximal path is efficient. It is also called Strong Pareto or Strong sensitivity. The axiom ensures that the social preferences are sensitive to utility increases of any one generation. The other basic ethical axiom on R imposes equal treatment of all generations by requiring that any social preferences must leave the social valuation of a utility path unchanged when the utility levels of any two generations along the path are permuted. Equity (of R) Axiom. If 1 u = (u 1 , u 2 , . . . ) and 1 v = (v1 , v2 , . . . ) are two utility paths with u s  = vs  and u s  = vs  for some s  , s  and u t = vt for all t = s  , s  , then 1 v I 1 u. The Equity axiom is sometimes also called Weak anonymity or Intergenerational neutrality. It can be considered a basic fairness norm as it ensures that everyone counts the same in social evaluation.5 In the intergenerational context the Equity axiom implies that it is not justifiable to discriminate against some generation only because it appears at a later stage on the time axis. Also in the intergenerational context Equity seems to fall within the category of principles that many endorse, at

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least in a world of certainty (as we assume here). This motivates a clarification of what this prevalent normative view implies for the acceptability of intergenerational utility paths in various technological environments. Note that both Efficiency and Equity are compatible with u t being only an ordinal measure that is level comparable to the utility level of any other generation. This in turn means that these ethical axioms do not entail that a certain decrease of utility for one generation will be compensated by the same increase of utility for another generation since changes in utility need not be comparable. In other words, Equity does not imply undiscounted utilitarianism. Obviously, the Efficiency and Equity axioms are not sufficient to determine a complete binary relation. It is of interest to consider the incomplete binary relation R ∗ that is generated by Efficiency and Equity, i.e., which is obtained when only these two axioms are assumed. Formally, we seek a reflexive and transitive binary relation R ∗ that satisfies Efficiency and Equity and has the property of being a subrelation6 to any reflexive and transitive binary relation R satisfying Efficiency and Equity. It turns out that such a binary relation R ∗ exists and coincides with the well-known Suppes–Sen Grading principle RS (cf. Sen, 1970; Suppes, 1966; and, e.g., Madden, 1996; and Svensson, 1980, in the intergenerational context). The binary relation RS deems two paths to be indifferent if the one is obtained from the other through a finite permutation, where a permutation π , i.e., a bijective mapping of {1, 2, . . . } onto itself, is called finite whenever there is a T such that π(t) = t for any t ≥ T . Definition 1: For any two utility paths 1 u = (u 1 , u 2 , . . . ) and 1 v = (v1 , v2 , . . . ), the relation RS holds if there is a finite permutation π of {1, 2, . . . } which has vπ(t) ≥ u t for all t. Let PS denote the asymmetric part of RS . Say that 1 v Suppes–Sen dominates 1 u if 1 v PS 1 u. By Definition 1, a utility path Suppes–Sen dominates an alternative path if a finite permutation of the former Pareto-dominates the latter. The following proposition states that the Suppes–Sen relation is indeed generated by Efficiency and Equity: Proposition 1: RS satisfies Efficiency and Equity and RS is a subrelation to any reflexive and transitive binary relation satisfying Efficiency and Equity. Thus, in the intergenerational context the Suppes–Sen Grading principle can be given an ethical foundation in terms of two focal normative postulates for social preferences. The proof (which is straightforward and hence deleted) is based on the observation that if a reflexive and transitive binary relation satisfies Equity, then two utility paths are indifferent if the one is obtained from the other by moving around the utility levels of a finite number of generations.7 In Sect. 4 we will justify sustainability by the use of the Suppes–Sen Grading principle. Since this justification is concerned with nondecreasing paths, the following characterization of the Suppes–Sen Grading principle turns out to be useful, where, for any path 1 u = (u 1 , u 2 , . . . ) and any time T , 1 uT = (u 1 , . . . , u T ) denotes the

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truncation of 1 u at T and 1 u˜ T denotes a permutation of 1 uT having the property that ˜ T is nondecreasing.8 1u Proposition 2: For any two paths 1 u = (u 1 , u 2 , . . . ) and 1 v = (v1 , v2 , . . . ), the relation 1 v RS 1 u holds if and only if there is a time T such that: (i) (ii)

T +1 v

˜T 1v

Pareto-dominates or is identical to T +1 u.

Pareto-dominates or is identical to 1 u˜ T .

Proof. (If) Obvious. (Only if) If 1 v RS 1 u, there is a time T at least as large as any period that is affected by the finite permutation of Definition 1. Then T +1 v Pareto-dominates or is identical to T +1 u, i.e., (i) holds, and a permutation of 1 vT Pareto-dominates or is identical to 1 uT . Suppose neither 1 v˜ T Pareto-dominates nor is identical to 1 u˜ T ; i.e., there is a period s ≤ T with v˜s < u˜ s . Consider a finite permutation π of {1, . . . , T } with v˜π(t) ≥ u˜ t for all t ∈ {1, . . . , T }, which exists by construction since a permutation 1 vT of Pareto-dominates or is identical to 1 uT , and hence, a permutation of 1 v˜ T Pareto-dominates or is identical to 1 u˜ T . There can be no s  > s with π(s  ) ≤ s as, 1 u˜ T and 1 v˜ T being nondecreasing, this would imply v˜π(s  ) ≤ v˜s < u˜ s ≤ u˜ s  , which contradicts that v˜π(t) ≥ u t for all t ∈ {1, . . . , T }. Thus π is even a permutation of the subset of periods t ∈ {1, . . . , s}. For all periods within this set, however, no v˜t exceeds v˜s , as v˜s = max{v˜t : t ≤ s}, which contradicts that v˜π(s) ≥ u s . Hence, if a permutation of 1 vT Pareto-dominates or is identical to 1 uT , then there is no period s with v˜s < u˜ s . Proposition 2 deals with the Suppes–Sen Grading principle in a general setting. However, to establish a link to the concept of sustainability, we will have to consider the implications of this principle within a relevant class of technologies. This amounts to defining a domain restriction for RS , and we now turn to this issue. 3. SUSTAINABLE PATHS AND PRODUCTIVE TECHNOLOGIES In order to define sets of feasible paths we assume that the initial endowment of generation t ≥ 1 is given by a n-dimensional (n < ∞) vector of capital stocks kt , which may include different forms of man-made capital as well as different types of natural and environmental re-source stocks. A generation t acts by choosing a utility level u t and a vector of capital stocks kt+1 which is bequeathed to the next generation t + 1. For every t the function Ft gives the maximum utility attainable for generation t if kt is inherited and kt+1 is bequeathed, i.e., u t ≤ Ft (kt , kt+1 ) has to hold for any feasible utility-bequest pair (u t , kt+1 ) of generation t. Furthermore, it is assumed that the utility level of each generation cannot fall below a certain lower bound u. This lower bound serves two purposes. First, u can be interpreted as the subsistence level of any generation. Moreover, since there are technological limitations on the accumulation of stocks in the course of one period, Ft (kt , kt+1 ) < u can be used to capture that the bequest kt+1 is infeasible given the inheritance kt . Hence, generation t’s utility-bequest pair (u t , kt+1 ) is said to be feasible at t given kt

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if u ≤ u t ≤ Ft (kt , kt+1 ). Assuming ordinal level comparability, nothing is changed if u, u t , and Ft are transformed by the same strictly increasing function φ so that the feasibility constraint reads as φ(u) ≤ φ(u t ) ≤ φ(Ft (kt , kt+1 )). The sequence 1 F = (F1 , F2 , . . . ) characterizes the technology of the economy under consideration. Given the technology 1 F = (F1 , F2 , . . . ), a utility path t u = (u t , u t+1 , . . . ) is feasible at t given kt if there exists a path t+1 k = (kt+1 , kt+2 , . . . ) such that, for all s ≥ t, generation s’s utility-bequest pair (u s , ks+1 ) is feasible at s given ks . If t u = (u t , u t+1 , . . . ) is feasible at t given kt , then the same holds true for any other path t v = (vt , vt+1 , . . . ) with u ≤ vs ≤ u s for each s ≥ t since u ≤ vs ≤ u s ≤ F(ks , ks+1 ) implies that (vs , ks+1 ) is feasible at s given ks . Before providing an ethical justification for sustainability in this technological framework we need to clarify what this concept means. As noted by Krautkraemer (1998, p. 2091), “[w]hile there is an abundance of definitions of sustainability, it basically gets at the issue of whether or not future generations will be at least as well off as the present generation.” Our definition is in line with this view: Definition 2: Generation t with inheritance kt is said to behave in a sustainable manner if it chooses a feasible utility-bequest pair (u t , kt+1 ) so that the constant utility path (u t , u t , . . . ) is feasible at t + 1 given kt+1 . The utility path 1 u = (u 1 , u 2 , . . . ) is called sustainable given k1 if there exists 2 k = (k2 , k3 , . . . ) such that every generation behaves in a sustainable manner along (1 k,1 u) = (k1 , (u 1 , k2 ), (u 2 , k3 ), . . . ). This definition corresponds closely to what is usually meant by sustainability, e.g., it can be shown that any path sustainable according to Definition 2 is also sustainable according to a definition of sustainability proposed by Pezzey (1997).9 Furthermore, Definition 1 satisfies a characterization of sustainability suggested by Asheim and Brekke (2002).10 Definition 2 does not entail that it will be desirable to follow any sustainable path. In particular, a generation may leave behind a wrong mix of capital stocks, leading to the realization of an inefficient path. Such inefficiency may be the result of a sequence of generations performing piece-wise planning rather than an omnipotent and benevolent social planner implementing an overall plan. However, even though it will not be desirable to follow any sustainable path, it might be the case that any “good” path is sustainable. Such a justification for sustainability is offered in the following section. For this purpose, we make use of a condition which is sufficient for sustainability of utility paths.11 Proposition 3: If 1 u = (u 1 , u 2 , . . . ) is a nondecreasing utility path that is feasible given k1 , then 1 u is sustainable. Proof. By feasibility there exists 1 k = (k1 , k2 , . . . ) so that u ≤ u t ≤ Ft (kt , kt+1 ) for all t ≥ 1. Hence, for all t, t+1 u = (u t+1 , u t+2 , . . . ) is feasible at t + 1 given kt+1 . If 1 u = (u 1 , u 2 , . . . ) is nondecreasing, then u t ≤ u t+1 , u t ≤ u t+2 , . . . , and it follows that (u t , u t , . . . ) is feasible at t + 1 given kt+1 , implying that any generation

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t behaves in a sustainable manner by choosing (u t , kt+1 ). Thus 1 u = (u 1 , u 2 , . . . ) is sustainable. The converse of Proposition 3 does not hold; i.e., it is not the case that only nondecreasing utility paths are sustainable. In particular, it is not in conflict with sustainability that generation t makes a large sacrifice to the benefit of future generations, leading to its own utility being lower than that of generation t − 1. To give a justification for sustainability we will show that, whenever utility paths are ethically justified according to Efficiency and Equity, they fulfil the sufficient condition for sustainability provided by Proposition 3; i.e., they are nondecreasing. This is not possible without imposing a restriction on the technology which, however, is not very demanding. In an intertemporal context one usually considers technologies that exhibit some kind of productivity. Such productivity can be based on the assumption that consumption can be costlessly postponed to later periods by transforming consumption sacrifices into stocks of man-made capital or by not depleting natural capital. This means that it will be possible to switch consumption between two periods when originally there is higher consumption in the earlier period. This is the starting point of our assumption of Immediate productivity, where productivity for a certain technology is defined, not in terms of consumption, but directly in terms of utility. Immediate productivity (of 1 F) Assumption. If t u = (u t , u t+1 , . . . ) is feasible at t given kt with u t > u t+1 , then (u t+1 , u t , u t+2 , . . . ) is feasible and inefficient at t given kt . By the postulated inefficiency of the permuted utility path, there is even a utility gain when the excess utility enjoyed by generation t in comparison to generation t + 1 is deferred one period. But even if Immediate productivity holds efficient sustainable paths need not exist. To ensure existence of such paths we make another assumption, which is fulfilled for technologies usually considered in the context of sustainability. Eventual productivity (of 1 F) Assumption. For any t and kt , there exists a feasible and efficient path with constant utility. Hence, if 1 F satisfies Eventual productivity, there is, for any t and kt , a utility level m t (kt ) such that the path (m t (kt ), m t (kt ), . . . ) is feasible and efficient at t given kt . Thus, the utility level m t (kt ) is the maximum sustainable utility that can be attained if capital kt is inherited in period t. Note that the assumptions of Immediate productivity and Eventual productivity are both invariant w.r.t. the same positive transformation of each Ft for t = 1, 2, . . . ; i.e., they are also compatible with u t being only an ordinal measure that is level comparable to the utility level of any other generation. The following examples show that the general framework described above includes many important classes of technologies as special cases. As a first example, which also shows the logical independence of Immediate productivity and Eventual

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productivity, consider the class of linear technologies, which have been used by, e.g., Epstein (1986). Example 1: A linear technology is defined by an exogenously given positive price path 1 p = ( p1 , p2 , . . . ).
A consumption path t c = (ct , ct+1 , . . . ) is feasible at t given kt if and only if ∞ s=t ps cs ≤ pt kt which is the intertemporal analogue of the standard budget constraint of a household. This explains the interpretation of pt as the “price” of consumption in period t. In the corresponding linear growth model the ratio pt / pt+1 is the gross rate of return 1 + rt on the part of the one-dimensional and non-negative inheritance kt that is not consumed at time t: 0 ≤ kt+1 ≤

pt pt+1 (kt

− ct ),

ct ≥ 0 .

If utility of consumption ct is described by a strictly increasing utility function u, a linear technology falls into the framework above, with u = u(0) and Ft (kt , kt+1 ) = u(ct ) = u(kt − pt+1 kt+1 / pt ). A linear technology satisfies Immediate productivity if and only if the exogenously given positive price path 1 p = ( p1 , p2 , . . . ) is strictly decreasing; i.e., if pt > pt+1 for all t ≥ 1, entailing that rt = pt / pt+1 − 1, the net rate of return on the part of the inheritance kt that is
not consumed at time t, is positive. Eventual productivity  (with m t (kt ) = u pt kt / ∞ ps ) is satisfied for a linear technology if and only s=t
if ∞ s=1 ps < ∞. As this assumption is compatible with pt < pt+1 for some time t, this example also shows that Eventual productivity does not imply Immediate productivity. Conversely, as a strictly decreasing price path 1 p = ( p1 , p2 , . . . ) with
∞ s=1 ps = ∞ exists (e.g., when pt = 1/t for any t), Immediate productivity does not imply Eventual productivity. Example 2: A second example is given by the one-sector model where f is a strictly increasing and concave production function, depending solely on the non-negative stock of man-made capital kt which is physically identical to the consumption good. Here, ct + kt+1 ≤ f (kt ) + kt , ct ≥ 0, kt+1 ≥ 0 , implying that u = u(0) and F(kt , kt+1 ) = u( f (kt ) + kt − kt+1 ), where u is again a strictly increasing utility function as above. Since the technology is not timedependent, the functional value of F depends only on inheritance and bequest. The one-sector model satisfies Immediate productivity since f is strictly increasing in kt . It satisfies Eventual productivity with m t (kt ) = u( f (kt )) since the constant utility path t u = (m t (kt ), m t (kt ), . . . ) is feasible at t given kt , while increasing utility in period t to a level u t > m t (kt ) but still having u s = m t (kt ) in the subsequent periods s > t will eventually tear the capital stock down to zero. Example 3: The third example is a discrete-time version of the Dasgupta–Heal– Solow-model (cf. Dasgupta and Heal, 1974; Solow, 1974), where production also depends on the extraction of an exhaustible natural resource. Here, kt = (ktm , ktn ), where ktm is the non-negative stock of man-made capital, and where ktn is the

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non-negative stock of the natural resource available in period t. By letting, in this example, f denote a production function depending on the input ktm of man-made n , we capital and the non-negative extraction rate of the natural resource ktn − kt+1 m n m n m get that ct + kt+1 ≤ f (kt , kt − kt+1 ) + kt . Hence, u = u(0) and F(kt , kt+1 ) = n ) + k m − k m ) for any strictly increasing utility function as u( f (ktm , ktn − kt+1 t t+1 above. The Dasgupta–Heal–Solow model satisfies Immediate productivity if the production function f is strictly increasing in its first variable. In this model, Cass and Mitra (1991) give a necessary and sufficient condition for the existence of a path with constant and positive consumption given an initial vector of positive stocks. β    n ) = km α kn − kn (In the Cobb-Douglas case where f (ktm , ktn − kt+1 holds, t t t+1 this condition is α > β; i.e., the elasticity of production of man-made capital has to exceed the elasticity of production of the natural resource input.) Dasgupta and Mitra (1983) show that this implies the existence of an efficient path with constant consumption so that Eventual productivity holds. There may also be examples where utility at time t is directly dependent on stocks of natural resources available at t. Immediate productivity can well be obtained in such an economy if it is based on investing man-made capital and if there is no autonomous depreciation of the stocks of natural capital. An efficient constant utility path which is required for Eventual productivity could be given either by substituting man-made goods for the utility value provided by the stocks of natural capital or, if such a substitution is not possible, by leaving the amounts of the natural resource stocks invariant. The technological framework used in this paper also captures this situation, which is the perspective of proponents for strong sustainability.

4. A JUSTIFICATION FOR SUSTAINABILITY Given any technology 1 F and any binary relation R, say that:

r The feasible utility path 1 u is R-maximal given k1 , if there exists no feasible path 1 v given k1 such that 1 v P 1 u.

r Is effective in 1 F if, for any k1 , there exists an R-maximal path given k1 . r An R-maximal path 1 u is time-consistent if, for any corresponding path of

capital stock vectors 1 k and for all t > 1, 1 u˜ is R-maximal given k˜1 in 1 F˜ where k˜1 = kt and for all s ≥ 1, u˜ s = u s+t−1 , and F˜s = Fs+t−1 .

We will establish that when social preferences R satisfying Efficiency and Equity are applied to technologies fulfilling the assumption of Immediate productivity, then any R-maximal is nondecreasing and thus sustainable. To provide such a justification for sustainability we will first determine the set of utility paths that are maximal w.r.t. Suppes–Sen Grading principle RS . The question of effectiveness will be treated in Proposition 6.

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Proposition 4: If the technology satisfies Immediate productivity, then the set of RS -maximal utility paths is equal to the set of efficient and nondecreasing paths, and every RS -maximal utility path is time-consistent. Proof. (i) Every RS -maximal path is efficient and nondecreasing: Efficiency is obvious by the definition of RS . Suppose that there is an RS -maximal path 1 u given k1 in period 1 which is not nondecreasing. Then there is a period t where u t > u t+1 . If kt is the capital vector in period t, then, by Immediate Productivity, (u t+1 , u t , u t+2 , . . . ) is feasible at t given kt , and it is Pareto-dominated by another path t v = (vt , vt+1 , . . . ) that is feasible at t given kt . This means that the utility path (u 1 , u 2 , . . . , vt , vt+1 , . . . ) is feasible given k1 and Suppes–Sen dominates 1 u, contradicting that 1 u is RS -maximal. Hence, 1 u is nondecreasing. (ii) Every efficient and nondecreasing path is RS -maximal: Suppose that a nondecreasing path 1 u = (u 1 , u 2 , . . . ) is efficient given k1 , and that a path 1 v = (v1 , v2 , . . . ) is feasible given k1 and Suppes–Sen dominates 1 u. Since by Suppes–Sen dominance there is a finite permutation of 1 v that Pareto-dominates 1 u, there exists a T such that T +1 v Pareto-dominates or is identical to T +1 u. Let 1 v˜ T be a permutation of 1 vT having the property that 1 v˜ T is nondecreasing. By Immediate productivity the path (1 v˜ T ,T +1 v) is feasible given k1 , as by a sequence of pairwise permutations it is possible to start a feasible utility path in period 1 with the minimum utility level of 1 vT , and so on. By Proposition 2 and the premises that 1 uT is nondecreasing and 1 v Suppes–Sen dominates 1 u, it follows that (1 v˜ T ,T +1 v) Pareto-dominates 1 u, which contradicts the efficiency of 1 u. Hence, 1 u is R-maximal. (iii) Every RS -maximal utility path 1 u = (u 1 , u 2 , . . . ) with a corresponding path 1 k = (k1 , k2 , . . . ) of capital stock vectors is nondecreasing and efficient by (i). Then for any time period t t u = (u t , u t+1 , . . . ) is nondecreasing and efficient at t given kt so that it is RS -maximal by (ii) if RS is applied to the set of all utility paths feasible at t given kt . This shows time-consistency. Combining this result with Propositions 1 and 3 gives an ethical justification for sustainability in the following sense: In any technology satisfying Immediate Productivity, only sustainable paths are maximal whenever Efficiency and Equity are endorsed as ethical axioms. This is the central result of the paper. Proposition 5: If the reflexive and transitive social preferences R satisfy Efficiency and Equity and the technology satisfies Immediate productivity, then only sustainable utility paths are R-maximal. Proof. If R is a reflexive and transitive binary relation satisfying Efficiency and Equity, then it follows from Proposition 1 that every R-maximal element is RS -maximal and thus by Proposition 4 that it is nondecreasing. By Proposition 3, however, any such path is sustainable. Proposition 5, which can be illustrated by Figure 3.1, means that every unsustainable utility path is unacceptable given any theory of justice within a broad class, as long as a weak productivity assumption is satisfied. The class of theories of justice for which our argument applies is broad, as we have accepted:

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Feasible

Sustainable Efficient Nondecreasing R S -maximal

R-maximal

Figure 3.1. Illustration of Proposition 5.

r Incompleteness of the social preferences. r Any informational framework as long as utility is (at least) ordinally measurable and level comparable.

r Any consequentialistic theory of distributive justice that satisfies Efficiency and Equity, which are requirements that many endorse. Our results resemble those obtained in an earlier work of Asheim (1991) where, however, the equality of intergenerational distributions of utility measured in the Lorenz sense provides the basis for social preferences, thereby requiring a cardinal measure of utility that is both unit and level comparable. It is an important feature of Proposition 5 that it makes the assumption of cardinal unit comparability dispensable. However, if cardinal unit comparability of the utility of different generations is assumed and, in addition, utilitarianism with zero intergenerational discounting – generalized to an infinite number of generations by means of the overtaking criterion12 – is adopted, then we obtain a special case of social preferences satisfying the Efficiency and Equity axioms. This in turn means that Propositions 1 and 4 give a generalization of the observation made by Dasgupta and Heal (1979, pp. 303–308) and Hamilton (1995, p. 407), namely that in the Dasgupta–Heal–Solow model the undiscounted utilitarian maximum will nowhere show decreasing utility. Propositions 4 and 5 do not address the question of effectiveness of RS , i.e., the existence of RS -maximal paths. Even if the technology satisfies Immediate productivity, utility paths which are both nondecreasing and efficient need not exist so that the set of RS -maximal paths may well be empty. This is indeed the case in the linear technology of Sect. 3 (Example 1) when the price path is strictly decreasing, but where the sum of the prices diverges. The following proposition, however, shows that effectiveness of RS can be established by assuming Eventual productivity.

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Proposition 6: If the technology satisfies Eventual productivity, then the Suppes– Sen Grading principle RS is effective. Furthermore, for any k1 there is an RS maximal path that is time-consistent and sustainable.13 Proof. If the technology satisfies Eventual productivity, then, for any given k1 , there exists a feasible and efficient path 1 u = (m 1 (k1 ), m 1 (k1 ), . . . ) with constant utility. Since 1 u has constant utility, the existence of an alternative feasible path 1 v Suppes– Sen dominating 1 u would contradict the efficiency of 1 u. Hence, 1 u is RS -maximal given k1 . This path is time-consistent, and by Proposition 3, it is also sustainable. Thus it is seen that in a relevant class of technologies, the Equity axiom is useful for intergenerational social evaluation, even in the case of an infinite number of generations. This conclusion is somewhat different from the message conveyed by the literature.

5. ON THE POSSIBILITY OF TREATING AN INFINITE NUMBER OF GENERATIONS EQUALLY In most of the literature since Koopmans (1960) the view prevails that Equity might be difficult to apply in the intergenerational context if there is an infinite number of generations. So, e.g., Diamond (1965, p. 170) purports to show “the impossibility of treating all time periods the same,” and for Dasgupta and Heal (1979, p. 280), when summarizing their discussion of the ethical foundation for resource economics, the “key point is that generations cannot be treated identically.”14 A main conclusion of this literature is that the ordinary procedure for establishing effectiveness is blocked when Efficiency and Equity are postulated in the context of an infinite number of generations. More precisely, the Weierstrass theorem cannot be applied in this case since, for relevant classes of technologies, there is no topology that makes the continuity of complete social preferences satisfying the axioms of Efficiency and Equity compatible with the compactness of the set of feasible paths (cf., e.g., Epstein, 1987).15 Based on this finding, a common message of the discussion in the literature is that some kind of impatience or discounting has to be imposed. In the extreme this amounts to saying that a rational evaluation of infinite utility streams will unavoidably lead to discriminating against future generations. In contrast, the present paper’s justification for sustainability indicates that the impression suggested by this literature – that generations cannot be treated equally – is exaggerated. Efficiency and Equity can well be applied to filter out the nonempty set of efficient and nondecreasing paths as maximal solutions as long as some fairly weak productivity assumptions hold. To establish this positive result we followed Epstein (1986, 1987) in changing the focus from the impossibility of having a continuous ordering on a compact set of feasible utility paths to the possibility of having social preferences that are effective (in the sense of having a nonempty set of maximal elements) in a relevant class of technologies.16

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However, even if one accepts this change in the perception of the problem, an objection might be that the filter provided by the derived incomplete ethical preferences is rather coarse and leads to a set of maximal paths within which no comparison can be made. Apart from the technical difficulties in ensuring effectiveness when completeness is imposed, there is also an ethical problem involved in comparing efficient and nondecreasing paths. Going beyond Efficiency and Equity is not compatible with our aim at establishing an overlapping consensus as any additional axiom for resolving distributional conflicts between different generations is likely to be controversial. However, as shown in Proposition 5, basing ethical preferences solely upon the two focal axioms of Efficiency and Equity proved to be fruitful insofar as it was completely sufficient to give a justification for sustainability. Nevertheless, the incomplete binary relation RS generated by Efficiency and Equity could still be deemed unsatisfactory if there were no possibility at all for comparing the RS -maximal elements in a way that is consistent with Efficiency and Equity. In that case the Efficiency and Equity axioms could never be reconciled with the desire to find a solution that is weakly preferred to any other feasible path. However, Proposition 7 below shows that under Eventual productivity there exists even a complete17 binary relation that satisfies Efficiency and Equity and yields a unique (and sustainable) maximal path. In looking for a complete binary relation that satisfies Efficiency and Equity, consider the leximin principle. In the case of infinite utility paths the leximin principle yields a complete binary relation on the class of nondecreasing paths: If 1 u and 1 v are nondecreasing, then 1 v is (strictly) preferred to 1 u (i.e., 1 v leximin-dominates 1 u) if there is a s ≥ 1 with vt = u t for all 1 ≤ t < s and vs > u s . It is possible to extend the domain of the leximin principle in the infinite case beyond the class of nondecreasing paths (cf. Asheim, 1991, p. 355). For a statement of this binary relation, for any 1 u = (u 1 , u 2 , . . . ) and any T ≥ 1, write 1 u˜ T for a permutation of ˜ T is nondecreasing. 1 uT = (u 1 , . . . , u T ) having the property that 1 u Definition 3: For any two utility paths 1 u = (u 1 , u 2 , . . . ) and 1 v = (v1 , v2 , . . . ), the relation 1 v R L 1 u holds if there is a T˜ ≥ 1 such that for all T ≥ T˜ , either 1 v˜ T = 1 u˜ T or there is a s ∈ {1, . . . , T } with v˜t = u˜ t for all 1 ≤ t < s and v˜s > u˜ s . The binary relation R L defined in this way is reflexive, transitive, and satisfies Efficiency and Equity, implying by Proposition 1 that the Suppes–Sen Grading principle RS is a subrelation to R L . On the class of nondecreasing paths the binary relation R L is complete and coincides with the above mentioned leximin principle, while R L may not be able to compare two paths if (at least) one is not nondecreasing. However, by invoking Svensson (1980, Theorem 2),18 there exists a complete, reflexive, and transitive binary relation R L which has R L and thus RS as a subrelation. Since R L ranks an efficient path with constant utility above any other feasible path, the following proposition can be established. Proposition 7: If the technology satisfies Eventual Productivity, then there exists a complete, reflexive, and transitive binary relation R L , satisfying Efficiency and

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Equity, that is effective. Furthermore, for any k1 there is a unique R L -maximal path. This utility path is time-consistent and, due to its constant utility, sustainable. Proof. R L has RS as a subrelation and thus satisfies Efficiency and Equity. If the technology satisfies Eventual productivity, then, for any k1 , there exists a feasible and efficient path 1 u = (m 1 (k1 ), m 1 (k1 ), . . . ) with constant utility. Since 1 u is efficient and has constant utility, 1 u PL 1 v (and hence, 1 u PL 1 v since R L is a subrelation to R L ) where 1 v is any alternative path that is feasible given k1 . Hence, 1 u is the unique R L -maximal path given k1 . This path is time-consistent, and by Proposition 3, it is also sustainable. In Proposition 7 RS is completed by means of the leximin principle. This is only one possibility for constructing complete social preferences that satisfy Efficiency and Equity. In a technology that satisfies Immediate Productivity, a completion of the overtaking criterion may also yield a sustainable path that is preferred to any other path. If we follow this alternative route, however, then we will have to go beyond the framework where utility is only an ordinal measure that is level comparable to the utility level of any other generation. The reason is that use of the overtaking criterion requires that one generation’s gain is comparable to another generation’s loss. Depending on how we construct a cardinal scale (i.e., how we assign cardinal value to gains and losses at different levels of ordinal utility), a wide diversity of paths can be maximal under the completed overtaking criterion and hence under complete social preferences satisfying Efficiency and Equity (see, e.g., Fleurbaey and Michel, 1999). In particular, the criterion does not necessarily entail “excessively” high savings rates leading to an unacceptable strain on the present generation (cf. Arrow, 1999, pp. 15–16). On the other hand, for a given cardinal scale there need not be any maximal path as the assumption of Eventual productivity is not sufficient to ensure that the overtaking criterion is effective. Another approach to making comparisons among RS -maximal paths is to let the choice of an RS -maximal path be a side-constraint in a maximization procedure that does not otherwise take into account ethical considerations (cf. Asheim, 1991; Pezzey, 1997). To fix ideas, consider maximizing the sum of discounted utilities subject to the constraint that the chosen path is efficient and nondecreasing. In the onesector model (cf. Example 2) the unconstrained maximum under discounted utilitarianism is nondecreasing for an initial capital stock that does not exceed the modified golden rule size, due to a sufficiently high and sustained productivity of man-made capital. In such circumstances there is no conflict between discounting utilities and the ethical preferences generated by Efficiency and Equity. Although Equity rules out social preferences based on discounted utilitarianism, this axiom does not necessarily rule out paths that are maximal under discounted utilitarianism. However, in other technological environments – like the Dasgupta–Heal–Solow model (cf. Example 3), where a sufficiently high productivity of man-made capital cannot be sustained even if Eventual productivity is satisfied – any maximal path under discounted utilitarianism (with a constant discount rate) is rejected by Efficiency and Equity.

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Thus, a requirement to choose an RS -maximal path necessarily becomes a binding side-constraint in this model. Some may claim that the nondecreasing paths resulting from the application of discounted utilitarianism in the former model appeal to ethical intuitions, while the maximal paths of this criterion in the latter model do not, as they impoverish generations in the distant future even though sustainable paths are feasible. If so, our analysis helps to explain these intuitions and provides a way to amend unacceptable paths by justifying sustainability as a side-constraint.19 6. CONCLUSION The sustainability requirement, which has come to be considered as an important guideline for environmental policy, is a genuinely ethical one as it at least implicitly draws much of its appeal from the desire to be fair toward future generations. It is, however, far less obvious what the precise relation is between intergenerational justice on the one hand and sustainability on the other. There is a long tradition in economics to define justice by referring to the degree of inequality of income distributions, measured, e.g., by Lorenz curves. In trying to give a justification for sustainability such an approach was developed by Asheim (1991). In this paper we have instead directly imposed that every generation be treated equally in intergenerational social preferences, which is tantamount to saying that discrimination against future generations is excluded. The Equity axiom corresponding to this prevalent ethical norm has a long history in the theory of evaluating intergenerational utility paths. The axiom, however, is considered to cause difficulty, because it might be in conflict with the demand for effectiveness. Here we have shown under weak productivity assumptions how Equity combined with the strong Pareto axiom (Efficiency) is compatible with effectiveness and can be used to justify sustainability in the following sense: Only sustainable paths are ethically acceptable whenever Efficiency and Equity are endorsed as ethical axioms. A further question might be how ethics based on only these two axioms can be extended in order to give clearer advice on how to resolve distributional conflicts between generations going beyond the sustainability question. Acknowledgments: We thank three anonymous referees, Kenneth Arrow, Aanund Hylland and seminar participants in Bergen, Oslo, Paris, Santiago, Ulvön and Zurich for helpful comments. Figure 1 was suggested by Minh Ha-Duong. Financial support from the Research Council of Norway (Ruhrgas grant) is gratefully acknowledged. NOTES 1 Most of the extensive social choice literature on the evaluation of infinite utility paths does not deal with

the issue of sustainability. Exceptions are Asheim (1991) and the prominent contributions by Chichilnisky (1996, 1997), as well as an informal treatment by Buchholz (1997) who suggested the idea that the present paper develops.

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2 The concern for effectiveness was emphasized by Koopmans (1960, Postulate 5) and Epstein (1986;

1987, p. 723), while the importance of limiting attention to a particular class of technologies was illustrated by, e.g., Asheim (1991). A problem with the social choice approach to sustainability suggested by Chichilnisky (1996, 1997) is that it is not effective even in relevant technologies (provided that the term in her maximand reflecting the sustainable utility level is not made redundant by a decreasing discount rate). For a specific discussion of this problem, see Asheim (1996); for a general investigation into the applicability of Chichilnisky’s criterion, see Heal (1998). 3 Sen (1970) is the basic reference on measurability and comparability assumptions in social choice theory. See Blackorby et al. (1984) for an instructive survey. 4 Reflexivity means that u R u for any u. Transitivity means that u R w if u R v and v R w. 1 1 1 1 1 1 1 1 1 5 Invoking impartiality in this way is the cornerstone of ethical social choice theory reaching far beyond intergenerational comparisons (see, e.g., d’Aspremont and Gevers, 1977; Hammond, 1976; Mongin and d’Aspremont, 1996; Roemer, 1996, p. 32; Sen, 1970, Chap. 5). In a setting with an infinite number of time periods, the Equity axiom was first introduced by Diamond (1965). Later it has been used in many contributions to formalize distributional concerns between an infinite number of generations (see, e.g., Svensson, 1980, where the term ethical preferences is associated with the Efficiency and Equity axioms). 6 R ∗ is said to be a subrelation to R if v R ∗ u implies v R u and v P ∗ u implies v P u, with P ∗ 1 1 1 1 1 1 1 1 denoting the asymmetric part of R ∗ . 7 van Liedekerke and Lauwers (1997) argue that moving around only a finite number of utility levels is not sufficient to ensure the impartial treatment of an infinite number of generations. But if Equity is strengthened to allow for the permutation of the utility levels of an infinite number of generations, then a reflexive and transitive binary relation cannot simultaneously satisfy Efficiency. See Vallentyne (1995) for a defence of the finite version of Equity. 8 Saposnik (183) proves a similar result for the finite number case. 9 His definition is: The path u = (u , u , . . . ) is sustainable given k if there exists k = (k , k , . . . ) 1 1 2 1 2 2 3 such that, for all t ≥ 1, there exists a constant path (u¯ t , u¯ t , . . . ) with u¯ t ≥ u t that is feasible at t given kt . 10 Their characterization is: Generation t behaves in a sustainable manner given k by choosing a feasible t pair (u t , kt+1 ) if and only if it is possible for generation t + 1 to choose (u t+1 , kt+2 ) with u t+1 ≥ u t even if generation t + 1 behaves in a sustainable manner given kt+1 . 11 Pezzey (1997, p. 451) refers to a nondecreasing utility path as sustained development. See Pezzey (1997, pp. 451–452) for a discussion of the distinction between sustainability and sustainedness. 12 See von Weizsäcker (1965) and, for a more recent discussion in the philosophical literature, Vallentyne (1993) and Vallentyne and Kagan (1997). 13 Note that within a technology satisfying Eventual productivity only, the Suppes–Sen Grading principle is not sufficient to rule out unsustainable utility paths. 14 Similar statements can also be found in the more recent social choice literature Epstein (1986); Lauwers (1997); Shinotsuka (1998). 15 The topology is not unambiguously given in the infinite number case. Hence, the question is whether there is a topology large enough to allow for continuity and small enough to make interesting sets of feasible utility paths compact. For a discussion of the relevance of the underlying topology for the continuity of social preferences, cf. Brown and Lewis (1981), Campbell (1985), Diamond (1965), Lauwers (1997), Shinotsuka (1998), and Svensson (1980). 16 Epstein (1987, p. 723) argues that, from a given perspective, “it seems more pertinent to investigate the link between effectiveness and impatience directly, without involving continuity which after all, is at best sufficient and definitely not necessary for existence of optimal paths. Thus, for example, a pertinent question is whether impatience (in some precise sense) is necessary for effectiveness in a relevant set of choice environments.” 17 Completeness means that v R u or u R v for any u and v. Hence, a complete binary relation is 1 1 1 1 1 1 able to compare any pair of paths. 18 Svensson’s (1980) Theorem 2 states that any reflexive and transitive binary relation that has the Suppes–Sen Grading principle as a subrelation is itself a subrelation to a complete, reflexive, and transitive binary relation (i.e., an ordering). In proving this result Svensson refers to a general mathematical Lemma by Szpilrajn (1930).

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19 Fleurbaey and Michel (1994) provide a criterion for balancing the interests of the different generations which explicitly depends on the underlying technology. One could also use their criterion for making a choice between nondecreasing paths.

REFERENCES Arrow, K.J. (1999), Discounting, morality, and gaming, in Portney, P.R. and Weyant, J.P. (eds.), Discounting and Intergenerational Equity. Resources for the Future, Washington, DC Asheim, G.B. (1991), Unjust intergenerational allocations, Journal of Economic Theory 54, 350–371 (Chap. 7 of the present volume) Asheim, G.B. (1996), Ethical preferences in the presence of resource constraints, Nordic Journal of Political Economy 23, 55–67 Asheim, G.B. and Brekke, K.A. (2002), Sustainability when capital management has stochastic consequences, Social Choice and Welfare 19, 921–940 Blackorby, C., Donaldson, D. and Weymark, J.A. (1984), Social choice with interpersonal utility comparisons: A diagrammatic introduction, International Economic Review 25, 327–356 Brown, D.G. and Lewis, L. (1981), Myopic economic agents, Econometrica 49, 359–368 Buchholz, W. (1997), Intergenerational equity, in Zylicz, T. (ed.), Ecological Economics, a Volume in the Series A Sustainable Baltic Region. Uppsala University Press, Uppsala Campbell, D.E. (1985), Impossibility theorems and infinite horizons planning, Social Choice Welfare 2, 283–293 Cass, D. and Mitra, T. (1991), Indefinitely sustained consumption despite exhaustible natural resources, Economic Theory 1, 119–146 Chichilnisky, G. (1996), An axiomatic approach to sustainable development, Social Choice and Welfare 13, 231–257 Chichilnisky, G. (1997), What is sustainable development?, Land Economics 73, 467–491 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Dasgupta, S. and Mitra, T. (1983), Intergenerational equity and efficient allocation of exhaustible Resources, International Economic Review 24, 133–153 d’Aspremont, C. and Gevers, L. (1977), Equity and the informational basis of collective choice, Review of Economic Studies 44, 199–209 Diamond, P. (1965), The evaluation of infinite utility streams, Econometrica 33, 170–177 Epstein, L.G. (1986), Intergenerational preference orderings, Social Choice and Welfare 3, 151–160 Epstein, L.G. (1987), Impatience, in Eatwell, J. et al. (eds.), The New Palgrave: A Dictionary of Economics. Macmillan, London Fleurbaey, M. and Michel, P. (1994), Optimal growth and transfers between generations, Recherches Economiques de Louvain 60, 281–300 Fleurbaey, M. and Michel, P. (1999), Quelques réflexions sur la croissance optimale, Revue Économique 50, 715–732 Hamilton, K. (1995), Sustainable development, the Hartwick rule and optimal growth, Environmental and Resource Economics 5, 393–411 Hammond, P.J. (1976), Equity, Arrow’s conditions, and Rawls’ difference principle, Econometrica 44, 753–803 Heal, G.M. (1998), Valuing the Future: Economic Theory and Sustainability. Columbia University Press, New York Koopmans, T.C. (1960), Stationary ordinal utility and impatience, Econometrica 28, 287–309 Krautkraemer, J. (1998), Nonrenewable resource scarcity, Journal of Economic Literature 36, 2065–2107 Lauwers, L. (1997), Continuity and equity with infinite horizons, Social Choice and Welfare 14, 345–356 Madden, P. (1996), Suppes–Sen dominance, generalised Lorenz dominance and the welfare economics of competitive equilibrium – some examples, Journal of Public Economics 61, 247–262 Mongin, P. and d’Aspremont, C. (1996), Utility theory and ethics, in Barbera, S., Hammond, P.J., and Seidl, C. (eds.), Handbook of Utility Theory. Kluwer, Boston

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Pezzey, J. (1997), Sustainability constraints versus “optimality” versus intertemporal concern, and axioms versus data, Land Economics 73, 448–466 Pigou, A.C. (1932), The Economics of Welfare. Macmillan, London Ramsey, F.P. (1928), A mathematical theory of saving, Economic Journal 38, 543–559 Rawls, J. (1971), Theory of Justice. Harvard University Press, Cambridge, MA Rawls, J. (1993), Political Liberalism. Columbia University Press, New York Roemer, J.E. (1996), Theories of Distributive Justice. Harvard University Press, Cambridge, MA Sandler, T. (1997), Global Challenges: An Approach to Environmental, Political, and Economic Problems. Cambridge University Press, Cambridge, UK Saposnik, R. (1983), On evaluating income distribution: Rank dominance, the Suppes–Sen grading principle of justice, and Pareto-optimality, Public Choice 40, 329–336 Sen, A.K. (1970), Collective Choice and Social Welfare. Oliver and Boyd, Edinburgh Sidgwick, H. (1907), The Methods of Ethics. Macmillan, London Shinotsuka, T. (1998), Equity, continuity, and myopia: A generalization of Diamond’s impossibility theorem, Social Choice and Welfare 15, 21–30 Solow, R.M. (1974), Intergenerational equity and exhaustible resources. Review of Economic Studies (Symposium), 29–45 Suppes, P. (1966), Some formal models of grading principles, Synthese 6, 284–306 Svensson, L.G. (1980), Equity among generations, Econometrica 48, 1251–1256 Szpilrajn, E. (1930), Sur l’extension de l’ordre partial, Fundamenta Mathematicae 16, 386–389 Toman, M.A., Pezzey, J. and Krautkraemer, J. (1995), Neoclassical economic growth theory and ‘sustainability’, in Bromley, D.W. (ed.), Handbook of Environmental Economics. Blackwell, Oxford, UK Vallentyne, P. (1993), Utilitarianism and infinite utility, Australasian Journal of Philosophy 71, 212–217 Vallentyne, P. (1995), Infinite utility: Anonymity and person-centredness, Australasian Journal of Philosophy 73, 413–420 Vallentyne, P. and Kagan, S. (1995), Infinite value and finitely additive value theory, Journal of Philosophy 94, 5–26 van Liedekerke, L. and Lauwers, L. (1997), Sacrificing the patrol: Utilitarianism, future generations and infinity, Economics and Philosophy 13, 159–174 von Weizsäcker, C.C. (1965), Existence of optimal program of accumulation for an infinite time horizon, Review of Economic Studies 32, 85–104 WCED (The World Commission on Environment and Development) (1987), Our Common Future. Oxford University Press, Oxford, UK

CHAPTER 4 RESOLVING DISTRIBUTIONAL CONFLICTS BETWEEN GENERATIONS

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern, NO-0317 Oslo, Norway Email: [email protected]

BERTIL TUNGODDEN Department of Economics Norwegian School of Economics and Business Administration Helleveien 30, NO-5045 Bergen, Norway Email: [email protected]

Abstract. We describe a new approach to the problem of resolving distributional conflicts between an infinite and countable number of generations. We impose conditions on the social preferences that capture the following idea: If preference (or indifference) holds between truncated paths for infinitely many truncating times, then preference (or indifference) holds also between the untruncated infinite paths. In this framework, we use such conditions to (1) characterize different versions of leximin and utilitarianism by means of equity conditions well known from the finite setting, and (2) illustrate the problem of combining Strong Pareto and impartiality in an intergenerational setting.

1. INTRODUCTION The Suppes–Sen grading principle captures both a concern for equal treatment of generations and the demand for efficiency, through the conditions of Weak Anonymity and Strong Pareto. It turns out that this is all that is needed in order to justify sustainable solutions within reasonable technological frameworks, as shown by Asheim et al. (2001). However, there are two problems with this approach. First, even if we accept this justification for sustainability, there exists the further problem about how to resolve Originally published in Economic Theory 24 (2004), 221–230. Reproduced with permission from Springer.

53 Asheim, Justifying, Characterizing and Indicating Sustainability, 53–62 c 2007 Springer


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distributional conflicts between generations that go beyond the sustainability question. Second, it has been argued that the Suppes–Sen grading principle cannot capture impartiality among an infinite and countable number of generations in a satisfactory manner van Liedekerke and Lauwers (1997). In the following, we consider both these problems. In Sects. 3–5, we go beyond the sustainability question by introducing conditions on (strict) preference that turn out to bring the infinite intergenerational setting into line with the framework for distributive justice in the finite setting. These conditions capture the idea that one infinite utility path should be considered strictly better than another path if the head of the former is considered strictly better than the head of the latter at infinitely many truncating times. Within this framework, we show how equity conditions well known from the finite setting can be applied to the debate on infinite intergenerational justice. Moreover, we provide characterizations of intergenerational versions of leximin and utilitarianism. In Sect. 6, we consider the possibility of extending impartiality among an infinite number of generations through the idea that one infinite utility path should be considered indifferent to another path if the head of the former is considered indifferent to the head of the latter at infinitely many truncating times. It turns out, however, that this approach does not move us beyond Weak Anonymity unless we are ready to reject Strong Pareto. The formal framework – including the conditions of Weak Anonymity and Strong Pareto – are introduced in Sect. 2. 2. THE FRAMEWORK There is an infinite number of generations t = 1, 2, . . . . The utility level of generation t is given by u t , which should be interpreted as the utility level of a representative member of this generation. A binary relation R over paths 1 u = (u 1 , u 2 , . . . ) starting in period 1 expresses social preferences over different intergenerational utility paths. Any such binary relation R is throughout assumed to be reflexive and transitive on the countably infinite Cartesian product R∞ of the set of real numbers R. The social preferences R may be complete or incomplete, with I denoting the symmetric part, i.e., indifference, and P denoting the asymmetric part, i.e., (strict) preference. For any utility path 1 u = (u 1 , u 2 , . . . ) and any time T , 1 uT = (u 1 , u 2 , . . . , u T ) denotes the truncation of 1 u at T , and 1 u˜ T is a permutation of 1 uT having the property that 1 u˜ T is nondecreasing. Refer to 1 uT as the T -head and T +1 u as the T -tail of 1 u. A path 1 v weakly Pareto-dominates another path 1 u if every generation is weakly better of in 1 v than in 1 u and some generation is strictly better off. Assume a technology determining a set of feasible paths. A feasible path 1 v is said to be efficient if there is no other feasible path that weakly Pareto-dominates it. A feasible path 1 v is said to be R-maximal, if there exists no feasible path 1 u such that 1 u P 1 v. A feasible path 1 v is said to be R-optimal, if 1 v R 1 u for any feasible path 1 u. Any R-optimal path is R-maximal, while the converse need not hold if R is incomplete.

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The following two conditions are imposed on the social preferences: Condition SP (Strong Pareto). For any 1 u and 1 v, if vt ≥ u t for all t and vs > u s for some s, then 1 v P 1 u. Condition WA (Weak Anonymity). For any 1 u and 1 v, if for some finite permutation π , vπ(t) = u t for all t, then 1 v I 1 u.1 The Suppes–Sen grading principle R S deems two paths to be indifferent if one is obtained from the other through a finite permutation, and one utility path to be preferred to another if a finite permutation of the former weakly Pareto-dominates the other. The binary relation R S is a subrelation2 to the social preferences R if and only if R satisfies SP and WA. 3. PREFERENCE CONTINUITY The relation generated by SP and WA – the Suppes–Sen grading principle, R S – is incomplete. In the following three sections, we pose the problem: how to resolve distributional conflicts between generations when comparing paths that are R S maximal, by extending (strict) preference beyond what the Suppes–Sen grading principle entails. We will impose conditions that establish a link to the standard finite setting of distributive justice, by transforming the comparison of any two infinite utility paths to an infinite number of comparisons of utility paths each containing a finite number of generations. We may then apply well-known equity conditions from the traditional literature on distributive justice. There are two options:3
≥ 1 such Condition WPC (Weak Preference Continuity). For any 1 u and 1 v, if ∃T
, (1 vT , T +1 u) P 1 u, then 1 v P 1 u. that ∀T ≥ T
Condition SPC (Strong Preference Continuity). For any 1 u and 1 v, if ∃T
, (1 vT , T +1 u) R 1 u, and ∀T
≥ 1, ∃T ≥ T
such that ≥ 1 such that ∀T ≥ T (1 vT , T +1 u) P 1 u, then 1 v P 1 u. These conditions can alternatively be formulated as follows. Write R :=

∞  ∞ 

{(1 v, 1 u)| (1 vT ,

T +1 u)

R 1 u} .


=1 T =T
T

If R is complete for comparisons between paths having the same tail, then R denotes
there exists no T such that the set pairs (1 v, 1 u) satisfying that beyond some T 1 u is preferred to (1 vT , T +1 u). Write PT := {(1 v, 1 u) ∈ R| (1 vT , T +1 u) P 1 u} and P∞ := {(1 v, 1 u)| 1 v P 1 u}. Then WPC means that lim inf of the sequence PT is included in P∞ , ∞  ∞ 
=1 T =T
T

PT ⊆ P∞ ,

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while SPC means that lim sup of the sequence PT is included in P∞ , ∞ ∞  

PT ⊆ P∞ .


=1 T =T
T

In the following we illustrate how the conditions of WPC and SPC can be used to characterize the intergenerational versions of the Rawlsian leximin principle and the utilitarian principle. All binary relations considered (in Definitions 1–4) are still incomplete. However, it follows from Svensson’s (1980) Theorem 2 that there exist completions of these binary relations.4 4. CHARACTERIZING LEXIMIN The Rawlsian leximin principle has been stated as follows in the infinite case (see, e.g., Asheim, 1991, p. 355), where “S” indicates that RSL will be shown to correspond to the Strong Preference Continuity: Definition 1 (S-Leximin): For any 1 u and 1 v, 1 v RSL 1 u holds if ∃Tˆ ≥ 1 such that
, either 1 v˜ T = 1 u˜ T or there is a s ∈ {1, . . . , T } with v˜t = u˜ t for all 1 ≤ t < ∀T ≥ T s and v˜s > u˜ s . L , that will be shown to Alternatively, there is a weaker formulation of leximin, RW correspond to the Weak Preference Continuity. L u holds if ∃ T
≥ 1 such that Definition 2 (W-Leximin): For any 1 u and 1 v, 1 v IW 1 L


, there is a ∀T ≥ T , 1 v˜ T = 1 u˜ T , and 1 v PW 1 u holds if ∃T ≥ 1 such that ∀T ≥ T s ∈ {1, ..., T } with v˜t = u˜ t for all 1 ≤ t < s and v˜s > u˜ s .

We start out by characterizing RSL . It is well known that the leximin principle covering finite cases can be characterized by the Suppes–Sen grading principle and the equity condition suggested by Hammond (1976, 1979): Condition HE (Hammond Equity). If 1 u and 1 v satisfy that there exist j, k such that u j > v j > vk > u k and u t = vt for all t = j, k, then 1 v R 1 u. It is not straightforward to translate this result into the infinite case.5 However, by applying SPC, we obtain the following characterization. Proposition 1: RSL is a subrelation to R if and only if R satisfies Strong Pareto, Weak Anonymity, Hammond Equity, and Strong Preference Continuity.
≥1 Lemma 1: Assume that R satisfies Weak Anonymity. For any 1 u and 1 v, if ∃T
such that ∀T ≥ T , 1 v˜ T = 1 u˜ T , then 1 v I 1 u.
, 1 v˜ T = 1 u˜ T and 1 v˜ T +1 = 1 u˜ T +1 , implying vT +1 = u T +1 . Proof. For all T ≥ T Hence, T
+1 v = T
+1 u. By WA, 1 v I 1 u. Proof of Proposition 1. (If) Assume that R satisfies SP, WA, HE, and SPC. According to the definition of a subrelation (cf. Note 2), we have to show that, for

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57

any 1 u and 1 v, 1 v ISL 1 u implies 1 v I 1 u and 1 v PSL 1 u implies 1 v P 1 u. This divides the if part of the proof into two subparts.
≥ 1 such (1) Consider any 1 u and 1 v such that 1 v ISL 1 u. By definition of RSL , ∃T
that ∀T ≥ T , 1 v˜ T = 1 u˜ T . By WA and Lemma 1, 1 v I 1 u. (2) Consider any 1 u and 1 v such that 1 v PSL 1 u. For T satisfying 1 v˜ T = 1 u˜ T , it follows from WA that (1 vT , T +1 u) I 1 u. For T satisfying that there is a s ∈ {1, . . . , T } such that v˜t = u˜ t for all 1 ≤ t < s and v˜s > u˜ s , we can construct a T -head 1 uˆ T by means of a sequence of steps involving conflicts between two generations. In particular, let ⎧ ⎪ for n = 0 ⎨ 1 u˜ T n ˜ s+n , s+n+1 u˜ T ) for n = 1, . . . , T − s − 1 (1 u˜ s−1 , u ns , s+1 w 1 uT = ⎪ ⎩ ( u˜ , u n , ˜ T ) = 1 uˆ T for n = T − s , 1 s−1 s s+1 w where, for n = 1, . . . , T − s, u ns = u˜ s + n(v˜s − u˜ s )/(T − s + 1) and, for t = s + 1, . . . , T , w˜ t = min{u˜ t , v˜t }. Then, for n = 1, . . . , T − s, (1 unT ,

T +1 u)

R (1 un−1 T ,

T +1 u)

by HE if v˜s+n < u˜ s+n (since u˜ s+n > w˜ s+n > u ns > u sn−1 ) and SP if v˜s+n ≥ u˜ s+n (since u˜ s+n = w˜ s+n and u ns > u sn−1 ). Hence, by transitivity, (1 uˆ T ,

T +1 u)

R (1 u˜ T ,

T +1 u) .

Since R satisfies SP, it follows that (1 v˜ T , T +1 u) P (1 uˆ T , T +1 u), while WA implies that (1 vT , T +1 u) I (1 v˜ T , T +1 u) and (1 u˜ T , T +1 u) I 1 u. Hence, by transitivity, (1 vT , L
≥ 1 such that ∀T ≥ T
, T +1 u) P 1 u. This shows, by the definition of RS , that ∃ T
≥ 1, ∃T ≥ T
such that (1 vT , T +1 u) P 1 u. Since R satis(1 vT , T +1 u) R 1 u, and ∀T fies SPC, it now follows that 1 v P 1 u. (Only if) Assume that RSL is a subrelation to R. It is trivial to establish that R
≥ 1 such satisfies SP, WA, and HE. To show that R satisfies SPC, assume that ∃T
, (1 vT , T +1 u) R 1 u, and ∀T
≥ 1, ∃T ≥ T
such that (1 vT , T +1 u) P 1 u. that ∀T ≥ T Since RSL is a subrelation to R and RSL is complete for comparisons between paths
≥ 1 such that ∀T ≥ T
, (1 vT , T +1 u) R L 1 u, having the same tail, this implies that ∃T S
≥ 1, ∃T ≥ T
such that (1 vT , T +1 u) P L 1 u. By definition of R L , this entails and ∀T S S that 1 v PSL 1 u, which in turn implies 1 v P 1 u since RSL is a subrelation to R. Thus, we have established that R satisfies SPC. This result deals with an objection to the Rawlsian leximin position – that the leximin principle is implausible because it assigns absolute priority to the interests of the worst off generation in cases where it is in conflict with the interest of an infinite number of future generations. Proposition 1 tells us that our view on intergenerational justice can be determined by considering a particular set of two-generation conflicts. If we agree on assigning absolute priority to the worse off in such a conflict, then we have to assign absolute priority to the worse off in general. Hence, our result provides

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GEIR B. ASHEIM AND BERTIL TUNGODDEN

a defense for the leximin principle in the infinite setting since it seems less difficult to accept the two-generation claim. L through a trivial modification of An analogous result can be established for RW the parts of the proof of Proposition 1 that involve SPC. L is a subrelation to R if and only if R satisfies Strong Pareto, Proposition 2: RW Weak Anonymity, Hammond Equity, and Weak Preference Continuity.

We must impose an assumption on the technological framework in order to ensure L , since R L is a subthat there exists a maximal path according to RSL (and thus RW W relation to RSL ). This provides the following complete justification for an egalitarian approach to intergenerational justice. Proposition 3: Assume a technology determining a set of feasible paths. If there exists a feasible and efficient path 1 v with constant utility, and R satisfies Strong Pareto, Weak Anonymity, Hammond Equity, and Weak Preference Continuity, then 1 v is the unique R-optimal path. Proof. Any alternative feasible path 1 u provides at least one generation with lower L u for any other feasible path. It follows from utility than does 1 v. Hence, 1 v PW 1 L Proposition 2 that RW is a subrelation to R. Therefore, 1 v P 1 u for any other feasible path, implying that 1 v is the unique R-optimal path. It follows that the feasible and efficient path with constant utility is preferred to any L is a subrelation; in other feasible path according to any binary relation to which RW L particular, this holds for RS . Hence, the egalitarian path is the unique optimal path also under the stronger version of leximin. 5. CHARACTERIZING UTILITARIANISM The utilitarian overtaking criterion, introduced by Atsumi (1965) and von Weizsäcker (1965), represents an important alternative approach to intergenerational justice. As with leximin, there are two versions to consider.
≥ 1 such that Definition 3 (Catching Up): For any 1 u and 1 v, 1 v RSU 1 u holds if ∃T T T
∀T ≥ T , t=1 vt ≥ t=1 u t . U u holds if ∃ T
≥ 1 such that Definition 4 (Overtaking): For any 1 u and 1 v, 1 v IW 1 T T U

,
∀T ≥ T , t=1 vt = t=1 u t , and 1 v PW 1 u holds if ∃T ≥ 1 such that ∀T ≥ T T T t=1 vt > t=1 u t .

As an illustration, compare 1 v = (2, 0, 2, 0, . . . ) and 1 u = (1, 1, 1, 1, . . . ). Here PSU 1 u since 1 u never catches up with 1 v, while the utility paths are incomparable U since v never overtakes u. Atsumi (1965, p. 128) and von Weizaccording to RW 1 1 säcker (1965, p. 85) define optimality by catching up (i.e., 1 v is optimal if 1 v RSU 1 u for any feasible path 1 u), while von Weizsäcker defines preference by overtaking U u). (i.e., 1 v is preferred to 1 u if 1 v PW 1

1v

RESOLVING DISTRIBUTIONAL CONFLICTS BETWEEN GENERATIONS

59

To provide characterizations of these utilitarian criteria, we appeal to a weak twogeneration version of an invariance condition.6 Condition 2UC (2-Generation Unit Comparability). For any 1 u and 1 v, if 1 v R 1 u and there exist j, k and (a j , ak ) ∈ R2 such that uˆ j = u j + a j , vˆ j = v j + a j , uˆ k = ˆ u k + ak , vˆk = vk + ak , and uˆ t = u t and vˆt = vt for all t = j, k, then 1 vˆ R 1 u. Lemma 2: Assume that R satisfies Weak Anonymity and 2-Generation Unit Comparability. If 1 u and 1 v satisfy that there exist j, k such that u j − v j = vk − u k and u t = vt for all t = j, k, then 1 v I 1 u. Proof. Set a j = −v j and ak = −u k and form 1 uˆ and 1 vˆ as follows: uˆ j = u j + a j , vˆ j = v j + a j , uˆ k = u k + ak , vˆk = vk + ak , and uˆ t = u t and vˆt = vt for all t = j, k. Clearly, uˆ k = vˆ j = 0 and, since u j − v j = vk − u k , uˆ j = vˆk , while uˆ t = vˆt for all ˆ and by 2UC, 1 v I 1 u. t = j, k. By WA, 1 vˆ I 1 u, By applying Lemma 2 we overcome an objection to the catching up and overtaking criteria, namely that these criteria allow a large number of smaller gains for many generations to outweigh a greater loss for a single generation. The following results show that this is only a consequence of considering two-generation conflicts where one generation’s gain equals the others loss. Proposition 4: RSU is a subrelation to R if and only if R satisfies Strong Pareto, Weak Anonymity, 2-Generation Unit Comparability, and Strong Preference Continuity. Lemma 3: Assume that R satisfies the following condition: If 1 u and 1 v satisfy that there exist j, k such that u j − v j = vk − u k and u t = vt for all t = j, k, then 1 v I 1 u. T Then it holds that for any 1 u, (1 uˆ T , T +1 u) I 1 u whenever uˆ s = t=1 u t /T for all 1 ≤ s ≤ T. Proof. We can construct the egalitarian T -head 1 uˆ T by means of a sequence of steps involving conflicts between two generations. In particular, let ⎧ ⎪ for n = 0 ⎨ 1 uT n (1 uˆ n , u˜ n+1 , n+2 uT ) for n = 1, . . . , T − 2 1 uT = ⎪ ⎩ ( uˆ , u˜ ) = uˆ for n = T − 1 , 1 n T 1 T n+1  where, for n = 1, . . . , T − 1, u˜ n+1 = t=1 u t − nt=1 uˆ t . Then it follows by the premise of the lemma that (1 unT , T +1 u) I (1 un−1 T , T +1 u) for n = 1, . . . , T − 1. Hence, by transitivity, (1 uˆ T , T +1 u) I 1 u. Proof of Proposition 4. (If) Assume that R satisfies SP, WA, 2UC, and SPC. According to the definition of a subrelation (cf. Note 2), we have to show that, for any 1 u and 1 v, 1 v ISU 1 u implies 1 v I 1 u and 1 v PSU 1 u implies 1 v P 1 u. This divides the if part of the proof into two subparts.
≥1 (1) Consider any 1 u and 1 v such that 1 v ISU 1 u. By definition of RSU , ∃T T T +1 T +1 T
such that ∀T ≥ T , t=1 vt = t=1 u t and t=1 vt = t=1 u t , implying vT +1 =

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GEIR B. ASHEIM AND BERTIL TUNGODDEN


, uˆ s = vˆs = u T +1 . Hence, T
+1 v = T
+1 u. Let, for all 1 ≤ s ≤ T T

t=1 vt /T . By WA, 2UC, and Lemmas 2 and 3, (1 vT
,


+1 u) T

I (1 vˆ T
,


+1 u) T

and (1 uˆ T
,


+1 u) T

T



=

t=1 u t /T

I 1u .

Since 1 vˆ T
= 1 uˆ T
and T
+1 v = T
+1 u, it follows by transitivity that 1 v I 1 u. T (2) Consider any 1 u and 1 v such that 1 v PSU 1 u. For T satisfying t=1 vt = T u , it follows, by adapting subpart (1), that ( v , u) I u. For T satis1 T T +1 t=1 t  T T 1 T vt > t=1 u t , let, for all 1 ≤ s ≤ T , uˆ s = t=1 u t /T and vˆs = fying that t=1 T v /T . By WA, 2UC, and Lemmas 2 and 3, t=1 t (1 v T ,

T +1 u)

I (1 vˆ T ,

T +1 u)

and (1 uˆ T ,

T +1 u)

I 1u .

Since R satisfies SP, it follows that (1 vˆ T , T +1 u) P (1 uˆ T , T +1 u). Hence, by transi
≥ 1 such that tivity, (1 vT , T +1 u) P 1 u. This shows, by the definition of RSU , that ∃T
≥ 1, ∃T ≥ T
such that (1 vT , T +1 u) P 1 u. Since
, (1 vT , T +1 u) R 1 u, and ∀T ∀T ≥ T R satisfies SPC, it now follows that 1 v P 1 u. (Only if) Assume that RSU is a subrelation to R. It is trivial to establish that R satisfies SP, WA, and 2UC. Arguments similar to those used in the only-if part of the proof of Proposition 1 establish that R satisfies SPC. U through a trivial modification of An analogous result can be established for RW the parts of the proof of Proposition 4 that involve SPC. U is a subrelation to R if and only if R satisfies Strong Proposition 5: RW Pareto, Weak Anonymity, 2-Generation Unit Comparability, and Weak Preference Continuity.

It is more difficult to establish conditions that guarantee that there exists an optimal (or maximal) path according to the catching up and overtaking criteria, and we leave such a task for another occasion. 6. INTERGENERATIONAL IMPARTIALITY According to van Liedekerke and Lauwers (1997, p. 163), formal impartiality is ensured by imposing the axiom of Strong Anonymity (entailing indifference to any permutation of utilities of an infinite number of generations). As the following example shows, this demand cannot be combined with SP: 1 v = (1, 0, 1, 0, 1, 0, . . . ) can be attained from 1 u = (0, 0, 1, 0, 1, 0, . . . ) by a permutation where generation 2 gets the utility of generation 1, generation t the utility of generation t + 2 when t is an odd number, and generation t + 2 the utility of generation t when t is an even number. Liedekerke and Lauwers suggest to establish a framework where an acceptable trade-off between the demands of impartiality and SP can be made. This implies a rejection of the Suppes–Sen grading principle, which is characterized by WA (entailing indifference to any permutation of utilities of only a finite number of generations) and SP. One might think that it should be possible to find some intermediate position,

RESOLVING DISTRIBUTIONAL CONFLICTS BETWEEN GENERATIONS

61

where impartiality is extended beyond WA (by entailing indifference to some – but not all – permutations of an infinite number of generations) within a framework satisfying SP, and hence, the Suppes–Sen grading principle. However, if one – in analogy with WPC – imposes that an infinite utility path should be considered indifferent to another infinite utility path if the head of the former is considered indifferent to the latter at every point in time beyond a certain initial phase, then it follows from Lemma 1 that no extension of WA is obtained. On the other hand, if one – in analogy with SPC – imposes that an infinite utility path should be considered indifferent to another infinite utility path if, at any point in time, there is a future point in time at which the head of the former is considered indifferent to the latter, then it is not possible to establish an equivalence relation without coming in conflict with SP: If 1 u = (0, 0, 1, 0, 1, 0, . . . ), 1 v = (1, 0, 1, 0, 1, 0, . . . ), and 1 w = (0, 1, 0, 1, 0, 1, . . . ), then such strong indifference continuity implies 1 v I 1 w and 1 w I 1 u, and by transitivity, 1 v I 1 u. This contradicts SP. These difficulties, associated with extending indifference beyond WA without coming into conflict with SP, illustrate the problem of constructing a complete binary relation satisfying SP and WA (cf. Note 4).

Acknowledgments: We thank Kaushik Basu, Marc Fleurbaey, David Miller, Tapan Mitra, Lars-Gunnar Svensson, and an anonymous referee for helpful comments. Asheim gratefully acknowledges the hospitality of the Stanford University research initiative on the Environment, the Economy and Sustainable Welfare, and financial support from the Hewlett Foundation through this research initiative. NOTES 1 A permutation, i.e., a bijective mapping of {1, 2, . . . } onto itself, is finite whenever there is a T such

that π(t) = t for any t > T .

2 R is said to be a subrelation to R

if (1) v I u implies v I

u and (2) v P u implies v P

u 1 1 1 1 1 1 1 1 with I and I

and P and P

denoting the symmetric and asymmetric parts of R and R

, respectively. 3 This approach is in spirit related to the concepts of “overtaking” Atsumi (1965); von Weizsäcker (1965)

and “agreeable plans” Hammond and Mirrlees (1973); Hammond (1975). WPC is implied by Axioms 3 and 4 of Brock (1970). WPC and SPC are different from Koopmans et al.’s (1964) conditions of weak and strong time perspective, which involve the evolution of welfare differences between (1 wT , T +1 vT ) and (1 wT , T +1 uT ) when T increases, where for all T, t ≥ 1, u TT +t = u t and vTT +t = vt .

4 Svensson (1980) invokes Szpilrajn’s (1930) Lemma, which is nonconstructive. It is an open question

whether one can construct a complete binary relation satisfying SP and WA; cf. Fleurbaey and Michel (2003). Basu and Mitra (2003) show that one cannot represent a complete binary relation satisfying SP and WA by an SWF. 5 Lauwers (1997) characterizes the maximin relation by a version of Hammond Equity within a framework where Strong Pareto is relaxed. 6 See Sen (1977), d’Aspremont and Gevers (1977), Roberts (1980), Basu (1983), and in the infinite setting, Basu and Mitra (2007).

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REFERENCES Asheim, G.B. (1991), Unjust intergenerational allocations, Journal of Economic Theory 54, 350–371 (Chap. 7 of the present volume) Asheim, G.B., Buchholz, W. and Tungodden, B. (2001), Justifying sustainability, Journal of Environmental Economics and Management 41, 252–268. (Chap. 3 of the present volume) Atsumi, H. (1965), Neoclassical growth and the efficient program of capital accumulation, Review of Economic Studies 32, 127–136 Basu, K. (1983), Cardinal utility, utilitarianism and a class of invariance axioms in welfare analysis, Journal of Mathematical Economics 12, 193–206 Basu, K. and Mitra, T. (2003), Aggregating infinite utility streams with inter-generational equity: the impossibility of being Paretian, Econometrica 71, 1557–1563 Basu, K. and Mitra, T. (2007), Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization, Journal of Economic Theory 133, 350–373 Brock, W.A. (1970), An axiomatic basis for the Ramsey-Weizsäcker overtaking criterion, Econometrica 38, 927–929 d’Aspremont, C. and Gevers, L. (1977), Equity and the informational basis of collective choice, Review of Economic Studies 46, 199–210 Fleurbaey, M. and Michel, P. (2003), Intertemporal equity and the extension of the Ramsey criterion, Journal of Mathematical Economics 39, 777-802 Hammond, P.J. (1975), Agreeable plans with many capital goods, Review of Economic Studies 42, 1–14 Hammond, P.J. (1976), Equity, Arrow’s conditions, and Rawls’ difference principle, Econometrica 44, 753–803 Hammond, P.J. (1979), Equity in two person situations – some consequences, Econometrica 47, 1127– 1135 Hammond, P.J. and Mirrlees, J.A. (1973), Agreeable plans. In: Mirrlees, J.A. and Stern, N.H. (eds.), Models of Economic Growth, Macmillan, London Koopmans, T.C., Diamond, P.A. and Williamson, R.E. (1964), Stationary utility and time perspective, Econometrica 32, 82–100 Lauwers, L. (1997), Rawlsian equity and generalised utilitarianism with an infinite population, Economic Theory 9, 143–150 Roberts, K.W.S. (1980), Interpersonal comparability and social choice theory, Review of Economic Studies 47, 421–439 Sen, A.K. (1977), On weights and measures: Informational constraints in social welfare analysis, Econometrica 45, 1539–1572 Svensson, L.G. (1980), Equity among generations, Econometrica 48, 1251–1256 Szpilrajn, E. (1930), Sur l’extension de l’ordre partial, Fundamenta Mathematicae 16, 386–389 van Liedekerke, L. and Lauwers, L. (1997), Sacrificing the patrol: Utilitarianism, future generations and infinity, Economics and Philosophy 13, 159–174 von Weizsäcker, C.C. (1965), Existence of optimal program of accumulation for an infinite time horizon, Review of Economic Studies 32, 85–104

CHAPTER 5 THE MALLEABILITY OF UNDISCOUNTED UTILITARIANISM AS A CRITERION OF INTERGENERATIONAL JUSTICE

GEIR B. ASHEIM Department of Economics, University of Oslo, P.O. Box 1095 Blindern, NO-0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics, University of Regensburg, DE-93040 Regensburg, Germany Email: [email protected]

Abstract. Discounting future utilities is often justified by the ethically motivated objective to protect earlier generations from the excessive saving that seems to be implied by undiscounted utilitarianism in productive economies. In this paper we question this justification of discounting by showing that undiscounted utilitarianism has sufficient malleability within important classes of technologies: Any efficient and nondecreasing allocation can be the unique optimum according to an undiscounted utilitarian criterion for some choice of utility function.

1. INTRODUCTION In the theory of economic growth it is quite common to determine optimal growth programs by means of a discounted utilitarian criterion, where the positive discount rate reflects pure time preference. Future utilities of consumption measured on a cardinal scale are through discounting transformed into present values, the sum of which one seeks to maximize. Thus, discounted utilitarianism provides a class of objective functions that are often used for evaluating intertemporal choice.

Originally published in Economica 70 (2003), 405–422. Reproduced with permission from Blackwell.

63 Asheim, Justifying, Characterizing and Indicating Sustainability, 63–82 c 2007 Springer


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Even though discounting has sometimes been considered as “irrational” in the context of individual decision making, such impatience has still been accepted as a natural part of exogenously given individual preferences. When discounting, however, refers to future generations, some fundamental ethical problems seem to arise. In the case of intergenerational discounting it is natural to question whether it is fair to value the utility of future generations less than that of the present one. This criticism against discounting has a long tradition in economics, dating back at least to Pigou (1932). It was revitalized in the ongoing debate on sustainable development, which – it is sometimes claimed – is in danger when discounting is applied. The proponents of intergenerational discounting, however, also turn to ethical reflections when they try to justify this procedure. Thus it appears that a deep ethical conflict is present in the debate of the discounting issue. It is the aim of this paper to evaluate the use of discounting in utilitarian social criteria for choice between intergenerational allocations. As a first step, we retrace the ethical arguments of the opponents and the proponents of intergenerational discounting. As a second step we show that this debate might be considered misplaced as there is not much room for a serious controversy on the discounting question. In particular, in important classes of technologies the undiscounted utilitarian criterion is sufficiently malleable to avoid excessive saving and thus allow for choices the proponents of intergenerational discounting seek to implement by discounting future utilities. 2. INTERGENERATIONAL DISCOUNTING AS AN ETHICAL PROBLEM The position of the opponents against intergenerational discounting is primarily based on the view that it is not a priori justified to give different generations unequal weight in social evaluations as they do not seem to be fundamentally different, at least if population size is constant and there is no uncertainty. The only obvious distinction between members of different generations is that they do not appear simultaneously on the time axis, which, however, does not provide an ethically compelling reason for unequal treatment. If, under otherwise identical circumstances, welfare comparisons are to be made in a static context with a finite number of agents living within the same generation, one would usually not deny that equal individuals should get equal consideration in social welfare functions. Therefore, it is quite standard in welfare economics to adopt an anonymity principle by which discrimination of particular agents is excluded. However, discounting does not only seem questionable from its normative basis but also from its consequences under certain technological assumptions. When, e.g., the level of production depends not only on man-made capital and labor, but also on the input of an exhaustible natural resource (like oil), applying a discounted utilitarian criterion for any time-invariant and strictly positive discount rate will, in the long run, force the consumption level to approach zero, even though positive and nondecreasing consumption is technically feasible. Thus, in such a Dasgupta–Heal–Solow technology constant and positive utility discounting leads to an outcome which does not appeal to commonly shared ethical intuitions, and which

THE MALLEABILITY OF UNDISCOUNTED UTILITARIANISM

65

is not compatible with sustainable development – i.e., with having no generation enjoy a level of well-being that cannot be shared by future generations. As appealing as these ethically motivated arguments against intergenerational discounting may look, there are also important arguments in favor of the position of the proponents of discounting. The first of these arguments is of a technical nature. If the world does not come to its end at some predetermined date, it is an ethical imperative to take all future generations into consideration, which is well in line with the advocates of sustainability. But it is just the fact that the number of generations are modeled to be infinite that creates specific problems in making welfare comparisons. In the infinite case the existence of a socially most preferred intergenerational allocation cannot be ensured in the same way as it is usually done in the finite case, where the Weierstrass theorem easily applies. This theorem says that on a compact domain a continuous real-valued function will have at least one maximum. In the finite case, not very demanding assumptions are needed to ensure that the premises of this theorem hold so that the existence of maximal elements is ensured. In the infinite case, however, it is not possible – under relevant technological assumptions – to find a topology that leads to compactness of the set of feasible allocations and, at the same time, continuity of social preferences that are sensitive to the interest of each generation and treat all generations equally (cf. Koopmans 1960; Diamond 1965; Svensson 1980 and for a general discussion, Lauwers 1997). Sensitivity here refers to the “Strong Pareto” axiom (which we will also call “Efficiency”), meaning that social preferences must deem one allocation superior to another if at least one generation is better off and no generation is worse off, while equal treatment refers to the “Weak anonymity” axiom (which we will also call “Equity”), meaning that social preferences must leave the social valuation of an allocation unchanged when the consumption levels of any two generations along the allocation are permuted. That this way of establishing existence cannot be generalized to the infinite case if “Equity” is postulated explains some of the scepticism people have with this normative precept in the infinite number case. From such a perspective adhering to this axiom may seem rather pointless if it is difficult to make use of it. The second argument, however, for the widespread refusal of the “Equity” axiom in the intergenerational context flows from ethical reservations. It is claimed that – even if future consumption is perfectly certain to be realized – equal treatment is in conflict with finding an acceptable balance between the interests of different generations. In particular, it is often suggested that the application of undiscounted utilitarianism leads to a distributional imbalance by impairing the earlier generations to an unacceptable degree. For example, Rawls (1971, p. 287) argues that “the utilitarian doctrine may direct us to demand heavy sacrifices of the poorer generations for the sake of greater advantages for the later ones that are far better off,” while Dasgupta and Mäler (1995, p. 2395) refer to calculations by Mirrlees (1967) and Chakravarty (1969) showing in plausible economic models that the present generation would be asked to save and invest around 50% of GNP under undiscounted utilitarianism. Thus, with a very productive economy, the danger exists that maximizing the sum of undiscounted utilities leads to a growth pattern that requires high savings rates

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initially and thereby imposes excessive hardships on earlier generations. Following such arguments, Rawls (1971, p. 297) reluctantly points out that “[t]his consequence can be to some degree corrected by discounting the welfare of those living in the future,” and Arrow (1999, p. 16) concludes “that the strong ethical requirement that all generations be treated alike, itself reasonable, contradicts a very strong intuition that it is not morally acceptable to demand excessively high savings rates of any one generation, or even of every generation.” To avoid consumption patterns that are too much to the disadvantage of earlier generations, the incorporation of a positive discount rate into the utilitarian criterion is thus considered inevitable even in the case of certainty. Stated in another way, this justification of utility discounting entails that discounting is not only required in order to be able to make choices at all, but also to ensure that the chosen distributions are ethically acceptable. The first of these arguments – namely the problem of existence when the axioms of “Efficiency” and “Equity” are imposed – has been considered in a previous work Asheim et al. (2001). There we show that the axioms of “Efficiency” and “Equity” are not incompatible with the existence of maximal allocations, given that one considers technologies that are productive in the sense of satisfying the following two conditions: “Immediate productivity,” which means that there are negative transfer costs from the present to the future if the future is worse off than the present, and “Eventual productivity,” which means that there exist efficient and completely egalitarian allocations. In fact, we show that any efficient and nondecreasing allocation is maximal if the social preferences over infinite intergenerational allocations are generated by the axioms of “Efficiency” and “Equity.” In the present paper we consider the second of these arguments. We show that (1) under a general assumption undiscounted utilitarianism is so flexible that any efficient and nondecreasing allocation can be the unique optimal intergenerational allocation under the utilitarian criterion provided that the utility function is appropriately chosen, and that (2) this assumption is satisfied within three important classes of productive technologies. This means that, for any member of these classes of technologies, any outcome consistent with “Efficiency” and “Equity” can be realized under undiscounted utilitarianism. Hence, the problem with undiscounted utilitarianism is neither that it does not allow for optimal allocations nor that it leads to unequal distributions imposing a too heavy burden on the present generation. Rather, the problem is that it – as a class of orderings – does not limit the set of optimal allocations more than the axioms of “Efficiency” and “Equity” do. Thus, undiscounted utilitarianism has no bite beyond these axioms. Throughout the well-being of any generation will be measured by a onedimensional indicator “consumption,” which comprises everything that affects a generation’s livelihood. It is appropriate to think of “consumption” as the money metric utility that a generation derives from the goods and services at its disposal, with the money metric utility function being identical for all generations. Even though we prove as our main result that any efficient and nondecreasing allocation is compatible with zero utility discounting, the underlying assumption requires that any such allocation is characterized by a positive consumption interest rate that cannot decrease too fast. Our analysis is therefore not an argument against positive

THE MALLEABILITY OF UNDISCOUNTED UTILITARIANISM

67

discounting in social cost–benefit analysis, even for decisions relating to long-term environmental policy (e.g., controlling the greenhouse effect). However, the consumption interest rate – which will reflect net capital productivity – will be of a magnitude and have a time structure1 so that the decisions taken will not undermine the livelihood of future generations. In the analysis of the present paper, there is no uncertainty about whether future consumption will be realized. Hence, u is not a von Neumann–Morgenstern utility function, where the corresponding concavity would be an expression of risk aversion. Rather, the concavity of the utility function u represents the aversion in social evaluation toward inequality between generations. The concerns about the unfavorable consequences for the earlier generations of using undiscounted utilitarianism can be seen as an expression of inequality aversion. It therefore seems appropriate to respond to these concerns by making the utility function more concave. If risk and uncertainty were introduced into our analysis, it would still be appropriate in principle to distinguish between inequality aversion and risk aversion (cf. Kreps and Porteus 1979), although this is often not done in practical applications. Furthermore, even if one follows Harsanyi (1953) in arguing that undiscounted utilitarianism derives its justification from hypothetical decisions under uncertainty in the original position, the risk aversion in his setting relates to decision problems that are not faced by actual decision makers. We therefore claim that the concavity of the utility function u in the undiscounted utilitarian criterion is not observable in a market economy. The paper is organized as follows. Section 3 contains a simple numerical illustration, while Sect. 4 presents the main result. Section 5 shows that the assumption underlying this result is satisfied in the three classes of technologies that we explicitly consider. Section 6 contains a discussion of the results, while analysis of more technical nature is included in the two appendices.

3. A NUMERICAL ILLUSTRATION The main argument presented in this paper is that, within certain classes of technologies, any efficient and nondecreasing allocation can be realized under undiscounted utilitarianism for some choice of utility function. We now illustrate this possibility within a simple setting where there are only two generations. Let c1 and c2 denote consumption (or well-being) of generations 1 and 2, and let the set of feasible allocations be given by: (c1 , c2 ) ≥ 0 is feasible if and only if c1 + 14 c2 ≤ 2 . This linear technology implies that the net capital productivity equals 3 (= 4 − 1). Let u be the time-invariant utility function and let δ (∈ (0, 1]) be the discount factor. This means that the utility discount rate equals 1/δ − 1. An allocation (c1 , c2 ) is a

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utilitarian optimum if and only if (c1 , c2 ) solves max

c1 ≥0,c2 ≥0

u(c1 ) + δu(c2 ) s.t. c1 + 14 c2 ≤ 2 .

This means that if (c1 , c2 )  0, then (c1 , c2 ) satisfies u  (c1 ) = δ4u  (c2 ). We can now verify that (c1 , c2 ) = (1, 4) is a utilitarian optimum under both of the following constellations of utility function and discount factor: u(c) = c1/2 and δ = 1/2 u(c) = ln c and δ = 1 . This illustrates how a more concave utility function can substitute for discounting along an allocation where consumption is strictly increasing. 4. MAIN RESULT In a general setting, assume that consumption ct in period t = 0, 1, 2, . . . is onedimensional. Write 0 c = (c0 , c1 , . . . ) and correspondingly for other sequences. Refer to 0 c as an allocation. An allocation 0 c is said to be nondecreasing if ct+1 ≥ ct for all t ≥ 0 and stationary if ct+1 = ct for all t ≥ 0. Given a set of feasible allocations, a feasible allocation 0 c = (c0 , c1 , . . . ) is said to be efficient if there exists no alternative feasible allocation 0 c = (c0 , c1 , . . . ) with ct ≥ ct for all t ≥ 0, with strict inequality for some t. One problem when applying the undiscounted utilitarianism to a situation with an infinite number of generations is that the sum of undiscounted utilities will generally diverge in the infinite horizon case. To resolve this, we invoke an “overtaking” criterion, under which one allocation is better than another if the lim inf, as T goes to infinity, of the sum up to time T of the difference between the utilities generated by the allocations is positive: 0c



Pu 0 c ⇔ lim inf T →∞

T
(u(ct ) − u(ct )) > 0 .

(P)

t=0

We will use this criterion to show how under a given technological assumption any efficient and nondecreasing allocation is the unique optimum according to undiscounted utilitarianism for some choice of utility function. The technological assumption – which as shown in Sect. 5 is satisfied for any member of three important classes of technologies – is the following: Assumption 1: If a feasible allocation 0 c = (c0 , c1 , . . . ) is efficient and nondecreasing, then there exists a sequence of consumption discount factors 0 p = ( p0 , p1 , . . . ) satisfying 0 < pt+1 < pt for all t ≥ 0 such that
∞
∞ pt ct ≥ pt ct ∞> t=0

t=0

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for any feasible allocation 0 c = (c0 , c1 , . . . ). We can now state the main result of this paper. Proposition 1: Under Assumption 1, if a feasible allocation 0 c = (c0 , c1 , . . . ) is efficient and nondecreasing, then there exists a continuous, concave, and nondecreasing utility function u : R+ → R such that 0 c Pu 0 c for any feasible allocation    0 c = (c0 , c1 , . . . ) that does not coincide with 0 c. Proof. By Assumption 1 there exists a sequence of consumption discount fac, p1 , . . . ) satisfying 0 < pt+1 < pt for all t ≥ 0 such that ∞ > tors ∞ 0 p = ( p0 ∞   p c ≥ t t t=0 t=0 pt ct for any feasible allocation 0 c . Construct a continuous, concave, and nondecreasing utility function u as follows. Let u 0 ∈ ( p0 , ∞)

and

u t ∈ ( pt , pt−1 ) for t ≥ 1 .

If 0 ≤ c < sup{ct |t ∈ N}, write := min{t ∈ N|ct ≥ c} (≥ 0), and let
t c u(c) = u t (ct − ct−1 ) − u t c (ct c − c) , tc

t=0

where c−1 := 0. Note that u is a piece-wise linear, continuous, and concave function, with u(0) = 0 and with the slope on each linear segment (ct , ct+1 ) lying strictly between pt+1 and pt . If sup{ct |t ∈ N} = ∞, then u has been constructed. Otherwise, write c¯ := sup{ct |t ∈ N}. Since u is continuous and nondecreasing on [0, c), ¯ limc↑c¯ u(c) exists; write u¯ := limc↑c¯ u(c). Complete the construction of u by setting u(c) = u¯ for c ≥ sup{ct |t ∈ N}. Note that u is constructed so that, for each t ≥ 0, u(ct ) − pt ct ≥ u(c) − pt c for any c ≥ 0, with the inequality being strict if c = ct . This implies that
∞
T lim infT →∞ (u(ct ) − u(ct )) ≥ pt (ct − ct ) ≥ 0 , t=0

t=0

for any feasible allocation 0 c , where the first inequality is strict if 0 c = 0 c and the second inequality follows from Assumption 1. This means that 0 c Pu 0 c if 0 c  = 0 c . The construction of the continuous and piece-wise linear utility function u is shown in Fig. 5.1, thereby illustrating the proof of Proposition 1. An extension of the preceding analysis shows that a differentiable utility function can be obtained if and only if 0 c is strictly increasing. As illustrated by Fig. 5.1, the utility function may have to be chosen such that marginal utility is zero beyond some consumption level. If so, undiscounted utilitarianism will not satisfy the axiom of “Efficiency” (or “Strong Pareto”; cf. Sect. 2) unless some amendment is introduced. In Appendix A we show how an extended undiscounted utilitarian criterion can be constructed, having the properties that it

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u slope

= p3   = p1 !! # ! slope " # !! = p2"" # " # slope




u(c)

slope

= p0












c0

c1 c2

c3 c4 .. .

c

Figure 5.1. Illustration of the Proof of Proposition 1

1.

Yields a (strict) preference for 0 c over 0 c whenever 0 c Pu 0 c ,

2.

Is complete in the sense of deeming two allocations indifferent whenever there is not a (strict) preference

3.

Satisfies the axioms of “Efficiency” and “Equity”

Note that the first property means that Proposition 1 holds if such an extended undiscounted utilitarian criterion is substituted for Pu as defined by (P). 5. HOW STRONG IS ASSUMPTION 1? Assumption 1 provides a general condition that if fulfilled yields a justification for applying undiscounted utilitarianism. In this section we illustrate the generality of Assumption 1 by considering three standard classes of technologies. We show that any technology within these classes satisfies Assumption 1. Hence, by invoking Proposition 1 this in turn implies that an arbitrary efficient and nondecreasing allocation for a given initial endowment in such a technology is the unique optimum according to undiscounted utilitarianism for an appropriate choice of utility function. We consider the classes of linear, Ramsey, and Dasgupta–Heal–Solow technologies. For the technologies of these classes we assume that gross output yt available at the beginning of period t = 0, 1, 2, . . . is one-dimensional. Furthermore, kt denotes invested man-made capital, i.e., the gross output not consumed in period t. Any member of the three classes of technologies satisfies the conditions of “Immediate

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productivity” and “Eventual productivity” (in the terminology of Asheim et al. 2001) under given assumptions. Immediate productivity means that if 0 c = (c0 , c1 , . . . , ct , ct+1 , . . . ) is feasible and ct > ct+1 , then 0 c = (c0 , c1 , . . . , ct+1 , ct , . . . ) is feasible and inefficient. Eventual productivity means that, for any initial endowment, there exists an efficient and stationary allocation 0 c = (c, c, . . . ). For further comparison, one can associate “Immediate productivity” with positive net capital productivity leading to decreasing consumption discount factors, and “Eventual productivity” with the existence of an ∞efficient and nondecreasing allocation having “finite consumption value” (i.e., t=0 pt ct < ∞). Only linear technologies explicitly incorporate technological progress. Technological progress corresponds to increasing net capital productivity and makes it easier to satisfy “Eventual productivity.” Dasgupta–Heal–Solow technologies illustrate the case where resource depletion plays an important role for future development and leads to decreasing net capital productivity. If production depends on inputs of depletable natural resources, Ramsey technologies can be seen to represent the case where technological progress just “makes up” for diminishing resource availability, thus avoiding the decline in net capital productivity that would otherwise occur. We assume throughout that population is constant. With a nonconstant population and interpreting ct as per capita consumption, “Immediate productivity” entails in the context of linear and Ramsey technologies that net capital productivity exceeds the rate of population growth. 5.1. Linear Technologies A technology is linear if, in any period, the ratio of gross output and invested manmade capital is fixed and equal to at . On the other hand, this gross productivity factor can vary between periods. We assume positive net capital productivity: at > 1

for all

t > 0.

(A.1)

This assumption means that the condition of “Immediate productivity” is satisfied. Let the transformation set Tt 1 in period t be given by Tt 1 = {(k, y)|0 ≤ y ≤ at k} . Let y > 0 be the exogenously given endowment of output available at the beginning of period 0. A program (0 y, 0 k) is said to be feasible given y if y0 = y ,

and

kt ≤ yt and (kt , yt+1 ) ∈ Tt 1 for all t ≥ 0 .

An allocation 0 c is said to be feasible if there is a feasible program (0 y, 0 k) such that ct = yt − kt for all t ≥ 0.2 A linear technology determines a unique sequence of consumption discount factors 0 p = ( p0 , p1 , . . . ) as follows: p0 = 1 and

pt+1 at = pt for all t ≥ 0 .

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Note that it holds for all t ≥ 0 that 0 < pt+1 < pt since at > 1. It follows from Lemma 1 of Appendix B that the set of efficient and nondecreasing feasible allocations is nonempty if the following condition holds
∞ pt < ∞ , (E.1) t=0   ∞ since then the stationary allocation 0 c = (c, c, . . . ) with c = y t=0 pt > 0 is feasible given y. If, on the other hand, (E.1) does not hold, then this set is empty. This means that a linear technology satisfies “Eventual productivity” if and only if (E.1) is satisfied. An increasing sequence of gross productivity factors – i.e., 0 a = (a0 , a1 , . . . ) satisfies at+1 > at for all t ≥ 0 – can be interpreted as exogeneous technological progress. Condition (E.1) clearly holds if a0 > 1 and 0 a is nondecreasing. 5.2. Ramsey Technologies A Ramsey technology (cf. Ramsey 1928) is determined by a stationary production function g : R+ → R+ that satisfies g is concave, continuous for k ≥ 0 , and twice differentiable for k > 0 . g(0) = 0 ,

g  > 0 for k > 0 ,



g (k) → ∞ as k ↓ 0

(A.2)

and



g (k) ↓ 0 as k → ∞ . This assumption implies that the condition of “Immediate productivity” is satisfied. Let the gross output function f be given by f (k) = g(k) + k for all k ≥ 0. The transformation set T 2 is time-invariant and is given by T 2 = {(k, y)|0 ≤ y ≤ f (k); k ≥ 0} . Let y > 0. A program (0 y, 0 k) is said to be feasible given y if y0 = y ,

and

kt ≤ yt and (kt , yt+1 ) ∈ T 2 for all t ≥ 0 .

An allocation 0 c is said to be feasible if there is a feasible program (0 y, 0 k), where 0 k is bounded above, such that ct = yt − kt for all t ≥ 0. That 0 k is bounded above means that, given 0 k, there exists k¯ such that kt ≤ k¯ for all t ≥ 0. This assumption is made in order to show in Lemma 2 that efficient and nondecreasing allocations have “finite consumption value” and thereby enable Proposition 2 below to be established. All programs promoted by proponents of discounted utilitarianism in Ramsey technologies are included, as under (A.2) 0 k is bounded above in any discounted utilitarian optimum. The set of efficient and nondecreasing feasible allocations is nonempty since the stationary allocation 0 c = (c, c, . . . ) where c > 0 solves y = f (y − c) is feasible given y. Hence, any Ramsey technology satisfies “Eventual productivity.”

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5.3. Dasgupta–Heal–Solow Technologies A Dasgupta–Heal–Solow technology (cf. Dasgupta and Heal 1974; Solow 1974) is determined by a stationary production function G : R3+ → R+ that satisfies G is concave, nondecreasing, homogeneous of degree one, and continuous for (k, r, ) ≥ 0 , G is twice differentiable and satisfies (G k , G r , G  )  0 for (k, r, )  0 .

(A.3)

G(k, 0, ) = 0 = G(0, r, ) Given any (k  , r  )  0, there is η > 0 such that for all (k, r ) satisfying k ≥ k  , 0 < r ≤ r  , we have [r G r (k, r, 1)]/G  (k, r, 1) ≥ η .

This assumption implies that the condition of “Immediate productivity” is satisfied. Note that both capital kt and resource extraction rt are essential in production. The available labor force is assumed to be stationary and equal to 1. Still, labor needs to be explicitly considered in order to state the latter part of (A.3), namely that the ratio of the share of the resource in net output to the share of labor in net output is assumed to be bounded away from zero. Let the gross output function F be given by F(k, r ) = G(k, r, 1) + k for all (k, r ) ≥ 0. The transformation set T 3 is timeinvariant and is given by T 3 = {[(k, m), (y, m  )]|0 ≤ y ≤ F(k, r ); 0 ≤ r = m − m  , (k, m  ) ≥ 0} . Let (y, m)  0, where m is the resource stock available at the beginning of period 0. A program (0 y, 0 m, 0 k) is said to be feasible given (y, m) if y0 = y kt ≤ yt

and

and

m0 = m

[(kt , m t ), (yt+1 , m t+1 )] ∈ T 3

for all t ≥ 0 .

An allocation 0 c is said to be feasible if there is a feasible program (0 y, 0 m, 0 k) such that ct = yt − kt for all t ≥ 0. Assumption (A.3) does not ensure the existence of a stationary allocation with positive consumption. Therefore assume the following condition. There exists given (y, m)  0 a feasible stationary allocation with positive consumption.

(E.3)

Dasgupta and Mitra (1983) show within the setting of Dasgupta–Heal–Solow technologies that this implies the existence of an efficient and stationary allocation. This means that a Dasgupta–Heal–Solow technology satisfies “Eventual productivity” and has a nonempty set of efficient and nondecreasing feasible allocations if (E.3) holds. We state (E.3) in its reduced form since it is outside the scope of the present paper to provide primitive technological conditions; instead we refer to Cass and Mitra (1991) who give necessary and sufficient conditions on the production function G for (E.3) to hold.

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The importance of Dasgupta–Heal–Solow technologies in a discussion of intergenerational justice derives from the property that under (A.3) net capital productivity G k (k, r, 1) converges to zero along any efficient and nondecreasing allocation.3 This occurs in a stationary technology setting due to the dwindling availability of the resource. 5.4. A Characterization Result for Efficient and Nondecreasing Allocations Consider a member of the classes of linear, Ramsey or Dasgupta–Heal–Solow technologies. Assume (A.1)–(A.3). In Appendix B we prove that if a feasible allocation 0 c = (c0 , c1 , . . . ) is efficient and nondecreasing, then there exists a sequence of consumption discount factors 0 p = ( p0 , p1 , . . . ) satisfying 0 < pt+1 < pt for all t ≥ 0 such that
∞
∞ pt ct ≥ pt ct ∞> t=0

t=0

for any feasible allocation 0 c = (c0 , c1 , . . . ). Proposition 2: Under (A.1)–(A.3) any linear, Ramsey or Dasgupta–Heal–Solow technology satisfies Assumption 1. 6. DISCUSSION We have earlier (in Asheim et al. 2001) showed that the rather uncontroversial ethical axioms of “Efficiency” and “Equity” rule out any allocation that is not efficient and nondecreasing in technologies that satisfy the conditions of “Immediate productivity” and “Eventual productivity.” All members of the three classes of technologies that we have considered in Sect. 5 satisfy “Immediate productivity” under the assumptions of (A.1)–(A.3), and they satisfy “Eventual productivity” provided that (E.1) and (E.3) are satisfied. Hence, “Efficiency” and “Equity” alone rule out any allocation that is not efficient and nondecreasing. Here, we have showed within these classes of technologies that any efficient and nondecreasing allocation can be a unique undiscounted utilitarian optimum for a given initial endowment, provided that the utility function is appropriately chosen. Hence, the class of undiscounted utilitarian criteria – any member of which satisfies “Efficiency” and “Equity” – is sufficiently malleable to allow for any efficient and nondecreasing allocation to be the unique choice for a given initial endowment. This may be taken to mean that no efficient and nondecreasing allocation can be ruled out solely from ethical consideration, at least in the three classes of technologies that we consider. This is a negative conclusion in the sense that adopting undiscounted utilitarianism as a class of social preferences does not indicate how to provide a sharper prescription for choice among feasible allocations – unless one has a specific ethical intuition concerning the concavity of the utility function which, however, seems hard to obtain on the basis of ethical axioms alone. In Ramsey technologies discounted utilitarianism with a positive and time-invariant discount rate might be seen as an

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Table 5.1. Results on Undiscounted and Discounted Utilitarianism.

Undiscounted utilitarianism Discounted utilitarianism

Ramsey technologies

D–H–S technologies

All allocations consistent with “Effic.” & “Equity” More equal alloc. consistent with “Effic.” & “Equity”

All allocations consistent with “Effic.” & “Equity” No allocation consistent with “Effic.” & “Equity”

alternative that – even as an entire class of criteria – yields sharper prescriptions. Suppose that the discount rate is chosen to be smaller or equal to g  (k(y)), where k(y) solves y = f (k(y)) for given y. Then discounted utilitarianism applied to the Ramsey model will, quite independently of the shape of the utility function, lead to an efficient and nondecreasing allocation. In fact, the stock of capital cannot grow ¯ where g  (k) ¯ equals the utility discount rate.4 This means to a size that exceeds k, ¯ and thus, the introduction of such a that consumption cannot grow beyond g(k), “small” utility discount rate excludes allocations with a large inequality between the present and unfortunate generations and the future and fortunate generations. Hence, even though discounted utilitarianism as a criterion for social decision making is inconsistent with the axiom of “Equity,” in Ramsey technologies it leads to outcomes that are consistent with “Efficiency” and “Equity” as long as the discount rate is sufficiently “small.” This attractive feature of discounted utilitarianism arises in Ramsey technologies since an efficient and stationary allocation has constant net capital productivity g  (k(y)). In a Dasgupta–Heal–Solow, however, the net capital productivity G k (k, r, 1) converges to zero along any efficient and nondecreasing allocation. This means that discounted utilitarianism will force consumption to approach zero in the long run, independently of how “small” the discount rate is, even if (E.3) is satisfied so that the set of efficient and nondecreasing allocations is nonempty. This in turn implies that discounted utilitarianism is what Page (1977, p. 198) calls a “fair weather criterion” that necessarily leads to outcomes inconsistent with the axioms of “Efficiency” and “Equity” in the more severe context of a Dasgupta–Heal–Solow technology. These results are summarized in Table 5.1. These conclusions mean that – although discounted utilitarianism appeals to our ethical intuitions in the simple context of Ramsey technologies – introducing positive discounting is not in general an appropriate way of achieving ethically desirable outcomes. In contrast, undiscounted utilitarianism is sufficiently malleable so that any ethically desirable outcome can be realized as long as such an allocation is consistent with “Efficiency” and “Equity.” Hence, in the context of utilitarianism there appears to be no ethical argument in favor of discounting if there is no uncertainty. In the end

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such a conclusion may appear uncontroversial, but – as documented in Sect. 2 – it contradicts views expressed by many influential scholars. Determining an intergenerational allocation amounts to nothing else than resolving the distributional conflict between the different generations. Following Atkinson (1970) it is quite familiar in economics to have distributional objectives incorporated in symmetric additive social welfare functions where the consumption (or income) of every economic agent is evaluated by some utility function u. Then the degree of inequality aversion contained in the social welfare function is expressed by the degree of concavity of the utility function u (cf. Cowell (2000) for a general overview as well as Collard (1994) and Fleurbaey and Michel (2001), for specific applications in the intergenerational context). In the static case this leads to a completely egalitarian distribution unless there are costs of redistribution. In the present intergenerational context, the condition of “Immediate productivity” means that there are negative transfer costs from the present to the future. Referring to Okun’s wellknown “leaking bucket,” Schelling (1995, p. 396) vividly describes such a situation by using the term “incubation bucket” in which “the good things multiply in transit so that more arrives at the destination than was removed from the origin.” Hence, if such productivity of the technology is assumed, any efficient and nondecreasing allocation may be consistent with maximization of a symmetric additive welfare function. This means that the familiar welfarist approach might be attractive to apply also for evaluating intergenerational allocations, even in the infinite case.5 That a more concave u can be used for skewing the intergenerational distribution in favor of earlier generations, thus slowing economic growth, has by interpreting the well-known Ramsey rule for optimal economic growth been observed for a long time (see e.g., Dasgupta and Heal (1979, p. 292), as well as Fleurbaey and Michel 1994, p. 294 and 1999, p. 273). Here, we have shown in the context of linear and Ramsey technologies that whatever decreased inequality is desirable it can be obtained under undiscounted utilitarianism by choosing a more concave utility function, since the utility function can always be adjusted so that a given efficient and nondecreasing allocation is obtained as the undiscounted utilitarian optimum. Such adjustment is even possible in Dasgupta–Heal–Solow technologies, where discounted utilitarianism, for any positive and constant discounting, leads to unacceptable treatment of generations in the distant future. Thus, this malleability of undiscounted utilitarianism makes it possible in the intergenerational context to determine optimal allocations that appeal to our ethical intuitions through applying complete and transitive social preferences that satisfy reasonable ethical axioms. The conclusion that discounting is not needed to be able to make ethically appealing choices in the intergenerational context, however, does not imply that intergenerational discounting is unjustified in any case. Discounting may well provide an instrument for taking uncertainty into account. This aspect makes generations inherently different and therefore has ethical relevance for intergenerational discounting. Referring to Sidgwick (1907, p. 412), Dasgupta and Heal (1979, p. 262) clearly state: “The point then is that one might find it ethically reasonable to discount future utilities at positive rates . . . because there is a positive chance that future generations will not exist.” A discussion of such more legitimate reasons for intergenerational

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discounting has not been the subject here. Rather, our purpose has been to clarify the ethical role of discounting in the case of certainty. APPENDIX A: EXTENDING UNDISCOUNTED UTILITARIANISM To construct an extended undiscounted utilitarian criterion that yields a (strict) preference for 0 c over 0 c whenever 0 c Pu 0 c (cf. (P) of Sect. 4), is complete in the sense of deeming two allocations indifferent whenever there is not a (strict) preference, and satisfies the axioms of “Efficiency” and “Equity,” we introduce the following components:

r A first-order utility function u : R+ → R that is continuous, concave, and nondecreasing, and which is applied in a “catching up” criterion, under which one allocation is as good as another if the lim inf, as T goes to infinity, of the sum up to time T of the difference between the utilities generated by the allocations is non-negative.

r A second-order utility function v : R+ → R that is continuous and strictly increasing, and which is applied lexicographically to resolve ties so that the axiom of “Efficiency” is satisfied. For this purpose we will assume throughout that v is given by v(c) = c.

r Given u, a complete and transitive binary relation R  that is a completion of u “catching up”-utilitarianism when these utility functions are used lexicographically, and which invokes a result due to Szpilrajn (1930). We first show how an incomplete reflexive and transitive binary relation Ru can be derived from u when “catching up”-utilitarianism is lexicographically extended. We then consider the issue of completing Ru to Ru . Let 0 c = (c0 , c1 , . . . ) and 0 c = (c0 , c1 , . . . ) be two allocations. Let u be given, and determine Ru as follows. ⎧ T ⎪ (u(ct ) − u(ct )) ≥ 0 , ⎨lim infT →∞ t=0 T      (R) 0 c Ru 0 c ⇔ and lim infT →∞ t=0 (v(ct ) − v(ct )) ≥ 0 ⎪ ⎩whenever ∞ (u(c ) − u(c )) = 0 . t t t=0 It is clear that Ru is reflexive and transitive. Let Pu and Iu denote the asymmetric and symmetric parts of Ru , respectively; i.e., Pu denotes (strict) preference, while Iu denotes indifference. It follows from (R) and (P) that 0 c Pu 0 c whenever 0 c Pu 0 c . To show that Ru satisfies “Efficiency” (or “Strong Pareto”), let 0 c be derived from         0 c by having cs = cs + ,  > 0, and ct = ct for t  = s, then 0 c Pu 0 c since – even     if u(cs ) = u(cs ) – we do have that v(cs ) > v(cs ). This means that Ru satisfies the axiom of “Efficiency”. To show that Ru satisfies “Equity” (or “Weak anonymity”), let 0 c be derived from 0 c by having cs  = cs , cs  = cs , and ct = ct for t = s  , s  ,  ∞     then 0 c Iu 0 c since ∞ t=0 (u(ct ) − u(ct )) = 0 and t=0 (v(ct ) − v(ct )) = 0. This  means that Ru satisfies the axiom of “Equity.”

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Say that R  is a subrelation to R  if (i) 0 c R  0 c implies 0 c R  0 c and (ii)        0 P 0 c implies 0 c P 0 c , with P and P denoting the asymmetric parts of   R and R , respectively. Svensson’s (1980) Theorem 2 states that any reflexive and transitive binary relation that satisfies the axioms of “Efficiency” and “Equity” is a subrelation to a complete and transitive binary relation (i.e., an ordering). In proving this result Svensson refers to a general mathematical lemma by Szpilrajn (1930). By invoking this result, there exists a complete and transitive binary relation Ru which has Ru as a subrelation. Note that it follows from the definition of a subrelation that also Ru satisfies the axioms of “Efficiency” and “Equity.” For the purpose of Proposition 1 we need not be concerned about how Ru is completed to Ru , because Proposition 1 shows that the considered efficient and decreasing allocation Pu -dominates (and thus Pu -dominates) any alternative feasible allocation for an appropriately chosen u. Then it follows that the considered efficient and decreasing allocation Pu -dominates any alternative allocation, independently of how Ru is completed to Ru . Hence, Proposition 1 obtains also if Pu is substituted for Pu . c

APPENDIX B: PROOF OF PROPOSITION 2 The proof of Proposition 2 is based on three lemmas. We start with Lemma 1 which states the following result for linear technologies. Lemma 1: An allocation 0 c in a linear technology is feasible given y if and only if ct ≥ 0 for all t ≥ 0 and ∞ t=0 pt ct ≤ y. Proof. Assume that 0 c is feasible. Then there exists a feasible program (0 y, 0 k) such that ct = yt − kt ≥ 0 for all t ≥ 0. In particular, yt+1 ≤ at kt for all t ≥ 0. T −1 T −1 pt+1 yt+1 ≤ t=0 pt+1 at kt . Since pt+1 at = pt , this implies This means that t=0  T −1 that t=0 pt (yt − kt ) ≤ p0 y0 − pT yT . We now obtain ∞ t=0 pt ct ≤ y as pt ct = pt (yt − kt ) ≥ 0, p0 y0 = y and pT yT ≥ 0. Assume that ct ≥ 0 for all t ≥ 0 and ∞ t=0 pt ct ≤ y. Construct (0 y, 0 k) by y0 = y, and kt = yt − ct and yt+1 = at kt for t ≥ 0. Then pt ct + pt+1 yt+1 = pt yt − all pt kt + pt+1 at kt = pt yt , implying that ∞ s=t ps cs ≤ pt yt . As ps cs ≥ 0 for all s ≥ t, it follows that yt ≥ 0 for arbitrary t, and (0 y, 0 k) is feasible. Turn now to the class of Ramsey technologies. A feasible program (0 y, 0 k) is competitive if there is a non-null sequence of non-negative prices 0 p such that for t ≥ 0, pt+1 yt+1 − pt kt ≥ pt+1 y − pt k

for all (k, y) ∈ T 2 .

In other words, along a competitive program intertemporal profits are maximized at each point in time. A competitive program is said to satisfy the transversality condition at the price sequence 0 p if limT →∞ pT k T = 0 .

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Lemma 2: Under (A.2), if a feasible and nondecreasing allocation 0 c in a Ramsey technology is efficient, then there exists a feasible program (0 y, 0 k) with ct = yt − kt for all t ≥ 0 that is competitive and satisfies the transversality condition at prices 0 p given by pt+1 f  (kt ) = pt

for t ≥ 0 .

Furthermore, it holds that, for all t ≥ 0, 0 < pt+1 < pt , and

∞

t=0

pt ct < ∞.

Proof. Assume that the feasible allocation 0 c is efficient and nondecreasing. Construct (0 y, 0 k) by y0 = y, and kt = yt − ct and yt+1 = f (kt ) for all t ≥ 0. Since 0 c is feasible and efficient, it follows that there exists k¯ such that 0 < kt < k¯ for all t ≥ 0. Since, in addition, 0 c is nondecreasing, it follows that kt+1 − kt ≥ 0 for all t ≥ 0, because otherwise, kt would have been reduced to zero in a finite ¯ for number of periods. This in turn implies that ct = g(kt−1 ) − (kt − kt−1 ) ≤ g(k) all t ≥ 1. Determine a sequence of consumption discount factors 0 p = ( p0 , p1 , . . . ) as follows: p0 = 1 and

pt+1 f  (kt ) = pt for all t ≥ 0 .

Note that it holds for all t ≥ 0 that 0 < pt+1 < pt since f  (kt ) = g  (kt ) + 1 > 1. By the concavity of g it follows that (0 y, 0 k) is competitive. Since kt < k¯ for all ¯ t ≥ 0, it follows that (0 y, 0 k) satisfies the transversality condition: pT k T ≤ δ T k,  (k). ¯ Finally, since c0 ≤ c1 and ct ≤ g(k) ¯ for all t ≥ 1, it follows that where δ := 1/ f ∞ t ∞ ¯ t=0 pt ct ≤ t=0 δ g(k) < ∞. Finally, consider the class of Dasgupta–Heal–Solow technologies. A feasible program (0 y, 0 m, 0 k) is competitive if there is a non-null sequence of non-negative prices (0 p, 0 q) such that for t ≥ 0, pt+1 yt+1 + qt+1 m t+1 − pt kt − qt m t ≥ pt+1 y + qt+1 m  − pt k − qt m for all [(k, m)(y, m  )] ∈ T 3 . A competitive program is said to satisfy the transversality condition at the price sequence (0 p, 0 q) if limT →∞ ( pT k T + qT m T ) = 0 . Lemma 3: Under (A.3), if a feasible and nondecreasing allocation 0 c in a Dasgupta– Heal–Solow technology is efficient, then there exists a feasible program (0 y, 0 m, 0 k) with ct = yt − kt for all t ≥ 0 that is competitive and satisfies the transversality condition at prices (0 p, 0 q) given by pt+1 Fk (kt , m t − m t+1 ) = pt qt = q > 0

and

for t ≥ 0

pt+1 Fr (kt , m t − m t+1 ) = q

Furthermore, it holds that, for all t ≥ 0, 0 < pt+1

for t ≥ 0 . ∞ < pt , and t=0 pt ct < ∞.

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Proof. Let s  ≥ 0 be the first period with positive consumption: cs  > 0 and ct = 0 for 0 ≤ t < s  . Then, by Proposition 2 in Dasgupta and Mitra (1983), kt+1 > kt > 0 and 0 < m t+1 < m t for t ≥ s  . Let s  , where 0 ≤ s  ≤ s  , be the first period with positive extraction: rs  = m s  − m s  +1 > 0 and rt = m t − m t+1 = 0 for 0 ≤ t < s  . If s  > 0, then (A.3) implies that the resulting allocation is inefficient, since a Paretodominating allocation can be constructed by having rs  −1 =  > 0 and rs  = rs  −  > 0 (for sufficiently small ) and reinvesting the additional output at time s  − 1. Thus, s  = 0. Hence, kt > 0 and rt = m t − m t+1 > 0 for t ≥ 0. The result now follows from Theorem 4.1 and Corollary 4.1 in Mitra (1978). Proof of Proposition 2. Part 1: Linear technologies. The result follows immediately fromLemma 1, since ∞ t=0 pt ct = y (< ∞) if 0 c is feasible given y and efficient,  ≤ y if c is feasible. and ∞ p c t 0 t t=0 Part 2: Ramsey technologies. Lemma 2 establishes the existence of a feasible program (0 y, 0 k) with ct = yt − kt for all t ≥ 0 that is competitive and satisfies the transversality condition at prices 0 p satisfying 0 < pt+1 < pt for all t ≥ 0 such  that ∞ t=0 pt ct < ∞. Furthermore,
T t=0

pt (ct − ct ) = ≤


T

pt [(yt − kt ) − (yt − kt )] t=0 p0 (y0 − y0 ) − pT (k T − k T ) ≤ pT k T

,

where 0 c is any feasible allocation with (0 y  , 0 k  ) as a corresponding feasible program, since (0 y, 0 k) is competitive, y0 = y0 = y, and pT k T ≥ 0. Finally, ∞  the transversality condition. Note that t=0 pt (ct − ct ) ≤ 0 since (0 y, 0 k) satisfies  it follows from Lemma 2 that ∞ t=0 pt (ct − ct ) ≤ 0 holds even if the feasible program (0 y  , 0 k  ) corresponding to 0 c does not satisfy that 0 k  is bounded above. Part 3: Dasgupta–Heal–Solow technologies. Lemma 3 establishes the existence of a feasible program (0 y, 0 m, 0 k) with ct = yt − kt for all t ≥ 0 that is competitive and satisfies the transversality condition at prices (0 p, 0 q), where 0 p satisfies 0 <  pt+1 < pt for all t ≥ 0, such that ∞ p t=0 t ct < ∞. Furthermore,
T t=0

pt (ct − ct ) = ≤


T

pt [(yt − kt ) − (yt − kt )] t=0 p0 (y0 − y0 ) + q0 (m 0 − m 0 ) − [ pT (k T − k T )] + qT (m T − m T )]

≤ pT k T + qT m T , where 0 c is any feasible allocation with (0 y  , 0 m  , 0 k  ) as a corresponding feasible program, since (0 y, 0 m, 0 k) is competitive, y0 = y0 = y, m 0 = m 0 = m, and   pT k T + qT m T ≥ 0. Finally, ∞ t=0 pt (ct − ct ) ≤ 0 since (0 y, 0 m, 0 k) satisfies the transversality condition.

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Acknowledgments: We have received many helpful comments from two referees, Snorre Kverndokk, Stef Prost, Eric Rasmusen, and Martin Weitzman, as well as seminar participants at Harvard, ZEW Mannheim, CESifo Munich, Regensburg, and Southampton. Financial support from the Research Council of Norway (Ruhrgas grant) is gratefully acknowledged. NOTES 1 It may to some extent decrease over time. This is called “slow” or “hyperbolic” discounting (cf. e.g.,

Heal 1998, pp. 62–63 and Weitzman 1998, 2001). 2 Writing j for gross investment, q for gross product, and d for the rate of depreciation, a linear techt t t nology can be described as follows: ct + jt = qt , 0 ≤ qt ≤ bt−1 kt−1 and 0 ≤ kt ≤ jt + (1 − dt−1 )kt−1 , where jt satisfies −(1 − dt−1 )kt−1 ≤ jt ≤ qt and where net capital productivity is positive at time t − 1 if and only if bt−1 > dt−1 . 3 To see that G (k , r , 1) → 0 as t → ∞, note that k ≥ k > 0 and r > 0 for all t ≥ 0 by the proof t t k t t 0 of Lemma 3, while rt → 0 as t → ∞ due to the finite resource stock m. Hence, 0 < G k (kt , rt , 1) ≤ G(kt , rt , 1)/kt ≤ G(k0 , rt , 1)/k0 since G(0, rt , 1) = 0 and G is concave, while G(k0 , rt , 1)/k0 → 0 as t → ∞ since G(k0 , 0, 1) = 0 and G is continuous. 4 Since the application of such a “small” discount rate leads to k being bounded above, the resulting t

allocation is feasible in the terminology of Sect. 5.2. Hence, Proposition 2 applies and the given discounted utilitarian criterion can be substituted by an undiscounted utilitarian criterion (cf. Joshi 1994, for a related but different statement in a special case of the Ramsey model). Thus, the introduction of pure time preference, which may appear to be without “intrinsic ethical appeal” and therefore “purely ad hoc” (Rawls, 1971, p. 298), is not needed to obtain appealing allocations in Ramsey technologies. 5 Following partly the same line of argument, Schelling (1995, p. 401) concludes: “The discount-rate question should disappear. In its place is what utility function to use in valuing future increments in other people’s consumption. This is a real question, not a matter of mathematical convenience.”

REFERENCES Arrow, K.J. (1999), Discounting, morality, and gaming, in P.R. Portney and J.P. Weyant, (eds.), Discounting and Intergenerational Equity. Resources for the Future, Washington, DC Asheim, G.B., Buchholz, W. and Tungodden, B. (2001), Justifying sustainability, Journal of Environmen˝ tal Economics and Management 41, 252U–268. (Chapter 3 of the present volume.) Atkinson, A. (1970), On the measurement of inequality, Journal of Economic Theory 2, 244–263 Cass, D. and Mitra, T. (1991), Indefinitely sustained consumption despite exhaustible natural resources, Economic Theory 1, 119–146 Chakravarty, S. (1969), Capital and Development Planning. MIT Press, Cambridge, MA Collard, D. (1994), Inequality aversion, resource depletion and sustainability. Economics Letters 45, 513– 515 Cowell, D. (2000), Measurement of inequality, in A.B. Atkinson and F. Bourguignon, (eds.), Handbook of Income Distribution, vol. I. Elsevier, Amsterdam Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Dasgupta, P.S. and Mäler, K.G. (1995), Poverty, institutions, and the environmental resource-base, in J. Behrman, and T.N. Srinavasan, (eds.), Handbook of Development Economics, vol. III. Elsevier, Amsterdam Dasgupta, S. and Mitra, T. (1983), Intergenerational equity and efficient allocation of exhaustible resources, International Economic Review 24, 133–153 Diamond, P. (1965), The evaluation of infinite utility streams, Econometrica 33, 170–177

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Fleurbaey, M. and Michel, P. (1994), Optimal growth and transfers between generations, Recherches Economiques de Louvain 60, 281–300 Fleurbaey, M. and Michel, P. (1999), Quelques réflexions sur la croissance optimale, Revue Économique 50, 715–732 Fleurbaey, M. and Michel, P. (2001), Transfer principles and inequality aversion, with an application to optimal growth, Mathematical Social Sciences 42, 1–11 Harsanyi, J.C. (1953), Cardinal utility in welfare economics and in the theory of risk-taking, Journal of Political Economy 61, 434–435 Heal, G.M. (1998), Valuing the Future: Economic Theory and Sustainability. Columbia University Press, New York Joshi, S. (1994), Duality between discounted and undiscounted models of economic growth, Economics Letters 44, 403–406 Koopmans, T.C. (1960), Stationary ordinal utility and impatience, Econometrica 28, 287–309 Kreps, D. and Porteus, E.L. (1979), Temporal von Neumann–Morgenstern and induced preferences, Journal of Economic Theory 20, 81–109 Lauwers, L. (1997), Continuity and equity with infinite horizons, Social Choice and Welfare 14, 345–356 Mirrlees, J.A. (1967), Optimal growth when technology is changing, Review of Economic Studies 34, 95–124 Mitra, T. (1978), Efficient growth with exhaustible resources in a neoclassical model, Journal of Economic Theory 17, 114–129 Page, T. (1977), Conservation and Economic Efficiency. The John Hopkins University Press, Baltimore Pigou, A.C. (1932), The Economics of Welfare. Macmillan, London Ramsey, F.P. (1928), A mathematical theory of saving, Economic Journal 38, 543–559 Rawls, J. (1971), Theory of Justice. Harvard University Press, Cambridge, MA Schelling, T. (1995), Intergenerational discounting, Energy Economics 23, 395–401 Sidgwick, H. (1907), The Methods of Ethics. Macmillan, London Solow, R.M. (1974), Intergenerational equity and exhaustible resources. Review of Economic Studies (Symposium), 29–45 Svensson, L.G. (1980), Equity among generations, Econometrica 48, 1251–1256 Szpilrajn, E. (1930), Sur l’extension de l’ordre partial, Fundamenta Mathematicae 16, 386–389 Weitzman, M.L. (1998), Why the far-distant future should be discounted at the lowest possible rate, Journal of Environmental Economics and Management 36, 201–208 Weitzman, M.L. (2001), Gamma discounting, American Economic Review 91, 260–271

CHAPTER 6 RAWLSIAN INTERGENERATIONAL JUSTICE AS A MARKOV-PERFECT EQUILIBRIUM IN A RESOURCE TECHNOLOGY

Abstract. The Rawlsian maximin criterion is combined with nonpaternalistic altruistic preferences in a nonrenewable resource technology. The maximin programme is shown to be time-inconsistent for a subset of initial conditions. A solution to this intergenerational conflict is found, under a given assumption, as a generically unique subgame-perfect equilibrium.

1. INTRODUCTION Several authors have shown that the Rawlsian maximin criterion (Rawls, 1971) has undesirable properties when applied to questions of intergenerational justice. On the one hand, without altruism (i.e., where the utility of each generation depends only on its own consumption) maximin implies that an initially poor economy is locked into perpetual poverty (Dasgupta, 1974b; Solow, 1974). On the other hand time inconsistency may arise when maximin is combined with a simple form of paternalistic altruism, where the utility of each generation is an additively separable function of its own consumption and the consumption of the next generation (Arrow, 1973; Dasgupta, 1974a; Leininger, 1985). A remedy suggested by Calvo (1978) and generalized by Rodriguez (1981) is to use a simple recursive form of nonpaternalistic altruism, where the utility of each generation is an additively separable function of its own consumption and the utility of the next generation. Then maximin produces optimal allocations which avoid perpetual poverty and ensure time consistency when applied to the neoclassical one-sector technology. Such nonpaternalistic altruism prevents time inconsistency in this technology by letting each generation recognize the altruism of its children and thereby removing a source of intergenerational conflict. This paper checks the robustness of the Calvo–Rodriguez result in a technology which includes both reproducible capital and a nonrenewable resource. Such a resource technology poses an interesting problem of intergenerational distribution since the use of the traditional utilitarian criterion with an arbitrarily small but positive discount rate may force consumption to eventually approach zero even if unbounded consumption growth is feasible (Dasgupta and Heal, 1974; Solow,

Originally published in Review of Economic Studies 55 (1988), 469–484. Reprinted with permission from Blackwell.

83 Asheim, Justifying, Characterizing and Indicating Sustainability, 83–101 c 2007 Springer


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1974). This undesirable property of the utilitarian criterion is caused by a dwindling capital productivity as the resource becomes scarcer. It turns out that this declining productivity gives rise to an additional source of intergenerational conflict such that time inconsistency reappears in this technology even though maximin is combined with the simple recursive form of nonpaternalistic altruism described above. The existence of time inconsistency provides the motivation for analyzing subgame-perfect Nash equilibria in an extensive form game, where any generation selects a strategy – describing its vector of consumption and resource extraction as a function of the history of the game – which is a best reply to the strategies of later generations. What appears to be new here compared, for example, to the gametheoretic analysis of the Arrow–Dasgupta model (Dasgupta, 1974b; Kohlberg, 1976; Lane and Leininger, 1984; Lane and Mitra, 1981; Leininger, 1986; Bernheim and Ray, 1983, 1987), is that the payoff of each generation is not its altruistic utility. Rather, generations retain their commitment to maximin as an ethical principle, implying that the payoff of any generation is the infimum over the altruistic utility of all remaining generations. Assuming that a time-consistent maximin programme is followed as soon as one exists, the existence of a generically unique subgame-perfect equilibrium is demonstrated (Theorem 2). The equilibrium strategies are Markov and stationary; i.e., the action of any generation is a stationary function of its inherited stocks as history-summarizing variables. The equilibrium programmes have the property of maximizing altruistic utility over the class of feasible programmes with nondecreasing consumption. Section 2 describes the model, and gives a necessary and sufficient condition for maximin to give rise to time inconsistency (Theorem 1). A solution to this intergenerational conflict is established in Sect. 3 through Theorem 2. Section 4 contains all proofs.

2. THE MODEL Following Dasgupta and Mitra (1983), let k denote the stock of reproducible capital, r the rate of resource extraction, and z labour input, and assume that the stationary production function G : R3+ → R+ satisfies (A.1)

G(k, r, z) is concave, homogeneous of degree one, and continuous for (k, r, z)
0; twice differentiable for (k, r, z)  0.1

(A.2)

G is nondecreasing in k, r , and z, for (k, r, z)
0; also, (G k , G r , G z )  0 for (k, r, z)  0.2

We assume that both capital and the resource are essential in production: (A.3)

G(k, 0, z) = 0 = G(0, r, z).

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The available labor force is assumed to be stationary and equal to 1. Given this labor force, we assume that the ratio of the share of resource in output to the share of labor in output is bounded away from zero: ˜ r˜ )  0, there is η˜ > 0 such that for all (k, r ) satisfying k
(A.4) Given any (k, ˜k, 0 < r  r˜ , we have [r G r (k, r, 1)]/G z (k, r, 1)
η. ˜ A total output function, F, can be defined by F(k, r ) ≡ G(k, r, 1) + k

for (k, r )
0.

For our main results, we assume that F satisfies (A.5)

F(k, r ) is strictly concave.

(A.6)

Fkr
0.

Let y denote total output and m the resource stock. Then production possibilities are described by a stationary technology set T of input–output pairs in the following way: T = {[(k, m), (y, m  )] : 0  y  F(k, r ); 0  r = m − m  ; (k, m  )
0}. Write t y = (yt , yt+1 , . . .) and correspondingly for other sequences. A programme (t y, t m, t k) is said to be (y, m)-feasible (or a (y, m)-programme) at time t if yt = y ks  ys

and

and

mt = m

[(ks , m s ), (ys+1 , m s+1 )] ∈ T

for all s
t.

Let c denote consumption. An allocation t c is said to be (y, m)-feasible (or a (y, m)allocation) at time t if there is a (y, m)-programme (t y, t m, t k) at time t such that cs = ys − ks for all s
t. Each generation lives for one period. The preferences of any generation t are described by an ancestor-insensitive and stationary altruistic utility function u : R∞ + → R+ defined by u(t c) ≡

∞


bs−t · v(cs ) = v(ct ) + b · u(t+1 c),

0 < b < 1,

s=t

where v : R+ → R+ is a stationary one-period felicity function satisfying (V.1) v(c) is strictly increasing, concave, and differentiable. (V.2) vc (c) → ∞ as c → 0, and where b is the felicity discount factor. Note that the utility function has both a paternalistic and a nonpaternalistic representation, the latter being of a simple recursive form. (See e.g., Ray (1987).)

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The payoff of generation t is given by an ancestor-insensitive and stationary social welfare function w : R∞ + → R+ defined by w(t c) ≡ inf u(s c). s
t

Each generation subscribes to maximin as an ethical principle. Hence given some (yt , m t ), generation t seeks to (W) maximize w(t c ) over the class of (yt , m t )-allocations at time t. Under our assumptions, a maximin programme from any (y, m) always exists. A maximin programme (0 y, 0 m, 0 k) from (y, m) at time 0 and the associated (y, m)allocation 0 c are time consistent, if for all t
0, t c solves (W). A (y, m)-allocation 0 c at time 0 is stationary if ct = ct+1 for all t
0. A (y, m)allocation 0 c at time 0 is efficient if there is no (y, m)-allocation 0 c at time 0 with ct
ct for all t
0, with strict inequality for some t. A (y, m)-programme (0 y, 0 m, 0 k) at time 0 is efficient if the associated allocation is efficient. Assumptions (A.1)–(A.4) are not sufficient to ensure the existence of a stationary allocation with positive consumption (Solow, 1974; Stiglitz, 1974; Cass and Mitra, 1979). Therefore assume the following condition: (E) There exists from any (y, m)  0 a stationary allocation with positive consumption. In a general model, a necessary and sufficient condition for (E) to hold is given in Cass and Mitra (1979). To set the stage for further analysis, we state Proposition 1: Under (A.1)–(A.4), there exists from any (y, m)  0 (and at any time t
0) an efficient stationary (y, m)-allocation t ce with positive consumption if and only if (E) holds. The associated programme (t y e , t m e , t k e ) is unique if (A.5) also holds. Refer to the programme of Proposition 1, (t y e , t m e , t k e ), as the egalitarian programme from (y, m) at time  t, and let itbe denoted by et (y, m). Given (y, m)  0 and 0 y e , 0 m e , 0 k e = e0 (y, m), define b(y, m) by b(y, m) · Fk (k0e , m − m e1 ) = 1. Clearly, b(y, m) is well defined and 0 < b(y, m) < 1. We now state our first main result. Theorem 1: Under (A.1)–(A.6), (E), and (V.1), a maximin programme from (y, m)  0 is time consistent if and only if b(y, m)
b. If a maximin programme from (y, m)  0 at time 0 is time consistent, it is given by e0 (y, m). Using e0 (y, m) = (0 y e , 0 m e , 0 k e ) as a reference programme, generation t is said to save if ct < cte . Furthermore, define saving as transitorily utility productive for generation t if b(yte , m et ) < b; i.e., the capital productivity along (0 y e , 0 m e , 0 k e ) at time t, Fk (kte , m et − m et+1 ) − 1, is greater than the felicity discount rate, (1/b) − 1.

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If b(y, m)
b, then e0 (y, m) is maximin and time consistent because (a) saving is not transitorily utility productive for any generation (due to a diminishing capital productivity along (0 y e , 0 m e , 0 k e ) since kte → ∞ and m et → 0 and t → ∞) and (b) dissaving is incompatible with maximin as an ethical principle. Conversely, let (0 y, 0 m, 0 k) be a maximin programme from (y, m) at time 0, with b(y, m) < b. Then there exists a first t such that b(yt , m t )
b. The problem of time inconsistency arises since it is transitorily utility productive for generation t − 1 to save in order to let generation t consume on its behalf. However, the ethical principle obliges the latter to share the return with later generations. As in the Arrow–Dasgupta model, myopia in the form of a low felicity discount factor “helps” time consistency: time inconsistency arises if and only if a generation (given an egalitarian programme as reference) prefers the next generation to consume on its behalf. 3. EQUILIBRIUM PROGRAMMES If the present generation cannot dictate that future generations follow its maximin programme, a possibility – suggested by Strotz (1955–1956) and Pollak (1968) – is for the present generation to behave in a sophisticated manner by choosing the best programme among those which later and equally sophisticated generations actually will carry out. This provides the motivation for considering subgame-perfect Nash equilibria where any generation’s equilibrium strategy – as a function of the history of the game – is a best reply to the equilibrium strategies of the succeeding generations (Selten, 1975; Goldman, 1980). Let Ht be the set of feasible histories at time t. Hence, H0 = {h 0 : h 0 = (y, m)
0}

and for t > 0,

Ht = {h t : there is a(y, m)
0 and a (y, m)-programme at time 0, (0 y, 0 m, 0 k), such that h t = (k0 , . . . , kt−1 ; y0 , . . . , yt ; m 0 , . . . , m t )}. Denote the bequest to generation t under history h t by (y h t , m h t ). A strategy for generation t is a function σt (h t ), where σt = (σtk , σt y , σtm ) : Ht → 3 R+ satisfies the feasibility condition σtk  y h t

and

[(σtk , m h t ), (σt y , σtm )] ∈ T

for all h t ∈ Ht .

A sequence of strategies t σ = (σt , σt+1 , . . .) and a history h t generate a (y h t , m h t )σ σ σ programme at time t, (t y t , t m t , t k t ), in the following way:  σ  σ tσ σ σ σ  for all s
t. h tt = h t and h ts+1 = h ts ; kst , ys+1 , m ts+1  tσ tσ    σ σ ks , ys+1 , m ts+1 = σs h ts for all s
t. σ

Denote the associated (y h t , m h t )-allocation at time t, t ct (h t ), making explicit the dependence on history.

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A sequence of strategies 0 σ is a subgame-perfect equilibrium at time 0 if at any time t
0 and for any h t ∈ Ht ,   σ   σ w t ct (h t )
w t ct (h t ) ,  where σt is any alternative strategy for generation t and where t σ  = (σt,t+1 σ ). Note once more that the payoff of generation t is the social welfare at time t, not the altruistic utility of generation t. Since altruism and maximin are ancestor insensitive, the vector (y h t , m h t ) summarizes the payoff-relevant information for the remaining subgame at time t; (y h t , m h t ) is the state of the system. A generation may therefore select a state-dependent strategy describing its input–output pair as a function of its inherited stocks. A strategy is Markov at time t if there is a function µt : R2+ → R3+ such that σt (h t ) = µt (y h t , m h t ) for all h t ∈ Ht . If a sequence of Markov strategies 0 µ is a subgameperfect equilibrium, then the equilibrium is said to be Markov-perfect. A sequence of Markov-strategies 0 µ is stationary if µt = µt+1 for all t
0. A trivial Markov-perfect equilibrium, yielding at any time the minimal social welfare of v(0)/(1 − b), is realized by letting all generations choose zero consumption and zero resource extraction (i.e. they all use the Markov-strategy µ(y, m) = (y, y, m)). Furthermore, it can be shown that, given some Markovperfect equilibrium, we can construct a new Markov-perfect equilibrium by assuming that all generations disregard some part of the resource stock. Finally, any programme can be realized as a subgame-perfect equilibrium programme if a deviation triggers the use of the strategy µ(y, m) = (y, y, m) by all remaining generations. In order to exclude such trivial and/or undesirable equilibria, and to refine the solution concept, assume the following condition: σ

σ

σ

(C) Along any subgame-perfect equilibrium programme (t y t , t m t , t k t ), a timeconsistent maximin programme must be followed as soon as one exists; i.e., from t on if b(y h t , m h t )
b. In the model of Calvo (1978), it can easily be demonstrated that, if capital is not initially overaccumulated (i.e., capital does not exceed the Golden Rule stock), the time-consistent maximin allocation at time t maximizes u(t c) over the class of feasible nondecreasing allocations at time t. This suggests the interest of the following result. Proposition 2: Under (A.1)–(A.4) and (V.1)–(V.2), there exists from any (y, m)  0 (and at any t
0) an efficient (y, m)-allocation t c j with positive and nondecreasing consumption maximizing u(t c) over the class of nondecreasing (y, m)-allocations at time t if and only if (E) holds. The associated programme (t y j , t m j , t k j ) is unique if (A.5)–(A.6) also holds. Refer to the programme of Proposition 2, (t y j , t m j , t k j ), as the just programme from (y, m) at time t, and let it be denoted by jt (y, m).

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Given this result we can establish the existence and generic uniqueness of a subgame-perfect equilibrium: Theorem 2: Under (A.1)–(A.6), (E), and (V.1)–(V.2), there exists a subgameperfect equilibrium satisfying (C). Assuming (C), the equilibrium strategies are unique, Markov, and stationary on {h t ∈ Ht : (y h t , m h t )  0}. For any h t such that (y h t , m h t )  0, the unique equilibrium programme in the subgame remaining at time t is given by jt (y h t , m h t ). Remark 1: For h t such that y h t = 0 or m h t = 0, the remaining subgame is trivial in the sense that any (y h t , m h t )-allocation yields all generations the minimal social welfare v(0)/(1 − b). Hence, for such h t , any feasible input–output pair is a best reply at time t. Specifying state-dependent replies for t
0 makes the equilibrium Markov-perfect. Given (0 y e , 0 m e , 0 k e ) = e0 (y, m), let a(y, m) denote an average of b(yse , m es ), s = 0, 1, . . . , defined by the following expression:

 t  ∞
   = 0. b yse , m es (a(y, m))t+1 − t=0

s=0

As explained in the remark preceding Lemma 4, 1/a(y, m) − 1 may be interpreted as the productivity of saving when the returns are constrained to be stationary, using (0 y e , 0 m e , 0 k e ) as a reference programme. Hence, we may refer to saving as permanently utility productive if a(y, m) < b. The diminishing capital productivity along (0 y e , 0 m e , 0 k e ) = e0 (y, m) (i.e., b(yte , m et ) is increasing) is the key to understanding the difference between transitory and permanent utility productiveness. In order to produce a stationary increase in consumption from time 1 on, relative to e0 (y, m), generation 0 has to save, generation 1 has to reinvest some of the return, and so forth. Due to the diminishing capital productivity, the internal rate of return on this investment programme (= 1/a(y, m) − 1) is smaller than the capital productivity at time 0 (= 1/b(y, m) − 1). Hence, a(y, m) > b(y, m), and therefore, permanent utility productiveness implies transitory utility productiveness, while the converse is not true. j j The just programme, (0 y j , 0 m j , 0 k j ) = j0 (y, m), follows et (yt , m t ) from t on if saving at time t is not permanently utility productive, while consumption is growing as long as it is. This is a desirable distributional property: the just allocation allows for consumption (and utility) growth when the economy is highly productive (a(y, m) < b), while protecting distant generations from the grave consequences of felicity discounting when capital productivity is low and diminishing (a(y, m)
b). Since a(y, m) > b(y, m), (0 y j , 0 m j , 0 k j ) satisfies (C). Moreover, if b(y, m) < b, there generally (disregarding the case with a(y j t−1 , m j t−1 ) < b and b(y j t , m j t )
b) exists a time t at which saving is transitorily, but not permanently utility productive: a(y j t , m j t )
b > b(y j t , m j t ). The just (i.e., equilibrium) programme follows the egalitarian programme from this t on. However, since saving at time t is transitorily utility productive, the altruistic utilities – and, in fact, the payoffs – of

90

GEIR B. ASHEIM

all generations s
t can be increased by having generation t save and generation t + 1 reinvest only a fraction of the return. Hence, the equilibrium does not generally produce Pareto-efficient allocations in the sense of there being no alternative (y, m)allocation increasing the payoff of some generation without decreasing the payoff of any other generation. The deviation from the equlibrium described above is not, however, profitable for generation t as long as generation t + 1 cannot be committed to consuming on behalf of an earlier generation; thus, the potential Pareto-improvement cannot be realized. A generation will save if and only if this increases its altruistic utility in spite of its children sharing the return with later generations, i.e., when saving is permanently utility productive. In the model of Calvo (1978), capital productivity is constant along an egalitarian programme. Therefore, saving is transitorily utility productive if and only if it is permanently utility productive, and thus, time inconsistency does not arise. Condition (C) converts an infinitely lasting extensive form game into a game that is essentially finished as soon as b(y h t , m h t )
b. Hence, with (C), an equilibrium can be found by backwards induction. This explains that for (y, m)  0, any (y, m)programme can be supported as an equilibrium without condition (C), while there is a unique equilibrium programme with (C). Still, it may be considered undesirable that condition (C) has to be assumed. Rather, the solution concept should imply that a time-consistent maximin programme is followed as soon as one exists. Even insisting on a Markov-perfect equilibrium does not ensure this. What is needed is a kind of renegotiation-proofness (see e.g., Farrell and Maskin (1987)). Allowing any generation to renegotiate with future generations – i.e., suggest a new equilibrium for the remaining subgame – surely implies (C). This idea can be captured in the present setting using an approach to game theory introduced by Greenberg (1990). Informally (see Asheim (1987) for a formal statement and a proof), let any generation be able to suggest a programme in (induce) the remaining subgame, given its inherited stocks. Then there exists a unique standard of behavior being both internally and externally stable, where a standard of behavior is a mapping assigning a set of programmes (a solution) to any time-state pair, and where “stability” is used in the sense of a von Neumann and Morgenstern abstract stable set. Furthermore, this stable standard of behavior assigns only one programme, νiz. jt (y, m), if (y, m)  0. This latter formulation corresponds closely to the above-mentioned notion of having the present generation choose the best programme among those which later generations will carry out. The internal stability implies that the just programme will be carried out by future generations, while the external stability implies that any programme yielding the present a higher payoff will not be carried out by future generations. The proofs lean heavily on the fact that, in the present technology, any efficient programme with positive and nondecreasing consumption has decreasing capital productivity. It is of course true in the neoclassical one-sector technology that capital productivity is nonincreasing along any efficient programme with nondecreasing consumption. This suggests a generalization that will not be pursued here: does

RAWLSIAN INTERGENERATIONAL JUSTICE

91

maximin combined with the simple recursive form of nonpaternalistic altruism yield as a solution an allocation maximizing u(0 c) over the class of feasible nondecreasing allocations, in a more general technology (which includes the one-sector technology and the present resource technology as special cases) having the property that nondecreasing consumption implies nonincreasing capital productivity along an efficient programme? It would also be of interest to establish conditions for time consistency in such a general technology. 4. PROOFS A (y, m)-programme (0 y, 0 m, 0 k) is interior if (kt , m t − m t+1 )  0 for t
0. Lemma 1: Under (A.1)–(A.3), if (0 y, 0 m, 0 k) is an efficient (y, m)-programme with positive and nondecreasing consumption, then kt+1 > kt and m t+1 < m t for t
0; furthermore, the programme is interior. Proof. Proposition 2 in Dasgupta and Mitra (1983). A (y, m)-programme (0 y, 0 m, 0 k) is called competitive if there is a non-null sequence of non-negative prices (0 p, 0 q) such that for t
0, pt+1 yt+1 + qt+1 m t+1 − pt kt − qt m t
pt+1 y + qt+1 m  − pt k − qt m for all [(k, m), (y, m  )] ∈ T .

(1)

In other words, the intertemporal profit maximization condition (1) is satisfied at each date. A competitive programme is said to satisfy the transversality condition at the price sequence (0 p, 0 q) if lim ( pt kt + qt m t ) = 0.

(2)

t→∞

Note that if the (y, m)-programme (0 y, 0 m, 0 k) is competitive  and satisfies the transversality condition at prices (0 p, 0 q), and furthermore, ∞ t=0 pt ct < ∞, then, for any alternative (y, m)-programme (0 y  , 0 m  , 0 k  ), ∞
t=0

pt (ct − ct ) =

∞


pt [(yt − kt ) − (yt − kt )]

t=0

 p0 (y0 − y0 ) + q0 (m 0 − m 0 ) = 0,

(3)

by (1) and (2) since (y0 , m 0 ) = (y0 , m 0 ) = (y, m). Lemma 2: Under (A.1)–(A.4), a (y, m)-programme (0 y, 0 m, 0 k) with positive and nondecreasing consumption is efficient iff it is competitive and it satisfies the transversality condition at prices (0 p, 0 q) given by pt+1 Fkt = pt qt = q > 0 and

for

t
0

pt+1 Frt = q

for

t
0,

92

GEIR B. ASHEIM

where Fkt = Fk (kt , m t − m t+1 ) and Fr1 = Fr (kt , m t − m t+1 ). Furthermore, ∞


pt ct < ∞

t=0

if (0 y, 0 m, 0 k) is efficient. Proof. By Lemma 1, if (0 y, 0 m, 0 k) is efficient, then (0 y, 0 k, 0 m) is interior and kt
k0 for t
0. The result follows from Theorem 4.1 and Corollary 4.1 in Mitra (1978), using part of the necessity proof of Proposition 5 in Dasgupta and Mitra (1983). Note in particular that efficiency implies Frt+1 = Frt · Fkt+1 for t
0. Lemma 3: Under (A.1)–(A.6), if (0 y, 0 m, 0 k) is an efficient (y, m)-programme with positive and nondecreasing consumption, then Fkt > Fkt+1 for t
0. Proof. By Lemma 1, kt+1 > kt > 0 and rt = m t − m t+1 > 0 for t
0. By Lemma 2 and (A.2), Frt+1 /Frt = Fkt+1 > 1 for t
0. By (A.5) F(kt+1 , rt+1 ) − F(kt , rt ) < Fkt · (kt+1 − kt ) + Frt · (rt+1 − rt ) F(kt , rt ) − F(kt+1 , rt+1 ) < Fkt+1 · (kt − kt+1 ) + Frt+1 · (rt − rt+1 ) such that     0 < Fkt − Fkt+1 (kt+1 − kt ) + Frt − Frt+1 (rt+1 − rt )     = Fkt − Fkt+1 (kt+1 − kt ) + Frt · 1 − Fkt+1 (rt+1 − rt ) and since kt+1 > kt   Fkt − Fkt+1 > Frt · Fkt+1 − 1 · (rt+1 − rt )/(kt+1 − kt ). Hence, if rt+1
rt , then Fkt > Fkt+1 . But if rt+1 < rt , then Fkt > Fkt+1 by (A.5) and (A.6) since kt+1 > kt . Proof of Proposition 1. Theorem 1 in and Mitra (1983). By Lemma 2, Dasgupta ∞ e , k e ) satisfies (1), (2), and e < ∞ at positive prices ( p, q). (0 y e , 0 m p c 0 t 0 0 t t=0 e By (3), ∞ t=0 pt (ct − ct )  0 if 0 c is an arbitrary (y, m)-allocation. If the associated (0 y, 0 m, 0 k) is not identical to (0 y e , 0 m e , 0 k e ), then ∞ (y, m)-programme e ) < 0 by (A.5), and c cannot be efficient and stationary since p (c − c 0 t t=0 t t cte = ce for t
0. Proof of Theorem 1. (The proof is due to Debraj Ray.) (Sufficiency) Let b(y, m)
b. It suffices to show that (0 y e , 0 m e , 0 k e ) is the unique maximin programme from (y, m)
0 at time 0, since, by Lemma 3, b(yte , m et ) > b for t
1. Suppose that a programme (0 y, 0 m, 0 k) that is not identical to (0 y e , 0 m e , 0 k e ) is maximin from

93

RAWLSIAN INTERGENERATIONAL JUSTICE

(y, m) at time 0. Then u(t c) − u(t ce ) =

∞


bs−t · [v(cs ) − v(ce )]
0

for t
0.

(4)

s=t

Using (V.1) [v(cs ) − v(ce )  vc (ce ) · (cs − ce )] and (4), ∞


bs (cs − ce )
0

for t
0.

(5)

s=t

 e Since (0 y, 0 m, 0 k) = (0 y e , 0 m e , 0 k e ), ∞ t=0 pt (ct − c ) < 0 by Lemma 2 (at 0 p described there), (3), and (A.5). This yields the contradiction ∞ ∞

e s e pt (ct − c ) = p0 · b (cs − c ) 0> t=0

+

∞
t=1

s=0



( pt /b − pt−1 /b t

t−1

)·

∞




b (cs − c ) s

e


0

s=1

by (5) because for t
1, pt /bt − pt−1 /bt−1 = ( pt−1 /bt )( pt / pt − 1 − b) = ( pt−1 / e , me e e e t bt )[1/Fk (yt−1 t − 1 − m t ) − b] = ( pt − 1 /b ) [b(yt −1 , m t−1 ) − b ]
0. Hence, (0 y e , 0 m e , 0 k e ) is the unique maximin programme from (y, m). (Necessity) Let b(y, m) < b and suppose (0 y, 0 m, 0 k) is maximin and time consistent from (y, m)  0 at time 0. Then there is a first time T such that (b(yT , m T )
b. (To see this, note that yt → ∞ as t → ∞ if (0 y, 0 m, 0 k) is maximin from (y, m)  0 (otherwise, lim inft→∞ u(t c) = v(0)/(1 − b)). By (A.1) and (A.3), there ˜  1/b. Choose a time t such that yt
F(k, ˜ m). exists a k˜
1 satisfying F(1, m/k) e e e e e ˜ Along (t y , t m , t k ) = et (yt , m t ), kt+1 > kt by Lemma 1. Hence, f (k, m)  yt = e e . Since m e − m e e = F(kte , m et − m et+1 ) since cte = ct+1 yte < yt+1 t t+1  m t = m t  m, (A.2) now implies that kte > k˜
1. Thus, Fk (kte , m et − m et+1 )  F(kte , m et − ˜  1/b; i.e., b(yt , m t ) > b (use m et+1 )/kte  F[1, (m et − m et+1 )/kte ] < F(1, m/k) (A.1)–(A.3)).) Hence, b(yT −1 , m T −1 ) < b, b(yT , m T )
b, and from time T on, the timeconsistent maximin programme must be identical to the egalitarian programme from (yT , m T ) at time T (see first part of the proof). Let c¯ be the stationary consumption level from time T on. Case 1. cT −1 > c. ¯ We contradict the fact that T −1 c is maximin from (yT −1 , m T −1 ) at time T − 1 as follows. Note that u(t c) < u(T −1 c) for t
T . Construct an alternative (yT −1 , m T −1 )-programme at time T − 1, (T −1 y  , T −1 m  , T −1 k  ), by choosing k T −1 ∈ (k T −1 , yT −1 − c) ¯ and letting (T y  , T m  , T k  ) be the egalitarian  (F(k T −1 , m T −1 − m T ), m T )-programme at time T (use Proposition 1). We have that u(t c ) > w(T −1 c) for t
T − 1. Case 2. cT −1 < c. ¯ In this case u(t c) > u(T −1 c) for t
T . We contradict maximin at time T − 1 as follows. Choose any t > T . Then b(yt , m t ) > b by Lemma 3.

94

GEIR B. ASHEIM

Construct a (yT −1 , m T −1 )-programme at time T − 1, (T −1 y  , T −1 m  , T −1 k  ), by  making a “small” transfer ε > 0 from t + 1 to t (i.e., kt = kt − ε, yt+1 =  F(kt , m t − m t+1 ), and otherwise identical to (T −1 y, T −1 m, T −1 k)). Thus, T −1 c =  ,c  (cT −1 , cT , . . . , ct−1 , ct , ct+1 t+2 , . . .) is a (yT −1 , m T −1 )-allocation with u(s c ) > u(T −1 c) = w(T −1 c) for s
T − 1. Case 3. cT −1 = c. ¯ Here u(t c) = u(T −1 c) for t
T . Construct a (yT −1 , m T −1 )programme at time T − 1, (T −1 y  , T −1 m  , T −1 k  ), where k T −1 = k T −1 + ε, yT = F(k T −1 , m T −1 − m T ), and k T = k T + δ, and where (T +1 y  , T +1 m  , T +1 k  ) is the egalitarian (F(k T , m T − m T +1 ), m T +1 )-programme at time T + 1. We interpret ε > 0 to be a “small” transfer from T − 1 to T . Generation T consumes “most” of this transfer, passing only δ > 0 to T + 1, thereby raising the stationary consumption from T + 1 onward. Since b(yT −1 , m T −1 ) < b, u(t c ) > u(t c) = w(T −1 c) for t
T − 1 if the two transfers are chosen carefully. Combining Cases 1, 2, and 3, completes the proof. Given (y, m)  0 and (0 y e , 0 m e , 0 k e ) = e0 (y, m), define a(y, m) by

 
∞  t  e e . b ys , m s a(y, m) = 1 1+1 t=0 s=0

ye,

me,

ke)

iscompetitive at prices (0 p, 0 q) such that By Lemma 2, (0 0 0 e , m e ) = 1/F = p e b(y / p and ( ∞ that a(y, m)[= k s+1 s t=0 pt ) · c < ∞. It follows  s∞ s ∞ ( t=1 pt )/( t=0 pt )] is well defined and 0 < a(y, m) < 1. Since ( ∞ t=1 pt )/ p0 is the efficiency price of a consumption annuity from time 1 in terms of consumption at  time 0, we may interpret 1/a(y, m) − 1[= p0 /( ∞ t=1 pt )] as the productivity of saving at time 0 when the returns are constrained to be stationary, using (0 y e , 0 m e , 0 k e ) as a reference programme. Note that a(y, m) > b(y, m) by Lemma 3. Lemma 4: Under (A.1)–(A.6), (E), and (V.1), let (0 y j , 0 m j , 0 k j ) be an efficient programme from (y, m)  0 at time 0 which associated positive and nondecreasing allocation 0 c j , with u(0 c j ) < ∞, maximizes u(0 c) over the class of nondecreasing (y, m)-allocations at time 0. Then ( a)

(0 y j , 0 m j , 0 k j ) is unique and time-consistent (for all t
0, t c j maximizes j j u(t c) over the class of nondecreasing (yt , m t )-allocations),

(b)

tc

j

j

j

is stationary iff a(yt , m t )
b,

(c) There is a λ0 such that if 0 c is an arbitrary (y, m)-allocation at time 0, with 1 c nondecreasing, then ∞
t=1

  j    j  br · υ(ct ) − υ ct  λ0 · υ c0 − υ(c0 ) ,

95

RAWLSIAN INTERGENERATIONAL JUSTICE

where λ0 = 1 if a(y, m) < b, where 0 < λ0  1 if a(y, m)
b, and where the inequality is strict if the associated programme (0 y, 0 m, 0 k) is not identical to (0 y j , 0 m j , 0 k j ). Proof. By Lemma 2, (0 y j , 0 m j , 0 k j ) satisfies (1) and (2) at the positive  j prices (0 p, 0 q) described there. Furthermore, ∞ t=0 pt ct < ∞. Define the positive sequence 0 λ by  j λt · vc ct = pt . (6) Then λt is the efficiency price of felicity at time t. By (V.1) and (3), lim sup

T


T →∞ s=t

j

∞   j 
 j λs · v(cs ) − v cs  ps · cs − cs  0

(7)

s=t

j

for any (yt , m t )-allocation t c at time t, with the latter inequality being strict if the associated programme (t y, t m, t k) = (t y j , t m j , t k j ). ∞ ∞ j j We have that t=0 λt < ∞. Because if ct = c for t
0, then t=0 λt = ∞  j j ( t=0 pt )/ve (c j ) < ∞. But if ct < ct+1 for some t, then ( ∞ λ s=t+1 s )/λt  j

b/(1 − b). Because otherwise a higher u(0 c) could be produced by increasing ct by a “small” amount at the cost of decreasing s+1 c j by a “small” and (in terms of felicity) stationary amount. Note that  ∞the altered allocation would still be nondecreasing. Hence, ( ∞ s=t+1 λs )/( s=r λs ) is well-defined and ⎞  ⎞  ⎛ ⎛ ∞

∞ ∞ ∞

0 < pt+1 / pt < ⎝ ps ⎠ ps  ⎝ λs ⎠ λs (8) s=t+1

s=t

s=t

s=t+1

∞

by Lemma 3 and (V.1). Since ( s=t+1 λs )/λt is the efficiency price of a felicity annuity from  time t + 1 in termsof felicity at time t, we may interpret  ∞ ∞ 1/[( ∞ λ )/( s s=t+1 s=t λs )] − 1 = λt /( s=t+1 λs ) as the productivity of saving felicity at time t when felicity returns are constrained to be stationary, using (0 y j , 0 m j , 0 k j ) as reference. j The possibility of changing ct by a “small” amount at the cost of changing j t+1 c by a “small” and (in terms of felicity) stationary amount, yields the following necessary conditions, ⎞  ⎛
∞ ∞
j j j ct−1 < ct < ct+1 ⇒ ⎝ λs ⎠ λs = b (9) ⎛ j ct−1

<

j ct

=

j ct+1

⇒⎝

s=t+1 ∞


s=t

⎞ λs ⎠

j ct−1

=

j ct

<

j ct+1

⇒⎝

∞


s=t+1

λs


b

(10)

 b,

(11)

s=t

s=t+1

⎛


∞

⎞ λs ⎠


∞ s=t

λs

96

GEIR B. ASHEIM

taking into account that the altered allocation be nondecreasing. j j j If ct = ct+1 for some t, then t c j is stationary. To see this, suppose ct = · · · = j

j

j

j

cT < cT +1 , where t = 0 or ct−1 < ct . Then

⎞  ⎛
∞ ∞
PT PT +1 pt+1 ⎠ ⎝ < ··· < < < ps ps pt PT −1 PT ⎛ ⎝

s=T +1

⎞

∞


λs ⎠

s=T



s=T +1

∞


λs

b

s=T j

j

by Lemma 3, (8), and (11). Furthermore, since ct = · · · = cT , λt+1 /λt < · · · < λT /λT −1 < b by (6). Hence, λT /λt < b T −t , and a higher u(0 c) could be produced by increasing cT j by a “small” amount at the expense of ct , while keeping the altered allocation nondej creasing. This contradicts that 0 c maximizes u(0 c) over the class of nondecreasing (y, m)-allocations. We may therefore characterize 0 c j as follows: there is a possibly infinite T
0 j j j j such that c0 < · · · < cT −1 < cT = cT +1 = · · · . Furthermore, ⎞  ⎛
∞ ∞
⎝ λs ⎠ λs = b for t < T j

⎛ ⎝ ⎛ ⎝

s=t

s=t+1 ∞


⎞ λs ⎠

λs


b

for t = T

>b

for t > T

(12)

s=t

s=t+1 ∞



∞

⎞ λs ⎠


∞

λs

s=t

s=t+1

∞ s−t ∞ = by (9), (10), and Lemma 3. Normalize prices such that ∞ s=t λs = s=t b ∞ 1/(1 − b). Then λt = (1/(1 − b))[1 − ( s=t+1 λs )/( s=t λs )], and by (12), λt = 1

if t < T

0 < λt  1

if t = T

0 < λt < 1

if t > T.

(13)

Furthermore, since T c j is stationary and λs = bs−t for t  s < T , ∞
 j (λs − bs−t ) · v cs = 0. s=t

(14)

RAWLSIAN INTERGENERATIONAL JUSTICE

97

Moreover, by Lemma 3 and (6), there is a τ , dependent on t, such that λs − bs−t  0 for t  s < τ , and λs − bs−t
0 for s
τ . For any nondecreasing allocation t c, let c¯ satisfy cs  c¯ for t  s < τ and cs
c¯ for s
τ . j j (Part (a)) If t c is any nondecreasing (yt , m t )-allocation, then lim sup

T


T →∞ s=t

T
 s−t   j  j  b bs−t · v(cs ) − v(cs ) = lim sup · v(cs ) − λs · v cs T →∞ s=t

by (14)  lim sup

T
[bs−t · v(cs ) − λs · v(cs )]

T →∞ s=t

by (7) with strict inequality if the associated programme (t y, t m, t k) = (t y j , t m j , t k j ) = lim sup

T
(bs−t − λs )[v(cs ) − v(c)] ¯

T →∞ s=t

as

∞

s=t (b

s−t

− λs ) = 0  0 by the construction of c. ¯

(Part (b)) If a(y, m)
b, then it follows from (6) that e0 (y, m) satisfies (12) with T = 0. By the proof of part (a), e0 (y, m) is the unique programme which associated allocation maximizes u(0 c) over the class of nondecreasing (y, m)j j allocations. If a(y, m) < b, then suppose (0 y j , 0 m j , 0 k j ) = e0 (y, m). Then c0 = c1 ∞ ∞ and s=1 λs / s=0 λs < b, contradicting (10).  ∞ s (Part (c)) Normalize such that ∞ s=1 λs = s=1 b = b/(1 − b). Then ∞ prices λ0 = (b/(1 − b))[( s=0 λs )/( ∞ s=1 λs ) − 1], and it still follows from (12) that (13) holds for t = 0. By part (b), λ0 = 1 if a(y, m) < b and 0 < λ0  1 if a(y, m)
b.  j s Furthermore, ∞ s=1 (λs − b ) · v(cs ) = 0, and finally, since 1 c is nondecreasing we may as above construct a c¯ such that (λs − bs )[v(cs ) − v(c)] ¯
0 for s
1. Hence, lim sup

T


T →∞ s=1

  j  bs · v(cs ) − v cs

= lim sup

T
 s  j  b · v(cs ) − λs · v cs

T →∞ s=1

T
  j   λ0 · v c0 − v(c0 ) + lim sup [bs · v(cs ) − λs · v(cs )] T →∞ s=1

98

GEIR B. ASHEIM

by (7) with strict inequality if (0 y, 0 m, 0 k) = (0 y j , 0 m j , 0 k j ) T
  j  (bs − λs )[v(cs ) − v(c)] ¯ = λ0 · v c0 − v(c0 ) + lim sup T →∞ s=1

  j  ¯  λ0 · v c0 − v(c0 ) by the construction of c. Proof of Proposition 2. (Necessity) Suppose there exists an efficient (y, m)allocation 0 c j with positive and nondecreasing consumption from (y, m)  0 at j time 0. Then 0 c = (c, c, . . .), where c = c0 > 0, is a stationary (y, m)-allocation with positive consumption. (Sufficiency) Under (A.1)–(A.4) and (V.1), there exists from any (y, m)  0 a nondecreasing (y, m)-allocation 0 c j maximizing u(0 c) over the class of nondecreasing (y, m)-allocations. (It is well known that u is continuous on the class of (y, m)-allocations in the topology of pointwise convergence. Moreover, the class of nondecreasing (y, m)-allocations is nonempty and compact in the same topology.) j j If ct = c0 for t
0, then (E) and Proposition 1 establish that 0 c j is efficient and j furthermore, ct > 0 for t
0. Therefore, suppose 0 c j is inefficient with T
1 j j being the first time at which consumption is growing (i.e. ct = c0 for 0  t < T and j j cT > c0 ). By Lemma 1 in Dasgupta and Mitra (1983), and the fact that Fk
1, there j j j is a nondecreasing (y, m)-allocation 0 c with cT
ct > ct for 0  t < T and ct = ct j for t
T . This contradicts that 0 c maximizes u(0 c) over the class of nondecreasing (y, m)-allocations, and establishes that 0 c j is efficient. If 0 c j is nondecreasing but j not stationary, then c0 > 0 by (V.2). (Uniqueness) Lemma 4(a). Define J−s by J−s = {(y, m)  0 : s is the first time such that j

j

b(ys , m s )
b, where (0 y j , 0 m j , 0 k j ) = j0 (y, m)}. By Proposition 2, J−s is well defined for s
0 and (J0 , J−1 , . . .) is a partition of +t R2++ . Define J−s , t  s, by +t = {(y, m) : there is a (y  , m  ) ∈ J−s such that J−s

(0 y j , 0 m j , 0 k j ) = j0 (y  , m  ) satisfies yt = y and m t = m}.  +t−1 +t R2+ by ⊂ J−s+1 ⊂ · · · ⊂ J−s+t . Define for t
1, Pt : ∞ s−t J−s → 2 j

+t Hence, J−s

j

Pt (y, m) = {(y  , m  ) : (0 y j , 0 m j , 0 k j ) = j0 (y  , m  ) satisfies yt = y and m t = m}. j

j

Proof of Theorem 2. (Following the just programme is a subgame-perfect equilibj j j rium satisfying (C).) Define µ j : R2++ → R3+ by µ j (y, m) = (k0 , y1 , m 1 ), where (0 y j , 0 m j , 0 k j ) = j0 (y, m). Let the strategy sequence 0 σ j be defined as follows:

99

RAWLSIAN INTERGENERATIONAL JUSTICE j

j

For t
0 and h t ∈ Ht , σt (h t ) = µ j (y h t , m h t ) if (y h t , m h t )  0, σt (h t ) is arbitrary otherwise. σ

Case 1. y h t = 0 or m h t = 0. Then, w[t ct (h t )] = v(0)/(1 − b) for any t σ , and j σt (h t ) is trivially a best reply to t+1 σ j . Case 2. (y h t , m h t )  0. Let σt be any alternative strategy for generation t, and write t σ = (σt , t+1 σ j ). We can assume σt (h t )  0, because otherwise, σ σ w[t ct (h t )] = v(0)/(1 − b). For σt (h t )  0, t+1 ct (h t ) is a nondecreasing allocation. By Lemma 4(a), t σ j and h t generate the just allocation from (y h t , m h t ) at time σ t, denoted t c j . Moreover, denote t ct (h t ) by t c. Then, since t c j is nondecreasing, j

w(t c) − w(t c j )  u(t c) − u(t c j ) = v(ct ) − v(ct ) +

∞


  j  bs−t · v(cs ) − v cs

s=t+1

  j   (1 − λt ) v(ct ) − v ct

(15)

by Lemma 4(c), where 0 < λt  1. j Case 2a. a(y h t , m h t ) < b. By Lemma 4(c), λt = 1, and σt (h t ) is a best reply to j t+1 σ . Case 2b. a(y h t , m h t )
b. By Lemma 4(b), t c j is stationary and w(t c) − w(t c j )  u(t+1 c) − u(t+1 c j ) =

∞


  j  bs−t−1 · v(cs ) − v cs

s=t+1

  j   λt · v ct − v(ct ) /b

(16)

by Lemma 4(c), where 0 < λt  1. Combining (15) and (16) implies that w(t c) − j w(t c j )  0. Hence, σt (h t ) is a best reply to t+1 σ j . Combining Cases 1 and 2 establishes 0 σ j as a subgame perfect equilibrium. This equilibrium satisfies (C) because (a) t σ j and h t generate jt (y h t , m h t ) for (y h t , m h t )  0, and (b) jt (y, m) = et (y, m) for (y, m)  0 such that b(y, m)
b. (Any subgame-perfect equilibrium satisfying (C) generates the just programme for any (y, m)  0.) If 0 σ is an equilibrium satisfying (C), and h t satisfies (y h t , m h t ) ∈ J0 , then 0 σ and h t generate jt (y h t , m h t ) with associated allocation t c j . Hence, σ

σ

[u(t ct (h t )) =] w(t ct (h t )) = w(t c j )[= u(t c j )], since any just allocation is nondecreasing. Assume that the following inequality holds if 0 σ is an equilibrium satisfying (C), and h t satisfies (y h t , m h t ) ∈ J−s : σ

σ

[u(t ct (h t ))
] w(t ct (h t ))
w(t c j )[= u(t c j )],

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GEIR B. ASHEIM

where t c j is the just allocation from (y h t , m h t ) at time t. Then, for any h t−1 satisfying (y h t−1 , m h t−1 ) ∈ J−s−1 , generation t − 1 can secure σ

σ

[u(t−1 ct−1 (h t−1 ))
] w(t−1 ct−1 (h t−1 ))
w(t−1 c j )[= u(t−1 c j )] j

j

j

by choosing σt−1 (h t−1 ) = (kt−1 , yt , m t ), where (t−1 y j , t−1 m j , t−1 k j ) = jt−1 (y h t−1 , m h t−1 ) with associated allocation t−1 c j . (Use Lemma 4(a).) By backwards induction, any subgame-perfect equilibrium satisfying (C) generates a social welfare as high as the just programme. Now, assume the just programme is followed from some (y, m) ∈ J−s at time t. If +1 (y, m) ∈ J−s \J−s−1 , then there is no equilibrium predecessor in R2++ \J0 since no such programme would give a social welfare at time t − 1 as high as the just programme (using the first part of this proof and Lemma 4(c)). By the same argument, +1 if (y, m) ∈ J−s−1 , then an equilibrium predecessor is an element of P1 (y, m). Assume h 0 = (y, m)  0 and let 0 σ be a subgame-perfect equilibrium satisfying  (C). Then either (y, m) ∈ J0 , and we are finished, or (y, m) ∈ ∞ J . Then there s=1 −s  σ σ σ is a first T such that yT0 , m 0T ∈ J0 (because otherwise, w[0 c0 (h 0 )] = v(0)/(1 − b)). But, as established in the previous paragraph, ifσ (y, m) is a T th equilibrium σ σ σ predecessor of (yT0 , m 0T ), then (y, m) ∈ PT (yT0 , m 0T ).

Acknowledgments: I am especially grateful for a substantial contribution by Debraj Ray (including the proof of Theorem 1). This research has benefited from my visit to Stanford University in the academic year 1985–1986 and was supported by the Norwegian Research Council for Science and the Humanities. NOTES 1 For any two n-vectors, α and β, α
β means α
β for i = 1, . . . , n; α > β means α
β but α  = i i β; α  β means αi > βi for i = 1, . . . , n. 2 For any twice differentiable function f (α, β), where α and β are scalars, write f = ∂ f /∂α, f = α β ∂ f /∂β, and f αβ = ∂ f 2 /∂α∂β.

REFERENCES Arrow, K.J. (1973), Rawls’ principle of just saving, Swedish Journal of Economics 75, 323–335 Asheim, G.B. (1987), Rawlsian Intergenerational Justice as a Markov-Perfect Equilibrium in a Resource Technology (Discussion Paper 0686, revised, Norwegian School of Economics and Business Administration) Bernheim, B.D. and Ray, D. (1983), Altruistic Growth Economics: I. Existence of Bequest Equilibria (Technical Report No. 419, Institute for Mathematical Studies in the Social Sciences, Stanford University) Bernheim, B.D. and Ray, D. (1987), Growth with altruism, Review of Economic Studies 54, 227–243 Calvo, G. (1978), Some notes on time inconsistency and Rawls’ maximin criterion, Review of Economic Studies 45, 97–102

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Cass, D. and Mitra, T. (1979), Persistence of Economic Growth Despite Exhaustion of Natural Resources (Working Paper 79–27, Center for Analytic Research in Economic and the Social Sciences, University of Pennsylvania) Dasgupta, P. (1974a), Some problems arising from Professor Rawls’ conception of distributive justice, Theory and Decision 4, 325–344 Dasgupta, P. (1974b), On some alternative criteria for justice between generations, Journal of Public Economics 3, 405–423 Dasgupta, P. and Heal, G. (1974), The optimal depletion of resources, Review of Economic Studies, Symposium 3–28 Dasgupta, S. and Mitra, T. (1983), Intergenerational equity and efficient allocation of exhaustible resources, International Economic Review 24, 133–153 Farrell, J. and Maskin, E. (1987), Renegotiation in Repeated Games (Working Paper 8759, Department of Economics, University of California, Berkeley) Goldman, S. M. (1980), Consistent plans, Review of Economic Studies 47, 533–537 Greenberg, J. (1990), The Theory of Social Situations. Cambridge University Press, Cambridge, UK Kohlberg, E. (1976), A model of economic growth with altruism between generations, Journal of Economic Theory 13, 1–13 Lane, J. and Leininger, W. (1984), Differentiable nash equilibria in altruistic economies, Zeitschrift für Nationalökonomie 44, 329–347 Lane, J. and Mitra, T. (1981), On nash equilibrium programs of capital accumulation under altruistic preferences, International Economic Review 22, 309–331 Leininger, W. (1985), Rawls’ maximin criterion and time-consistency: Further results, Review of Economic Studies 52, 505–513 Leininger, W. (1986), The existence of perfect equilibria in a model of growth with altruism between generations, Review of Economic Studies 53, 349–367 Mitra, T. (1978), Efficient growth with exhaustible resources in a neoclassical model, Journal of Economic Theory 17, 114–129 Pollak, R.A. (1968), Consistent planning, Review of Economic Studies 35, 201–208 Rawls, J. (1971), A Theory of Justice. Harvard University Press, Cambridge, MA Ray, D. (1987), Nonpaternalistic intergenerational altruism, Journal of Economic Theory 41, 112–132 Rodriguez, A. (1981), Rawls’ maximin criterion and time consistency: A generalization, Review of Economic Studies 48, 599–605 Selten, R. (1975), Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory 4, 25–55 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies, Symposium 29–45 Stiglitz, J.E. (1974), Growth with exhaustible natural resources: Efficient and optimal growth paths, Review of Economic Studies, Symposium 123–137 Strotz, R.H. (1955–1956), Myopia and inconsistency in dynamic utility maximization, Review of Economic Studies 23, 165–180

CHAPTER 7 UNJUST INTERGENERATIONAL ALLOCATIONS

Abstract. An intergenerational allocation is defined to be unjust if there is a feasible allocation with more total consumption and less relative inequality. Unjust allocations are characterized in technologies satisfying certain regularity conditions. After ruling out unjust allocations, the consequences of letting generations choose according to a standard form of altruistic preferences are explored into particular classes of technologies. A connection between excluding unjust allocations and maximizing the welfare of the worst off generation is established in these technologies.

1. INTRODUCTION It has been the purpose of several writers (see, e.g., Page, 1977, for verbal arguments and Ferejohn and Page, 1978, for an axiomatic analysis) to point out that the utilitarian criterion with positive discounting may not be an appropriate criterion for intergenerational justice. A particularly disturbing outcome occurs in natural resource models where the criterion for any positive discount rate may force consumption to eventually approach zero even if unbounded consumption growth is feasible (see Dasgupta and Heal, 1974, 1979). Ferejohn and Page (1978, p. 274) write: “Our result suggests that the research for a “fair” rate of discount is a vain one. Instead of searching for the “right” number. “the” social rate of discount, we must look to broader principles of social choice to incorporate ideas of intertemporal equity. Once found, these principles might be used as side conditions in a discounting procedure to rule out gross inequities that can arise with discounting, even with a “low” discount rate.”

The present paper follows this program by employing a quasiordering attributed to Sen (1973) by Blackorby and Donaldson (1977) to exclude allocations of consumption that are not desirable candidates for a social choice. We call such allocations unjust. Loosely speaking, an allocation is unjust if there exists another feasible allocation with more total consumption and less relative inequality. Here we define this quasiordering for infinite consumption sequences and demonstrate that in productive technologies (implying that waiting is productive) only efficient and nondecreasing allocations remain after ruling out allocations that are unjust.

Originally published in Journal of Economic Theory 54 (1991), 350–371. Reproduced with permission from Elsevier.

103 Asheim, Justifying, Characterizing and Indicating Sustainability, 103–122 c 2007 Springer


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GEIR B. ASHEIM

Generations are assumed to choose according to a simple recursive form of nonpaternalistic altruistic preferences, where the welfare of each generation is an additively separable function of its own utility and the welfare of the next generation. This corresponds to the traditional utilitarian criterion with positive discounting. However, generations are here assumed to be required to subscribe to an overriding ethical principle, deeming unjust allocations as socially unacceptable. The implications of combining the exclusion of unjust allocations with this standard form of altruism is explored in two particular classes of technologies, viz. the usual one-sector technology as well as a resource technology in which the unrestricted use of the utilitarian criterion with discounting leads to undesirable outcomes. Both technologies are shown to be productive, and consequently, the selected allocations are efficient and nondecreasing. Furthermore, in both cases the optimal allocations correspond to outcomes that would arise if generations as an ethical principle instead of excluding unjust allocations had maximized the welfare of the worst off generation. This approach that conceptions of intergenerational justice should be evaluated by their implications in specific economic environments is in principle supported by Koopmans (1967), Koopmans (1977), and Dasgupta and Heal (1979, pp. 308– 311) as well as Rawls (1971, p. 20). It is argued in this paper that combining the exclusion of unjust allocations with altruism yields desirable implications in the two chosen classes of technologies. In particular, a trade-off exists between present and future consumption so that some degree of economic development is allowed without leading to any gross inequalities. A dilemma posed by Epstein (1986b) (that an economy has to choose between development and equity; it cannot have both) is thereby apparently resolved. Moreover, in the two classes of technologies considered, we obtain allocations in congruence with a view expressed by Dasgupta and Heal (1979, p. 311), viz. that trading off present consumption for future consumption is more appropriate for poorer societies, while equality considerations should dominate for richer ones. One may argue that the above mentioned quasiordering is uncontroversial only if each generation is egoistic in the sense that its welfare depends solely on its own consumption. Here, in contrast, each generation is altruistic: its welfare depends in part on the welfare of the next generation. However, there is an argument to be made in favor of distinguishing the conception of justice applied in a society from the forces that are instrumental in attaining it. Hence, the present paper may be seen to discuss whether altruism as a motivating force is able to implement a weak conception of justice (by not leading to unjust allocations) in the classes of technologies considered. This resembles the distinction, made by Rawls (1988), between a political conception of justice (“the right”) and a religious, philosophical, or moral doctrine (“the good”), where the right is assumed to set the limits within which the good may operate. Fitting this distinction to the present analysis, each generation’s consumption is to be interpreted as an indicator of its objective well-being. The adopted conception of justice (the right) is concerned with the attainment of an equitable distribution of such well-being and draws the limit by excluding allocations that are unjust.1

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Each generation’s altruistic welfare, on the other hand, is to be interpreted as a representation of its subjective preferences in which the doctrine that each generation should care about the welfare of its immediate successors (the good) has been internalized.2 Still, the above mentioned correspondence between excluding unjust allocations and maximizing minimal welfare means that this particular conception of justice can be reformulated as a restriction on the distribution of altruistic welfare, provided attention is confined to the kind of altruism and technologies considered. In summary: the original contribution of this paper is to apply a weak conception of justice as an ethical restriction in problems of intergenerational distribution, and to show that it combined with altruism leads to equitable outcomes in two important classes of technologies. The paper is organized as follows: The quasiordering defining unjust allocations is discussed in Sect. 2. The consequences of excluding unjust allocations in productive economies are explored in Sect. 3. After introducing altruistic preferences (Sect. 4) these results are then applied to a one-sector technology (Sect. 5) and a resource technology (Sect. 6). 2. THE QUASIORDERING Consider a constant population economy where each generation lives for one period. Let cs be a nonnegative scalar denoting the consumption of generation s. Write s c = (cs , cs+1 , . . . ) and correspondingly for other sequences. Refer to s c as an allocation at time s, let s ct denote a truncated allocation (i.e.,
s ct = (cs , . . . , ct )), and let s µt represent the mean consumption of s ct (i.e., s µt = tσ =s cσ /(t − s + 1)). The quasiordering we introduce in Definition 3 ranks one allocation as high as another it compares favorably both w.r.t. total consumption and w.r.t. relative inequality. Hence, we need a quasiordering that ranks allocations according to total consumption as well as one that ranks according to relative inequality. For total consumption the comparison relies on von Weizsäcker’s (1965) overtaking criterion. Definition 1: s c C s c (s c catches up with s c in finite time) if there is a t˜ such that for all t ≥ t˜, s µt ≥ s µt : For relative inequality the comparison relies on weak Lorenz-domination (see below). Definition 2: For s c , s c > 0,3 s c E s c (s c is as egalitarian as s c ) if there is a t˜ such that for all t ≥ t˜ there is a bistochastic4 (t − s + 1) × (t − s + 1) matrix At such that s ct /s µt = At ·s ct /s µt . In order to interpret Definition 2, let s c˜ t denote the permutation of s ct ordered according to increasing size (i.e., c˜i ≤ c˜i , i = s, . . . , t − 1, where the index i does not refer to time). Also, write s c E s c when the quasiordering E ranks s c strictly above s c (i.e., s c E s c iff s c E s c , but not s c E s c ) and correspondingly for other quasiorderings. We can now give three equivalent formulations of Definition 2.

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GEIR B. ASHEIM

Lemma 1: For s c , s c > 0, s c E s c is equivalent to any of the following three conditions: There is a t˜ such that for all t ≥ t˜, (i) (c˜s + · · · + c˜j )/s µt ≥ (c˜s + · · · + c˜j )/s µt for all j ∈ {s, . . . , t}; (ii)

(iii)

˜ t /s µt sc

can be obtained from s c˜ t /s µt by a finite sequence of transformations of the form


t

σ =s

xi+1 = xi + e ≤ x j

j >i,

x +1 = x j + e ≥ xi j

e > 0 ,

xn+1 = xn

if n = i, j ;

u(cσ /s µt ) ≥


t

σ =s

u(cσ /s µt ) for any concave function u : R+ → R.

Moreover, the above equivalence is valid for s ct E s ct if, in addition, there are infinitely many t such that (i) holds with strict inequality for at least one j, the sequence of transformations in (ii) is nonempty, and (iii) holds with strict inequality for any strictly concave function u. Proof. Lemma 2, Dasgupta et al. (1973). By Lemma 1 (i), Definition 2 entails that for large t, the Lorenz-curve associated with    s ct lies nowhere outside the corresponding curve for s ct ; i.e., s ct weakly Lorenz   dominates s ct . By Lemma 1 (ii), this is equivalent to s ct /s µt being obtainable from   s ct /s µt by a finite number of pairwise transfers (from richer to poorer generations). We are now able to define the quasiordering which will be our main concern. Definition 3: For s c , s c > 0, s c J s c (s c is as just as s c ) iff s c C s c and   s c E s c . s c J 0 for all s c ≥ 0; s c J 0 if s c > 0. Both C and E as well as J are easily seen to be reflexive and transitive; thus, they are quasiorderings. Some results are immediate. Lemma 2: (i) Form s c from s c > 0 by multiplying s c by a scalar κ > 1 (s c = κ ·s c ). Then s c J s c . (ii) Form s c from s c by transferring e > 0 from generation τ to generation t, where cτ − ct > e. Then s c J s c . (iii) Form s c from s c by interchanging the consumption of a finite number of generations. Then s c J s c and s c J s c Proof. (i) s c C s c and s c E s c . (ii) s c C s c and s c E s c by Lemma 1(ii).

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(iii) By Lemma 1 (i) since there is a t˜ such that for all t > t˜ s c˜ t = s c˜ t . Property (ii) is often called the (strong) principle of transfers and was originally formulated by Dalton (1920). Birchenhall and Grout (1979) pointed out that for the principle of transfers to be valid in the presence of an infinite sequence of generations, it is necessary to define the quasiordering by a procedure inspired by von Weizsäcker’s (1965) overtaking criterion, as we have done through Definitions 1–3. Property (iii) shows that the quasiordering J is equitable in the sense of Svensson (1980) and Epstein (1986b). Let us further explore the welfare implications of the quasiordering J by introducing two important quasiorderings of infinite consumption sequences. Definition 4 (Lexicographic maximin): s c L s c iff there is a t˜ such that for all t ≥ t˜, either there is a j ∈ {s, . . . , t} such that c˜j > c˜j and c˜i = c˜i for all s ≤ i < j, or s c˜ t = s c˜ t .   Definition 5 (Utilitarianism): s c U s c iff there is a t˜ such that for all t ≥ t˜,
t
t   σ =s u(cσ ) ≥ σ =s u(cσ ), where u : R+ → R is an increasing and strictly concave function.

In order to analyze the relation between J and these quasiorderings, introduce the notion of a subrelation. Definition 6: Consider two quasiorderings, R1 and R2 . R1 is a subrelation of R2 iff (s c R1 s c ⇒ s c R2 s c ) and (s c R 1 s c ⇒ s c R 2 s c ). Proposition 1: The quasiordering J is a subrelation of each of the quasiorderings L and U (for any increasing and strictly concave function u). Proof. Case 1: s c and s c are both semipositive. Assume s c J s c . Then, for all t exceeding some t˜, there is a s ctt and a bistochastic matrix At such that  s µt

≥ s µtt = s µt

and

  s ct /s µt

= s ctt /s µtt = At · s ct /s µt .

By Lemma 1 (i) [Lemma 1 (iii)], s c L s c [s c U s c ]. Assume in addition that    t 5 s c J s c . Then, either s µt > s µt for infinitely many t, or At is not a permutation     matrix for infinitely many t. By Lemma 1 (i) [Lemma 1 (iii)], s c L s c [s c U s c ]. Case 2. s c = 0 so that s c J s c , with s c J s c only if s c > 0. That s c L 0 [s c U 0] for all s c ≥ 0, with s c L 0 [s c U 0] if s c > 0, follows directly from Definition 4 [Definition 5]. Proposition 1 states that if J ranks two allocations, then L and U (for any increasing and strictly concave function u) will rank the allocations accordingly. Proposition 1 therefore implies that J is fairly uncontroversial. The quasiordering J is clearly not complete. In particular, it does not rank two allocations if the one has less relative inequality, but does not catch up with the other. In this sense, no trade-off between relative inequality and total consumption is

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allowed. Hence, the ordering cannot be used to select an optimal allocation relative to a feasibility constraint; i.e., we cannot find a s c such that s c J s c for all feasible allocations s c. However, we can determine a set of quasioptima, each element of which has the property that there is no feasible allocation ranked strictly higher by J . Given a set of feasible allocations, F, and an arbitrary quasiordering, Q, such quasioptima will be referred to as Q-optima. Formally, Definition 7: s c ∈ F is a Q-optimum relative to F if there is no s c ∈ F such that  sc Q sc . The notion of a quasioptimum may be used to exclude undesirable allocations. Definition 8: s c ∈ F is unjust relative to F if s c is not a J -optimum relative to F. Proposition 1 implies the following corollary. Corollary 1: If s c is an L-optimum, or a U -optimum (for some increasing and strictly concave u), then s c is a J -optimum. This corollary entails that if egoistic generations were to choose the distribution of consumption in ignorance of their own position (i.e., before knowing in which sequence they would live, any sequence being equally probable), then generations being risk-averse with respect to the level of their own consumption would choose a J -optimal allocation – no matter their degree of risk-averseness – provided the von Neumann-Morgenstern axioms are satisfied (see Vickrey, 1945, p. 329; Harsanyi, 1953; 1955). In Sect. 3, we completely characterize the set of allocations that are not unjust in a class of technologies satisfying certain regularity conditions.

3. PRODUCTIVE TECHNOLOGIES A cake-eating technology is defined as follows: Let F 0 (ys ) denote the set of feasible allocations at time s when the size of the cake
at time s is given by a nonnegative scalar ys . We have that F 0 (ys ) = {s c ≥ 0 : ∞ σ =s cσ ≤ ys }. Proposition 2: In a cake-eating technology, there exists a J -optimum relative to F 0 (ys ) iff ys = 0. Proof. (Sufficiency.) Trivial since F 0 (0) = {0}. (Necessity.) We have to show that all feasible allocations are unjust if ys > 0. First, note that 0 is unjust by Definition 3. If F 0 (ys ) s c > 0, there exist τ and t such that cτ > ct . Form s c ∈ F 0 (ys ) by cσ = (1/2)(cτ + ct ) for σ = τ , t, and cσ = cσ otherwise. By Lemma 2 (ii), s c J s c , which shows that s c is unjust. A cake-eating technology is unproductive: The cake cannot be invested yielding positive net returns. Proposition 2 shows that in such a technology, available consumption cannot be allocated to an infinite number of generations in a just manner (except for the trivial case where there is no consumption available for any generation). We

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109

therefore turn to a class of technologies which – due to their productiveness – have nontrivial J -optima. Let the vector ys ∈ Rn+ denote the n-dimensional state of the economy at time s; n ≥ 1. Interpret ys as the vector of stocks available at the beginning or period s. In the example of Sect. 5, the state is the one-dimensional output produced in the previous period; in the example of Sect. 6, the state is two-dimensional, consisting of output as well as a resource stock. The set of feasible allocations at time s, denoted Fs (ys ), is determined by the state ys at time s and a correspondence Fs : Rn+ → R∞ + . Let a sequence of correspondences 0 F = (F0 , F1 , . . . ) be referred to as a technology6 if for any feasible allocation s c ∈ Fs (ys ) at time s there exists a corresponding sequence of states s y ∈ Rn×∞ , with ys = ys , such that the following holds at each +   t > s: (a) t c ∈ Ft (yt ) and (b) t c ∈ Ft (yt ) ⇒ (s ct−1 , t c ) ∈ Fs (ys ). Consider the following conditions. Condition 1: ( Costless storage; by inefficiency, costless augmentation of initial consumption). For any s c ∈ Fs (ys
), there is a δ ≥ 0, strictly positive iff s c is 7 inefficient, such that (s c ≥ 0 and tσ =s (cσ − cσ ) ≤ δ for all t ≥ s) implies s c ∈ Fs (ys ). Condition 2: ( Costly acceleration of consumption). Let s c ∈ Fs (ys ) be efficient, with
corresponding sequence of states s y  0. Then there is no s c ∈ Fs (ys ) such that tσ =s (cσ − cσ ) ≥ 0 for all t ≥ s, with strict inequality for at least one t. Condition 3: ( Existence of an efficient and stationary8 allocation). If ys ∈ Rn++ , then there is an efficient, positive, and
stationary allocation s c ∈ Fs (ys ). If ys ∈ Rn+ \Rn++ , then s c ∈ Fs (ys ) implies that tσ =s cσ is finite. In order to explain Conditions 1 and 2, form s c from s c ∈ Fs (ys ) by transferring e > 0 from generation τ˜ to generation τ . First, assume τ˜ < τ so
that consumption is stored. If Fs satisfies Condition 1, then s c ∈ Fs (ys ) since tσ =s (cσ − cσ ) =
−e for t ∈ {τ˜ , . . . , τ − 1} and tσ =s (cσ − cσ ) = 0 otherwise. Hence, Condition 1 entails that consumption can be postponed without cost. Next, assume τ < τ˜ so that consumption is accelerated. If Fs satisfies Condition 2 and s c is an efficient allocation with corresponding sequence of states
s y  0, then s c ∈ / Fs (ys )
since tσ =s (cσ − cσ ) = e for t ∈ {τ, . . . , τ˜ − 1} and tσ =s (cσ − cσ ) = 0 otherwise. Hence, Condition 2 entails that the transfer of consumption from a later to an earlier generation is costly in the sense that the later generation has to give up more than the earlier one receives. A technology 0 F satisfying Conditions l–3 for any s ≥ 0 will be referred to as a productive technology. A cake-eating technology does not satisfy Conditions 2 and 3. For illustrative purposes, consider technologies satisfying only two out of the three conditions above. In order to facilitate these examples, introduce the concept of a linear technology 0 F λ defined by an given positive price sequence 0 p.
exogenously  ≤ p y }, where y is a nonnegative We have that Fsλ (ys ) = {s c ≥ 0 : ∞ p c s s s σ =s σ σ   ) scalar and where s y is defined recursively by ys = ys and pt yt = pt−1 (yt−1 − ct−1

110

GEIR B. ASHEIM

Figure 7.1.

for t > S. This technology is productive iff pt > pt+1 for all t ≥ 0 and pσ < ∞. Example 1: A linear technology 0 F 1 with pt > pt+1 for all t ≥ 0 and diverging, satisfies only Conditions 1 and 2.


∞

σ =0


∞

σ =0

pσ

Example 2: A linear technology 0 F 2 with p0 = p1 , pt > pt+1 for all t ≥ 1 and
∞ σ =0 pσ ≤ ∞, satisfies only Conditions 1 and 3. Example 3: Let 1 F 3 be a linear technology with pt = apt+1 for all t ≥ 1; a > 1. Let T03 (y0 ) denote the following transformation set (illustrated in Fig. 7.1):      T03 (y0 ) = (k, y) : 0 ≤ y ≤ a · k for k ∈ 0, ya0 ∪ yα0 , y0 and      0 ≤ y ≤ y0 + α · k − ya0 for k ∈ ya0 , yα0 ; k ≥ 0, a > α > 1 .

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Define F03 (y0 ) by F03 (y0 ) =



c ≥ 0 : ∃(k0 , y1 ) s.t. c0 = y0 − k0 , (k0 , y1 ) ∈ T03 (y0 )  and 1 c ∈ F13 (y1 ) 0

The resulting technology 0 F 3 satisfies Conditions 2 and 3, but not Condition 1 since, given an inefficient allocation with c0 = y0 − y0 /α, one need not be able to augment the consumption of generation 0 without reducing the consumption of a later generation. The following three propositions fully characterize the set of J -optima in productive technologies. Proposition 3: If s c ∈ Fs (ys ) is a J -optimum relative to Fs (ys ), and 0 F is productive, then s c is nondecreasing.9  Proof. Suppose ct > ct+1 for some t. By Condition 1, there is a s c ∈ Fs (ys ) formed  ) for σ = t, t + 1, and c = c otherwise. By  from s c by cσ = (1/2)(ct + ct+1 σ σ  Lemma 2(ii), s c J s c , which contradicts that s c is a J -optimum.

By the proof of Proposition 3, if s c were not nondecreasing, a carefully chosen postponement of consumption would decrease relative inequality without reducing total consumption, thereby producing a J -improvement. Hence, by Condition 1, in a productive technology a J -optimal allocation is nondecreasing. This result also implies Proposition 4, stating that in a productive technology with a positive initial state a J -optimal allocation is efficient. Because if it were not, by Condition 1, the consumption of the first least favored generation(s) could be augmented without cost, yielding less relative inequality and more total consumption. Proposition 4: Let ys ∈ Rn++ . If s c ∈ Fs (ys ) is a J -optimum relative to Fs (ys ), and 0 F is productive, then s c is efficient. Proof. Suppose s c is a J -optimum that is not efficient. By Proposition 3, s c is nondecreasing. By Condition 3 and Lemma 2 (i), s c is not stationary. Hence, s c   has a finite set {s, . . . , t} of least favored generations: cs = cs+1 = · · · = ct−1 =   ct < ct−1 . By Condition 1, for any ε ∈ (0, δ/(t − s + 1)), given some δ > 0, there is a s c ∈ Fs (ys ) formed from s c by cσ = cσ + ε for σ ∈ {s, . . . , t} and cσ = cσ  ) since otherwise. By Lemma 1 (ii), s c J s c (for any ε > 0 satisfying ct + ε < ct−1 for τ > t, s µτ > s µτ , and s cτ /s µτ can be obtained from s cτ /s µτ by a finite number of pairwise transfers. This contradicts that s c is a J -optimum. Conversely, if s c is efficient and nondecreasing, reducing relative inequality requires that consumption be transferred from a later to an earlier generation. However, by Condition 2, in a productive technology this can only be achieved by reducing total consumption. Hence, as stated in Proposition 5, in a productive technology with

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a positive initial state there exists no feasible J -improvement of an efficient and nondecreasing allocation. Proposition 5: Let ys ∈ Rn++ . If s c ∈ Fs (ys ) is efficient and nondecreasing, and  0 F is productive, then s c is a J -optimum relative to Fs (ys ). Proof. By Condition 3, there is a τ˜ and an ε > 0 such that cτ ≥ ε for all τ ≥ τ˜ , and thus, yσ ∈ Rn++ for all σ ≥ s. Hence, Condition 2 applies. Suppose s c ∈ Fs (ys )

j satisfies s c J s c . Form s c˜ t and s c˜ t for any t ≥ s. Then σj =c cσ ≥ i=c c˜i for all    j ∈ {s, . . . , t}, and s ct = s c˜ t since s c is nondecreasing. It therefore follows from
Definition 3 and Lemma 1 (i) that there is a t˜ such that t > t˜ implies σj =s (cσ − cσ ) ≥ 0 for all j ∈ {s, . . . , t}. By Condition 2, s c ∈ Fs (ys ) implies s c = s c , which contradicts s c J s c . We have proven that if the technology is productive and the initial state ys is positive, then the set of J -optima is identical to the set of efficient and nondecreasing allocations. Note that Condition 3 implies that this set is nonempty; it contains at least an efficient and stationary allocation. Example 1 above does not satisfy Condition 3. A proof similar to the one of Proposition 2 shows that there exists no J -optimum relative to F01 (y0 ) if y0 > 0. In the case of Example 2 (which does not satisfy Condition 2) a J -optimum relative to F02 (y0 ) requires that c1 = c2 ; i.e., not all efficient and nondecreasing allocations are J -optima. In the case of Example 3 (which does not satisfy Condition 1), there are inefficient J -optima. In particular, 0 c ∈ F03 (y0 ) with c0 = y0 − y0 /α and ct = c ∈ ([a − 1] · (y0 /α) · [1 − ((a − α))/a)2 ], [a − 1] · (y0 /α)) for all t ≥ 1 is inefficient, but still a J -optimum, since a reduction in relative inequality, involving increased consumption at time 0, can be achieved only by reducing total consumption. In productive linear technologies, the exclusion of unjust allocations is related to Epstein’s (1986a) Efficiency and Equity axioms: s c ∈ Fsλ (ys ) satisfies Efficiency iff
∞    p σ =s σ cσ = ps ys , and Equity iff for all τ , t ≥ s, pτ > pt ⇒ cτ ≤ ct . Clearly, when 0 F λ is productive, s c satisfies Efficiency and Equity iff s c is efficient and nondecreasing; i.e., iff s c is J -optimal. However, when 0 F λ is not productive, the Efficiency and Equity axioms are not sufficient for J -optimality because they admit s c with cτ > ct , for τ , t satisfying pτ = pt . Such s c are not J -optimal relative to Fsλ (ys ).10 For ys  0 in a productive technology, the continuation at time t (> s) of a J optimal allocation s c ∈ Fs (ys ), t c , is J -optimal relative to Fs (yt ), since yt  0 and t c is efficient and nondecreasing. However, by choosing an alternative J -optimal allocation t c ∈ Ft (yt ) at time t, generation t may destroy the J -optimality relative to Fs (ys ) of the resulting allocation (s ct−1 , t c ) ∈ Fs (ys ). This will occur when  ct−1 > ct . We will return to this problem of time-inconsistency. Having characterized the set of J -optima in productive technologies as the set of efficient and nondecreasing allocations, one question remains: Does the imposition of J -optimality lead to the selection of more desirable allocations? Such an evaluation of J -optimality as an ethical principle is provided by the next three sections

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where the consequences of combining J -optimality with a simple recursive form of altruistic preferences are explored in two classes of productive technologies. 4. ALTRUISTIC PREFERENCES Assume in the sequel that the subjective preferences of any generation s are altruistic and can be represented by an ancestor-insensitive and stationary altruistic welfare function w : R∞ + → R+ defined by: w(s c) ≡

∞ 

bσ −s · u(cσ ) = u(cs ) + b · w(s+1 c) ,

0 < b < 1,

σ =s

where u : R+ → R+ is a stationary one-period utility function satisfying (u.1)

u is continuous, strictly increasing, and concave; it is continuously differentiable at any c > 0.

(u.2) du/dc → ∞ as c ↓ 0. and where b is the utility discount factor. Note that the welfare function has both a paternalistic and a nonpaternalistic representation, the latter being of a simple recursive form (see, e.g., Ray, 1987). If each generation maximizes w constrained only by the technology, the result corresponds to the traditional utilitarian criterion with positive discounting. As mentioned in the introduction, this criterion prescribes ethically questionable intergenerational allocations, especially in resource technologies. This provides a motivation for requiring generations to subscribe to J -optimality as an overriding ethical principle, deeming unjust allocations socially unacceptable.11 Hence, if generation s inherits ys in the technology 0 F , it seeks to (W) maximize w(s c) over the class of J -optimal s c ∈ Fs (ys ). Let 0 c denote the choice of generation 0 with 0 y as the corresponding sequence of states. The allocation 0 c is said to be time-consistent relative to (W), if, for all s ≥ 0,  s c solves (W). If the present generation cannot dictate the future to follow its chosen allocation, time-consistency is essential. Otherwise, a game-theoretic approach is called for. In the two classes of productive technologies we consider in Sects. 5 and 6, productivity of waiting (defined as the marginal rate of technical transformation between consumption in one period and the next) is nonincreasing along any efficient and nondecreasing allocation. This ensures time-consistency relative to (W).12 Sen (1977, p. 1559) denotes by welfarism “[t]he general approach of making no use of any information about the social states other than that of personal welfares generated in them . . . .”13 Note that respecting the subjective preferences of each generation, only within the restricted class of J -optimal allocations, is not in accordance with welfarism: The ethical principle of imposing J -optimality is based directly on the intergenerational allocations of consumption and works by excluding allocations

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that are unjust. Hence, it does not reflect the subjective preferences of the generations involved. As an alternative, consider the case where each generation is required to subscribe to maximin as an ethical principle; i.e., given some bequest ys , generation s seeks to ( ) maximize π(s c) over all s c ∈ Fs (ys ), with π : R∞ + → R+ defined by π(s c) ≡ infσ ≥s w(σ c). Here welfarism holds since the ethical principle depends solely and in an invariant way on the subjective preferences as represented by w. A chosen allocation 0 c (with corresponding sequence of states 0 c ) is said to be time-consistent relative to ( ) if, for all s ≥ 0, s c solves ( ). Note that maximin combined with altruism allows for consumption growth (see Calvo, 1978; Asheim, 1988). It turns out that in the productive technologies considered in the two subsequent sections J -optimality combined with altruism (i.e., (W)) gives rise to the same allocations as maximin combined with altruism (i.e., ( )) provided that time-consistency relative to ( ) is imposed. 5. A ONE-SECTOR MODEL First, consider a technology where total output y is split between consumption c and capital k, the latter producing the total output available in the next period. Assume that the stationary production function g : R+ → R+ satisfies (g.1)

g is continuous, strictly increasing, and strictly concave; it is continuously differentiable at any k > 0,

(g.2)

g(0) = 0; dg/dk → ∞ as k ↓ 0, dg/dk ↓ 0 as k → ∞

A total output function f is defined by f (k) ≡ g(k) + k. Production possibilities are described by a stationary transformation set T µ of input–output pairs: T µ = {(k, y) : 0 ≤ y ≤ f (k); k ≥ 0} . The stationary technology F µ , describing the set of feasible allocations, is defined by F µ (ys ) = {s c ≥ 0 : ∃(s y, s k), ∀t ≥ s, ct = yt − kt and (kt , yt+1 ) ∈ T µ } . The pair (s y, s k) is called the associated program. Lemma 3: The technology F µ satisfies Conditions 1–3; i.e., it is productive. Proof. See the Appendix. Consider maximizing w(0 c) over the class of J -optimal 0 c ∈ F µ (y0 ). With y0 = 0, this problem is trivial since F µ (0) = {0}. Therefore turn to the case with y0 > 0. Define k∞ by b · d f (k∞ )/dk ≡ 1, y∞ by y∞ ≡ f (k∞ ), and c∞ by c∞ ≡ y∞ − k∞ . It is well known that for any y0 > 0, the modified Ramsey program (associated with the allocation maximizing w(0 c) over all 0 c ∈ F µ (y0 )) converges to (y∞ , k∞ ).

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Proposition 6: For any y0 > 0, there is an allocation 0 c ∈ F µ (y0 ) maximizing w(0 c) over the class of J -optimal 0 c ∈ F µ (y0 ). Moreover, 0 c is unique and time consistent relative to (W). For y0 ∈ (0, y∞ ), ct < ct+1 for all t ≥ 0, with ct ↑ c∞  as t → ∞. For y0 ≥ y∞ , ct = c0 ≡ y0 − k0 for all t ≥ 0 (where k0 is defined by y0 ≡ f (k0 )). Proof. Case 1: y0 ∈ (0, y∞ ). It is well known (see Beals and Koopmans, 1969) that there is an allocation 0 c ∈ F µ (y0 ) with associated program (0 y , 0 k ) such that for all s ≥ 0, s c uniquely maximizes w(s c) over all s c ∈ F µ (ys ). Furthermore,  ct < ct+1 and 0 < yt < y∞ for all t ≥ 0. By Proposition 5 and Lemma 3, s c is J optimal for all s ≥ 0, since s c is efficient and nondecreasing. Hence, for all s ≥ 0,  µ  s c uniquely maximizes w(s c) over the class of J -optimal s c ∈ F (ys ).  µ Case 2: y0 ≥ y∞ . Define the stationary allocation 0 c ∈ F (y0 ) by ct = c0 (∀t ≥ 0) with associated program (0 y , 0 k ) given by (yt , kt ) = (y0 , k0 ) (∀t ≥ 0). Write β≡

1 , d f (k0 )/dk

λt ≡

pt ≡ λt · du(c0 )/dc

1−β · β t−s , 1−b

and

(∀t ≥ s)

so that ∞ 

∞

λσ =

σ =s

 1 = bσ −s 1 − b σ =s

 pt+1 yt+1 − pt kt ≥ pt+1 y − pt k ,

∀t ≥ s ,

(1) ∀[k, y] ∈ T µ ,

(2)

 ]. Hence, for any s ≥ 0 and for any c ∈ with strict inequality if [k, y] = [kt , yt+1 s µ   F (ys ) with associated program (s y , s k ), t  σ =s

λσ · [u(cσ ) − u(cσ )] ≤ =

t  σ =s t  σ =s

pσ · [cσ − cσ ]

by (u.1) since cσ = c0 (∀σ ≥ s)

pσ · [(yσ − kσ ) − (yσ − kσ ]

≤ pt · (kt − kt ) − ps · (ys − ys )

by (2),

with strict inequality if (s y , s k ) = (s y , s k ). Since pt ↓ 0 as t → ∞, kt = k0 , and kt ≥ 0 (∀t ≥ 0), and ys = ys = y0 , it follows that s c is efficient: lim sup

t 

t→∞ σ =s

λσ · [u(cσ ) − u(cσ )] ≤ 0 ,

(3)

with strict inequality if (s y , s k ) = (s y , s k ). As for Case 1, s c is J -optimal. It remains to be shown that s c uniquely maximizes w(s c) over the class of J -optimal s c ∈ F µ (ys ). By Proposition 3, it suffices to show

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that s c uniquely maximizes w(s c) over the class of nondecreasing s c ∈ F µ (ys ). If  µ    s c ∈ F (ys ) is nondecreasing with associated program (s y , s k ), then lim sup

t 

t→∞ σ =s

= lim sup

t 

t→∞ σ =s

≤ lim sup

t 

t→∞ σ =s

= lim sup

t 

t→∞ σ =s

bσ −s · [u(cσ ) − u(cσ )] [bσ −s · u(cσ ) − λσ · u(cσ )]

by (1) since cσ = c0 (∀σ ≥ s)

[bσ −s · u(cσ ) − λσ · u(cσ )]

by (3) (< if (s y , s k ) = (s y , s k ))

(bσ −s − λσ ) · [u(cσ ) − u(c)] ¯

by (1) for any constant c¯

≤0 if c¯ is chosen such that cτ−1 ≤ c¯ ≤ cτ , and where τ satisfies bt−s − λt ≥ 0 for t < τ and bt−s − λt ≤ 0 for t ≥ τ .14 The proof of Proposition 6 shows that 0 c , denoting the allocation in F µ (y0 ) that uniquely maximizes w(0 c) over the class of J -optimal allocations in F µ (y0 ), follows a modified Ramsey program if this is nondecreasing; otherwise 0 c is efficient and stationary. It it in this latter case that the process of disallowing unjust allocations affects the selection made by the utilitarian criterion with positive discounting: Along a modified Ramsey program when y0 > y∞ , the current generation enjoys a binge at the expense of all future generations. Compared to the efficient stationary allocation, this leads to more relative inequality and less total consumption. It is of interest to note that substituting maximin combined with altruism (i.e., ( )) for J -optimality combined with altruism (i.e., (W)) is of no consequence in this particular technology: For any y0 > 0, 0 c of Proposition 6, in addition to maximizing w(0 c) over the class of J -optimal allocations in F µ (y0 ), also uniquely maximizes π(0 c) over all allocations in F µ (y0 ) (see Calvo, 1978). In Sect. 6 we show that this result carries over (subject to the imposition of time-consistency relative to ( )) to a resource technology in which the altruistic preferences of Sect. 4 implement ethically questionable intergenerational allocations when not combined with an ethical principle.

6. A RESOURCE TECHNOLOGY Following Dasgupta and Mitra (1983), consider a technology where capital k, resource extraction r , and labor z produce the total output y available in the next period. Assume that the stationary production function G : R3+ → R+ satisfies

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117

(G.1)

G(k, r, z) is continuous, nondecreasing, concave, and homogeneous of degree one; it is continuously twice differentiable at any (kr, z)  0.

(G.2)

G(0, r, z) = 0 = G(k, 0, z); (G k , G r , G z )  0 for (kr, z)  0.15

The labor force is assumed to be stationary (= 1). Given z = 1, let the ratio of the resource share in output to the labor share in output be bounded away from zero: (G.3)

˜ r˜ )  0, there is η˜ > 0 such that for all (k, r ) satisfying k ≥ k, ˜ Given any (k, 0 < r ≤ r˜ , we have [r G r (k, r, 1)]/G z (k, r, 1) ≥ η. ˜

A total output function F, defined by: F(k, r ) ≡ G(k, r, 1) + k, satisfies (F)

F(k, r ) is strictly concave; Fkr ≥ 0.

Let m denote the resource stock. Then production possibilities are described by a stationary transformation set T ν of input–output pairs: T ν = {[(k, m), (y, m +1 )] : 0 ≤ y ≤ F(k, r ); 0 ≤ r = m − m +1 ; (k, m +1 ) ≥ 0} . The stationary technology F ν , describing the set of feasible allocations, is defined by: F ν (ys , m s ) = {s c ≥ 0 : ∃(s y, s m, s k), ∀t ≥ s, ct = yt − kt and [(kt , m t ), (yt+1 , m t+1 )] ∈ T ν } . The triple (s y, s m, s k) is called the associated program. (G.l)–(G.3) are not sufficient to ensure the existence of a stationary allocation with positive consumption Solow (1974); Cass and Mitra (1991). Therefore assume (A)

 sc

∈ F ν (ys , m s ) with ct = cs > 0 (∀t ≥ s) exists if (ys , m s )  0.

Cass and Mitra (1991) give a necessary and sufficient condition for (A) to hold. Lemma 4: The technology F ν satisfies Conditions 1–3; i.e., it is productive. Proof. See the Appendix. Consider maximizing w(0 c) over the class of J -optimal 0 c ∈ F ν (y0 , m 0 ). With y0 = 0, this problem is trivial since F ν (0, m 0 ) = {0}. With m 0 = 0, we are faced with a cake-eating problem for which no J -optimum exists if y0 > 0 (see Proposition 2). Therefore turn to the case with (y0 , m 0 )  0. Proposition 7: For any (y0 , m 0 )  0, there is an allocation 0 c ∈ F ν (y0 , m 0 ) maximizing w(0 c) over the class of J -optimal 0 c ∈ F ν (y0 , m 0 ). Moreover, 0 c is unique and time-consistent relative to (W). The allocation 0 c is nondecreasing with an eventual stationary phase. For some (y0 , m 0 ), this eventual phase is preceded by an initial phase with increasing consumption.

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Proof. By Asheim (1988, Proposition 2 and Lemma 4(a)), there is an efficient and nondecreasing allocation 0 c ∈ F ν (y0 , m 0 ) with associated program (0 y , 0 m ,   0 k ) such that for all s ≥ 0, s c uniquely maximizes w(s c) over the class of nonν   decreasing s c ∈ F (ys , m s ). By Proposition 5 and Lemma 4, s c is J -optimal for all s > 0. By Proposition 3, it suffices to consider the class of nondecreasing ν    s c ∈ F (ys , m s ). The two phase structure of s c is shown in the proof of Lemma 4 of Asheim (1988). The allocation 0 c of Proposition 7 has the following desirable distributional properties (discussed in Asheim, 1988, p. 474): It allows for consumption (and welfare) growth in an economy that is highly productive due to a small capital stock and a large resource stock, while the eventual stationarity protects distant generations from the grave consequences of utility discounting when the productivity of waiting is low and diminishing. Not restricting the maximization of w(0 c) to J -optimal allocations would have forced consumption to eventually approach zero. Compared to a stationary allocation with positive consumption, this leads to an allocation with more relative inequality and less total consumption. Hence, such an allocation is unjust. Since the restriction to J -optimal allocations is not in accordance with welfarism,  0 c of Proposition 7 need not be Pareto-efficient in the sense of there being no alternative allocation in F ν (y0 , m 0 ) increasing the welfare w(s c) of some generation s without decreasing the welfare of any other generation. It can be shown (see Asheim, 1988, p. 475) that, for a class of initial states, 0 c of Proposition 7 is in fact Pareto-inefficient, being Pareto-dominated by an allocation which is not nondecreasing. This provides a motivation for considering maximin combined with altruism (i.e., ( )) as an alternative to J -optimality combined with altruism (i.e., (W)). Denote by  ν ν 0 c an allocation in F (y0 , m 0 ) maximizing π(0 c) over all allocations in F (y0 , m 0 ).  It turns out that 0 c is not nondecreasing and, hence, not time-consistent relative to ( ) for the same class of initial states as the one for which 0 c of Proposition 7 is Pareto-inefficient (Asheim, 1988, Theorem 1). Moreover, if generation 0 maximizes π(0 c) over the class of allocations in F ν (y0 , m 0 ) that future generations subscribing to ( ) actually will carry out, then 0 c of Proposition 7 is its unique optimal choice (Asheim, 1988, Theorem 2).16 In this sense, (W) and ( ) result in the same allocations, also in this technology. 7. CONCLUDING REMARKS In this paper, we have demonstrated that combining the exclusion of unjust intergenerational allocations of consumption (J -optimality) with a simple recursive form of altruistic preferences yields desirable outcomes in two important classes of technologies. In particular, while this form of altruism alone may cause gross inequities, and maximin combined with egoistic preferences may perpetuate poverty,17 the combination of J -optimality with altruism allows for development when the economy is highly productive, ensuring equality otherwise. It is noteworthy that this is achieved

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also in the resource technology considered, since this technology poses an awkward problem of intergenerational distribution when standard criteria are used. We have also shown, for the two classes of technologies, that J -optimality combined with altruism (i.e., (W)) give rise to the same allocations as maximin combined with altruism (i.e., ( )) given that time-consistency relative to ( ) is imposed. It is natural to ask whether this result carries over to more general classes of technologies. That (W) and ( ) are not equivalent is most easily demonstrated by considering a cake-eating technology: For a positive initial stock, no allocation is J -optimal while all allocations are maximin.18 It is my conjecture, though, that (W) and ( ) will lead to the same allocations in a more general class of productive technologies (which includes the one-sector technology and the resource technology as special cases) having the property that the productivity of waiting is nonincreasing along any efficient and nondecreasing allocation. APPENDIX Proof of Lemma 3. Condition 1. (a) Consider s c ∈ F µ (ys ) associated with
t   (s y , s k ). Let s c ≥ 0 satisfy σ =s (cσ − cσ ) ≤ 0 for all t ≥ s. Construct (s y, s k): kt = kt −

t  σ =s

(cσ − cσ ) [≥ kt ≥ 0]

and

yt = ct + kt

(∀t ≥ s) .

 − kt ≤ f (kt ) − kt ≤ Hence, s c ∈ F µ (ys ) since for all t ≥ s, 0 ≤ yt+1 − kt = yt+1 f (kt ) − kt . (b) If s c ∈ F µ (ys ) is inefficient, we proceed to show that there is a s c ∈ F µ (ys ) such that cs = cs + δ (δ > 0), cσ = cσ otherwise, and then repeat part (a), considering s c .   Condition 2. Let s c ∈ F µ (ys ) be efficient
t with (s y , s k )  0. Suppose s c ∈ µ F (ys ) associated with (s y, s k) satisfies σ =s (cσ − cσ ) ≥ 0 for all t ≥ s with strict inequality for at least one t. W.1.o.g. we may assume cs > cs :. Hence,  ks − ks = cs − cs > 0 since ys = ys . The efficiency of s c ∈ F µ (ys ) implies yt+1 =  f (kt ) for all t ≥ s. Thus,  − ys+1 ≥ f (ks ) − f (ks ) ≥ (cs − cs ) · d f (ks )/dk > cs − cs , ys+1

since g(k) = f (k) − k is strictly increasing and concave. Hence, there is a s c ∈  F µ (ys ) such that cs = cs , cs+1 = cs+1 + (cs − cs ) · d f (ks )/dk, cσ = cσ otherwise.  = c  By Condition 1, there is a s c ∈ F µ (ys ) such that cs = cs , cs+1 s+1 + (cs −
      cs ) · [d f (ks )/dk − 1], cσ = cσ otherwise, since cs = cs and tσ =s (cσ − cσ ) ≥
t     0 implies σ =s (cσ − cσ ) ≤ 0 for t ≥ s. Since (cs − cs ) · [d f (ks )/dk − 1] > 0,  ∈ F µ (y ) contradicts the efficiency of c . c s s s Condition 3. For any ys > 0, it is easy to verify that the stationary allocation s c ∈ F µ (ys ), satisfying for all t ≥ s, ct = cs ≡ ys − ks (with ks defined by ys ≡ f (ks )), is efficient (see proof of Proposition 6). If ys = 0, then F µ (ys ) = {0}.

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Proof of Lemma 4. Condition 1. (a) Consider s c ∈ F ν (ys , m s ) associated with (s y , s m , s k ). Define f t by f t (k) ≡ F(k, m t − m t+1 ). We can now repeat part l(a) of the proof of Lemma 3 since f t (k) − k is nondecreasing. (b) If s c ∈ F ν (ys , m s ) is inefficient, we can correspondingly repeat part I(b) of the proof of Lemma 3. Condition 2. Let s c ∈ F ν (ys , m s ) be efficient with (s y , s m , s k )  0. We need to show that m t > m t+1 for all t > s. Suppose m t−1 = m t for some t. Since efficiency implies m τ ↓ 0 as τ → ∞, we may w.l.o.g. assume that m t > m t+1 . Consider     s c associated with (s y, s m, s k) : m t = (1/2)(m t−1 + m t+1 ), yt = F(kt−1 , m t−1 −      m t ), kt satisfies F(kt , m t − m t+1 ) = yt+1 ; (yσ , m σ , kσ ) = (yσ , m σ , kσ ) for all σ = t. Hence, s c is inefficient since s c ∈ F ν (ys , m s ), ct = yt − kt > ct by (Gl) and (G2), and cσ = cσ for all σ = t. Thus, m t > m t+1 for all t ≥ s, and f t defined by f t (k) ≡ F(k, m t − m t+1 ) is concave and f t (k) − k strictly increasing. Now, repeat part 2 of the proof of Lemma 3. Condition 3. For (ys , m s )  0, see Theorem 1 of
Dasgupta and Mitra (1983). If ys = 0 or m s = 0, then s c ∈ F ν (ys , m s ) implies that ∞ σ =s cσ ≤ ys . Acknowledgments: This research was initiated during a visit to Stanford University, 1985–1986. An earlier version of the paper was presented at the European Public Choice Society and European Economic Association meetings in 1988. I thank an associate editor and a referee as well as Bjørn Sandvik. H.A.A. Verbon, and Bengt-Arne Wickström for helpful comments. Financial support by the Norwegian Research Council for Science and the Humanities is gratefully acknowledged. NOTES 1 The result that only efficient and nondecreasing allocations are not unjust in productive technologies

means that this conception of justice may be looked at as a normative basis for the present-day goal of sustainability WCED (1987). 2 Rawls (1971, p. 129) can be interpreted as supporting the view that altruism should not enter into the conception of justice: “There is no inconsistency, then, in supposing that once the veil of ignorance is removed, the parties find that they have ties of sentiment and affection, and want to advance the interests of others and to see their ends attained. But the postulate of mutual disinterest in the original position is made to insure that the principles of justice do not depend upon strong assumptions. Recall that the original position is meant to incorporate widely shared and yet weak conditions. A conception of justice should not presuppose, then, extensive ties of natural sentiment.” 3 An allocation c is nonnegative (≥ 0) if c > 0 for all t ≥ S, positive ( 0) if c > 0 for all t ≥ s, and s t t semipositive (> 0) if s c ≥ 0, with ct > 0 for some t. 4 A square matrix is bistochastic if all its entries are nonnegative and each of its rows and columns sums to one. 5 A permutation matrix is a bistochastic matrix with entries either 0 or 1. 6 If F is stationary (i.e., F = F for all t ≥ 0), then F is also referred to as a technology. t 0 7 An allocation c ∈ F (y ) is inefficient relative to F (y ) if there is an allocation c ∈ F (y ) such s s s s s s s s that s c > s c (i.e., s c ≥ s c with ct > ct for some t ≥ s). An allocation is efficient iff it is not inefficient. 8 An allocation c is stationary if c = c s t t+1 for all t ≥ s. 9 An allocation c is nondecreasing if c ≤ c s t t+1 for all t ≥ s.

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10 In particular, in the case of F 2 all efficient allocations c ∈ F 2 (y ) with c ≤ c , c ≤ c , and 0 0 0 2 1 2 0 0 ct ≤ ct+1 for all t ≥ 2 satisfy Efficiency and Equity; i.e., we need not have c0 = c1 . 11 Harsanyi (1955, p. 315) uses the term ethical preferences (here choosing J -optimal allocations) as

opposed to subjective preferences (here represented by the function w).

12 In a productive technology, J -optimality constrains generation 0 to choose c ∈ F (y ) with c ≥ 0 0 0 s  cs−1 for any s > 0. However, generation s does not face this constraint. It can be shown that generation     s will not choose s c ∈ Fs (ys ) with cs < cs−1 if the productivity of waiting at time s is nonincreasing.

See Asheim (1988, proof of Lemma 4) for a formal demonstration of this point.

13 It should be noted that Sen (1977, pp. 1559–1562) presents arguments against welfarism. 14 Such a τ exists by (1) since y ≥ y implies that k ≥ k and β ≥ b. ∞ ∞ 0 0 15 Write G ≡ ∂G/∂k, F , ≡ ∂ 2 F/∂k∂r , and so forth. k kr 16 In Asheim (1988), I assume that an allocation that is time-consistent relative to ( ) will be followed as soon as one exists. Then 0 c of Proposition 7 is the unique subgame-perfect equilibrium allocation in

an intergenerational game. 17 See Solow (1974) for this criticism. 18 For any feasible allocation c. we have π( c) = u(0)/(1 − b). Hence, they are all maximin. 0 0

REFERENCES Asheim, G.B. (1988), Rawlsian intergenerational justice as a Markov-perfect equilibrium in a resource technology, Review of Economic Studies 55, 469–483 (Chap. 6 of the present volume) Beals, R. and Koopmans, T.C. (1969), Maximizing stationary utility in a constant technology, SIAM Journal of Applied Mathematics 17, 1001–1015 Birchenhall, C.R. and Grout, P. (1979), On equal plans with an infinite horizon, Journal of Economic Theory 21, 249–264 Blackorby, C. and Donaldson, D. (1984), Utility vs. equity: Some plausible quasi-orderings, Journal of Public Economics 7, 365–381 Calvo, G. (1978), Some notes on time inconsistency and RawlsŠ maximin criterion, Review of Economic Studies 45, 97–102 Cass, D. and Mitra, T. (1991), Indefinitely sustained consumption despite exhaustible natural resources, Economic Theory 1, 119–146 Dalton, H. (1920), The measurement of the inequality of incomes, Economic Journal 30, 348–361 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Dasgupta, P.S., Sen, A. and Starrett, D. (1973), Notes on the measurement of inequality, Journal of Economic Theory 6, 180–187 Dasgupta, S. and Mitra, T. (1983), Intergenerational equity and efficient allocation of exhaustible Resources, International Economic Review 24, 133–153 Epstein, L.G. (1986a), Intergenerational consumption rules: An axiomatization of utilitarianism and egalitarianism, Journal of Economic Theory 38, 280–297 Epstein, L.G. (1986b), Intergenerational preference orderings, Social Choice and Welfare 3, 151–160 Ferejohn, J. and Page, T. (1978), On the foundation of intertemporal choice, American Journal of Agricultural Economics 60, 269–275 Harsanyi, J.C. (1953), Cardinal utility in welfare economics and in the theory of risk-taking, Journal of Political Economy 61, 434–435 Harsanyi, J.C. (1955), Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility, Journal of Political Economy 63, 309–321 Koopmans, T.C. (1967), Intertemporal distribution and optima1 aggregate economic growth, in Ten Economic Studies in the Tradition of Irving Fisher. Wiley, New York Mishan, E. (1977), Economic criteria for intergenerational comparisons, Futures 9, 383–403 Page, T. (1977), Conservation and Economic Effiency. John Hopkins University Press, Baltimore, MD Rawls, J. (1971), Theory of Justice. Harvard University Press, Cambridge, MA

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Rawls, J. (1988), The priority of right and ideas of the good, Public Affairs 17, 251–276 Ray, D. (1987), Nonpaternalistic intergenerational altruism, Journal of Economic Theory 41, 112–132 Sen, A.K. (1973), On Economic Inequality. Oxford University Press, London/New York Sen, A.K. (1977), On weights and measures: Informational constraints in social welfare analysis, Econometrica 45, 1539–1572 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Svensson, L.G. (1980), Equity among generations, Econometrica 48, 1251–1256 Vickrey, W. (1945), Measuring marginal utility by reactions to risk, Econometrica 13, 319–333 von Weizsäcker, C.C. (1965), Existence of optimal program of accumulation for an infinite time horizon, Review of Economic Studies 32, 85–104 WCED (The World Commission on Environment and Development) (1987), Our Common Future. Oxford University Press, Oxford, UK

Part II CHARACTERIZING SUSTAINABILITY

CHAPTER 8 THE HARTWICK RULE: MYTHS AND FACTS

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern, NO-0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics, University of Regensburg DE-93040 Regensburg, Germany Email: [email protected]

CEES WITHAGEN Department of Economics, Tilburg University P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands Department of Economics, Free University De Boelelaan 1105, 1081 HV Amsterdam The Netherlands Email: [email protected]

Abstract. We shed light on the Hartwick rule for capital accumulation and resource depletion by providing semantic clarifications and investigating the implications and relevance of this rule. We extend earlier results by establishing that the Hartwick rule does not indicate sustainability and does not require substitutability between man-made and natural capital. We use a new class of simple counterexamples (1) to obtain the novel finding that a negative value of net investments need not entail that utility is unsustainable, and (2) to point out deficiencies in the literature.

1. INTRODUCTION In resource economics two intertemporal allocation rules have attracted particular attention: the Hotelling rule and the Hartwick rule. The Hotelling rule is the fundamental no-arbitrage condition that every efficient resource utilization path has to Originally published in Environmental and Resource Economics 25 (2003), 129–150. Reproduced with permission from Springer.

125 Asheim, Justifying, Characterizing and Indicating Sustainability, 125–145 c 2007 Springer


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meet. In its basic form it implies that the net price of an exhaustible resource must grow at a rate that equals the interest rate. Although the Hotelling rule is in principle relevant for all models of nonrenewable resource use, its simplest application is that of a cake-eating economy where consumption results from depleting a given resource stock. The Hartwick rule, in contrast, was formulated for a production economy where consumption at any point of time depends not only on the resource extraction but also on the stock of man-made capital available at that point in time. In such a Dasgupta–Heal–Solow model, Hartwick (1977) showed that a zero value of net investments entails constant consumption over time, provided the Hotelling rule holds as a condition for local efficiency. This result was the heart of what later on was called the Hartwick rule. Hartwick’s result reinforced a basic message of neoclassical resource economics (cf. Solow, 1974): Man-made capital can substitute for raw material extracted from a nonrenewable resource in such a way that resource depletion does not harm future generations. Hence substitutability between natural and man-made capital may, in spite of the exhaustibility of natural resources, allow for equitable consumption for all generations, and Hartwick (1977) seemed to have found the investment policy that would bring about sustainability. Doubts have since been raised concerning the true status of Hartwick’s results and thus of the Hartwick rule. Following Asheim (1994); Pezzey (1994) it has been claimed that the Hartwick rule is, contrary to the first impression, not a prescriptive but rather a descriptive rule (cf. Toman et al., 1995, p. 147). The original formulation of the Hartwick rule sounds, however, more like a prescription than a description. And even if one tends to see the Hartwick rule as a description, it is still not clear – 25 years after Hartwick’s pioneering work – what exactly is described by it. The ambiguous status of the Hartwick rule has led to false beliefs concerning the content of the rule. There are two myths on the Hartwick rule that are pertinent in the literature. Myth 1: The Hartwick rule indicates sustainability. This myth was already suggested by Hartwick (1977, pp. 973–974) himself when he stated that “investing all net returns from exhaustible resources in reproducible capital . . . implies intergenerational equity,” but it lives on in recent contributions. Myth 2: The Hartwick rule requires substitutability between man-made and natural capital. This myth is implicit in many contributions on the Hartwick rule. An explicit formulation can be found in, e.g., Spash and Clayton (1997, p. 146): “. . . the. . . Hartwick rule depends upon man-made capital . . . being a substitute for, rather than a complement to, natural capital.” We will demonstrate that neither of these two assertions is true, showing that an adequate understanding of the Hartwick rule is still pending. The structure of our argument will be as follows. After introducing the general technological framework in Sect. 2, we give some semantic clarifications in Sect. 3 where we distinguish

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among the Hartwick investment rule, the Hartwick result, and its converse. In Sects. 4 and 5, we will deal separately with the two myths described above. In Sect. 4, we use the Dasgupta–Heal–Solow model to illustrate that consumption may exceed or fall short of the maximum sustainable level even if capital management is guided by the Hartwick investment rule in the short run. By considering a new class of simple counterexamples (1) we settle an open question, showing that a negative value of net investments need not indicate that the current consumption is unsustainable, and (2) we point to deficiencies in Hamilton’s (1995) analysis of the Hartwick rule. In Sect. 5, we show how the Hartwick rule applies even in models with no possibility for substitution between man-made and natural capital. Based on the analysis of the previous sections we then discuss in Sect. 6 whether the Hartwick rule should be viewed as a prescription or a description.

2. THE SETTING While Hartwick (1977) had used the Dasgupta–Heal–Solow model to formulate his rule, Dixit et al. (1980) applied a general framework to establish its broad applicability. We adopt their more general approach here and use the following notation. At time t (≥ 0) the vector of consumption flows is denoted c(t), the vector of ˙ capital stocks is denoted k(t), and the vector of investment flows is denoted k(t). Here, consumption includes both ordinary material consumption goods, as well as environmental amenities, while the vector of capital stocks comprises not only different kinds of man-made capital, but also stocks of natural capital and stocks of accumulated knowledge. Let k0 denote the initial stocks at time 0. We describe the technology by a time-independent set F. The triple (c(t), k(t), ∞ ˙ ˙ ˙ k(t)) is attainable at time t if (c(t), k(t), k(t)) ∈ F, and the path {c(t), k(t), k(t)} t=0 ˙ is feasible given k0 if k(0) = k0 and (c(t), k(t), k(t)) is attainable at all t ≥ 0. We assume that:

r The set F is a smooth, closed, and convex. r Consumption flows are non-negative: (c, k, k) ˙ ∈ F implies c ≥ 0. r Capital stocks are non-negative: (c, k, k) ˙ ∈ F implies k ≥ 0. r Free disposal of investment flows: (c, k, k) ˙ ∈ F and k˙  ≤ k˙ imply (c, k, k˙  ) ∈ F.

The latter assumption means, e.g., that stocks of environmental resources are considered instead of stocks of pollutants. We assume that there is a constant population, where each generation lives for one instance. Hence, generations are not overlapping nor infinitely lived, implying that any intertemporal issue is of an intergenerational nature. Issues concerning distribution within each generation will not be discussed. The vector of consumption

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goods generates utility, u(c), where u is a time-invariant, strictly increasing, concave, and differentiable function. Write u(t) = u(c(t)) for utility at time t. We assume that there are market prices for all consumption goods and capital goods. The discussion of the Hartwick rule is facilitated by using present value prices; i.e., deflationary nominal prices that correspond to a zero nominal interest rate. Hence, prices of future deliveries are measured in a numeraire at the present time. The vector of present value prices of consumption flows at time t is denoted p(t), and the vector of present value prices of investment flows at time t is denoted ˙ q(t). It follows that −q(t) is the vector of rental prices for capital stocks at time t, ˙ + q(t)k(t) ˙ entailing that p(t)c(t) + q(t)k(t) can be interpreted as the instantaneous profit at time t. Competitiveness of a path is defined in the following way. ∞ is competitive ˙ Definition 1: Let T > 0 be given. A feasible path {c(t), k(t), k(t)} t=0 T T during (0, T ) at discount factors {µ(t)}t=0 and prices {p(t), q(t)}t=0 if, for all t ∈ (0, T ), µ(t) > 0, (p(t), q(t)) ≥ 0, and the following conditions are satisfied:

Instantaneous utility is maximized: c(t) maximizes µ(t)u(c) − p(t)c .

(1a)

˙ Instantaneous profit is maximized: (c(t), k(t), k(t)) ˙ ∈F. ˙ maximizes p(t)c + q(t)k˙ + q(t)k subject to (c, k, k)

(1b)

In the sequel, we will refer to (1a) and (1b) as the competitiveness conditions. Competitive paths have the following property that is at the heart of the analysis of the Hartwick rule. ∞ is competitive during ˙ Lemma 1: Let T > 0 be given. Suppose {c(t), k(t), k(t)} t=0 T and {p(t), q(t)}T . Then: (0, T ) at {µ(t)}t=0 t=0

1.

For all t ∈ (0, T ), µ(t)∂u(c(t))/∂ci = pi (t) if ci (t) > 0.

2.

˙ For all t ∈ (0, T ), p(t)˙c(t) + d(q(t)k(t))/dt = 0.

Proof. Part 1 follows directly from (1a). For the proof of part 2, we follow Dixit et al. (1980). Since F is time-invariant, (1b) implies that ˙ + t) + q(t)k(t ˙ p(t)c(t + t) + q(t)k(t + t) ˙ + q(t)k(t) ˙ ≤ p(t)c(t) + q(t)k(t) . Divide by t, and let t → 0 from both directions. This yields ¨ + q(t) ˙ ˙ ˙ k(t) 0 = p(t)˙c(t) + q(t)k(t) = p(t)˙c(t) + d(q(t)k(t))/dt as the right-hand derivative cannot lie above zero and the left-hand derivative cannot lie below zero and both have to coincide. Some results on the Hartwick rule require that the path is not only competitive, but also regular.

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∞ is regular if it is competitive dur˙ Definition 2: A feasible path {c(t), k(t), k(t)} t=0 ∞ ing (0, ∞) at discount factors {µ(t)}t=0 and prices {p(t), q(t)}∞ t=0 , and the following conditions are satisfied:
∞ µ(t)u(c(t))dt < ∞ , (2a) 0

q(t)k(t) → 0

t → ∞. (2b) ∞ It can be shown that a regular path maximizes 0 µ(t)u(c(t))dt over all feasible paths, implying that any regular path is efficient. In the real world environmental externalities are not always internalized. This is one of many causes that prevent market economies from being fully efficient. Furthermore, for many capital stocks (e.g., stocks of natural and environmental resources or stocks of accumulated knowledge) it is hard to find market prices (or to calculate shadow prices) that can be used to estimate the value of such stocks. Since the Hartwick rule is formulated in terms of efficiency prices, we must abstract from these problems in our analysis. The time-independency of the set F is an assumption of constant technology. It means that all technological progress is endogenous, being captured by accumulated stocks of knowledge. If there is exogenous technological progress in the sense of a time-dependent technology, we may capture this within our formalism by including time as an additional stock. Since the time-derivative of time equals 1, this can be ˙ done as follows: The triple (c(t), k(t), k(t)) is attainable at time t if as

˙ (c(t), (k(t), t), (k(t), 1)) ∈ F . This formulation, which is used by e.g., Cairns and Long (2006) and Pezzey (2004), does, however, lead to the challenge of calculating the present value price associated with the passage of time. Vellinga and Withagen (1996) show how this price in principle can be derived through a forward-looking term. 3. WHAT IS THE HARTWICK RULE? The term “the Hartwick rule” has been used in different meanings. E.g., Dixit et al. (1980) in their first paragraph (p. 551) associated this term with both the investment rule of keeping “the total value of net investment under competitive pricing equal to zero” and the result that following such an investment rule “yields a path of constant consumption.” It will be clarifying to differentiate between

r The Hartwick investment rule, which is associated with the prescription of ˙ holding the value of net investments q(t)k(t) (also known as “genuine savings”) constant and equal to zero, and

r The Hartwick result that we will associate with the finding that following such a prescription leads to constant utility.

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In this section, we formally state the definitions that we will suggest, present the results that follow from the analysis of Sect. 2, and provide a partial review of the relevant literature. Definition 3: Let T > 0 be given. The Hartwick investment rule is followed along ∞ for t ∈ (0, T ) if the path is competitive during (0, T ) ˙ a path {c(t), k(t), k(t)} t=0 T T , and q(t)k(t) ˙ and prices {p(t), q(t)}t=0 = 0 for all at discount factors {µ(t)}t=0 t ∈ (0, T ). We first show the result that Hartwick (1977) originally showed in a special model, but which – as established by Dixit et al. (1980) – carries over to our general setting. Proposition 1 (The Hartwick Result): Let T > 0 be given. If the Hartwick investment rule is followed for t ∈ (0, T ) in an economy with constant population and constant technology, then utility is constant for all t ∈ (0, T ). Proof. For all t ∈ (0, T ) we have that µ(t)u(t) ˙ = p(t)˙c(t) ˙ = −d(q(t)k(t))/dt =0

(by Lemma 1(i)) (by Lemma 1(ii)) ˙ (since q(t)k(t) = 0),

noting that prices of consumption flows that remain equal to zero, and thus are constant, do not matter for the first equality. Dixit et al. (1980) made the observation that the Hartwick result can be generalized. For the statement of this more general result we first need to define “the generalized Hartwick investment rule,” which is the prescription of holding the present value of ˙ constant, but not necessarily equal to zero. net investments q(t)k(t) Definition 4: Let T > 0 be given. The generalized Hartwick investment rule is ∞ for t ∈ (0, T ) if the path is competitive ˙ followed along a path {c(t), k(t), k(t)} t=0 T and prices {p(t), q(t)}T , and q(t)k(t) ˙ during (0, T ) at discount factors {µ(t)}t=0 t=0 is constant for all t ∈ (0, T ). Proposition 2 (The Generalized Hartwick Result): Let T > 0 be given. If the generalized Hartwick investment rule is followed for t ∈ (0, T ) in an economy with constant population and constant technology, then utility is constant for all t ∈ (0, T ). ˙ Proof. The proof of Proposition 1 applies even if q(t)k(t) = ν for all t ∈ (0, T ), with ν constant. Dixit et al. (1980) posed the question of whether the converse of the Hartwick result can be established. It is instructive to observe that the converse of the (ordinary) Hartwick result is not correct.

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Incorrect Claim 1 (The Converse of the Hartwick Result): Let T > 0 be given. If utility is constant for all t ∈ (0, T ) in an economy with constant population and constant technology, then the Hartwick investment rule is followed for t ∈ (0, T ). We provide a counterexample using the Ramsey model. Here there is a single consumption good, and one capital good. To denote these scalars we use symbols in italics instead of boldface. The stock of the aggregate capital good (k(t)) leads to production f (k(t)) that can either contribute to the quality of life of generation t ˙ or be used to accumulate capital. Hence, (c(t), k(t), k(t)) is attainable if and only ˙ ≤ f (k(t)). The initial stock equals k 0 . The production function f is if c(t) + k(t) twice continuously differentiable, with f  > 0 and f  < 0. Furthermore, f (0) = 0, limk→0 f  (k) = ∞, and limk→∞ f  (k) = 0. For this model the competitiveness ˙ = f (k(t)), p(t) = q(t), and q(t) f  (k(t)) = condition (1b) implies that c(t) + k(t) −q(t). ˙ Hence, omitting the time variable, ˙ p c˙ = q c˙ = −q k¨ + q f  (k)k˙ = −q k¨ − q˙ k˙ = −d(q k)/dt . Suppose there exist T > 0 with c(t) ˙ = 0 for all t ∈ (0, T ). This is compatible with ˙ = ν = 0 for all t ∈ (0, T ). In particular, if ν < 0, then c = c(t) > f (k(t)), q(t)k(t) which is feasible in the short run. However, as shown by Dixit et al. (1980), the converse of the generalized Hartwick result can be established. Proposition 3 (The Converse of the Generalized Hartwick Result): Let T > 0 be given. If utility is constant for all t ∈ (0, T ) in an economy with constant population and constant technology, then the generalized Hartwick investment rule is followed for t ∈ (0, T ). Proof. Since (1a) and (1b) imply that ˙ µ(t)u(t) ˙ = p(t)˙c(t) = −d(q(t)k(t))/dt , as shown in the proof of Proposition 1, it follows from the constancy of utility that ˙ is constant. q(t)k(t) Applying these results at all times along infinite horizon paths yields some observations concerning the relationship between the (generalized) Hartwick result and the concept of sustainable development, as a precursor to the discussions of Sects. 4 and 5. For the statement of these results, we introduce the following definition. Definition 5: A utility path {u(t)}∞ t=0 is egalitarian if utility is constant for all t. The following two results are direct consequences of Propositions 1 and 2 established above. Corollary 1 (The Hartwick Rule for Sustainability): If the Hartwick investment rule is followed for all t in an economy with constant population and constant technology, then the utility path is egalitarian.

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Corollary 2 (The Generalized Hartwick Rule for Sustainability): If the generalized Hartwick investment rule is followed for all t in an economy with constant population and constant technology, then the utility path is egalitarian. One may wonder whether Corollary 2 is an empty generalization of Corollary 1, in the sense that any feasible competitive path with constant utility does in fact satisfy the (ordinary) Hartwick investment rule. This is not the case since in the Ramsey model there exist feasible competitive paths with constant utility for which ˙ = ν > 0 for all t, provided that ν < q(0) f (k 0 ). Then c = c(t) < f (k(t)) q(t)k(t) for all t, so that the path is inefficient since capital is over-accumulated. It is, however, true – as suggested by Dixit et al. (1980) and shown under general assumptions by Withagen and Asheim (1998) – that the (ordinary) Hartwick investment rule must be satisfied for all t if the egalitarian utility path is efficient. This is stated next. Proposition 4 (The converse of the Hartwick rule for sustainability): If the utility path is egalitarian along a regular path in an economy with constant population and constant technology, then the Hartwick investment rule is followed for all t. The proof by Withagen and Asheim (1998) is too extensive to be reproduced here. ˙ )→0 The result means that a regular path with constant utility satisfies q(T )k(T as T → ∞.1 Combining this transversality condition with the results of Lemma 1 ∞ ˙ means that q(t)k(t) = t µ(t)u(t)dt ˙ (cf. Aronsson et al., 1997, p. 105). Thus, the value of net investments at time t measures the present value of future changes in utility. From this it can be easily seen that the Hartwick investment rule is satisfied for all t if the utility path is egalitarian. The fact that there exist egalitarian, but inefficient, utility paths in the Ramsey model means that Proposition 4 does not hold if regularity is not assumed. If only the competitiveness conditions (1a) and (1b) are assumed to hold at any t, then the following weaker result – due to Dixit et al. (1980) – follows from Proposition 3. Corollary 3 (The Converse of the Generalized Hartwick Rule for Sustainability): If the utility path is egalitarian along a competitive path in an economy with constant population and constant technology, then the generalized Hartwick investment rule is followed for all t. When discussing the significance and applicability of the Hartwick rule, the results on sustainability (i.e., Corollaries 1–3 and Proposition 4) are of particular interest. In Sects. 4, we will discuss what significance the Hartwick rule may have for sustainability along two dimensions. First, we note that these results are weak since they are based on strong premises involving the properties of the entire paths. In Sect. 4, we therefore pose the question: can stronger results be obtained by weakening the premises – i.e., by relating sustainability of a path to only the current value of net investment – thereby addressing Myth 1. Second, in Sect. 5 we discuss whether

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the Hartwick rule for sustainability requires substitutability between man-made and natural capital, thereby addressing Myth 2. 4. MYTH 1: THE HARTWICK INVESTMENT RULE INDICATES SUSTAINABILITY What makes Hartwick’s investment rule so appealing in the framework of resource economics is its alleged relationship with intergenerational fairness. Hartwick himself purported to have found a prescription how “to solve the ethical problem of the current generation short-changing future generations by “overconsuming” the current product, partly ascribable to current use of exhaustible resources” (Hartwick, 1977, p. 972). By invoking Hartwick’s result the Hartwick investment rule then seemed to provide a sufficient condition for intergenerational justice. Such an interpretation carries over to some recent text books, e.g., Tietenberg (2001, p. 91) and Hanley et al. (2001, p. 137). Although the result proven by Hartwick (1977) is undoubtedly correct, it does not follow that one can draw a close link between Hartwick’s result and intergenerational equity without taking notice of additional conditions. There are more or less sophisticated versions of such precipitate interpretations. The first one only makes weak assumptions on the path under consideration and is rather easy to refute. Incorrect Claim 2 (Trivial Version): Let T > 0 be given. Suppose a path {c(t), ∞ is competitive during (0, T ) in an economy with constant population ˙ k(t), k(t)} t=0 ˙ is non-negative for and constant technology. If the value of net investments q(t)k(t) t ∈ (0, T ), then, for any t ∈ (0, T ), u(c(t)) can be sustained forever given k(t). Whether this claim, which combines short-term considerations with long-term results, is correct or incorrect crucially depends on the underlying technology. It is certainly correct in case of the Ramsey model. The claim, however, is not true in the Dasgupta–Heal–Solow model (see, e.g., Dasgupta and Heal, 1974, and Solow, 1974). In this model there are two capital stocks: man-made capital, denoted by kM , and a nonrenewable natural resource, the stock of which is denoted by kN . So, k = (kM , kN ). The initial stocks are given by 0 , k 0 ). The technology is described by a Cobb-Douglas production function k0 = (kM N a (−k˙ )b depending on two inputs, man-made capital k and the F(kM , −k˙N ) = kM N M raw material −k˙N that can be extracted without cost from the nonrenewable resource. The output from the production process is used for consumption and for investments ˙ in man-made capital k˙M . Hence, (c(t), k(t), k(t)) is attainable at time t if and only if c(t) + k˙M (t) ≤ kM (t)a (−k˙N (t))b where a > 0, b > 0 and a + b ≤ 1 , and c(t) ≥ 0, kM (t) ≥ 0, kN (t) ≥ 0, and −k˙N (t) ≥ 0. With r (t) := −k˙N (t) denoting the flow of raw material, these assumptions entail
∞ 0 r (t)dt ≤ kN and r (t) ≥ 0 for all t ≥ 0 . 0

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The competitiveness condition (1b) requires that c(t) + k˙M (t) = kM (t)a r (t)b ,

(3a)

r (t) + k˙N (t) = 0,

(3b)

p(t) = qM (t),

(3c)

qM (t) · b · kM (t)a r (t)b−1 = 1,

(3d)

qM (t) · a · kM (t)a−1r (t)b = −q˙M (t) ,

(3e)

where (3d) follows from qM (t) · b · kM (t)a r (t)b−1 = qN (t) and 0 = q˙N (t) by choosing extracted raw material as numeraire: qN (t) ≡ 1. Note that (3d) and (3e) entail that the growth rate of the marginal product of raw material equals the marginal product of man-made capital; thus, the Hotelling rule is satisfied. Assume that a > b > 0. Then there is a strictly positive maximum constant rate of consumption c∗ that can be sustained forever given k0 (see, e.g., Dasgupta and Heal, 1979, p. 203). It is well known that this constant consumption level can be implemented along a competitive path where net investment in man-made capital k˙M (t) is at a constant level i ∗ = bc∗ /(1 − b). To give a counterexample to the claim above, fix a consumption level c > c∗ . Set i = bc/(1 − b) and define T by
T 0 0 (i/b)1/b (kM + it)−a/b dt = kN . (4) 0

For t ∈ (0, T ), consider the path described by k(0) = k0 and c(t) = c, k˙M (t) = i, 0 + it)−a/b , −k˙N (t) = r (t) = (i/b)1/b (kM

which by (4) implies that the resource stock is exhausted at time T . This feasible path is competitive during (0, T ) at prices p(t) = qM (t) = r (t)/i and qN (t) = 1, implying that the value of net investments qM (t)i − qN (t)r (t) is zero, and thus the Hartwick investment rule is followed. Hence, even though the competitiveness condition (1b) is satisfied (while (1a) does not apply) and the value of net investments is non-negative during the interval (0, T ), the constant rate of consumption during this interval is not sustainable forever. Hartwick (1977) does not say much about efficiency requirements going beyond competitiveness conditions, i.e., the Hotelling rule, other than remarking that the entire stock of the nonrenewable resource has to be used up in the long run to achieve an optimal solution. It seems appropriate, however, to consider efficiency requirements going beyond competitiveness on a finite interval when looking for counterexamples. The path described above for the Dasgupta–Heal–Solow model is in fact 0 + i T , has not efficient. At time T a certain stock of man-made capital, kM (T ) = kM

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been accumulated, while the flow of extracted raw material falls abruptly to zero due the exhaustion of the resource. In our Cobb–Douglas case the marginal productivity of r is a strictly decreasing function of the flow of raw material for a given positive stock of man-made capital. This implies that profitable arbitrage opportunities can be exploited by shifting resource extraction from right before T to right after T , implying that the Hotelling rule is not satisfied at that time. As the path in this counterexample is inefficient, it might be possible that the Hartwick investment rule does not indicate sustainability in the example due to this lack of efficiency. However, this is not true either. The claim above does not become valid even if we refer to regular – and thus efficient – paths for which competitiveness holds throughout and transversality conditions are satisfied. Incorrect Claim 3 (Sophisticated Version): Let T > 0 be given. Suppose a path ∞ is regular in an economy with constant population and constant ˙ {c(t), k(t), k(t)} t=0 ˙ technology. If the value of net investments q(t)k(t) is non-negative for t ∈ (0, T ), then, for any t ∈ (0, T ), u(c(t)) can be sustained forever given k(t). Again, counterexamples can be provided in the framework of the Dasgupta–Heal– Solow model. Asheim (1994) and Pezzey (1994) independently gave a counterexample by considering paths where the sum of utilities discounted at a constant utility discount rate is maximized. If, for some discount rate, the initial consumption level along such a discounted utilitarian optimum exactly equals the maximum sustainable 0 and k 0 , then there exists an initial interval during which consumption level given kM N the value of net investments is strictly positive, while consumption is unsustainable given the current capital stocks kM (t) and kN (t). It is not quite obvious, however, that the premise of this statement can be fulfilled, i.e., that there exists some discount rate such that initial consumption along the optimal path is barely sustainable. This was subsequently established for the Cobb–Douglas case by Pezzey and Withagen (1998). The fact that their proof is quite intricate indicates, however, that this is not a trivial exercise. Consequently, we wish to provide another type of counterexample here, which is also within the Dasgupta–Heal–Solow model and resembles the one given above to refute Incorrect Claim 2. We will show that a path identical to that described in our first counterexample during an initial phase can always be extended to an efficient path. Moreover, this second counterexample can be used to show that there exist regular paths with a non-negative value of net investments during an initial phase even if a ≤ b, entailing that a positive and constant rate of consumption cannot be sustained indefinitely. The example, illustrated in Fig. 8.1, consists of three separate phases with constant consumption, constructed so that there are no profitable arbitrage opportunities at any time, not even at the two points in time, T1 and T2 , where consumption is not continuous. Both capital stocks are exhausted at T2 , implying that consumption equals zero for (T2 , ∞). 0 is given, while k 0 is treated as a parameter. In the construction of the example, kM N Fix some consumption level c1 > 0 and some terminal time T1 of the first phase of

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consumption c2 c1

max. sust. cons. given k0

T1

time

T2

Figure 8.1. Counterexample to Incorrect Claim 3

the path. In the interval (0, T1 ) the path is – as in our first example – described by k(0) = k0 and c(t) = c1 , k˙M (t) = i 1 , 0 + i t)−a/b , −k˙N (t) = r (t) = (i 1 /b)1/b (kM 1

where i 1 = bc1 /(1 − b), but with the difference that the resource stock will not be exhausted at time T1 . As in the first example, the Hartwick investment rule applies during this phase. The second phase starts at time T1 . Consumption jumps upward discontinuously to c2 > c1 , but we ensure that the flow of raw material is continuous to remove profitable arbitrage opportunities. Consumption is constant at the new and higher level c2 , and Proposition 3 implies that there exists ν2 < 0 such that, for all t ∈ (T1 , T2 ), the generalized Hartwick investment rule, qM (t)k˙M (t) = r (t) + ν2 , is observed. By (3a) and (3d), this rule may (for any c and ν) be written as: kM (t)a r (t)b − c = b · kM (t)a r (t)b−1 (r (t) + ν) =  c = (1 − b) · kM (t)a r (t)b 1 −

(cf. Hamilton, 1995). As

a rb kM

−b

a r b−1 r · kM

a r b, (1 − b) · kM

ν b · 1 − b r (t)



(5)

this implies .

(6)

Since both kM (t) and r (t) are continuous at time T1 , we can now use (6) to determine ν2 as follows:   b ν2 a b . (7) c2 = (1 − b) · kM (T1 ) r (T1 ) 1 − · 1 − b r (T1 ) By choosing c2 > kM (T1 )a r (T1 )b (> c1 ) and fixing ν2 according to (7), the generalized Hartwick investment rule combined with (3a)–(3b) determines a competitive

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path along which investment in man-made capital becomes increasingly negative.2 Determine T2 as the time at which the stock of man-made capital reaches 0, and 0 such that the resource stock is exhausted simultaneously. With both determine kN stocks exhausted, consumption equals 0 during the third phase (T2 , ∞). The Hotelling rule holds for (0, T1 ) and (T1 , T2 ), and by the construction of ν2 , a jump of the marginal productivity of the natural resource at T1 is avoided such that the Hotelling rule obtains even at T1 . Thus, the path is competitive throughout. By letting u(c) = c and, for all t ∈ (0, T2 ), µ(t) = p(t), it follows that both regularity conditions (2a) and (2b) are satisfied, implying that the path is regular. Note that the above construction is independent of whether a > b. If a ≤ b, so that no positive and constant rate of consumption can be sustained indefinitely, we have thus shown that having a non-negative value of net investments during an initial phase of a regular path is compatible with consumption exceeding the sustainable level. However, even if a > b, so that the production function allows for a positive level of sustainable consumption, we obtain a counterexample as desired. For this purpose, increase c2 beyond all bounds to that ν2 becomes more negative. Then T2 decreases and converges to T1 , and the aggregate input of raw material in the interval (T1 , T2 ) – being bounded above by r (T1 ) · (T2 − T1 ) since r (t) is decreasing (cf. Note 2) – converges to 0. This in turn means that, for large enough c2 , c1 cannot be sustained 0 needed to achieve exhaustion of the resource at forever given the choice of kN time T2 . This example shows that a non-negative value of net investments on an open interval is not a sufficient condition for having consumption be sustainable. However, it has up to now been an open question whether it is a necessary condition: Does a negative value of net investments during a time interval imply that consumption exceeds the sustainable level? The following result due to Pezzey (2004) shows such an implication when utilities are discounted at a constant rate. ∞ is regular in ˙ Proposition 5: Let T > 0 be given. Suppose a path {c(t), k(t), k(t)} t=0 an economy with constant population and constant technology, with the supporting utility discount factor satisfying, for all t, µ(t) = µ(0)e−ρt . If the value of net ˙ investments q(t)k(t) is negative for t ∈ (0, T ), then, for any t ∈ (0, T ), u(c(t)) cannot be sustained forever given k(t).

˙ Proof. It follows from Lemma 1 that µ(t)u(t) ˙ + d(q(t)k(t))/dt = 0, implying ˙ d(µ(t)u(t))/dt = µ(t)u(t) ˙ − d(q(t)k(t))/dt. When this observation is combined ˙ ) → 0 as T → ∞ (cf. Note 1), Weitzman’s with µ(t) = µ(0)e−ρt and q(T )k(T (1976) main result can be established:
∞
∞   ˙ µ(s) u(c(t)) + q(t)k(t)/µ(t) ds = µ(s)u(c(s))ds . (8) t

∞

t

Since the path is regular, it maximizes t µ(s)u(c(s))ds over all feasible paths. This combined with (8) implies that the maximum sustainable utility level given k(t) ˙ ˙ cannot exceed u(c(t)) + q(t)k(t)/µ(t). Suppose q(t)k(t) < 0 for t ∈ (0, T ). Then

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˙ u(c(t)) > u(c(t)) + q(t)k(t)/µ(t). Hence, u(c(t)) exceeds the maximum sustainable utility level and cannot be sustained forever given k(t). It is not, however, a general result that a negative value of net investments implies nonsustainability, as we establish below. ∞ is ˙ Incorrect Claim 4: Let T > 0 be given. Suppose a path {c(t), k(t), k(t)} t=0 regular in an economy with constant population and constant technology. If the value ˙ is negative for t ∈ (0, T ), then, for any t ∈ (0, T ), u(c(t)) of net investments q(t)k(t) cannot be sustained forever given k(t).

Also in this case we will provide a counterexample in the framework of the Dasgupta–Heal–Solow model. Assume that a > b so that the production function allows for a positive level of sustainable consumption. Again, the example (which is illustrated in Fig. 8.2) consists of three separate phases with constant consumption, constructed so that there are no profitable arbitrage opportunities at any time, not even at the two points in time, T1 and T2 , where consumption is not continuous. 0 is given, while k 0 is treated as a parameter. Fix some consumption As before, kM N level c1 > 0 and some terminal time T1 of the first phase of the path. Construct a path that has constant consumption c1 and obeys the generalized Hartwick investment rule (5) with ν1 < 0 in the interval (0, T1 ), where T1 is small enough to ensure that kM (T1 ) > 0. Let the path have, as its second phase, constant consumption c2 > 0 and a constant (present) value of net investments ν2 > 0 in the interval (T1 , T2 ). To satisfy the Hotelling rule at time T1 , c2 and ν2 must fulfill (7); hence, by choosing c2 < (1 − b) · kM (T1 )a r (T1 )b it follows that ν2 > 0. Let kM (T2 ) and r (T2 ) be the stock of man-made capital and the flow of raw material, respectively, at time T2 . At this point in time the path switches over to the third phase where the (ordinary) Hartwick investment rule is followed with c3 = (1 − b) · kM (T2 )a r (T2 )b . Since a > b, the production function allows for a positive level of sustainable 0 that ensures resource consumption, and there exists an appropriate choice of kN consumption c3 max. sust. cons. given k0

c1

c2 T1

T2

Figure 8.2. Counterexample to Incorrect Claim 4

time

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exhaustion as t → ∞ and makes the path regular. This initial resource stock depends on T1 and T2 , but it is finite in any case. Keep T1 fixed and increase T2 . As T2 goes 0 needed will also tend to infinity.3 The same holds true to infinity, then the stock kN 0 and the for the maximum sustainable consumption level c∗ that is feasible given kM 0 initial resource stock kN determined in this way. Hence, by shifting T2 far enough into the future, it follows that c1 < c∗ . Thus, a regular path can be constructed which has a first phase with a negative value of net investments even though the rate of consumption during this phase is sustainable given the initial stocks. Both our counterexamples are consistent with the result for regular paths that the value of net investments measures the present value of all future changes in of utility, which is a consequence of Lemma 1. It follows directly from that result that if along an efficient path utility is monotonically decreasing/increasing indefinitely, then the value of net investments will be negative/positive, while utility will exceed/fall short of the sustainable level. The value of net investments thus indicates sustainability correctly along such monotone utility paths. Hence, the counterexamples above are minimal by having consumption (and thus utility) constant except at two points in time. Moreover, paths with piecewise constant consumption would not yield counterexamples if constant consumption is associated with a constant consumption interest rate (as it is in the Ramsey model). In the Dasgupta–Heal–Solow model, however, the consumption interest rate, − p(t)/ ˙ p(t), which equals the marginal productivity of man-made capital along a competitive path, is decreasing whenever consumption is constant. It is therefore the nonmonotonicity of the paths, combined with the property that the consumption interest rate is decreasing when consumption is constant, that leads to the negative results established above concerning the connection between the value of net investments (the “genuine savings”) and the sustainability of utility. It is also worth emphasizing the point made in Asheim (1994) and elsewhere that the relative equilibrium prices of different capital stocks today depend on the properties of the whole future path. The counterexamples above show how the relative price of natural capital depends positively on the consumption level of the generations in the distant future. Thus, the future development – in particular, the distribution of consumption between the intermediate and the distant future – affects the value of net investments today and, thereby, the usefulness of this measure as an indicator of sustainability today. Hence, to link the (generalized) Hartwick investment rule to sustainability we cannot avoid letting this rule apply to investment behavior at all points in time. We present a correct claim concerning the value of net investments and the sustainability of utility by restating the generalized Hartwick rule for sustainability (Corollary 2). ∞ is competitive during (0, ∞) ˙ Correct Claim 1: Suppose a path {c(t), k(t), k(t)} t=0 in an economy with constant population and constant technology. If the value of net ˙ is constant for all t ∈ (0, ∞), then the rate of utility realized at investments q(t)k(t) any time t can be sustained forever given k(t).

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Proof. From the generalized Hartwick rule for sustainability, it follows that the utility path is egalitarian. Hence, at any time, utility is sustainable. If the path is regular, it follows from Proposition 4 that an egalitarian utility path ˙ implies that q(t)k(t) equals zero for all t. In the Ramsey model, it is feasible ˙ but not efficient to have q(t)k(t) constant and positive for all t. It follows from Note 3 that this is not even feasible in the Dasgupta–Heal–Solow model, since the integral of extracted raw material would become infinite as time goes to infin˙ ity. In both models, feasibility rules out q(t)k(t) being constant and negative for all t. Hamilton (1995) also analyses paths satisfying the generalized Hartwick rule (i.e., our (5)) in different versions of the Dasgupta–Heal–Solow model. For the version that overlaps with the one treated here (σ = 1), he incorrectly claims (Hamilton, 1995, pp. 397–398 and Table 1) that if ν > 0, then the rate of consumption has to become negative at a finite point in time, which contradicts Proposition 2. This as well as many other inaccuracies seem to be caused by his implicit and inappropriate assumption that variables are continuous functions of time throughout, even in the case when a constant consumption path cannot be sustained indefinitely. A competitive path with constant consumption and positive and constant (present) value of net investments (i.e., ν > 0) can be sustained up to the point when the resource stock has been exhausted. The path from then on must be a completely different path, which cannot be governed by the generalized Hartwick rule with ν > 0. E.g., it is not correct, as claimed by Hamilton (1995, pp. 397–398), that resource extraction goes continuously to zero as the stock of natural capital approaches exhaustion. It is an open question whether Correct Claim 1 can be strengthened to the following statement for competitive paths in an economy with constant population and ˙ constant technology: “If q(t)k(t) is non-negative for all t ∈ (0, ∞), then, for any t ∈ (0, T ), u(c(t)) can be sustained forever given k(t).” We cannot prove this under general assumptions, but neither do we have a counterexample.

5. MYTH 2: THE HARTWICK RULE FOR SUSTAINABILITY REQUIRES SUBSTITUTABILITY BETWEEN MAN-MADE AND NATURAL CAPITAL Hartwick (1977) concentrated his attention on economies where substitution of manmade capital and resource extraction is possible. In the wake of his contribution an impression appears to have been formed to the effect that the Hartwick rule for sustainability requires that man-made capital can substitute for natural capital; i.e., that the production possibilities are consistent with the beliefs held by the proponents of ‘weak sustainability’ (cf. the citation from Spash and Clayton, 1997, reproduced in the Introduction). If, on the other hand, natural capital has to be conserved in order for utility to be sustained (i.e., the world is as envisioned by the proponents of “strong sustainability”), then – it is claimed – the Hartwick rule for sustainability does not apply.

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The relevance of the Hartwick rule for sustainability is related to the question of whether a constant utility path exists. Since a false premise does not falsify an implication, the Hartwick rule for sustainability as an implication is true even if, ˙ in some specific model, there does not exist any path where q(t)k(t) equals zero for all t. What the Hartwick rule for sustainability entails is that if no constant ˙ utility path exists, then there cannot exist any path where q(t)k(t) equals zero for all t. Still, even though the nonexistence of an egalitarian path does not falsify the Hartwick rule for sustainability, it is interesting to discuss in what kind of technologies there exists an egalitarian utility path, implying that the result is relevant (i.e., not empty). It turns out that substitutability is not necessary for the relevance of the Hartwick rule for sustainability. Incorrect Claim 5: The Hartwick rule for sustainability is relevant only if manmade capital can substitute for natural capital. We provide a counterexample in a model which combines the restriction of the Ramsey model that available output must be produced, c(t) + k˙M (t) ≤ f (kM (t)), with the restriction that available output requires raw material in fixed proportion, c(t) + k˙M (t) ≤ r (t). Hence, we refer to it as the complementarity model.4 The production function f satisfies the assumptions of the Ramsey model (cf. Sect. 3), and the raw material is extracted without cost from a stock of a renewable natural   resource, kN (t), with a rate of natural regeneration that equals kN (t) · k¯N − kN (t) , where k¯N > 0. Together with the restrictions that c(t) ≥ 0, r (t) ≥ 0, kM (t) ≥ ˙ 0, and kN (t) ≥ 0, this determines what triples (c(t), k(t), k(t)) are attainable at time t, where k(t) = (kM (t), kN (t)). The initial stocks are given by k0 = 0 , k 0 ). (kM N As long as output does not exceed the maximal rate of natural regeneration, (k¯N )2 /4, this model behaves as the Ramsey model. However, if one tries to sustain production above such a level, then the resource stock will be exhausted in finite time, undermining the productive capabilities. The competitiveness condition (1b) implies that c(t) + k˙M (t) = min{ f (kM (t)), r (t)},

(9a)

  r (t) + k˙N (t) = kN (t) · k¯N − kN (t) ,

(9b)

p(t) = qM (t),

(9c)

(qM (t) − qN (t)) · f  (kM (t)) = −q˙M (t) ,

(9d)

  qN (t) · k¯N − 2kN (t) = −q˙N (t) .

(9e)

Any competitive path with constant consumption forever will satisfy the (ordinary) Hartwick investment rule by having the stock of man-made capital remain constant and the value of investment in natural capital equal to zero. Hence, constant

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∗ ), implying consumption along a competitive path is characterized by c∗ = f (kM that k˙M (t) = 0, while qN (t)k˙N (t) = 0. If, along such a path, the resource stock converges to a size larger than the one corresponding to the maximal level of natural regeneration, then qN (t) ≡ 0 and the productivity of man-made capital measures ∗ ) = −q˙ (t)/q (t). If, on the other hand, the the consumption interest rate: f  (kM M M resource stock is constant and smaller than the size corresponding to the maximal  ∗ ) = r ∗ = k ∗ · k¯ − k ∗ and q (t) > level of natural regeneration, then c∗ = f (kM N N N N 0. And the productivity of natural regeneration measures the consumption interest ∗ ) > −q˙ (t)/q (t) = −q˙ (t)/q (t) = k¯ − 2k ∗ . In this latter case, the rate: f  (kM M M N N N N application of the Hartwick investment rule leads to a feasible egalitarian path by keeping both stocks constant. Hence, the model is consistent with the world as envisioned by the proponents of ‘strong sustainability’; still, the Hartwick rule for sustainability applies. To state a correct claim concerning the relevance of the Hartwick rule for sustainability, we define the concept of “eventual productivity.”

Definition 6: A model satisfies eventual productivity given the initial stocks k0 if starting from k0 there exists a regular path with constant utility forever. Correct Claim 2: The Hartwick rule for sustainability is relevant in an economy with constant population and constant technology if eventual productivity is satisfied given the initial stocks k0 . Proof. From eventual productivity and the converse of the Hartwick rule for sustain˙ being constant ability (Proposition 4), it follows that there exists a path with q(t)k(t) and equal to zero for all t. The question of whether man-made capital can substitute for natural capital is important for the relevance of the Hartwick rule for sustainability only to the extent that a lack of such substitutability means that eventual productivity cannot be satisfied. 6. PRESCRIPTION OR DESCRIPTION? The preceding analysis leads to the following questions: Can the Hartwick investment rule be used as a prescription? Or is the Hartwick rule for sustainability (and its converse) a description of an egalitarian utility path; i.e., a characterization result? The new class of simple counterexamples presented in Sect. 4 yields the finding that (1) a generation may well obey the Hartwick investment rule but nevertheless consume more than the maximum sustainable consumption level, as well as the novel result that (2) a generation with a negative value of net investments will not necessarily undermine the consumption possibilities of its successors. The analysis of Sect. 4 thus reinforces the message of Toman et al. (1995, p. 147), namely that the Hartwick investment rule cannot serve as a prescription for sustainability. It is not enough to know whether the current investment in man-made capital in value makes up for the

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current depletion of natural capital, since the Hartwick result (Proposition 1) only says that following the Hartwick investment rule will entail constant consumption for an interval of time. This is neither sufficient nor necessary for development to be sustainable. Rather, a judgement on whether short-run behavior is compatible with sustainable development must be based on the long-run properties of the path and the technological environment. By the generalized Hartwick rule for sustainability (Corollary 2) these long-run properties are: 1.

Competitiveness conditions. The generalized Hartwick rule for sustainability requires that the economy realizes a perfectly competitive equilibrium indefinitely. In particular, this entails that all externalities will be internalized. How can we know now that competitiveness conditions will be followed at any future point in time?

2.

Constant present value of net investments. The generalized Hartwick rule for ˙ is constant indefinitely. It is not sufficient sustainability requires that q(t)k(t) to have current price-based information about the path in order to prescribe sustainable behavior; rather such information has to be available at all future ˙ will be constant for all t? points in time. How can we know now that q(t)k(t)

3.

Feasibility. The generalized Hartwick rule for sustainability is relevant only if positive and constant consumption can be sustained indefinitely. How can we know now that a path with constant consumption during some interval of time can be sustained forever? The Dasgupta–Heal–Solow model illustrates these problems; e.g., our counterexample to Incorrect Claim 2 shows how feasibility breaks down simply due to an overestimation of the resource stock.

4.

No exogenous technological progress. The generalized Hartwick rule for sustainability is valid only if all technological progress is endogenous, being captured by accumulated stocks of knowledge. How can we know now that we will be able to attribute any future technological progress to accumulated stocks of knowledge? Similar problems arise for an open economy facing changing terms-of-trade (e.g., a resource exporter facing increasing resource prices).

Moreover, if all this information about the long-run properties of paths as well as the future technological environment were available, and a constant consumption path is desirable, then the price-based information entailed in Hartwick’s rule would hardly seem necessary nor convenient for social planning. Therefore, it is our opinion that the Hartwick investment rule has little prescriptive value for decision-makers trying to ensure that development is sustainable. The Hartwick investment rule is, however, of interest when it comes to describing an efficient path with constant utility. It follows from the converse of the Hartwick rule for sustainability (Proposition 4) that any such egalitarian path will be characterized by the Hartwick investment rule being satisfied at all points in time. Note that the

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importance of this result is not that it tells decision-makers anything concerning how to steer the economy along such a path; rather, it describes how the path would look if it were followed. Hence, in line with the view of Toman et al. (1995, p. 147), it seems ˙ more natural to consider q(t)k(t) = 0 for all t as a descriptive result, characterizing an efficient and egalitarian utility path. What we have added here is to point out that this characterization result follows from the converse of Hartwick’s rule for sustainability and that it is general: While its relevance relies on the assumption of eventual productivity and its validity on the assumption that all technological progress can be attributed to accumulated stocks of knowledge, it does not impose any particular requirements on the possibility of substitution between man-made and natural capital, as was seen in Sect. 5. The Dasgupta–Heal–Solow model is only one application among many. Acknowledgments: We thank John Hartwick, David Miller, and Atle Seierstad as well as two referees for helpful discussions and comments and gratefully acknowledge financial support from the Hewlett Foundation (Asheim) and the Research Council of Norway (Ruhrgas grant; Asheim and Buchholz). NOTES 1 This follows from optimality (and hence, from regularity) of the path if there is a constant discount rate; i.e., if µ(t) = µ(0)e−ρt (cf. Dasgupta and Mitra, 1999). However, if µ(t) is not an exponentially

decreasing function, then it is not immediate that regularity implies this condition. 2 By differentiating k (t)a r (t)b − c = b · k (t)a r (t)b−1 (r (t) + ν ) w.r.t. time and observing that c M M 2 2 2 is constant, it follows that the growth rate of the marginal product of r equals the marginal product of kM , i.e., the Hotelling rule is satisfied and the path is competitive during this phase. By totally differentiating the same equation, it can be seen that a falling kM leads to a falling r and thus a falling rate of output and – due to the constant c2 – a falling k˙M . 3 It follows from (6) and c > 0 that r (t) > bν /(1 − b) (> 0) for all t ∈ (T , T ). 2 2 1 2 4 Variants of this model appear in Asheim (1978) and Hannesson (1986).

REFERENCES Aronsson, T., Johansson, P.-O. and Löfgren, K.-G. (1997), Welfare Measurement, Sustainability and Green National Accounting. Edward Elgar, Cheltenham, UK Asheim, G. B. (1978), Renewable Resources and Paradoxical Consumption Behavior. Ph.D. dissertation, University of California, Santa Barbara Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Cairns, R. D. and Long, N. G. (2006), Maximin: A direct approach to sustainability, Environment and Development Economics 11, 275–300 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P. S. and Heal, G. M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Dasgupta, S. and Mitra, T. (1999), On the welfare significance of national product for economic growth and sustainable development, Japanese Economic Review 50, 422–442 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556

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Hamilton, K. (1995), Sustainable development, the Hartwick rule and optimal growth, Environmental and Resource Economics 5, 393–411 Hanley, N., Shogren, J. F. and White, B. (2001), Introduction to Environmental Economics, Oxford University Press, Oxford. Hannesson, R. (1986), The effect of the discount rate on the optimal exploitation of renewable resources, Marine Resource Economics 3, 319–329. Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Pezzey, J. (1994), The optimal sustainable depletion of nonrenewable resources, Discussion Paper, University College London Pezzey, J. (2004), One-sided sustainability tests with amenities, and changes in technology, trade and population, Journal of Environment Economics and Management 48, 613–631 Pezzey, J. and Withagen, C. (1998), The rise, fall and sustainability of capital-resource economies, Scandinavian Journal of Economics 100, 513–527 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Spash, C. L. and Clayton, A. M. H. (1997), The maintenance of national capital: Motivations and methods’, in Light, A. and Smith, J. M. (eds.), Philosophy and Geography I: Space, Place and Environmental Ethics. Rowman & Littelfield, Lanham, Boulder, New York, and London Tietenberg, T. (2001), Environmental Economics and Policy, Third Edition. Addison-Wesley, Boston Toman, M. A., Pezzey, J. and Krautkraemer, J. (1995), Neoclassical economic growth theory and ‘sustainability’, in Bromley, D. W. (ed.), Handbook of Environmental Economics. Blackwell, Oxford and Cambridge, MA Vellinga, N. and Withagen, C. (1996), On the concept of green national income, Oxford Economic Papers 48, 499–514 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162. Withagen, C. and Asheim, G.B. (1998), Characterizing sustainability: The converse of Hartwick’s rule, Journal of Economic Dynamics and Control 23, 159–165 (Chap. 11 of the present volume)

CHAPTER 9 HARTWICK’S RULE IN OPEN ECONOMIES

Abstract. To sustain constant consumption the Hartwick rule prescribes reinvesting all resource rents in reproducible capital. This paper shows that the rule does not apply to open economies, since the underlying stationary technology assumption is violated when gains from trade are taken into account. A correct analog to the Hartwick rule for open economies is developed and applied to a model of capital accumulation and resource depletion. La règle de Hartwick dans les èconomies ouvertes. Pour maintenir la consommation constante, la règle de Hartwick prescrit qu’il faut réinvestir toutes les rentes des ressources dans du capital reproductible. Ce mémoire montre que la règle ne s’applique pas dans le cas d’économies ouvertes parce que le postulat d’une technologie stationnaire n’est pas satisfait quand on tient compte des gains engendrés par le commerce international. L’auteur développe une version correcte pour les économies ouvertes dans l’esprit de la règle de Hartwick et l’applique à un modèle d’accumulation de capital et d’épuisement des ressources.

1. INTRODUCTION The (ordinary) Hartwick rule (Hartwick, 1977; Dixit et al., 1980) gives a sufficient condition for constant consumption (or utility) in a closed economy with constant population and a stationary technology: Reinvest all rents from the flow of resource depletion in reproducible capital. Some writers seem to indicate that the Hartwick rule is also relevant to an open economy whose reproducible capital is defined to include foreign assets: Hartwick (1977) refers to it as a ‘Saudi Arabian’ rule, while Hoel (1981) obtains it for a single resource-exporter. It is the purpose of this paper to show that the Hartwick rule does not apply to open economies, since the stationary technology assumption is violated when gains from trade are taken into account in a general equilibrium setting. In particular, a resource-rich economy need not reinvest all resource rents in domestic and foreign assets in order to sustain constant consumption. We start out in the following section by developing a correct analog to the Hartwick rule for open economies. Sections 3 and 4 apply this result to a model of capital accumulation and resource depletion that Solow (1974) analysed. Proofs are contained in an appendix.

Originally published in The Canadian Journal of Economics/Revue canadienne d’ Economique, 19 (3) (1986), 395–402. Reproduced with permission from Blackwell.

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2. THE GENERAL FRAMEWORK We consider a constant population economy. Its stationary technology is described ˙ where consumption by a convex cone Y consisting of feasible triples (x, k, k), x is a non-negative scalar, and where k denotes a non-negative vector of capital stocks. Since production possibilities show constant-returns-to-scale (CRS), we shall assume that k includes human capital components’ being employed as labour. If human capital cannot be accumulated or depleted, the corresponding components of k˙ are zero. The components of k that represent non-renewable resources can only be depleted, and the corresponding components of k˙ must be non-positive.  ∞ A feasible path (xt  , kt  , k˙t )∞ t=0 has competitive present value prices ( pt , qt )t−0 iff P

 for each t, pt > 0 and (xt  , kt  , k˙t ) maximizes instantaneous profit pt x

˙ ∈ Y. + q˙ t k + qt k˙ subject to (x, k, k) The imputed rents to the assets are equal to −q˙ t , measuring the marginal productivity of the capital stocks. For resources that are unproductive as stocks, the corresponding components of −q˙ t are zero (i.e., the Hotelling rule). The corresponding components of qt equal the profits or rents that arise when resources are depleted. They measure the marginal productivity of the resource flows. Note that as a consequence of CRS pt xt = −d (qt kt  ) dt. m Following Burmeister and Hammond (1977), (xt m , kt m , k˙t )∞ t=0 prices ( pt , qt )∞ t=0 is called a regular maximin path (RMP) iff M1 xt m = x m > 0 (constant) for all t

(1) with competitive

M2 qt kt m → 0 as t → ∞. An RMP is efficient and maximizes inft xt over the collection of feasible paths (see Burmeister and Hammond, Theorem 2). Condition
∞
∞ M2 along with (1) and M1 imply that qt kt m equals t ps ds · x m , where 0 < t ps ds < ∞. Hence,  ∞ x m = qt kt m / ps ds. (2) t m Dixit et al. (1980) shows that RMP satisfy a generalized Hartwick rule (qt k˙t is constant). They also demonstrate that with free disposal and an absence of ‘stock m reversa’, RMP obey even the ordinary Hartwick rule: qt k˙t = 0 for all t. With CRS, the ordinary Hartwick rule implies that consumption is equal to the productive contributions of the capital stocks:

x m = −q˙ t kt m / pt .

(3)

If the RMP is stationary (k˙t = 0 for all t), (2) and (3)
are identical in the sense
∞ ∞ that −q˙ t / pt = qt / t ps ds, or equivalently −q˙ t = ( p/ t ps ds)qt . The reason is that such a path defines a price system with one constant interest rate equalling m

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− p˙ t / pt , the instantaneous rate of consumption return. Thus, −q˙ t = (− p˙ t / pt )qt , and
furthermore, since pt is an exponentially decreasing function, − p˙ t / pt equals ∞ pt / t ps ds. (The latter term may be interpreted as the rate of constant consump
∞ tion return, since t ps dspt is the value of a consumption annuity in terms of current consumption.) However, in the interesting case where the economy depletes resources, the RMP will not be stationary. Hence, the own rates of interest will vary
between different capital stocks, and furthermore, − p˙ t / pt will not equal ∞ pt / t ps ds. Consequently, (2) and (3) are not in general identical – as the model of the following section will serve to illustrate – even though in closed economies they give rise to the same RMP. That (2) applies on a disaggregated level is stated in the following theorem. Theorem 1: Within the framework of this section, an RMP can be realized in a decentralized competitive economy, where all agents (defined as subpopulations of constant size) choose efficient maximin paths, if every agent i observes the following savings rule:  S

For each t, x = i

qt kit /

∞

ps ds, t

where x i and kit are agent i’s positive and constant consumption and semi-positive vector of capital stocks. To see why (3) – which is the Hartwick rule in a CRS economy – cannot apply on a disaggregated level, consider an agent who is endowed solely with resources that are unproductive as stocks. By (3) this agent would not be allowed positive consumption. By S, however, positive and constant consumption is feasible if the agent’s wealth is positive.

3. A MODEL OF CAPITAL ACCUMULATION AND RESOURCE DEPLETION Consider a model in which the flow of a non-renewable resource, −k˙r , is combined with constant human capital, kh , and reproducible capital, kc , in order to produce a consumption good, x. By letting its technology be described by y = x + k˙c
(kh¯ )(1−a−b) (kc )a (−k˙r )b , a > b > 0, a + b < 1, this model fits into the framework of the preceding section. For given positive initial stocks, an efficient path with positive and constant consumption is feasible (Solow 1974), and satisfies the ordinary Hartwick rule, (Hartwick 1977). This path is an RMP along which kc continuously substitute kr . It is characterized by pt = qct > 0,

(4)

(1 − a − b)y m = −q˙ht kh m / pt ,

(5)

ay = −q˙ct kct / pt ,

(6)

m

m

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by m = −qr k˙r t / pt , m

(7)

y m = x m /(1 − b) = k˙c /b > 0,

(8)

qht → 0

(9)

m

and

krmt

→ 0 as

t → ∞,

where variables without subscript t are constant. The competitive conditions (4)–(7) are implied by P, while (8) and (9) obtain from M1 and M2. In this model an intuitive interpretation of the Hartwick rule is as follows: Consume the productive contributions, (5) and (6), of human and reproducible capital. While the share of gross output, (7), attributable to the flow of resource depletion, is not produced by the current generation and should be reinvested in reproducible capital. From (4), (6), and (8) it follows that: 

∞

pt /

ps ds = (a − b)y m /kct m < ay m /kct m = − p˙ t / pr

(10)

t

Thus, since the continuous transformation of kr into kc implies a falling rate of productivity of kc , the rate of constant consumption return is smaller than the instantaneous rate of consumption return, reflecting the dwindling reinvestment possibilities.
∞ We now see that (2) and (3) are not identical, since −q˙ t = ( pt / t ps ds)qt :   −q˙ct = (− p˙ t / pt )qct > pt −q˙r t = 0 <

  pt

∞

 ps ds qct ,

t

∞

 ps ds qr t .

t

Hence, it follows from the theorem that the Hartwick rule does not apply to the individual agents, if the modelled economy is decentralized, with each agent choosing an efficient maximin path. Proposition 1: Let the RMP in the model described in this section be realized in a decentralized competitive economy by each agent observing S. Then   i qt k˙ t = bpt y m kct i /kct m − kr t i /kr t m .

(11)

Hence, agents whose shares of the total stock of the resource do not coincide with their shares of the total stock of reproducible capital will choose to violate the Hartwick rule. Suppose each agent owns only one kind of capital. Then Proposition 1 enables us to partition the agents into three permanent classes – workers (H ), capitalists (C),

HARTWICK’S RULE IN OPEN ECONOMIES

151

and resource owners (R) – with each class choosing an efficient maximin path:  m i
i∈H kt i = kh m , 0, 0 ⇒
i∈H qt k˙t = 0 = qht k˙ht , since kht m = kh m (all t).

 m i
i∈C kt i = 0, kct m , 0 ⇒
i∈C qt k˙t = bpt y m = qct k˙ct by (4) and (8).  i
i∈R kt i = 0, 0, kr t m ⇒
i∈R qt k˙t = −bpt y m = qr k˙rmt by (7). Workers and resource owners consume their constant shares of gross output. Capitalists, however, can only consume a fraction of their stock’s productive contribution if their consumption is to be held constant. The remaining fraction, b/a, must be used for net accumulation of reproducible capital in order to compensate for its falling rate of productivity. Hence, in contrast with the intuition that the Hartwick rule provides, it is the capitalists’ – not the resource owners’ – responsibility to reinvest resource rents if the economy at large adopts a maximin criterion. 4. TRADE BETWEEN ECONOMIES WITH DIFFERING RESOURCE ENDOWMENTS Let the agents in the closed economy modelled in Sect. 3 correspond to open economies in a competitive world economy. Each open economy has to its disposal the same CRS technology as the aggregate economy and must employ its constant human capital, kh i , domestically. Its asset management problem is to choose paths for kct i and kr t i for given initial holdings. Competitive world markets insure overall productive efficiency. Owing to identical CRS technologies, such efficiency requires that kh i /kh m equal the relative size of the domestic economy where kct m kh i /kh m m of reproducible capital and −k˙r t kh i /kh m of resource flow are combined with kh i in order to produce y m kh i /kh m . We may therefore interpret kct i − kct m kh i /kh m i m as financial assets on an international capital market and −k˙r t + k˙r t kh i /kh m as resource exports. It follows from Proposition 1 that a resource-rich economy (defined by the property that kr t i /kr t m > kct i /kct m ) following a maximin criterion will choose to violate the Hartwick rule. The reason is that the rising resource price, qr / pr , implied by the Hotelling rule, improves its terms of trade. From a resource-rich economy’s point of view, the ‘technology’ – including gains from trade – facing future generations will be more favourable than the one facing the current generation. The current generation will therefore not undermine future generations’ consumption possibilities by neglecting to reinvest all resource rents in domestic and foreign assets. This explains intuitively why the Hartwick rule does not apply to open economies: The stationary technology assumption underlying the Hartwick rule is not satisfied when gains from trade are taken into account in a general equilibrium setting. These conclusions are highlighted by investigating an open economy with zero financial assets on the international capital market. Then allowable consumption consists of the economy’s net domestic product plus the value of its resource exports

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(positive if and only if the economy is resource-rich) (12), while capital accumulation in value equals the resources it consumes (13). Proposition 2: Consider an open economy choosing an efficient maximin path in a competitive world economy following the RMP described in the preceding section. If kct i = kct m kh i /kh m for all t, then  i m (12) x i = (1 − b)y m kh i /kh m + (qr / pt ) − k˙r t + k˙r t kh i /kh m , where the latter term is positive iff kr t i /kr t m > kct i /kct m . Furthermore,  i m k˙c = (qr /qct ) − k˙r t kh i /kh m .

(13)

Hence, in a world economy adhering to a maximin criterion, it is the resourceconsuming economies’ – not the resource-producing economies’ – responsibility to transform a declining resource base into reproducible capital. A resource-rich economy can therefore – within this specific framework – allow itself to use revenues arising from resource exports to finance additional consumption. APPENDIX m Proof of Theorem 1. Let (xt m , kt m , k˙t )∞ t=0 be an RMP with competitive prices i ∞ i ∞ i ˙ ( pt , qt ) . Let (xt , kt , kt ) be determined by S and t=0

t=0

pt xt i = −q˙ t kt i − qt k˙ it ,

(A1)

given some k0
k0 We have that xt is constant:    ∞ dxt i /dt = d qt kt i ps ds dt by S i

m.

i

t

 

  dt + qt kt i pt = d qt kt i

  

∞ t

  t i = − pt x t + pt x t

∞

∞

ps ds

ps ds t

ps ds = 0 by (A1) and S.

t i The path (x i , kt i , k˙t )∞ t=0 is therefore efficient and maximizes inft x t since lim
t i inft→∞ 0 ps (xs − x t )ds
0 for any path (xt , kt , k˙ t )∞ t=0 satisfying k0 = k0 and the
budget constrainst p
t xt
−q˙ t kt − qt k˙ t : t t i j 0 ps (x s − x )ds
0 [d(qs ks − qs ks )/ds]ds by (A1) and the constraint i i
qt kt , since k0 = k0 , kt  0, and, with free disposal, qt  0.  t ps (xs − x i )ds
limt→∞ qt kt i lim inft→∞ 0



= limt→∞ x





∞

j

ps ds t

= 0 by S.

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HARTWICK’S RULE IN OPEN ECONOMIES

Provided
j k0 i = k0 m , the RMP can thus be realized as a competitive equilibrium, since
j x i = x m by (2) and S, and the agents can manage their asset portfolios so that
j kt i = kt m (all t) by (1), (2), (A1), and S. Proof of Proposition 1. From (5), (7), and (9) it follows that  ∞  qht kh m = (−q˙hs )ds · kh m = (1 − a − b)y m t



qr kr t m = qr

∞

ps ds t



∞

(−kr s m )ds = by m

t

∞

ps ds. t

Thus, by applying S as well as (4) and (10), we obtain   x i = y m (1 − a − b)kh i /kh m + (a − b)kct j /kct m + bkr t i /kr t m .

(A2)

By (A1) and (A2), (11) follows, since (5), (6), and qr t = qr (all t) imply −q˙ t kt i = pt y m [(1 − a − b)kh i /kh m + akct i /kct m ]. i = k˙ m k i /k m and Proof of Proposition 2. Since kct i = kct m kh i /kh m (all t), k˙ct ct ct ct i m qct k˙ct = qct k˙ct kct i /kct m = bpt y m kct i /kct m

(A3)

by (4) and (8), while (11), kht i = kh i (all t), and (7) imply qr k˙ri t = −bpt y m kr t i /kr t m = qr k˙rmt kri t /kr t m .

(A4)

Furthermore, if kct i = kct m kh i /kh m (all t) is combined with (A2), we obtain x i = (1 − b)y m kh i /kh m + by m (kr t i /kr t m − khi /khm ). Applying (A4) and (7) to this expression yields (12), where sgn (−k˙ri t + k˙rmt kh i /kh m ) = sgn (kr t i /kr t m − kct i /kct m ), since −k˙ri t = −k˙rmt kr t i /kr t m , by (A4), i = k˙ i (all t) follows from (A3) and k i = k m k i /k m (all and −k˙rmt > 0. That k˙ct ct ct h h c t), which combined with (7) imply (13). REFERENCES Burmeister, E. and P. Hammond (1977) ‘Maximin paths of heterogeneous capital accumulation and the instability of paradoxical steady states.’ Econometrica 45, 853–70 Dixit, A., P. Hammond, and M. Hoel (1980) ‘On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion.’ Review of Economic Studies 47, 551–6 Hartwick, J. (1977) ‘Intergenerational equity and the investing of rents from exhaustible resources.’ American Economic Review 66, 972–4 Hoel, M. (1981) ‘Resource extraction by a monopolist with influence over the rate of return of nonresource assets.’ International Economic Review 22, 147–57 Solow, R. (1974) ‘Intergenerational equity and exhaustible resources.’ Review of Economic Studies (Symposium), 29–45

CHAPTER 10 CAPITAL GAINS AND NET NATIONAL PRODUCT IN OPEN ECONOMIES

Abstract. It is shown within a given framework that, with the world economy following an egalitarian path, the aggregate capital gains being positive is equivalent to the interest rate tending to decrease. This result is of importance for the concept of net national product in open economies. In particular, with positive aggregate capital gains, an open economy cannot sustain consuming the return on its capital stocks, since some part of the return must be used to augment the country’s national wealth. It is established that a country’s share of world-wide sustainable consumption equals its share of worldwide wealth.

1. INTRODUCTION In the aftermath of the World Commission on Environment and Development (WCED), there has been forthcoming a line of theoretical contributions1 suggesting how to correct the notion of net national product (NNP) so that depletion of natural and environmental resources are taken properly into account. Most, if not all, of these contributions are based on Weitzman (1976). Weitzman shows that NNP can serve as an indicator of welfare in a closed economy with a constant population and with no exogenous technological progress. If xt is consumption (being an indicator of the quality of life), kt is a vector of capital stocks, and Qt are competitive prices of the capital stocks in terms of current consumption, then this welfare indicator is given by xt + Qt k˙ t . Hence, NNP includes current consumption and the value of net ˙ t kt , are not included. investments; capital gains, Q In line with the emphasis of the WCED on the concept of a sustainable development, it would seem, however, desirable that the concept of NNP could serve as an indicator of sustainability. This would amount to, following Hicks (1946, Chap. 14), requiring that the NNP should measure what can be consumed in the present period without reducing future consumption possibilities.2 In other words, the NNP should equal the maximum consumption level that can be sustained. In Asheim (1994), I argue that even with the facilitating assumptions of a closed economy, a constant population, and no exogenous technological progress, NNP defined as xt + Qt k˙ t is not in general an exact indicator of sustainability, except in the uninteresting case with only one capital good.

Originally published in Journal of Public Economics 59 (1996), 419–434. Reproduced with permission from Elsevier.

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At the least, however, one would like the NNP to equal consumption if consumption happens to be constant and at any time equal the maximum level that can be sustained. That NNP := xt + Qt k˙ t has this property in a closed economy with a constant population and with no exogenous technological progress, follows from the Hartwick rule Hartwick (1977); Dixit et al. (1980) since Qt k˙ = 0 at any time along such an egalitarian path. Hence, even though NNP := xt + Qt k˙ t does not in general indicate the maximum sustainable consumption level, it does equal the maximum sustainable level along an efficient consumption path that happens to be egalitarian. If the Weitzman–Hartwick concept of NNP is being used in an open economy, then it implies that an economy living solely by harvesting nonrenewable resources has NNP equal to zero:3 xt = Qt (−k˙ t ), where (−k˙ t ) is the vector of extraction. Dasgupta (1990, footnote 24) claims that NNP in such an economy is equal to zero, but admits that this is paradoxical. The reason why this is paradoxical is that, with increasing resource prices on the world market, the economy’s “technology” (when taking into account its terms-of-trade) will not be constant.4 The economy may therefore be able to sustain a positive stationary level of consumption, and – if NNP is to serve as an indictor of sustainability – may have a positive NNP. Hence, capital gains cannot be excluded when the closed world economy is split into the open economies that the separate countries represent. At an opposite extreme, each country could include capital gains fully by letting ˙ t kt .5 This means that in an economy living solely by NNP be given by xt + Qt k˙ t + Q harvesting nonrenewable resources (i.e., xt = Qt (−k˙ t )), NNP would exactly equal ˙ t kt . If the NNP is being consumed (i.e., xt = xt + Qt k˙ t + Q ˙ t kt ) the capital gains, Q ˙ ˙ it follows that the national wealth is constant ((d/dt)[Qt kt ] = Qt kt + Qt kt = 0). This points to a more general result: If the goal is to keep the national wealth nondecreasing, then a concept of NNP which includes capital gains would indicate the maximum allowable level of consumption. The above yields a paradox. If each country wants to keep its national wealth constant, consumption equals a measure of NNP that includes capital gains. However, if we add all countries together to form a closed world economy and assume that the purpose is to keep the level of consumption constant, then consumption equals a measure of NNP that does not include capital gains. It is the purpose of this paper to resolve this paradox. In Sect. 3, it is established as Propositions 1 (and 2) that if the aggregate capital gains are positive along an efficient (and egalitarian) path, then the interest rate tends to decrease. This means that it is necessary for each country to accumulate national wealth in order to keep the consumption constant. In particular, an economy living solely by harvesting nonrenewable resources can consume only a part of the capital gains if it wants to sustain its consumption level; the rest must be used to accumulate national wealth in order to compensate for the decreasing rate of return. As the central result of the paper, Proposition 3 presents a measure-based on current prices (and price changes) – of any country’s sustainable consumption in a world economy that implements an egalitarian path by having each country keep consumption constant. This result is

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illustrated in Sect. 4 by a constant population, stationary technology model of capital accumulation and resource depletion analyzed by Solow (1974). It is an interesting feature of Solow’s (1974) model that along an egalitarian path, world-wide wealth is increasing, reflecting positive aggregate capital gains as well as a decreasing interest rate. Using this model, a numerical example is provided in Sect. 5 by considering a two-country world. The formal analysis – being based on Dixit et al. (1980) and Asheim (1986) – as well as all proofs are relegated to an appendix. Section 2 provides introductory intuition, while Sect. 6 concludes by relating the present results to the analyzes of other contributions. 2. INTUITION The following provides an intuition for the general result (of Propositions 1 and 2) that the existence of positive aggregate capital gains is equivalent to the interest rate tending to decrease. Assume that output y depends on manmade capital kc and a resource flow −k˙r and can be split into consumption x and investment in manmade capital k˙c : y = x + k˙c = F(kc , −k˙r ). Assume that F exhibits constant-returns-toscale (CRS), and that the (nonrenewable) resource is unproductive as a stock. Let the consumption (or rather, composite) good serve as numeraire. In a competitive equilibrium, the resource price Q r equals the marginal productivity of the resource flow, F2 (kct , −k˙r t ) = Q r t , while the interest rate i measures the marginal productivity of manmade capital and equals, with no profitable arbitrage opportunities, the growth rate of the resource price, F1 (kct , −k˙r t ) = i t = Q˙ r t /Q r t . With the consumption good as numeraire, aggregate capital gains equal Q˙ r t kr t . It follows that, for aggregate capital gains to be positive, it is sufficient that the marginal productivity of manmade capital and the value of the resource stock both be positive. Let c(i, Q r ) denote the minimum cost of producing one unit of output, c(i, Q r ) := min{ikc + Q r (−k˙r ) | F(kc , −k˙r ) = 1}. Given the CRS, and with the consumption good as numeraire, it follows that c(i, Q r ) = 1 in a competitive equilibrium. This is a zero-profit condition for the competitive firms of the CRS economy. The equation c(i, Q r ) = 1 defines a factor price contour, which – by a standard result of duality theory – is strictly decreasing given that the cost minimizing pair of capital stock and resource flow is strictly positive. Hence, an increasing Q r t , is equivalent to a decreasing i t . It follows that, in the particular model considered, a competitive equilibrium entails both positive aggregate capital gains and a decreasing interest rate. Within a similar framework, this conclusion is supported by van Geldrop and Withagen (1993). While retaining the assumption of CRS, the equivalence of positive aggregate

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capital gains and a decreasing interest rate will be shown under general technological assumptions in Propositions 1 and 2. The simple model of the present section serves as a two-fold warning: – For the analysis of a competitive equilibrium in a resource economy, one should not carelessly assume a constant interest rate since this assumption may well be inconsistent with the competitiveness of the economy. The formal analysis of the present paper is therefore performed in a framework allowing for a nonconstant interest rate. – An open economy that trades in resources should be analyzed in a general equilibrium setting. As a consequence, the present paper discusses the concept of NNP in open economies using general equilibrium prices. 3. RESULTS The general framework in which the analysis is embedded is presented formally in the appendix, where it is assumed – in addition to a constant population and no exogenous technological progress – that the technology of the closed world economy shows CRS. This is appropriate here since it enables: (1) Consumption to be split into the productive contributions of the factors of production. (2)

The world economy to be split into open economies (countries) with subpopulations of constant size.

Each of these countries obtains a fraction of the total consumption that equals the productive contributions of the factors of production it owns.6 The world economy is assumed to be competitive. For each country’s national wealth to comprise the full productive capabilities of its factors of production, it is necessary to treat all factors of production, also labor, as capital goods, the market prices of which correspond to the present value of future earnings. That is, labor corresponds to a vector of human capital, which may include accumulated knowledge from learning and research activities. Let (xt∗ , k∗t , k˙ ∗t )∞ t=0 denote a competitive equilibrium of the world economy with capital prices (Qt )∞ t=0 , where the consumption good serves as numeraire. The CRS imply that consumption exactly suffices to pay capital owners the marginal products of their capital stocks plus the profits (resource rents) that arise stocks are depleted: xt∗ = Rt k∗t + Qt (−k˙ ∗t ), where Rt is a vector measuring the marginal productivities of the capital goods as stocks.7 Consequently, in aggregate, the Weitzman–Hartwick concept of NNP exactly suffices to pay the capital owners the marginal products of their capital stocks: xt∗ + Qt k˙ ∗t = Rt k∗t . From the arbitrage equation, i t Qt = ˙ t , where i t denotes the instantaneous consumption interest rate, it follows that Rt + Q the instantaneous return on the capital stocks equals the sum of marginal products ˙ t k∗t . Hence, with of the stocks plus the aggregate capital gains: i t Qt k∗t = Rt k∗t + Q

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˙ t k∗t > 0, i t Qt k∗t > Rt k∗t . This observation reflects that, with positive aggregate Q capital gains, the instantaneous interest rate tends to decrease in the precise sense given below. Proposition 1: Within the given framework, if the competitive world economy implements an efficient path (xt∗ , k∗t , k˙ ∗t )∞ prices (Qt )∞ t=0 with capital t=0 , then

∞ ∞ ∗ ∗ ∗ ∗ ˙ Qt kt > 0 is equivalent to (i t Qt kt =) i t · t ps xs ds > t i s ps xs ds (= Rt k∗t ), where ( pt )∞ t=0 is the (implicit) consumption discount factor. If the world economy at any time consumes the maximum sustainable level, it will ˙m ∞ be assumed that such a maximin path is efficient.8 Let (x m , km t , kt )t=0 denote the m m efficient maximin path. The Hartwick rule implies that x = x + Qt k˙ m t , the CRS m , and the arbitrage equation implies that R km = = R k imply that x m + Qt k˙ m t t t t t m m ˙ m [Rt km t /(Rt kt + Qt kt )] · i t Qt kt . Hence, the maximum sustainable consumption m m ˙ m level x m = [Rt km t /(Rt kt + Qt kt )] · i t Qt kt falls short of the instantaneous return m ˙ t km on the capital stocks i t Qt kt if and only if the aggregate capital gains Q t are positive. As the following specialization of Proposition 1 shows, this observation also reflects the result that interest rates tend to decrease whenever aggregate capital gains are positive. Proposition 2: Within the given framework, if the competitive world economy ∞ ˙m ∞ implements an efficient path (x m , km t , kt )t=0 with capital prices (Qt )t=0 , then the instantaneous consumption interest rate i t exceeds the infinitely long term consump˙ t km tion interest rate9 if and only if the aggregate capital gains Q t are positive. Since, with the world economy implementing an efficient maximin path, the aggregate capital gains being positive is equivalent to the interest rate tending to decrease, a country j choosing an efficient maximin path cannot allow itself fully to consume the j j ˙ t ktj , if the aggregate instantaneous return on its capital stocks, i t Qt kt = Rt kt + Q capital gains are positive. This is stated as the next proposition. Proposition 3: Within the given framework, a worldwide efficient maximin path can be implemented in a decentralized economy where all countries choose efficient maximin paths, if and only if the consumption of each country j is given by: j

m For a.e. t, x j = Rt km t · (Qt kt /Qt kt ) j m ˙ m ˙ j = [Rt km t /(Rt kt + Qt kt )] · (Rt kt + Qt kt ) , j

where x j and kt are country j’s positive and constant consumption and semipositive vector of capital stocks. Proposition 3 supports the intuition that Hicksian income from a country’s ownership of capital depends on the value of its capital only; portfolio changes not influencing the value of its capital do not matter.10 In fact, since x m = Rt km t , Proposition 3 shows that a country’s share of worldwide sustainable consumption equals its share of worldwide wealth. Moreover, for given aggregate capital gains there is no reason

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for each separate country to differentiate between the return on capital caused by the productivity of its stocks and the return caused by capital gains. Proposition 3 implies that if an efficient maximin path is implemented and the aggregate capital gains are positive, then some part of the instantaneous return on a country’s capital stocks must be used to augment the country’s national wealth. Proposition 4: Within the given framework, if a worldwide maximin path is implemented in a decentralized economy by having all countries choose efficient maximin j paths, then each country j’s national wealth Qt kt is increasing if and only if the m ˙ t kt are positive. aggregate capital gains Q This result reflects the fact that, with positive aggregate capital gains, the interest rate tends to decrease; hence, this diminishing rate of return must be compensated by accumulating national wealth. It follows, however, from Proposition 3 that, since m ˙ m [Rt km t /(Rt kt + Qt kt )] > 0, even a country that is endowed solely with resources that are unproductive as stocks can sustain a positive level of consumption. 4. ILLUSTRATION In order to show that the results of Sect. 3 are not vacuous, a much used model of manmade capital accumulation and natural capital (resource) depletion – in which the aggregate capital gains are positive along an efficient maximin path – is presented as an illustration. Consider a model in which a flow of a nonrenewable resource −k˙r , is combined with constant human capital kh and manmade capital kc in order to produce a consumption good x. By letting its technology be described by y := x + k˙c ≤ (kh )1−a−b (kc )a (−k˙r )b , b < a < a + b < 1, this model fits into the given framework. As the analysis of Solow (1974) shows, a positive and constant consumption can be sustained indefinitely by letting accumulated manmade capital m m m substitute for a diminishing resource extraction. Let km t = (k h , kct , kr t ) denote the capital vector along such an efficient maximin path, with x m being the corresponding consumption level, where variables without subscript t are constant. The investment in manmade capital turns out to be constant along this path, implying that total output y m is constant and is split between consumption x m = (1 − b)y m and investment in manmade capital k˙cm = by m . The world economy can be divided into open economies (countries) by letting each country have at its disposal the same CRS technology as the aggregate economy. Assume no labor mobility, implying that each country j must employ its human j j capital kh domestically. Its asset management problem it to choose paths for kct and j kr t for given initial holdings. Competitive world markets with perfect capital mobility and free trade in the resource and the consumption good ensure overall productive j efficiency. Owing to identical CRS technologies, such efficiency requires that kh /khm m k j /k m of manmade capital equal the relative size of the domestic economy where kct h h j j j and −k˙rmt kh /khm of resource flow are combined with kh in order to produce y m kh /khm .

CAPITAL GAINS AND NET NATIONAL PRODUCT IN OPEN ECONOMIES j

161

j

m k /k m is j’s financial assets on an international In this multi-country world, kct − kct h h j j capital market −k˙r t + k˙rmt kh /khm is j’s resource exports. What is the consumption of each country j when a worldwide maximin path is implemented by having all countries choose efficient maximin paths? By combining (A2), (5), (6), and (7) of Asheim (1986), it is obtained that the consumption of j j j j country j owning the vector kt = (kh , kct , kr t , ) at time t is given by: j

j

x j = Rh k h +

a−b k j · Rct kct + Q r t (−k˙rmt ) rmt . a kr t

(*)

Hence, country j, on an efficient maximin path, will be consuming the marginal product of its human capital and resource rents in proportion to its stock of the resource, but only a fraction (a − b)/a of the marginal product of its manmade capj j j j ital. Since the CRS imply y j := x j + k˙ct = Rh kh + Rct kct + Q r t (−k˙r t ), it follows j j j j from (*) that k˙ct = (b/a) · Rct kct + Q r t (−k˙r t + k˙rmt kr t /krmt ). Thus, each country ends up reinvesting resource rents in proportion to its ownership of the total stock of manmade capital, provided that countries extract the resource in proportion to their m = by m = Q (−k˙ m ), a country owning the stocks. In particular, since (b/a) · Rct kct rt rt whole stock of manmade capital would be investing in manmade capital at a level equal to the sum of worldwide resource rents. Alternatively, a country owning the whole resource stock and no manmade capital would be using all resource rents for consumption. These conclusions will be verified by the simple two-country example of Sect. 5. Proposition 3 above sheds new light on these results. In the model considered, the instantaneous interest rate equals the marginal productivity of manmade capital; m . Furthermore, the interest rate is decreasing along the efficient hence, i t = ay m /kct maximin path since total output y m is constant and manmade capital is accumulated: m = k m + by m t. Consequently, by Proposition 2, the term [R km /(R km + Q ˙ t km kct t t t t t )] c0 11 of Proposition 3 is smaller than 1; in fact, it equals (a − b)/a. Hence, if country j j j j j owns the vector kt = (kh , kct , kr t , ) at time t in a world economy implementing a worldwide maximin path by having all countries choose efficient maximin paths, then xj =

a−b ˙ t )ktj , · (Rt + Q a

(**)

with calculations (e.g., based on (4)–(10) of Asheim, 1986) showing that Rh = (1 − a − b)y m /khm , Q˙ ht = [b/(a − b)](1 − a − b)y m /khm (i.e., [(a − b)/a] · (Rh + m, Q ˙ ct = 0, Rr t = 0, and Q˙ r t = Q˙ ht ) = (1 − a − b)y m /khm = Rh ), Rct = ay m /kct m m [a/(a − b)] · by /kr t . Although this conclusion derived from Proposition 2 confirms the results given by (*), new interpretations can be made: (1)

The owners of human capital can consume exactly the marginal product of their stocks since the capital gains exactly make up for the declining interest rate.

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GEIR B. ASHEIM

(2)

The owners of manmade capital enjoy no capital gains and can consequently consume only a fraction (a − b)/a of the marginal product of their stocks.

(3)

The resource has no marginal productivity as a stock; the owners of the resource can still consume a fraction (a − b)/a of the capital gains.

The last observation reiterates a point made earlier, viz. that a country endowed solely with a resource that is unproductive as a stock (e.g., oil) can sustain a positive consumption level indefinitely by consuming a fraction of the capital gains at each date. 5. A TWO-COUNTRY EXAMPLE Consider a two-country numerical example where human capital is distributed evenly (kh1 = kh2 = 12 khm ), while only country 1 is endowed with the resource (kr1t = krmt , kr2t = 0). By choosing a = 0.20 and b = 0.16, it follows that (a − b)/a = 15 . Furthermore, [(a − b)/a] · (Rh + Q˙ ht ) = Rh = 0.64y m /khm , (Rct + Q˙ ct ) = Rct = m , (R + Q ˙ r t ) = Q r t = 0.80y m /krmt , and Q r t = 0.16y m /(−k˙rmt ). Table 1 0.20y m /kct rt can now be constructed. Note that maximin consumption varies to a small degree with ownership of manmade capital since (**) implies that the owners of manmade capital can consume only a fraction (a − b)/a = 15 of the marginal product of their stocks. Note also that a country owning the whole resource stock and no manmade capital (country 1 of Case 3) is increasing its financial debt along a maximin path. 6. CONCLUDING REMARKS National accounting seeks to measure income based on current prices (and price changes). Such an undertaking can easily be dismissed by the fact that all prices (in particular, prices for natural and environmental resources) are not available.12 This paper has sought to investigate some problems remaining even after assuming the availability of all prices. One such problem is discussed in Asheim (1994), where it is shown that the Weitzman–Hartwick concept of NNP does not in general serve as an indicator of sustainability. However, its ignoring of aggregate capital gains is appropriate for measuring consumption along a world-wide egalitarian path. Therefore, assume furthermore that a world-wide egalitarian path is implemented by each country keeping consumption constant at its national level. Under these assumptions, how can current prices (and price changes) be used to measure each country’s sustainable consumption? This is the central question posed by the present paper. The question is answered through Proposition 3 which establishes the following: (1) If there are no aggregate capital gains, it is appropriate for each country to fully include its own individual capital gains (arising for example from changing terms-of-trade) in the measure of sustainable consumption. There is no reason

0.68 0.52 0.16 0.50 1 m 2 kct 0.08 1 ˙m 2 (−kr t ) −0.10

Country 1

0.32 0.32 0 0.50 m − 12 kct −0.08 − 12 (−k˙rmt ) 0.10

Country 2

Case 1 1 = km 2 =0 kct kct ct 0.58 0.50 0.08 0.50 0 0 1 (− k˙rmt ) 2 −0.08

Country 1

j

0.42 0.34 0.08 0.50 0 0 − 12 (−k˙rmt ) 0.08

Country 2

Case 2 1 = 1 km 2 = 1 km kct kct 2 ct 2 ct

j

0.48 0.48 0 0.50 m − 12 kct −0.08 1 ˙m 2 (−kr t ) −0.06

Country 1

j

m + 1 · 0.80k /k m . Line (2) follows from (**) since x j /y m = 0.64kh /khm + 15 · 0.20kct /kct rt rt 5 Line (3) is obtained since (1) ≡ (2) + (3). j Line (4) follows since productive efficiency requires that kh /khm (= 12 for j = 1, 2) equal the relative size of the domestic economy. For the same reason, half of the stock of manmade capital must be employed in each country (implying the financial assets of line (5)), half of the investment in manmade capital must occur in each country (implying that the financial investment of line (6) follows from line (3)), and half of the extracted resource must be used in each country (implying the resource exports of line (7)). Line (8) follows from the identity (4) ≡ (2) + ((3) − (6)) + (8), or alternatively, (6) ≡ (8)+ Q r t (7) + Rct (5).

0.52 0.36 0.16 0.50 1 m 2 kct 0.08 − 12 (−k˙rmt ) 0.06

Country 2

Case 3 1 =0 2 = km kct kct ct

j j m + 0.16(−k˙ j )/(−k˙ m ). Line (1) follows from CRS since y j /y m = 0.64kh /khm + 0.20kct /kct rt rt

(1) y j /y m (2) x j /y m j (3) k˙c /y m (4) Domestic output/y m (5) Financial assets (6) Fin. investments/y m (7) Resource exports (8) Cons. good exp./y m

Table 10.1. CAPITAL GAINS AND NET NATIONAL PRODUCT IN OPEN ECONOMIES

163

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GEIR B. ASHEIM

for each separate country to differentiate between the return on capital caused by the productivity of its stocks and the return on capital caused by capital gains. (2) The latter observation holds even if there are aggregate capitals gains. However, then, the fraction of the return on capital that can be used for consumption along an egalitarian path must be adjusted for the aggregate capital gains. (3) In any case, a country’s share of world-wide sustainable consumption equals its share of world-wide wealth. These results seem to be novel. An antecedent is Asheim (1986), which, however, is not a contribution on national accounting, since sustainable consumption along an egalitarian path is not characterized by current prices (and price changes) only. Weitzman (1976), Solow (1986), Hartwick (1990), and Mäler (1991), who lay a foundation for a Weitzman–Hartwick concept of NNP that is adjusted for the depletion of natural and environmental resources, do not discuss the problems associated with applying this concept of NNP in open economies which are confronted with changing termsof-trade. Vellinga and Withagen (1996) analyze problems associated with defining NNP in an open economy with changing terms-of-trade. Looking at improving terms-of-trade as a kind of exogenous (or anticipated) technological progress also makes the analysis of Aronsson and Löfgren (1993) relevant. These contributions do not, however, present general equilibrium analyzes, in contrast to the present paper. van Geldrop and Withagen (1993) present a general equilibrium analysis of a multi-country world, but they are not concerned with national accounting. The paper that is most closely related to the present one is Sefton and Weale (1996) who discusses the national accounting of open economies in a general equilibrium setting with nonconstant interest rates. However, their main concern goes beyond the scope adopted here. The present paper defines NNP in open economies as the maximum sustainable consumption and measures this notion using the prices that exist if consumption at any time equals the maximum sustainable level. It does not address the question of how NNP can be measured in open economies that do hold consumption constant. As argued in the introduction, observable prices will not, in this case, yield a notion of NNP that satisfies the Hicksian (1946, Chap. 14) foundation by serving as an exact indicator of sustainability. An alternative would be to analyze how Weitzman’s (1976) welfare foundation of NNP can be extended to the case of open economies interacting within a world economy in a general equilibrium setting. It is towards the solution of this problem that Sefton and Weale (1996) make an interesting contribution. APPENDIX Based on Dixit et al. (1980) and recalling Asheim (1986), consider the following constant population economy. Its stationary technology is described by a smooth ˙ where consumption x (being convex cone Y consisting of feasible triples (x, k, k),

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an indicator of the quality of life) is a nonnegative scalar, and where k denotes a nonnegative vector of capital stocks. Assume that the set Y satisfies free disposal ˙ ∈ Y and k˙  ≤ k, ˙ then (x, k, k˙  ) ∈ Y . The comof investment flows; i.e., if (x, k, k) ponents of k that represents nonrenewable natural resources can only be depleted, and the corresponding components of k˙ must be nonpositive. For environmental resources that positively influence the quality of life the assumption of free disposal of investment flows means that these positively valued components of k can freely be destroyed; hence, this assumption implies that negatively valued waste products can freely be generated, not freely be disposed of. Since production possibilities show CRS, assume that k includes human capital components being employed as labor. Such components may measure accumulated knowledge from learning and research activities. Hence, as noted by Weitzman (1976), the assumption of a stationary technology does not exclude endogenous technological progress. ∞ A feasible path (xt∗ , k∗t , k˙ ∗t )∞ t=0 has competitive present value prices ( pt , qt )t=0 if and only if for each t, (xt∗ , k∗t , k˙ ∗t ) maximizes instantaneous profit pt x + qt k˙ + q˙ t k ˙ ∈Y. subject to (x, k, k)

(P)

The imputed rents to the assets are equal to −q˙ t , measuring the marginal productivity ˙ t is obtained by letting of the capital stocks. The arbitrage equation i t Qt = Rt + Q i t := − p˙ t / pt denote the instantaneous consumption interest rate and by letting Qt := qt / pt and Rt := −q˙ t / pt denote the prices and the marginal productivities, respectively, of the capital goods in terms of current consumption. For capital goods that are unproductive as stocks (e.g., nonrenewable natural resources), the corresponding components of −q˙ t are zero (i.e., the Hotelling rule). The corresponding components of qt equal the profits or rents that arise when such resources are depleted. (P) combined with the assumption of free disposal of investment flows implies that the vector qt is nonnegative. Note that as a consequence of CRS pt xt∗ = −d(qt k∗t )/dt .

(A1)

∞ A path (xt∗ , k∗t , k˙ ∗t )∞ t=0 with competitive prices ( pt , qt )t=0 is called regular if and only if

qt k∗t → ∞ as t → ∞ .

(N)

It is easily shown that a regular path is efficient. Following Burmeister and Hammond ˙m ∞ (1977), a regular path (x m , km t , kt )t=0 is a regular maximin path (RMP) if and only if xtm = x m > 0 (constant) for all t .

(M)

An RMP is efficient and maximizes inft xt over the collection of feasible paths (see Burmeister and Hammond, 1977, Theorem 2). Condition (N) along with (A1) and

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GEIR B. ASHEIM

(M) imply that qt km t equals


∞ t

ps ds · x m , where 0 <


∞ t

ps ds < ∞. Hence,

qt km xm =
∞ t . t ps ds

(A2)

Asheim (1986) establishes that (A2) applies on a disaggregated level: Lemma 1: Within the framework of the appendix, an RMP can be implemented in a decentralized competitive economy where all agents (defined as subpopulations of constant size) choose efficient maximin paths if and only if the consumption of each agent j is given by: j

qt k For a.e. t, x j =
∞ t , t ps ds

(S)

j

where x j and kt are agent j’s positive and constant consumption and semipositive vector of capital stocks. Proof. The “if ” part follows from the theorem of Asheim (1986). The “only if ” j j part follows from the proof of the theorem of Asheim (1986) since (x j , kt , k˙ t )∞ t=0 j j j defined by (S) and the budget constraint pt xt = −q˙ t kt − qt k˙ t is efficient and yields j constant consumption. Hence, if xt = xt for a nontrivial subset of time, then agent j does not choose an efficient maximin path. Dixit et al. (1980) show that an RMP satisfies a generalized Hartwick rule (qt k˙ m t is constant). They also demonstrate that under weak assumptions an RMP even obeys the ordinary Hartwick rule: qt k˙ m t = 0 for all t. With CRS, the ordinary Hartwick rule implies that consumption is equal to the productive contributions of the capital stocks: x m = −q˙ t km t / pt .

(A3)

To see why (A3) cannot apply on a disaggregated level, consider an agent who is endowed solely with resources that are unproductive as stocks. By (A3) this agent would not be allowed positive consumption. By (S), however, positive and constant consumption is feasible if the agent’s wealth is positive. These preliminaries suffice for proving Propositions 1–4. ˙ t = d(qt / pt )/dt = q˙ t / pt − ( p˙ t / pt ) · (qt / pt ); Proof of Proposition 1. Note that Q ˙ t k∗t > 0, (− p˙ t / pt ) · qt k∗t / pt > q˙ t k∗t / pt . By (N) and (A1), qt k∗t = hence, with Q
∞
∞ ∗ ∗ t ps x s ds; thus, the l.h.s. equals (− p˙ t / pt ) · t ps x s ds/ pt . With i t = − p˙ t / pt (equal to the instantaneous
∞ consumption interest rate), it remains to be shown that the r.h.s. equals t (− p˙ s / ps ) ps xs∗ ds/ pt . By (A1), −q˙ t k∗t = pt xt∗ + qt k˙ ∗t . Since pt x˙t∗ = −d(qt k˙ ∗t )/dt (from Dixit et al., 1980, proof of Theorem 1), it follows that d( pt xt∗ )/dt = p˙ t xt∗ − d(qt k˙ ∗t )/dt, d( pt xt∗ + qt k˙ ∗t )/dt = p˙ t xt∗ , and

CAPITAL GAINS AND NET NATIONAL PRODUCT IN OPEN ECONOMIES

167


∞ pt xt∗ + qt k˙ ∗t = t − p˙ s xs∗ ds, provided pt xt∗ + qt k˙ ∗t → 0 as t → ∞.13 Hence,
∞ −q˙ t k∗t / pt = ( pt xt∗ + qt k˙ ∗t )/ pt = t (− p˙ s / ps ) ps xs∗ ds/ pt . ˙ t km Proof of Proposition 2. The proof of Proposition 1 implies that Q t > m m 0 is
equivalent to (− p˙ t / pt ) · qt kt > −q˙ t kt . Since by (A2) and (A3), ∞ ˙ t km ˙ t km follows that with Q ( pt / t ps ds) · qt km t
= −q t , it now t > 0, i t = − p˙ t / pt >
∞ ∞ m m −q˙ t kt /qt kt = pt / t ps ds/ pt . As t ps ds is the price of a perpetual consump
∞ tion annuity in terms of current consumption, pt / t ps ds is the infinitely long term consumption interest rate.
∞ Proof of Proposition 3. (A2) and (A3) imply that pt / t ps ds =
m = R km /Q km ; hence, by Lemma 1, x j = ( p / ∞ p ds) · −q˙ t km t t t t t s t /qt kt t j m ) · Q k j or x j = R km · (Q k j /Q km ). From the /Q k (qt kt / pt ) = (Rt km t t t t t t t t t t t j m ˙ t , it follows that (Rt km arbitrage equation, i t Qt = Rt + Q t /Qt kt ) · Qt kt = j j j m m m ˙ m ˙ (Rt km t /i t Qt kt ) · i t Qt kt = [Rt kt /(Rt kt + Qt kt )] · (Rt kt + Qt kt ). j

j

Proof of Proposition 4. From country j’s budget constraint, xt = (−q˙ t kt − j j j qt k˙ t )/ pt = Rt kt − Qt k˙ t , it follows that the time derivative of country j’s j j ˙ t ktj + Qt k˙ tj = Rt ktj + Q ˙ t ktj − xtj . By wealth, Qt kt , is given by: d(Qt kt )/dt = Q j j j j m ˙ t kt − xt = [Q ˙ t km ˙ m ˙ j Proposition 3, Rt kt + Q t /(Rt kt + Qt kt )] · (Rt kt + Qt kt ); i.e., j ˙ t km d(Qt kt )/dt > 0 is equivalent to Q t > 0. Acknowledgments: I am grateful for comments and suggestions by John Hartwick, James Sefton and a referee as well as participants at an international symposium on Models of Sustainable Development, Paris, This research is partly financed by the Norwegian Ministry of Environment and by the Norwegian Research Council for Applied Social Science through the Norwegian Research Centre in Organization and Management. NOTES 1 See for example Hartwick (1990) and Mäler (1991). 2 “[I]t would seem that we ought to define a man’s income as the maximum value which he can consume

during a week, and still expect to be as well off at the end of the week as he was in the beginning” (Hicks, 1946, p. 172). 3 This holds if the rents that arise by extracting the nonrenewable resources are solely Hotelling rents. As implied by the analysis of Hartwick (1991), any Ricardian rents that arise will contribute to a positive NNP. 4 This observation has been made by Asheim (1986) and, more recently, by Sefton and Weale (1996). See also Svensson (1986). 5 This has been the point of departure in the Statistics Norway’s attempts to calculate the Hicksian income from the Norwegian petroleum resources (see for example, Aslaksen et al., 1990). In these efforts, the main objective has been, however, to analyze how such calculations can be made when petroleum prices are stochastic.

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6 In line with usual meaning of NNP, it is the ownership of factors that matters, not where they are

employed.

7 Note that the assumption of CRS entails that the profits that arise when nonrenewable resources are

depleted are solely Hotelling rents; any Ricardian rents that arise if the marginal cost of extraction exceeds the average cost accrue to other factors of production. 8 Furthermore, the price of a perpetual consumption annuity is assumed to be positive and finite along such a path. This corresponds to what in the appendix is called a regular maximin path, following the terminology of Burmeister and Hammond (1977). 9 If the (implicit) consumption discount factor decreases exponentially as a function of time, then there is one constant consumption interest rate. Otherwise, there is a term structure of consumption interest rates. The very short term interest rate is i, the instantaneous consumption interest rate. The other extreme is the infinitely long term consumption interest rate. The inverse of the latter equals the price of a perpetual consumption annuity in terms of current consumption. It can be shown that the infinitely long term interest rate is strictly decreasing if and only if the instantaneous interest rate exceeds the infinitely long term interest rate. 10 In contrast, if the Weitzman–Hartwick concept of NNP is being used by a country living solely by harvesting nonrenewable resources, then its NNP changes from zero to a positive value if its in situ resources are sold in return for ownership in foreign productive capital stocks. 11 By the proof of Proposition 3, the term in question is the infinitely long term interest rate , the instantaneous interest rate m. which equals (a − b)/a since the numerator can be shown to be (a − b)y m /kct 12 This does not rule out that practical steps for adjusting national accounts can be suggested; see for

example, El Serafy (1989). 13 Given (A1) and (N), a sufficient condition for this convergence is that the growth rate of each capital stock and price is bounded above. Proof of this claim is available on request from the author.

REFERENCES Aronsson, T. and Löfgren, K.-G. (1993), Welfare consequences of technological and environmental externalities in the Ramsey growth model, Natural Resource Modeling 7, 1–14 Asheim, G.B. (1986), Hartwick’s rule in open economies, Canadian Journal of Economics 19, 395–402 [Erratum 20, (1987) 177] (Chap. 9 of the present volume) Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Aslaksen, I., Brekke, K.A., Johansen, T.A. and Aaheim, A. (1990), Petroleum resources and the management of national wealth, in Bjerkholt, O., Olsen, O. and Vislie, J. (eds.), Recent modelling approaches in applied energy economics. Chapman and Hall, London Burmeister, E. and Hammond, P. (1977), Maximin paths of heterogeneous capital accumulation and the Instability of Paradoxical Steady States, Econometrica 45, 853–870 Dasgupta, P.S. (1990), The environment as a commodity, Oxford Review of Economic Policy 6, 51–67 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 El Serafy, S. (1989), The proper calculation of income from depletable natural resources, in Ahmad, Y.J. et al. (eds.), Environmental accounting for sustainable development, The World Bank, Washington, DC Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Hartwick, J.M. (1990), National resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291–304 Hartwick, J.M. (1991), Investing exhaustible resource rents for constant Hicksian income and wealth, Queen’s University Hicks, J. (1946), Value and capital, 2nd edition. Oxford University Press, Oxford Mäler, K.-G. (1991), National accounts and environmental resources, Environmental and Resource Economics 1, 1–5

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Sefton, J.A. and Weale, M.R. (1996), The net national product and exhaustible resources: The effects of foreign trade, Journal of Public Economics 61, 21–47 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Solow, R.M. (1986), On the intergenerational allocation of natural resources, Scandianavian Journal of Economics 88, 141–149 Svensson, L.E.O. (1986), Cormment on R.M. Solow, Scandinavian Journal of Economics 88, 153–155 van Geldrop, J. and Withagen, C. (1993), General equilibrium and international trade with exhaustible resources, Journal of International Economics 34, 341–357 Vellinga, N. and Withagen, C. (1996), On the Concept of Green National Income, Oxford Economic Papers 48, 499–514 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162

CHAPTER 11 CHARACTERIZING SUSTAINABILITY: THE CONVERSE OF HARTWICK’S RULE

CEES WITHAGEN Department of Economics, Tilburg University P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands Department of Economics, Free University, De Boelelaan 1105 1081 HV Amsterdam, The Netherlands Email: [email protected]

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern, NO-0317 Oslo, Norway Email: [email protected]

Abstract. This note offers a general proof of the converse of Hartwick’s rule, namely that – in an economy with stationary instantaneous preferences and a stationary technology – an efficient constant utility path is characterized by the value of net investments being zero at each point in time. In a one consumption economy with two stocks – a stock of a natural resource and a stock of man-made capital – this means that if consumption remains constant at the maximum sustainable level, then the accumulation of man-made capital always exactly compensates in value for the depletion of the natural resource.

1. INTRODUCTION A requirement of sustainability entails that no generation should allow itself a level of utility that cannot also be shared by all future generations. In the present paper we pose the following question: What characterizes a development that yields any generation the maximum utility level that can be sustained also by future generations? This maximum sustainable level is the economywide analog to the notion of income suggested by Hicks (1946, p. 174): an individual’s income “must be defined Originally published in Journal of Economic Dynamics and Control 23 (1998), 159–165. Reproduced with permission from Elsevier.

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as the maximum amount of money which the individual can spend this week, and still expect to be able to spend the same amount in real terms in each ensuing week.” So, extending this concept to an economy as a whole, income would represent the maximum well being that can be enjoyed in a given period, leaving the economy with the capacity to generate the same well being in each ensuing period. Hence, we seek to characterize a path along which utility as an indicator of instantaneous well being remains equal to Hicksian income in this generalized sense. Consider an economy with stationary instantaneous preferences and a stationary technology. That is to say that the representation of the preferences at each instant of time is invariant with respect to time. Moreover, it is assumed that exogenous technical change is absent (note that endogenous technical change through the accumulation of human capital is allowed for). Hartwick’s rule states that if on an efficient path the value of net investments is zero at each point in time, then utility is constant. This rule was established for a very general class of models in an elegant and important piece of work by Dixit et al. (1980). In a one consumption good economy endowed with two stocks – a stock of an exhaustible nonrenewable resource and a stock of man-made capital – Hartwick’s rule means that if the accumulation of man-made capital always exactly compensates in value for the resource depletion, then consumption remains constant at the maximum sustainable level. In the seminal work of Solow (1974) such a two-stock economy is analyzed. The natural resource is exploited at no cost, and the raw material extracted (R) is, together with capital (K ), used as an input in a production process with a Cobb– Douglas technology (with factor elasticities of α1 for K and α2 for R). Output of this process is used for consumption purposes (C) and accumulation of capital ( K˙ ). The question Solow addresses is whether in this framework there exists a positive consumption level that can be maintained indefinitely. It is shown that the answer is in the affirmative if α1 > α2 . The maximal constant level of consumption can be derived explicitly. It is easily seen that – along the path with maximal constant consumption – net investment equals the value of the ‘revenues’ from exploitation (marginal product of the raw material times the input of the raw material) at each point in time. The fact that such exact compensation for resource depletion implies constant consumption in this setting was first pointed out by Hartwick (1977), after whom the rule is called Hartwick’s rule. The rule can easily be derived from a general neoclassical production function F. A necessary condition for efficiency in the economy is that Hotelling’s rule holds, saying that marginal productivity of capital equals the rate of change in the marginal productivity of the raw material (FK = F˙ R /FR ; here we assume differentiability of F). It follows from K˙ = F(K , R) − C that K¨ = ˙ If K˙ = FR R, then also K¨ = F˙ R R + FR R˙ = FK FR R + FR R˙ = FK K˙ + FR R˙ − C. ˙ ˙ FK K + FR R. Hence, C˙ = 0. Not only Hartwick’s rule holds in the model considered by Solow (1974), the converse of Hartwick’s rule holds as well: If consumption remains constant at the maximum sustainable level, then the accumulation of man-made capital always exactly compensates in value for the resource depletion. This is also shown for the Solow model by Hamilton (1995) and for the Ramsey model (without exhaustible

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resources and with a nonconstant utility discount rate) by Aronsson et al. (1995). A question that naturally arises is whether the converse of Hartwick’s rule holds in general in an economy with stationary instantaneous preferences and a stationary technology: Does an efficient constant utility path imply that the value of net investments equals zero at each point in time? Dixit et al. (1980) also attempted to establish the converse of Hartwick’s rule. However, they succeeded to do so only under an additional assumption that is related to a “capital deepening” condition employed by Burmeister and Turnovsky (1972). This assumption is hard to interpret, and it is not an attractive primitive foundation on which to base the analysis. Hence, it seems worthwhile to od’er a proof that does not rely on it. Consider an efficient constant utility path that is supported by positive utility discount factors having the property that the integral of the discount factors exists, i.e., the path is a regular maximin path in the sense of Burmeister and Hammond (1977). Then this constant utility path solves the problem of maximizing the integral of utilities discounted by these discount factors, subject to the feasibility constraints. Analyze this problem by optimal control theory. Provided that the Hamiltonian converges to zero as time approaches infinity, the converse of Hartwick’s rule follows from a result established by Dixit et al. (1980, Theorem 1), namely that the value of net investments is constant in present value prices if and only if utility is constant. Michel (1982) has shown that the Hamiltonian converges to zero as time approaches infinity if there is a constant utility discount rate. However, to assume a constant utility discount rate is too restrictive here; in particular, such an assumption is incompatible with constant utility (or consumption) in Solow’s (1974) model discussed above. Hence, one route for proving the converse of Hartwick’s rule would be to extend Michel’s result to the case without a constant utility discount rate and combine this extended result with Dixit et al.’s (1980) Theorem 1. In response to the current note, Seierstad (private communication) has indicated that such an extension could be shown by using an additional state variable to make the problem autonomous. Below we have chosen to follow an alternative route. We establish a direct and comprehensive proof of the converse of Hartwick’s rule in a very general setting without having to make any additional assumptions. Thereby our note makes a contribution to the characterization of sustainability. 2. STATEMENT AND PROOF Consider an optimal control problem with n state variables, denoted by x := (x1 , x2 , . . . , xn ), and r instruments, denoted by u := (u 1 , u 2 , . . . , u r ). Obviously n ∈ N and r ∈ N. Let X be an open connected subset of Rn and let U be a subset of Rr . There are given functions ( f 0 , f ) = ( f 0 , f 1 , f 2 , . . . , f n ) : X × U → Rn+1 . These variables and functions are interpreted as follows:

r The vector of state variables (x1 , x2 , . . . , xn ) is interpreted as a vector of stocks, which consists of different kinds of man-made capital (including accumulated knowledge) as well as natural and environmental resources.

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r The vector of instruments (u 1 , u 2 , . . . , u r ) determines jointly with the vector of stocks the output of consumption goods and environmental amenities, the input of various types of labor, and the accumulation (or depletion) of the stocks.

r f 0 (x, u) is the instantaneous utility that is derived from the vector of stocks x when the vector of instruments equals u.

r f j (x, u), 1 ≤ j ≤ n, is the time-derivative of stock j when the vector of stocks equals x and the vector of instruments equals u.

r The vector of instruments u is feasible if u ∈ U . That f 0 is time-independent, means that the instantaneous preferences are stationary. That f j for all j = 1, . . . , n are time-independent, means that the technology is stationary. A stationary technology entails that any technological progress is endogenous, implying that such progress is captured through accumulated stocks of knowledge. Fix some x 0 ∈ X and T ≥ 0. The set F(x 0 , T ) of feasible paths, given that the vector of stocks at time 0 equals x 0 and that the final time is T , is defined as follows: (x, u) ∈ F(x 0 , T ) if and only if x : [0, T ] → X is absolutely continuous, x(0) = x 0 , u : [0, T ] → U is measurable, and x(t) ˙ = f (x(t), u(t)) for all t ∈ [0, T ]. For given x 0 ∈ X and T ≥ 0, we say that (x, ˆ u) ˆ is optimal w.r.t. the utility discount factors π : [0, T ] → R++ if (x, ˆ u) ˆ ∈ F(x 0 , T ) and


T




T

π(t) f 0 (x(t), ˆ u(t))dt ˆ ≥

0

π(t) f 0 (x(t), u(t))dt

0

for all (x, u) ∈ F(x 0 , T ). The premise of the converse of Hartwick’s rule entails that there exists a constant utility path that is supported by positive utility discount factors. Hence, suppose there exist (a) (x, ˆ u) ˆ ∈ F(x 0 , ∞) and a constant ψ with f 0 (x(t), ˆ u(t)) ˆ = ψ for all t ∈ [0, ∞), and (b) π : [0, ∞) → R++ , such that ( x, ˆ u) ˆ is optimal w.r.t. π given x 0 ∈ X ∞ and T = ∞. Note that this implies that 0 π(t)dt is finite. Fix some arbitrary τ (> 0) and consider the problem of maximizing


T

π(t)[ f 0 (x(t), u(t)) − ψ]dt

0

over F(x 0 , T ) with the additional constraint that x(T ) = x(τ ˆ ), and where the maximization takes place with respect to T as well. So, we have a free final time optimal control problem. Proposition 1: A solution to the free final time optimal control problem stated above is: T = t and (x(t), u(t)) = (x(t), ˆ u(t)) ˆ for t ∈ [0, T ].

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Proof. First note that the proposed solution is clearly feasible. Suppose that there exist T ∗ and (x ∗ , u ∗ ) ∈ F(x 0 , T ∗ ) with x ∗ (T ∗ ) = x(t), ˆ which yields an higher value of the objective functional. Then
τ
T∗ ∗ ∗ π(t)[ f 0 (x (t), u (t)) − ψ]dt > π(t)[ f 0 (x(t), ˆ u(t)) ˆ − ψ]dt = 0 . 0

0

(x ∗ (t), u ∗ (t))

Consider the path (x(t), u(t)) = for t ∈ [0, T ∗ ] and (x(t), u(t)) = ∗ ∗ ∗ (x(t ˆ − T + τ ), u(t ˆ − T + τ )) for t ∈ [T , ∞). Due to the fact that f is timeindependent, this path is feasible in the sense that (x, u) ∈ F(x 0 , ∞). But
∞
∞
∞ π(t) f 0 (x(t), u(t))dt > π(t)ψdt = π(t) f 0 (x(t), ˆ u(t))dt ˆ , 0

0

contradicting that (x, ˆ u) ˆ is optimal w.r.t. π given

0

x0

∈ X and T = ∞.

A more general approach would entail the inclusion in the optimal control problem of mixed constraints of the form gi (x, u) ≥ 0 for all i = 1, . . . , m, where g = (g1 , g2 , . . . , gm ) : X × U → Rm (some m ∈ N). Hence, to be an element of the set F(x 0 , T ) of feasible paths, the condition g(x(t), u(t)) ≥ 0 for all t ∈ [0, T ] would have to be satisfied as well. The inclusion of such mixed constraints would not change Proposition 1 as long as gi for all i = 1, . . . , m are time-independent. Proposition 2: Assume that the maximum  ∞principle holds for the infinite horizon optimal control problem of maximizing 0 f 0 (x(t), u(t))dt over F(x 0 , ∞), and λˆ := (λˆ 1 , λˆ 2 , . . . , λˆ n ) be the unique vector of costate variables corresponding to ˆ · x˙ˆ = 0 for all t ∈ (0, ∞). (x, ˆ u). ˆ Then λ(t) Proof. Fix some arbitrary τ (>0). Then (x(t), u(t)) ˆ u(t)) ˆ for t ∈ [0, τ ]  τ = (x(t), solves the fixed final time problem of maximizing 0 f 0 (x(t), u(t))dt (or equivaτ lently, 0 π(t)[ f 0 (x(t), u(t)) − ψ]dt) over F(x 0 , τ ) with τ as finite horizon and x(t) ˆ as final stock. The unique costate variables of the infinite horizon problem, restricted to [0, τ ], are the unique costate variables of this problem. Moreover, as established in Proposition 1, T = τ and (x(t), u(t)) = (x(t), ˆ u(t)) ˆ for t ∈ [0, T ] T solve the free final time problem of maximizing 0 π(t)[ f 0 (x(t), u(t)) − ψ]dt over ˆ ) as the F(x 0 , T ) with the end point in [T1 , T2 ], where T1 < τ < T2 , and with x(τ final stock. Since τ is an optimal final time, the unique costate variables of the fixed final time problem coincide with the costate variables of this problem. Hence, the unique costate variables of the infinite horizon problem, restricted to [0, τ ], coincide with the costate variables of the free final time problem. As a necessary condition (see Seierstad and Sydsæter, 1987, Chapter 2, Theorem 11) we have that the Hamiltonian of the free final time problem equals zero at time τ : ˙ˆ ) . 0 = H(x(τ ˆ ), u(τ ˆ ), τ, λˆ (t)) := π(τ )[ f 0 (x(τ ˆ ), u(τ ˆ )) − ψ] + λˆ (τ ) · x(τ ˆ ), u(τ ˆ )) equals ψ, and τ can be chosen arbitrary, the result is estabSince f 0 (x(τ lished.

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The uniqueness of the costate variables of the infinite horizon problem is assumed to eliminate the possibility that they do not coincide with the costate variables of the free final time problem. To illustrate the rationale behind this assumption, consider an optimal control problem where the state variables do not appear in the objective functional and the instruments (u) are not present in the system of differential equations describing the evolution of the state variables. Then there is a continuum of vectors of costate variables that satisfy the necessary conditions, and it is not difficult in general to find one which does not attribute a zero value to net investments. This is also the case if the solution to the optimal control problem is unique. Note that our uniqueness assumption does not require uniqueness of a solution. It requires that to any solution there corresponds a unique vector of costate variables; stated differently, any other vector of costate variables will generate another u as the maximizer of the Hamiltonian. It is well known that with constraints of the form g(x, u) ≥ 0 the necessary conditions for optimality presuppose a constraint qualification (see, e.g., Seierstad and Sydsæter, 1987, Chapter 4, Theorem 3). If we are willing to make this additional assumption Proposition 2 still holds, after the straightforward modification of the set F(x 0 , T ). If the constraint qualification is not satisfied – e.g., in the case of pure state constraints – matters become much more complicated. The reader is referred to Seierstad and Sydsæter (1987, pp. 397–398). The unique vector of costate variables (λˆ 1 , λˆ 2 , . . . , λˆ n ) is interpreted as the vector of present value prices associated with the vector of stocks. Hence, Proposition 2 states that on a constant utility path that is supported by positive utility discount factors, the value of net investments equals zero at each point in time. This means that Proposition 2 is a statement of the converse of Hartwick’s rule, and that we – by way of the proof of Proposition 2 – have shown this result in a very general setting. Note that the assumptions of stationary instantaneous preferences and a stationary technology are needed in order to establish the converse of Hartwick’s rule. In particular, with exogenous technological progress, the continuation from time T ∗ on of the path (x, u) constructed in the proof of Proposition 1 may not be feasible if T ∗ < τ . Hence, with exogenous technological progress, the optimal T in the free final time problem may well be earlier than τ . This in turn indicates that – when evaluated along the path (x, ˆ u) ˆ – the Hamiltonian of the free final time problem may ˙ˆ ) < 0. This is in line with what well be negative at time τ , entailing that λˆ (τ ) · x(τ one would expect. Acknowledgments: We thank Atle Seierstad for valuable discussions and a referee for helpful comments. REFERENCES Aronsson, T., Johansson, P.O., and Löfgren, K.-G. (1995), Investment decisions, future consumption and sustainability under optimal growth, Umeå Economic Studies 371 Burmeister, E. and Hammond, P. (1977), Maximin paths of heterogeneous capital accumulation and the instability of paradoxical steady states, Econometrica 45, 853–870

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Burmeister, E. and Turnovsky, S.J. (1972), Capital deepening response in an economy with heterogeneous capital goods, American Economic Review 62, 842–853 Dixit, A., Hammond, P., and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Hamilton, K. (1995), Sustainable development, the Hartwick rule and optimal growth, Environmental and Resource Economics 5, 393–411 Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Hicks, J. (1946), Value and capital, 2nd edition. Oxford University Press, Oxford Michel, P. (1982), On the transversality condition in infinite horizon optimal control problems, Econometrica 50, 975–985 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Seierstad, A. and Sydsæter, K. (1987), Optimal Control Theory with Economic Applications. NorthHolland, Amsterdam

CHAPTER 12 ON THE SUSTAINABLE PROGRAM IN SOLOW’S MODEL

CEES WITHAGEN Department of Economics, Tilburg University P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands Department of Economics, Free University, De Boelelaan 1105 1081 HV Amsterdam, The Netherlands Email: [email protected]

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern, 0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics, University of Regensburg 93040 Regensburg, Germany Email: [email protected]

Abstract. We show that our general result Withagen and Asheim (1998) on the converse of Hartwick’s rule also applies for the special case of Solow’s model with one capital good and one exhaustible resource. Hence, the criticism by Cairns and Yang (2000) of our paper is unfounded.

1. INTRODUCTION What characterizes a maximin path in a capital-resource model with one consumption good? The converse of Hartwick’s rule answers this question in the following manner: A necessary condition for an efficient constant consumption path is that Originally published in Natural Resource Modeling 16 (2003), 219–231. The reproduced version has been corrected. Reproduced with permission from the Rocky Mountain Mathematics Consortium.

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the revenues from resource depletion are used for the accumulation of man-made capital. In a more general setting it amounts to the result that a necessary condition for an efficient constant utility path is that the value of net investments is equal to zero at all times. The necessity of Hartwick’s rule has been addressed earlier by Dixit et al. (1980) and Withagen and Asheim (1998) as well as Mitra (2002) and Buchholz et al. (2005) in a rather general setting. Cairns and Yang (2000) concentrate on Solow’s model (cf. Solow, 1974), which describes a two-sector economy with one sector exploiting a natural non-renewable resource and the other one using the raw material from that resource, together with capital, to produce a commodity that can be consumed and invested. In reference to our paper Cairns and Yang argue that we “explicitly posit positive utility-discount functions. Discounting utility in this context is contrived and inconsistent with the motivation of sustainability analysis.” They thus suggest that Solow’s model – which is the basic model in which Hartwick’s rule for sustainability was originally derived – falls outside the realm for the main result in Withagen and Asheim (1998). This view, however, is based on a misunderstanding that stems from confounding discounted utilitarianism as a primary ethical objective with having supporting utility or consumption discount factors in a model where intergenerational equity is the objective.1 The main result in Withagen and Asheim (1998) states that under certain conditions Hartwick’s rule is necessary for sustainability. In the present note we establish in detail how the main result in Withagen and Asheim (1998) (here reproduced as Proposition 1 in the current one-consumption good setting) can be used to obtain the converse of Hartwick’s rule in Solow’s model. Thereby we show that the criticism of Cairns and Yang is unfounded. Proposition 1 states that if a constant consumption path maximizes the sum of discounted consumption for some path of supporting consumption discount factors, then the value of net investments is equal to zero at all times. We here supplement Proposition 1 by showing that any maximin path in Solow’s model has constant consumption and maximizes the sum of discounted consumption for some path of supporting discount factors. This means that the premise of our general result on the converse of Hartwick’s rule is satisfied in the case of Solow’s model. We start in Sect. 2 by giving a formal presentation of Solow’s model, defining the concept of a maximin path, and reproducing Withagen and Asheim’s (1998) result as Proposition 1 in the context of Solow’s model. We then in Sect. 3 show that (a) the premise of Proposition 1 is satisfied for any maximin program that is interior and regular, and (b) that any maximin program in Solow’s model indeed is interior and regular provided that the infimum of consumption along the maximin program is positive. We conclude in Sect. 4 by proving our main results and commenting on Cairns and Yang’s analysis. 2. THE MODEL The Solow model describes a two sector economy. One sector exploits a nonrenewable resource, the size of which at time t is denoted by s(t). The initial stock is given and denoted by s0 . The raw material (r ) from the resource is used as an

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input in the other sector, together with capital (k). The production function in this sector is denoted by f , with subscripts denoting partial derivatives. Output is used for consumption (c) and net investments (i). The initial capital stock is k0 . There is no depreciation. We follow Cairns and Yang (2000) in their assumptions concerning the production function; however, we add that raw material be important (Mitra, 1978, p. 121), meaning that fr (k, r )r/ f (k, r ) ≥ α > 0 for all k > 0 and r > 0: Assumption 1: The production function f is concave, nondecreasing and continuous for non-negative inputs, and it is increasing and twice differentiable for inputs in the interior of the positive orthant. Both inputs are necessary and raw material is important. Finally, f k (∞, r ) = 0 for r > 0 and fr (k, 0) = ∞ for k > 0. A quintuple (c, i, r, k, s) is said to attainable if c ≥ 0,

i ≤ f (k, r ) − c ,

r ≥ 0,

k ≥ 0,

s ≥ 0.

{c(t), i(t), r (t), k(t), s(t)}∞ t=0

is said to feasible if (1) c(t), i(t) and r (t) A program are Lebesgue integrable functions of t, (2) k(t) and s(t) are absolutely continuous functions of t satisfying k(0) = k0 > 0 ,

s(0) = s0 > 0 ,

and (3) (c(t), i(t), r (t), k(t), s(t)) is attainable for all t and satisfies ˙ = i(t) , k(t)

s˙ (t) = −r (t) .

for a.e. t. A feasible program is said to be interior if, for all t, the quintuple is in the interior of the positive orthant. A feasible program is said to be efficient if there is no feasible program with at least as much consumption everywhere and larger consumption on a subset of the time interval with positive measure. A feasible program {c(t), i(t), r (t), k(t), s(t)}∞ ¯ t=0 is said to be maximin if inft c(t) ≥ inft c(t) ¯ ¯ for all feasible programs {c(t), ¯ i(t), r¯ (t), k(t), s¯ (t)}∞ . t=0 In Solow’s model, it may not be possible to maintain consumption above a positive lower bound forever, even if the initial stocks are positive. Here we simply assume the existence of a maximin program that sustains positive consumption, and refer to Cass and Mitra (1991) for a discussion of sufficient and necessary conditions in terms of the underlying technology. Assumption 2: There is a maximin program {c(t), i(t), r (t), k(t), s(t)}∞ t=0 with inft c(t) = c∗ > 0. It follows that any maximin program satisfies inft c(t) = c∗ > 0. We end this section by stating our general result on the converse of Hartwick’s rule in the setting of Solow’s model. Proposition 1 (Withagen and Asheim, 1998, Proposition 2): Assume that there are positive consumption discount factors {π(t)}∞ t=0 such
that maintaining con∞ sumption constant and equal to c∗ forever maximizes 0 π(t)c(t)dt over all feasible paths, that the maximum principle holds for the corresponding infinite

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horizon optimal control problem, and that the path of corresponding costate variables {λ(t), µ(t)}∞ t=0 is unique. Then, for all t, λ(t)i(t) = µ(t)r (t). This re-formulation of the main result in Withagen and Asheim (1998) shows that an important step in the following analysis will be to find consumption discount factors for which a maximin program can be implemented as a discounted utilitarian optimum. We will now show how this can be done. 3. MAIN RESULTS In this section, we use the concept of a “regular maximin program” to show that our general result in Withagen and Asheim (1998) (restated as Proposition 1) on the converse of Hartwick’s rule can be applied to demonstrate that along any maximin path in Solow’s model the revenues from resource depletion are used for accumulation of man-made capital. Since the concept of a “regular maximin program” requires the concept of a “competitive program,” we start by introducing the latter. A feasible program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is said to be competitive at positive consumption discount factors {π(t)}∞ t=0 and non-negative and competitive prices {λ(t), µ(t)}∞ t=0 (where π(t) is Lebesgue integrable and λ(t) and µ(t) are absolutely continuous) if, for a.e. t, ∗ π(t)c∗ (t) + λ(t)i ∗ (t) − µ(t)r ∗ (t) + λ˙ (t)k ∗ (t) + µ(t)s ˙ (t)

˙ ≥ π(t)c + λ(t)i − µ(t)r + λ(t)k + µ(t)s ˙

(1)

for all attainable quintuples (c, i, r, k, s). A program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), ∞ ∞ s ∗ (t)}∞ t=0 that is competitive at {π(t)}t=0 and {λ(t), µ(t)}t=0 is said to be a regular maximin path (cf. Dixit et al., 1980) if c∗ (t) = c∗ (constant)  ∞ π(t)dt < ∞

(2) (3)

0

λ(t)k ∗ (t) + µ(t)s ∗ (t) → 0 as t → ∞ .

(4)

It is essential to observe that the path of positive consumption discount factors {π(t)}∞ t=0 solely reflects the rate at which consumption at one point time can be transformed into consumption at some other point in time. In particular, it has no ethical significance since it is derived from the regular maximin program as a price support of the constant consumption path. We first show as Proposition 2 that the premise of Proposition 1 is satisfied for any maximin program that is interior and regular. Proposition 2: If {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is an interior and regular maximin program at consumption discount factors {π(t)}∞ t=0 and competitive prices {λ(t), µ(t)}∞ , then the premise of Proposition 1 is satisfied. t=0

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Second, we establish as Proposition 3 that, under Assumptions 1 and 2, any maximin program in Solow’s model indeed is interior and regular. Note that Proposition 3 is similar to results established by Dasgupta and Mitra (1983). Proposition 3: Any maximin program in Solow’s model is interior and regular at some appropriately chosen consumption discount factors {π(t)}∞ t=0 and competitive prices {λ(t), µ(t)}∞ , provided that Assumptions 1 and 2 are satisfied. t=0 Together these two main results – which are proven in the following section – demonstrate that Proposition 1 can be applied to show that the converse of Hartwick’s rule holds for Solow’s model. We have thus established the usefulness of our previous result on the converse of Hartwick’s rule, also in the context of Solow’s model. 4. PROOFS Proposition 2 is proven through the following two lemmas. First, we observe that ∞ ∗ if {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is a regular maximin program, then {c (t)}t=0 ∞ maximizes the sum of consumption discounted by {π(t)}t=0 . Lemma 1 (Dixit et al. (1980)): If a program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is a regular maximin program at {π(t)}∞ and {λ(t), µ(t)}∞ , then it maximizes t=0 t=0
∞ 0 π(t)c(t)dt over all feasible paths.
∞ Proof. Note that (2) and (3) imply that 0 π(t)c∗ (t)dt < ∞. It is sufficient to show that  T   lim supT →∞ π(t) c(t) − c∗ (t) dt ≤ 0 0

for all feasible programs {c(t), i(t), r (t), k(t), s(t)}∞ t=0 .  T   π(t) c(t) − c∗ (t) dt 0

 ≤

T

0

 =



    λ(t) i ∗ (t) − i(t) − µ(t) r ∗ (t) − r (t)

 ∗   ∗  ˙ + λ(t) k (t) − k(t) + µ(t) ˙ s (t) − s(t) dt by (1) T

    d λ(t)(k ∗ (t) − k(t)) + µ(t)(s ∗ (t) − s(t)) /dt dt

0

˙ = i(t) and s˙ (t) = −r (t) since k(t)

 = λ(T )(k ∗ (T ) − k(T )) + µ(T )(s ∗ (T ) − s(T ))   − λ(0)(k ∗ (0) − k(0)) + µ(0)(s ∗ (0) − s(0)) 

≤ λ(T )k ∗ (T ) + µ(T )s ∗ (T ) since k ∗ (0) = k(0) = k0 , s ∗ (0) = s(0) = s0 , λ(T ) ≥ 0, µ(T ) ≥ 0, k(T ) ≥ 0, and s(T ) ≥ 0 . By (4), the result follows.

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Note that, since the consumption discount factors {π(t)}∞ t=0 are positive by definition, Lemma 1 implies that a regular maximin path is efficient. Second, we show that for any interior and competitive program, the maximum principle holds and the path of costate variables is unique. Lemma 2: If an interior program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is a regular ∞ , then the maximum principle for maximin program at {π(t)}∞ and {λ(t), µ(t)} t=0 t=0 problems with Lebesgue measurable control holds for the problem of maximizing
∞ 0 π(t)c(t)dt and the path of corresponding costate variables is unique and equals 2 {λ(t), µ(t)}∞ t=0 . Proof. Since {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is interior and competitive it follows from (1) that, for a.e. t,   (c∗ (t), r ∗ (t)) maximizes π(t)c + λ(t) f (k ∗ (t), r ) − c − µ(t)r (5) over all non-negative (c, r ) λ(t) f k (k ∗ (t), r ∗ (t)) + λ˙ (t) = 0

(6)

µ(t) ˙ = 0.

(7)

Since the program is interior and f is smooth, it follows from (5) that, for a.e. t, π(t) − λ(t) = 0

(8)

∗

(9)

∗

λ(t) fr (k (t), r (t)) − µ(t) = 0 .

In fact, (9) holds for all t since λ(t) and µ(t) are absolutely continuous functions, while (8) shows that w.l.o.g. π(t) can be assumed to be an absolutely continuous function, entailing that also (8) holds for all t. Since {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 is a regular maximin program and thus,
∞ by Lemma 1, maximizes 0 π(t)c(t)dt over all feasible paths, it follows that (5)–(7) are necessary condition for optimality, where   H(k, s, c, r, λ, µ) = π(t)c + λ f (k, r ) − c − µr is the corresponding Hamiltonian function, with, for all t, (λ(t), µ(t)) being uniquely determined from π(t) by (8) and (9). Proof of Proposition 2. This is a direct consequence of Lemmas 1 and 2. The proof of Proposition 3 is based on one observation and three lemmas. We first make the following observation. Observation 1: If {c(t), i(t), r (t), k(t), s(t)}∞ t=0 is a feasible path with c(t) ≥ c > 0 and i(t) ≥ 0 for a.e. t, then  ∞ 1 dt < ∞ . f (k(t), r (t)) r 0

ON THE SUSTAINABLE PROGRAM IN SOLOW’S MODEL

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Proof. Since raw material is important and f (k(t), r (t)) ≥ c(t) + i(t) ≥ c(t) ≥ c for a.e. t, it follows that k(t) > 0, r (t) > 0, and 1 1 1 1 r (t) = r (t) ≤ r (t) ≤ fr (k(t), r (t)) fr (k(t), r (t))r (t) α f (k(t), r (t)) αc
∞
∞ hold for a.e. t. Therefore, 0 (1/ fr (k(t), r (t)))dt ≤ (1/αc) 0 r (t)dt < ∞. The three lemmas consider the problem of minimizing resource use subject to, for all t, c(t) ≥ c∗ :  ∞ min r (t)dt subject to c(t) ≥ c∗ for all t 0

over the set of feasible programs. Assumption 2 ensures
∞ that this problem is not vacuous (since there exist feasible programs satisfying 0 r (t)dt ≤ s0 and c(t) ≥ c∗ for all t), but it does not guarantee a solution. In Lemmas 3–5, we derive necessary conditions for the minimum resource use problem, while in the proof of Proposition 3, we show that this problem has a solution. Thus, suppose that the problem has a solution, which we will denote {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 . Since it is w.l.o.g. to set i ∗ (t) = f (k ∗ (t), r ∗ (t)) − c∗ (t) and c∗ (t) = c∗ for all t, the Hamiltonian function corresponding to the minimum resource use problem can be written   H(k, r, λ; c∗ ) = −r + λ f (k, r ) − c∗ from which we can derive the following necessary conditions: For a.e. t, r ∗ (t) maximizes − r + λ(t) f (k ∗ (t), r ) over all non-negative r,

(10)

˙ −λ(t) = λ(t) f k (k ∗ (t), r ∗ (t)) ,

(11)

where λ(t) is absolutely continuous. Note that also r ∗ (t) and i ∗ (t) are absolutely continuous, implying that k˙ ∗ (t) = i ∗ (t) and s˙ ∗ (t) = −r ∗ (t) for all t. Lemma 3: If a program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 solves the minimum resource use problem subject to c(t) ≥ c∗ , then it has constant consumption, and is interior and competitive. Proof. Clearly, c∗ (t) = c∗ > 0 for all t. Furthermore, since λ(τ ) ≤ 0 would imply λ(t) ≤ 0, r ∗ (t) = 0, and f (k ∗ (t), r ∗ (t)) = 0 for all t ≥ τ , contradicting that c∗ (t) = c∗ and k ∗ (t) ≥ 0 for all t, it follows from (10) and (11) that, for all t, λ(t) > 0 and r ∗ (t) > 0. Suppose there exist t1 and t2 with t1 < t2
such that k ∗ (t1 ) ≥ k ∗ (t2 ). Then ∞ ∗ ∗ {c (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=t1 minimizes t1 r (t)dt subject to c(t) ≥ c for ∗ ∗ all t over the set of feasible programs from
(k (t1 ), s (t1 )),
∞
∞even though there exist ∞ feasible programs satisfying t1 r (t)dt ≤ t2 r ∗ (t)dt < t1 r ∗ (t)dt and c(t) ≥ c∗ for all t ≥ t1 . This is a contradiction and implies that k ∗ (t) is strictly increasing (although not necessarily that i ∗ (t) = k˙ ∗ (t) > 0 for all t).

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¯ Then it holds for all t that Suppose that k ∗ (t) has an upper bound k. ¯ r ∗ (t)) , 0 < c∗ ≤ c∗ + i ∗ (t) = f (k ∗ (t), r ∗ (t)) ≤ f (k, ¯ r ) = c∗ . This leads to Hence, r ∗ (t) ≥ r for all t, where r > 0 is determined by f (k, ∗ the contradiction that resource use is unbounded. Hence, k (t) → ∞ as t → ∞. Since k ∗ (t) is strictly increasing and unbounded, there exists a value function V : [k0 , ∞) → R+ corresponding to minimum resource use and satisfying for all t, 

∗

V (k (t)) =

∞

r ∗ (τ )dτ .

(12)

t

We have that, for all t, d V (k ∗ (t))/dk = −λ(t). Combined with (12), this means that, for all t, λ(t)k˙ ∗ (t) = −dV (k ∗ (t))/dt = r ∗ (t) > 0, implying that i ∗ (t) = k˙ ∗ (t) > 0 and k ∗ (t) ≥ k0 > 0. Finally, for all t, s˙ ∗ (t) = −r ∗ (t) < 0 and s ∗ (t) ≥ 0, implying that s ∗ (t) > 0. Hence, any program that solves the minimum resource use problem subject to c(t) ≥ c∗ has constant consumption and is interior. It remains to be shown that any program solving the minimum resource use problem is competitive. To show this, set π(t) = λ(t) and µ(t) = 1, for all t. It is straightforward to check that the concavity of f implies that (1) is then satisfied for all t. Lemma 4: If a program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 solves the minimum resource use problem subject to c(t) ≥ c∗
, then it exhausts the resource and the path ∞ of the costate variable {λ(t)}∞ t=0 satisfies 0 λ(t)dt < ∞. Proof. Suppose that {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 solves the minimum resource use problem subject to c(t) ≥ c∗ . By Lemma 3, c∗ (t) = c∗ > 0 and i ∗ (t) > 0 for all t. By Observation 1 and (10), 

∞ 0



∞

λ(t)dt = 0

1 fr

(k ∗ (t), r ∗ (t))

dt < ∞ ,

(13)

keeping in mind that r ∗ (t) > 0 by Lemma 3. Suppose that {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 does not exhaust the resource,
∞ ∗ i.e., we have that 0 r (t)dt < s0 . Then it follows from (13) that it is possible to
∞ construct a feasible program with inft c(t) > c∗ , since 0 (1/ fr (k ∗ (t), r ∗ (t)))dt is the marginal resource cost of a uniform increment to consumption. However, the existence of a feasible program with inft c(t) > c∗ contradicts the definition of c∗ . Hence, the solution to the minimal resource problem exhausts the resource. Lemma 5: If a program {c∗ (t), i ∗ (t), r ∗ (t), k ∗ (t), s ∗ (t)}∞ t=0 solves the minimum resource use problem subject to c(t) ≥ c∗ , then λ(t)k ∗ (t) → 0 as t → ∞, where {λ(t)}∞ t=0 is the path of the costate variable.

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ON THE SUSTAINABLE PROGRAM IN SOLOW’S MODEL

Proof. In view of the concavity of the production function f , the value function V is convex in k, implying that, for all t, V (k) − V (k ∗ (t)) ≥

dV (k ∗ (t)) · (k − k ∗ (t)) dk

(14)

for all k ≥ k0 . Moreover, dV (k ∗ (t))/dk = −λ(t) < 0 and ∗



lim V (k (t)) = lim

t→∞

t→∞ t

∞

r ∗ (τ )dτ = 0 .

Since k ∗ (t) → ∞ as t → 0, it follows that V (k) → 0 as k → ∞. Let t be determined by k ∗ (t) = 2k0 . By (14) it holds for all t ≥ t that V ( 12 k ∗ (t)) − V (k ∗ (t)) ≥ 12 λ(t)k ∗ (t) . The left-hand side goes to zero as t → ∞. The right-hand side is non-negative and therefore goes to zero as well. Proof of Proposition 3. By Assumption 2, there exists a maximin program ∗ {c(t), i(t), r (t), k(t), s(t)}∞ t=0 with c(t) ≥ c > 0. W.l.o.g. we can set c(t) = c∗ and i(t) = f (k(t), r (t)) − c∗ for all t. Construct the program {c (t), i  (t), r  (t), k  (t), s  (t)}∞ t=0 by setting, for all t, k  (t) = max0≤τ ≤t k(τ ) and determining c (t), i  (t), r  (t) and s  (t) as follows: c (t) = c(t) = c∗  solves f (k  (t), r  (t)) = c∗ if k  (t) > k(t)  r (t) = r (t) otherwise,

i  (t) = k˙  (t) (≥ 0)  t r  (τ )dτ . s  (t) = s0 − 0

It can be checked that c (t), i  (t) and r  (t) are Lebesgue integrable, and k  (t) and s  (t) are absolutely continuous. To establish feasibility of
∞ {c (t), i  (t), r  (t), k  (t), s  (t)}∞ , it remains to be shown that 0 r  (t)dt ≤ t=0
t2 
t2
∞ 0 r (t)dt (≤ s0 ). For this it is sufficient to show that t1 r (t)dt ≤ t1 r (t)dt for every maximal interval (t1 , t2 ) on which k  (t) > k(t). Consider any such interval (t1 , t2 ). Note that k  (t) and r  (t) are constant on (t1 , t2 ), implying that f k (k  (t), r  (t)) = a > 0 and fr (k  (t), r  (t)) = b > 0 on (t1 , t2 ). Furthermore,

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k(t1 ) = k  (t1 ) = k  (t2 ) = k(t2 ). Hence, 0 = [k  (t2 ) − k  (t1 )] − [k(t2 ) − k(t1 )] =  =  ≥

t2

(i  (t) − i(t))dt

t1



t2

 f (k  (t), r  (t)) − f (k(t), r (t)) dt

t1 t2

f k ((k  (t), r  (t))[k  (t) − k(t)]dt +

t1



=a



t2

[k  (t) − k(t)]dt + b

t1





t2

fr ((k  (t), r  (t))[r  (t) − r (t)]dt

t1 t2

[r  (t) − r (t)]dt .

t1

Since is positive, the second must be negative, thereby establishing that
∞  the first
term ∞ r (t)dt ≤ r (t)dt (≤ s0 ). 0 0 By Observation 1 it now follows that:  ∞ 1 dt < ∞ .  fr (k (t), r  (t)) 0
∞
∞  Suppose 0 r  (t)dt < s0 . Since 0 1/ fr (k  (t), r  (t)) dt is the marginal resource cost of a uniform increment to consumption, this contradicts that {c (t), i  (t), r  (t), k  (t) s  (t)}∞ t=0 is a maximin path. Hence,  ∞  ∞ r  (t)dt ≤ r (t)dt ≤ s0 , s0 = 0

0

establishing that each maximin program {c(t), i(t), r (t), k(t), s(t)}∞ t=0 exhausts the resource. Suppose that {c(t), i(t), r (t), k(t), s(t)}∞ does not solve the minimum t=0 resource use problem subject to c(t) ≥ c∗ . This leads to the contradiction that there exists a maximin program not exhausting the resource. Hence, each maximin program {c(t), i(t), r (t), k(t), s(t)}∞ t=0 solves the minimum resource use problem subject to c(t) ≥ c∗ from which it follows that Lemmas 3–5 can be applied. By Lemma 3, any program solving the minimum resource use problem is interior. Furthermore, it is a regular maximin program since it is competitive with, for all t, π(t) = λ(t) and µ(t) = 1 (by Lemma 3) and satisfies (2) (by Lemma 3), (3) (by Lemma 4), and (4) (by Lemmas 5 and 4). One of the steps taken in this paper is to show that, along a program solving the minimum resource use problem, the minimum resource use coincides with the resource stock initially available (s0 ) in the original problem. Our proof of this in Lemma 4 is based on Observation 1, which in turn invokes the assumption that the resource is important. Cairns and Yang also provide an argument to show this. Essentially they attempt to apply the smoothness of the production function to establish the following implication: Since a uniform decrement to consumption is associated with a finite decrease in the resource stock, it follows that a finite increase in the resource stock can give rise to a uniform increment to consumption. However, neither their argument nor the similar argument we gave in the proof of Lemma 4 included in the

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original published version of this paper is correct, unless a further assumption on the asymptotic curvature of the production function is imposed, like the assumption that the resource be important. A second problem with the analysis by Cairns and Yang is their proof that Hotelling’s rule, f˙r / fr = f k , holds along a program with maximal constant consumption. The proof relies on a set of first-order approximations. This method is an excellent tool, in particular in the case at hand, to illustrate what Hotelling’s rule is actually saying – namely that there are no subintervals of time where the constant rate of consumption can be maintained, and at the same time the program ends up with larger capital and resource stocks than in the original program. However, such an argument cannot serve as a formal proof. Acknowledgments: We thank David Miller for helpful comments. Asheim gratefully acknowledges the hospitality of the research initiative on the Environment, the Economy and Sustainable Welfare at Stanford University and financial support from the Hewlett Foundation. For the present corrected version we thank Tapan Mitra for discussions, while Asheim acknowledges the hospitality of Cornell University and financial support from the Research Council of Norway. NOTES 1 Cairns and Yang also refer to our paper elsewhere. They argue that we “do not show that following

Hartwick’s rule leads to a unique outcome, much less a maximal level of consumption.” Since we were dealing with the necessity of Hartwick’s rule, we did not investigate uniqueness, while it was our premise that the program is maximin. 2 Cf. Seierstad and Sydsæter (1987, Footnote 9, pp. 132–133).

REFERENCES Buchholz, W., Dasgupta, S. and Mitra, T. (2005), Intertemporal equity and HartwickŠs rule in an exhaustible resource model, Scandinavian Journal of Economics 107, 547–561 Cairns, R.D. and Yang, Z. (2000), The converse of Hartwick’s rule and uniqueness of the sustainable path, Natural Resource Modeling 13, 493–502 Cass, D. and Mitra, T. (1991), Indefinitely sustained consumption despite exhaustible natural resources, Economic Theory 1, 119–146 Dasgupta, S. and Mitra, T. (1983), Intergenerational equity and efficient allocation of exhaustible Resources, International Economic Review 24, 133–153 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Mitra, T. (1978), Efficient growth with exhaustible resources in a neoclassical model, Journal of Economic Theory 17, 114–29. Mitra, T. (2002), Intertemporal equity and efficient allocation of resources, Journal of Economic Theory 107, 356–376 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium) 41 29–45 Seierstad, A. and Sydsæter, K. (1987), Optimal Control Theory with Economic Applications, NorthHolland, Amsterdam Withagen, C. and Asheim, G.B. (1998), Characterizing sustainability: The converse of Hartwick’s rule, Journal of Economic Dynamics and Control 23, 159–165 (Chap. 11 of the present volume)

CHAPTER 13 MAXIMIN, DISCOUNTING, AND SEPARATING HYPERPLANES

CEES WITHAGEN Department of Economics, Tilburg University P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands Department of Economics, Free University, De Boelelaan 1105 1081 HV Amsterdam, The Netherlands Email: [email protected]

GEIR B. ASHEIM Department of Economics, University of Oslo, P.O. Box 1095 Blindern, 0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics, University of Regensburg 93040 Regensburg, Germany Email: [email protected]

Is Hartwick’s rule a necessary condition for efficient and constant consumption in Solow’s (1974) model? Until recently this has been an open question; this is surprising given the prominence of the model. Cairns and Yang (2000) as well as Withagen et al. (2003) claim that the answer is in the affirmative and claim to provide a formal proof. The latter team argues that the proof by the former is not correct and provides an alternative proof, based on Withagen and Asheim (1998). Although Cairns and Yang (2000) assert that the methodology of Withagen and Asheim (1998) is “contrived,” our proof in this issue is not in dispute. This settles the question: Hartwick’s rule is necessary in Solow’s model. Originally published in Natural Resource Modeling 16 (2003), 213–217. Reproduced with permission from the Rocky Mountain Mathematics Consortium.

191 Asheim, Justifying, Characterizing and Indicating Sustainability, 191–194 c 2007 Springer


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Nevertheless there is continued controversy. The main point in Cairns’ reply in this issue refers to discounting. In Withagen and Asheim it is assumed (in a very general setting) that an efficient constant utility path is supported by positive utility discount factors having the property that the integral of the discount factors exists. In the application to Solow’s model this assumption needs to be checked. In our contribution in this issue we show that indeed it holds. Here we aim to clarify our approach in some detail and relate it to the concept of separating hyperplanes. 1. THE DISCOUNT FACTORS CORRESPOND TO A HYPERPLANE Modern microeconomic theory, as originating with Arrow and Debreu (Arrow, 1951; Debreu, 1951; Arrow and Debreu, 1954), is based on the following result: If technology and individual preferences are both convex, then there exists a hyperplane, containing a feasible allocation, that separates all feasible allocations from those that are preferable. Malinvaud (1953) introduced this mathematical tool to the study of dynamic infinite-horizon discrete-time economies. Inspired by Koopmans (1951), Arrow and Debreu led a noncalculus revolution in microeconomic theory, entailing that a separating hyperplane may not describe technology and preferences even locally. If one considers a hyperplane that separates the set of feasible allocations of utility across consumers from those utility allocations that are socially preferred, then utility allocations on the hyperplane are equally good in social evaluation, only if social welfare is a linear function of the individual utilities. However, such a Samuelson– Bergson welfare function need not be linear in utilities, implying that the economy’s primitive objective need not be to maximize a weighted sum of utilities. E.g., if the economy’s welfare judgements are based solely on the Pareto-criterion, then different utility allocations on the separating hyperplane are incomparable. In the contributions by Cairns and Yang (2000) and ourselves (this issue), the Samuelson–Bergson welfare function is maximin. In an economy with a finite number of consumers, maximin leads to an efficient allocation with equal utility for all consumers if a utility sacrifice by one consumer can be transformed into an equal utility gain for all other consumers. The extension of this condition to an economy with a continuum of consumers over an infinite time horizon, means that there are positive rates of utility transformation at different points in time, π(t), such that
∞ π(t)dt < ∞ . 0

This is the essential condition in the definition of a regular maximin path, due to Burmeister and Hammond (1977) and Dixit el al. (1980). There are no general results for determining whether regularity is a necessary condition for the solution to a maximin problem in the continuous-time infinite-horizon framework (cf. Mitra, 2002). There are nonregular maximin paths: If the initial capital stock in a one-sector model is at least as large as the Golden Rule size, then no maximin path is regular. However, in specific models it can be shown that any maximin path is regular. Indeed, in our paper in this issue, we show that any maximin

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path in Solow’s (1974) model is regular. We do so under assumptions that are more general than those that previously have been used for this purpose. The convexity and smoothness of Solow’s technology mean that there exists a hyperplane in utility space that is unique and separates any feasible utility path from those that are socially preferable according to the maximin criterion. The discount factors that determine this hyperplane are derived from the solution to the maximin program.1 Hence, they are endogenous. E.g., keeping the maximin objective fixed, a different vector of initial stocks or a different technology leads to a different path of discount factors. Any point on the hyperplane that differs from the maximin program is socially strictly less preferred as compared to the maximin program. Thus, any maximin path in Solow’s model is necessarily associated with supporting discount factors, implying that the premise of Withagen and Asheim (1998) is necessary for any efficient constant consumption path in this model. What is necessary cannot simultaneously be “contrived and inconsistent” as claimed by Cairns and Yang (2000). On the contrary, the necessity means that any path that does not satisfy the premise of Withagen and Asheim (1998) is inconsistent with efficient and constant consumption in Solow’s model. 2. CAIRNS’S MISINTERPRETATION OF OUR WORK Instead of acknowledging the mathematical fact that any maximin path in Solow’s model is supported by positive utility discount factors – discussed above in the tradition of Arrow, Debreu, and Malinvaud – Cairns in his response in this issue claims that, contrary to us, Cairns and Yang (2000) “avoided” discounting in their analysis of efficient constant consumption paths in Solow’s model. It is, however, a proven fact that any maximin path (and thus, any efficient constant consumption path) in Solow’s model maximizes the sum of discounted utility (or consumption) for appropriately chosen discount factors (cf. Withagen et al., 2003, Prop. 3). Cairns argues that “[i]n Withagen and Asheim’s objective the maximin problem is assumed to have been solved and the utility-discount factors . . . are assumed to be given, i.e., exogenous”, and writes: “In my view, by maximizing [the sum of discounted utilities] with exogenous discount factors, they posit a utilitarian objective.” The discount factors are, however, endogenous, being derived (as we do in the Proposition 3 of Withagen et al., 2003) from the maximin path. Subsequent to expression (4) of our paper we make this abundantly clear. And establishing that a maximin path corresponds to maximizing discounted utilities at appropriately chosen discount factors, does not entail “[a] willingness to exchange utility at different times” according to these discount factors (as suggested by Cairns). Cairns claims that a discussion of a regular maximin path need make no mention of a competitive path. However, as defined by Burmeister and Hammond (1977) and Dixit el al. (1980), any regular maximin path is competitive.2 He also maintains that Solow’s method of posing the minimum resource use problem “was appropriate in discussing the implications of an energy crisis, but could not be applied in a more general investigation.” However, we reconfirm that the method works in Solow’s

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model. One must be allowed to use this mathematical method even in times when an energy crisis is not imminent. Finally, he asserts that there is “no . . . link [between NNP and the value in the economy] in a maximin problem.” However, Asheim and Buchholz (2004) show that there is indeed such a link in the maximin case. Acknowledgments: We thank K. Arrow, A. Dixit, P. Hammond, and M. Hoel for discussions. The usual disclaimer applies. Asheim gratefully acknowledges the hospitality of the research initiative on the Environment, the Economy and Sustainable Welfare at Stanford University and financial support from the Hewlett Foundation. NOTES 1 There is a mathematical difficulty. The convex sets in utility space that are separated are infinite

dimensional due to the fact that we employ a continuum of consumers over an infinite time horizon. The sets can be seen as subsets of the space of (essentially) bounded Lebesgue measurable functions. The support (the discount factors) of the separating hyperplane is in the dual of this space. The dual is not the space of Lebesgue integrable functions. So, it remains to be proven that the support is indeed integrable, and this is done in our paper in this issue. As one would expect, the proof is not trivial. 2 Note that (M.1), (M.2), (M.4), and (M.5) are the competitiveness conditions in Burmeister and Hammond (1977).

REFERENCES Arrow, K.J. (1951), An extension of the basic theorems of welfare economics, in Neyman, J. (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley and Los Angeles Arrow, K.J. and Debreu, G. (1954), Existence of an equilibrium for a competitive equilibrium, Econometrica 22, 265–290 Asheim, G.B. and Buchholz, W. (2004), A general approach to welfare measurement through national income accounting, Scandinavian Journal of Economics 106, 361–384 (Chap. 18 of the present volume) Burmeister, E. and Hammond, P.J. (1977), Maximin paths of heterogeneous capital accumulation and the Instability of Paradoxical Steady States, Econometrica 45, 853–870 Cairns, R.D. and Yang, Z. (2000). The converse of Hartwick’s rule and uniqueness of the sustainable path, Natural Resource Modeling 13, 493–502 Debreu, G. (1951), The coefficient of resource utilization, Econometrica 19, 273–292 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556. Koopmans, T.C. (1951), Analysis of production as an efficient combination of activities, in Koopmans, T.C. (ed.), Activity Analysis of Production and Allocation, Cowles Commission Monograph 13. Wiley, New York Malinvaud, E. (1953). Capital accumulation and efficient allocation of resources, Econometrica 21, 233– 268 Mitra, T. (2002), Intertemporal equity and efficient allocation of resources, Journal of Economic Theory 107, 356–376 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Withagen, C. and Asheim, G.B. (1998), Characterizing sustainability: The converse of Hartwick’s rule, Journal of Economic Dynamics and Control 23, 159–165 (Chap. 11 of the present volume) Withagen, C., Asheim, G.B. and Buchholz, W. (2003), On the sustainable program in Solow’s model, Natural Resource Modeling 16, 219–231 (Chap. 12 of the present volume)

Part III INDICATING SUSTAINABILITY

CHAPTER 14 GREEN NATIONAL ACCOUNTING FOR WELFARE AND SUSTAINABILITY: A TAXONOMY OF ASSUMPTIONS AND RESULTS

Abstract. This paper summarizes assumptions made and results obtained in parts of the literature on welfare and sustainability accounting. I consider five different assumptions that can be imposed independently of each other, producing 32 different combinations. This taxonomy is used to organize results in welfare and sustainability accounting. The analysis illustrates how stronger results require stronger assumptions and thereby impose harder informational requirements.

1. INTRODUCTION During the more than 25 years since Martin Weitzman published his seminal paper Weitzman (1976) on the significance for dynamic welfare of comprehensive national accounting aggregates, there have been many important theoretical contributions on welfare and sustainability accounting. This literature shows how national accounting aggregates can be used to measure differences in welfare, both over time and across different economies, and to indicate whether development is sustainable. It is, however, often not very transparent under what assumptions different results on welfare and sustainability accounting will hold. On this background, I treat the topic systematically in this paper and summarize assumptions made and results obtained in major parts of this literature. I consider five different assumptions that can be imposed independently of each other, producing altogether 32 different combinations of assumptions. This taxonomy will be used to organize the different results. The most general analysis answering the “simplest” problems and imposing the weakest assumptions will be addressed in Sect. 3. Analysis requiring stronger assumptions, but answering more “complicated” questions will be addressed in Sects. 4 and 5. The presentation emphasizes the assumptions needed in order for results to be of interest for practical estimation, thereby organizing the discussion of informational problems that must be faced when doing empirical analysis. A discussion of methods to overcome informational constraints is contained in Sect. 6. Two tables yield an overview of assumptions and results.1

Originally published in Scottish J Political Economy 50 (2003), 113–130. Reproduced with permission from Blackwell.

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It is a prerequisite for most of the results that the list of goods and services included in national accounting aggregates is comprehensive. The national accounts are “comprehensive” if all variable determinants of current well-being are included in the vector of consumption flows, and if all variable determinants of current productive capacity are included in the vector of capital stocks. E.g., compared to NNP as normally measured, one must “green” the national accounts by introducing natural resource depletion and environmental degradation into the national accounts by (1) including such depletion and degradation of natural capital as negative components to the vector of investment goods, and (2) adding flows of environmental amenities to the vector of consumption goods.

2. MODEL, ASSUMPTIONS, AND NOTATION This section presents the general model that will be used throughout the paper, lists the five assumptions that will be considered, and introduces notation. 2.1. Model Consider a setting where population is constant2 and where the current instantaneous well-being at time t depends on the vector of commodities C(t) = (C1 (t), . . . , Cm (t)) consumed at time t. To concentrate on the issue of intertemporal distribution, we abstract from how the goods and services consumed at time t are distributed among the population. Thereby we may associate the instantaneous well-being at time t with the utility U (C(t)) that is derived from the vector of consumption flows, C(t), at time t, where U is a time-invariant, increasing, and sufficiently differentiable function. Current consumption is presumed to be observable, along with its associated vector of accounting prices. For some of the results one must have that U is concave. That U is time-invariant means that all variable determinants of current well-being are included in the vector of consumption flows. In particular, nonconstant flows of environmental amenities flows derived from nonconstant stocks of natural capital are represented by components of the extended consumption vector C. If labor supply is not fixed, then supplied labor corresponds to negative components of the vector C. Thus, changes in instantaneous well-being will be measured net of the cost of turning leisure into labor effort. The vector of capital goods K(t) = (K 1 (t), . . . , K n (t)) available at time t includes not only the usual kinds of man-made capital stocks, but also stocks of natural resources, environmental assets, human capital (like education and knowledge capital accumulated from R&D-like activities), and other durable productive assets. Corresponding to the stock of capital of type j at time t, K j (t), there ˙ is a net investment flow: I j (t) := K˙ j (t). Hence, I(t) = (I1 (t), . . . , In (t)) = K(t) denotes the vector of net investments. A consumption–investment pair (C(t), I(t)) at time t is attainable if and only if (C(t), I(t)) ∈ S(K(t), t), where S is a sufficiently smooth set that describes society’s productive capacity. Current

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net investments are presumed to be observable, along with the associated vector of accounting prices. Some of the results require that S(K(t), t) is a convex set. Assume that society’s actual decisions are taken according to a resource allocation mechanism that assigns an attainable consumption–investment pair to any vector of capital stocks K and time t. Hence, for any vector of capital stocks K and time t, the resource allocation mechanism determines the consumption and investment flows. The investment flows in turn maps out the development of the capital stocks. The resource allocation mechanism thereby implements a feasible path of consumption flows, investment flows, and capital stocks, for any initial vector of capital stocks and any initial time. 2.2. Assumptions Consider a society with social preferences over infinite horizon utility paths. Let dynamic welfare be an index that represents these social preferences, meaning that if one path yields higher dynamic welfare than another, then it is (strictly) preferred in social evaluation. In this context, results of welfare and sustainability accounting can be classified according to which of the following five assumptions are being adopted. Hence, these specific assumptions are considered because they enable us to construct the taxonomy of results presented in Sects. 3–5. Assumption OPT (Optimality). Society has an optimal resource allocation mechanism implementing a price-supported3 – and thus efficient – path that maximizes dynamic welfare. Assumption DU (Discounted Utilitarianism). Dynamic welfare at time t is given by
∞ U (C(s))e−ρ(s−t) dt , t

where ρ is a positive utility discount rate. Notice that OPT can be satisfied without DU and vice versa. E.g., consider throughout this paragraph a situation where society in fact implements an efficient path with constant utility, but where the implementation of an optimal path according to DU would have lead to nonconstant utility. If, on the one hand, society’s dynamic welfare is given by infs≥t U (C(s)), then OPT is satisfied (since the implemented path and maximizes dynamic welfare), while DU is not satisfied (since  ∞ is efficient −ρ(s−t) dt does not represent the social preferences). If, on the other U (C(s))e t ∞ hand, society’s dynamic welfare is given by t U (C(s))e−ρ(s−t) dt, then DU is satisfied, while OPT is not satisfied (since the implemented constant utility path does not maximize dynamic welfare). Weitzman (1976) assumes both OPT and DU. Dasgupta and Mäler (2000), Dasgupta (2001), and Arrow et al. (2003a) assume DU without assuming OPT, while Solow (1974) assumes OPT without assuming DU.

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Assumption ST (Stationary Technology). The set S does not depend directly on t. This assumption is usually identified with comprehensive accounting and means that any variable determinant of current productive capacity is included in the vector of capital stocks. Weitzman (1976) along with most of the subsequent literature makes this assumption. To make accounting comprehensive in this way is a major challenge for empirical estimation. There are several reasons why ST may not be satisfied: technological progress not captured by augmented stocks, unaccountedfor stocks of natural capital, and open economies with changing terms-oftrade. Assumption CRS (Constant Returns to Scale). S as a set valued function of K is homogeneous of degree 1. This is seldom made as an explicit assumption; neither Weitzman (1976) nor most of the subsequent literature makes it (although I have used it in some of my own papers, e.g., in Asheim, 1996). The assumption is often invoked in illustrating examples. Combined with ST it means that also fixed determinants of current productive capacity must be included; the fixed amount of land is a prime example of this. In a world where natural and environmental resources are important, trying to satisfy this assumption turns empirical estimation into a very demanding task. In particular, the assumption of CRS necessitates that consumption flows and capital stocks are measured along scales where 0 is defined. For flows like environmental amenities and stocks like knowledge it is unclear what this entails. Notice that CRS can be satisfied without ST and vice versa. If, on the one hand, the ˙ > 0, then CRS is satisfied, while technology is given by C + I ≤ A(t)K with A(t) ST is not satisfied. If, on the other hand, the technology is given by C + I ≤ f (K ) with f exhibiting decreasing returns to scale, then ST is satisfied, while CRS is not satisfied. Assumption LH (Linear Homogeneity). U as a function of C is homogeneous of degree 1. This is a generalization of an assumption made by Weitzman (1976) to multiple consumption goods. Throughout the years, Dasgupta and Mäler have argued that national accounting should not be based on this assumption. 2.3. Notation The following notation will be used: p :

nominal consumption prices

P :

real consumption prices

q :

nominal investment prices

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201

real investment prices

Y = PC + QI :

real NNP .

It will be explained along the way what exactly is meant by “real” prices. 3. RESULTS UNDER ST AND DU OR OPT Assume ST and DU. By ST one can assume that the possibly inefficient resource allocation mechanism in the economy is Markovian and stationary, in the sense that the implemented consumption-investment pair is a time-invariant function of the vector of capital stocks (cf. Arrow et al., 2003a). This means that the consumptioninvestment pair (C(t), I(t)) at any time t is determined by the vector of capital stocks at time t, and does not depend directly on t. Hence, if {C(s)}∞ s=t is the implemented path given the initial stock K(t) = K, then the dynamic welfare of this path,
∞ V (K) = U (C(s))e−ρ(s−t) ds , (1) t

is a function solely of K. In particular, dV (K(t))/dt = ∇V (K(t))I(t) > 0 means that dynamic welfare is increasing at time t, where ∇ denotes a vector of partial derivatives, and where we follow Arrow et al. (2003a) by assuming that V is differentiable. Assume furthermore that this vector of partial derivatives can be calculated up to the choice of numeraire; hence, a vector of accounting prices q(t) =

∇V (K(t)) λ(t)

is observable, where λ(t) > 0 is the price of the numeraire in terms of utils.4 Then the sign of the observable entity q(t)I(t) indicates whether dynamic welfare is increasing. Notice that q(t)I(t) represents the value of net investments, and it is often referred to as the “genuine savings indicator” (cf. Hamilton, 1994, p. 166). Its sign is of course independent of the numeraire in which q(t) is measured. By differentiating (1) w.r.t. time and using the property that the resource allocation mechanism is Markovian, we obtain
∞ dV (K(t)) ∇V (K(t))I(t) = =ρ U (C(s))e−ρ(s−t) ds − U (C(t)) dt t or U (C(t)) + ∇V (K(t))I(t) = ρV (K(t)) . Differentiating once more w.r.t. time yields: ˙ ∇U (C(t))C(t) +

d∇V (K(t))I(t) = ρ∇V (K(t))I(t) . dt

(2)

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If we follow Asheim and Weitzman (2001) and let p(t) =

∇U (C(t)) λ(t)

denote calculated consumption prices in terms of the numeraire, we obtain ˙ p(t)C(t) +

dq(t)I(t) = r (t)q(t)I(t) , dt

(3)

˙ is the interest rate associated with the numeraire. By letting where r (t) = ρ − λ/λ real prices {P(t), Q(t)} be determined locally-in-time using a Divisia consumption price index (cf. Asheim and Weitzman, 2001; Sefton and Weale, 2006), so that ˙ P(t)C(t) = 0, it follows that: d(P(t)C(t) + Q(t)I(t)) Y˙ (t) = = R(t)Q(t)I(t) , (4) dt where Y (t) is real NNP and R(t) is the real interest rate. Since Q(t) is proportional to ∇V (K(t)), (4) implies that also Y˙ (t) > 0 indicates welfare improvement. Proposition 1. Under ST and DU, welfare improvement can be indicated by a positive value of net investments (qI > 0), or by growth in real NNP (Y˙ > 0). Assume now ST and OPT. As before ST means that the resource allocation mechanism in the economy is Markovian and stationary. By OPT the implemented path is price-supported, and it follows from optimal control theory that there are investment prices
(t) in terms of utility, such that d
(t)I(t) = ρ(t)
(t)I(t) , dt where ρ(t) is the supporting utility discount rate at time t, and where, as shown by Asheim and Buchholz (2004), dynamic welfare is improving if and only if
(t)I(t) > 0. By assuming that the efficiency prices
(t) are observable up to the choice of numeraire, and by repeating Asheim and Weitzman’s (2001) argument above – so again (4) follows when real prices are determined by a Divisia index – we obtain the following result. ˙ ∇U (C(t))C(t) +

Proposition 2. Under ST and OPT, welfare improvement can be indicated by a positive value of net investments (qI > 0), or by growth in real NNP (Y˙ > 0). The fundamental equation in both these results is (4), stating that change in real NNP = real interest rate · the real value of net investments. It is this equation that allows the “ ‘futurity’ in any welfare evaluation of any dynamic situation” (Samuelson, 1961, p. 53) to be captured by current national accounting aggregates.5 Say that development is sustainable at the current time, if the utility derived from the current vector of consumption flows can potentially be sustained forever.

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What does Propositions 1 and 2 tell us about the following question: Is the value of net investments (or, equivalently, real NNP growth) an indicator of sustainable development? The answer depends on the circumstances. Assume that ST is combined with OPT, and that the social preferences take sustainability into account, e.g., through the constraint that, at any time, current utility should not exceed the maximum sustainable utility level given the current capital stocks. By OPT, the agents in society expect that development will indeed be sustainable, and these expectations will be reflected by the relative investment prices. In such circumstances, nondecreasing welfare may well correspond to development being sustainable. Hence, since ST is satisfied, it follows from Prop. 2 that a nonnegative value of net investments (or equivalently, non-negative rate of real NNP growth) may serve as an exact indicator of sustainability. In Asheim and Buchholz (2004, Sect. 6.2) we provide an explicit example of this within the context of the model of capital accumulation and resource depletion introduced by Dasgupta and Heal (1974) and Solow (1974). In the setting of this model, we show that the growth rate of real NNP decreasing towards zero indicates that unconstrained development is no longer sustainable. Hence, the information on welfare changes offered by the growth rate of real NPP (or equivalently, the sign of the value of net investments) can be useful for the management of society’s assets, given that unsustainable paths are deemed socially unacceptable. If instead society adheres to DU, then – even if ST and OPT hold – sustainability need not be indicated in this manner, since DU does not necessarily lead to sustainable development and the ratio of investment prices may be affected by this. In context of the Dasgupta–Heal–Solow model it was established by Asheim (1994) and Pezzey (1994) that the value of net investments can be positive at the same time as utility exceed the maximum sustainable level. However, Pezzey (2004) has recently established a one-sided sustainability test under ST, OPT, and DU: It is a necessary condition for sustainable development that the value of net investments (or, equivalently, real NNP growth) is non-negative. The following result is a version of Pezzey (2004, Proposition 2). Proposition 3. Under ST, OPT and DU, the current level of utility cannot be sustained forever if the value of net investments is negative (qI < 0), or if growth in real NNP is negative (Y˙ < 0). To see this, notice that if ST and DU are assumed, then it follows from (2) that U (C(t)) + ∇V (K(t))I(t) is a Hicks (1946)–Weitzman (1976) stationary equivalent of future utility
∞   U (C(t)) + ∇V (K(t))I(t) e−ρ(s−t) ds = V (K(t)) (5) since

∞ t

t

e−ρ(s−t) ds = 1/ρ. Moreover, if OPT is added, then
∞ V (K(t)) ≥ U¯ · e−ρ(s−t) ds t

(6)

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where U¯ is the maximum level of utility that can be sustained forever from time t on, given the initial stocks at time t. It follows from (5)and (6) that U (C(t)) exceeds the maximum sustainable level if, at t, qI = ∇V (K)I /λ < 0, or equivalently, Y˙ < 0. Finally, notice that if ST and DU – but not OPT – hold, then the value of net investments and real NNP growth are quite unreliable indicators of sustainability. Consider, e.g., a society where traditional growth is promoted through high investment in reproducible capital goods, but where incorrect (or lack of) pricing of natural capital leads to depletion of natural and environmental resources that is excessive both from the perspective of short-run efficiency and long-run sustainability. Then utility growth in the short to intermediate run will, if the discount rate ρ is large enough, lead to current growth in dynamic welfare. Hence, both the value of net investments and real NNP growth will be positive. At the same time, the resource depletion may seriously undermine the long-run livelihood of future generations, so that current utility far exceeds the level that can be sustained forever.6 The local-in-time character of these results means as NNP as a linear index (cf. Hartwick, 1990) has significance for welfare and sustainability even though no linearity assumptions (like CRS and LH) are made. The next two sections will present stronger assumptions, on the basis of which national accounting aggregates can be used for global welfare comparisons. The major information problem that must be faced to utilize the results on this section, is how to make accounting comprehensive when – at the outset – not all capital goods that contribute to increased productive capacity are included, i.e., the technology is not stationary. To apply Proposition 1, one must in addition be able to calculate the vector of partial derivatives of the welfare function, V , to determine accounting prices. If, instead, OPT holds, then the vector of relative investment prices, q(t), correspond to actual market prices or can be calculated as efficiency prices using standard techniques. 4. RESULTS UNDER ST, DU, AND LH Turn now to global welfare comparisons:

r Either in one society over time, where K = K(t  ) is the vector of capital stocks at time t  and K = K(t  ) is the vector of capital stocks at time t 

r Or across different societies, where K is the vector of capital stocks in the one society and K is the vector of capital stocks in the other society.

Which of the vectors of capital stocks, K or K , corresponds to higher welfare? It follows from the previous section that, under ST and DU,
K   ∇V (K)dK V (K ) − V (K ) = K

is a measure of welfare differences that is independent of the path between K and K . So if q corresponding to different values of K can be measured in a

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numeraire that is in a fixed proportion to utils, then a global measure of welfare differences would be available. However, to be useful for empirical estimation this essentially requires that utils are measurable. As the following argument suggests, LH is sufficient for utils to be measurable. If the utility function U is homothetic, then a Divisia consumption price index is path independent, so that real prices can be determined globally.7 Moreover, if LH is satisfied, so that U is linearly homogeneous, then these real prices are measured in a numeraire that is in a fixed proportion to utils. W.l.o.g. we may set the factor of proportionality equal to one, so that P = ∇U (C) and Q = ∇V (K), and implying that U (C) = ∇U (C)C = PC . It now follows from (2) that: Y = PC + QI = ρV (K) .

(7)

Furthermore, 



V (K ) − V (K ) =




K

K

Q dK .

This yields the following result. Proposition 4. Under ST, DU and LH, a positive welfare difference can be indi K cated by a positive real value of stock differences ( K Q dK > 0), or by a positive difference in real NNP (Y  − Y  > 0). The fundamental equation in this result is (7), stating that real NNP = real interest rate · the present value of future consumpton. This is Weitzman’s (1976) main result, which we here have established without invoking OPT, but instead assuming that the vector of partial derivatives of V can be calculated. Notice that the r.h.s. of (7) is not wealth in the sense of the current value of stocks, unless we make further assumptions (see below). Notice also that  K K Q dK is not a difference in wealth, but rather a “wealth-like magnitude,” to use Samuelson’s (1961) term. A major information problem that must be faced to utilize the result on this section, is how to measure utility if the utility function is not linearly homogeneous. The addition of LH does not help when it comes to indicating sustainability: Under ST and DU the value of net investments and real NNP growth are quite unreliable indicators of sustainability. 5. RESULTS WITHOUT ST If ST is not satisfied, then not all variable determinants of current productive capacity are included in the vector of capital stocks. Still, by imposing the linearity conditions CRS and LH in addition to OPT and DU, positive results can be obtained.

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GEIR B. ASHEIM

Assume OPT and CRS. Then it follows that the wealth, in the sense of the current value of stocks, equals the present value of future consumption:
∞ s q(t)K(t) = p(s)C(s)e− t r (τ )dτ ds , t

where r (s) is the nominal discount rate at time s. The result is simple to establish in the framework of Dixit et al. (1980), and has a clear intuition: CRS means that all flows of future earnings can be treated as currently existing capital. Notice that OPT is needed for this result. To see this, consider Solow’s (1974) maximin path in the Dasgupta–Heal–Solow model of capital accumulation and resource depletion when CRS is satisfied, and let society’s dynamic welfare be given by DU. Then the present value of future consumption as evaluated by DU is constant, while the value of the stocks increases since the value of net investments is zero (cf. Hartwick, 1977) and there are positive anticipated capital gains on the resource. When we add the assumptions of DU and LH, so that one can determine real prices in terms of utils and the value of consumption equals utility as discussed in Sect. 4, it follows from this result that:
∞
∞ Q(t)K(t) = P(s)C(s)e−ρ(s−t) ds = U (C(s))e−ρ(s−t) ds = V (K(t)). (8) t

t

Moreover, by time-differentiation we get that
∞ ˙ ˙ Q(t)K(t) + Q(t)K(t) =ρ U (C(s))e−ρ(s−t) ds − U (C(t)) t

˙ or (applying that U (C) = PC and I = K) ˙ ˙ Y (t) + Q(t)K(t) = P(t)C(t) + Q(t)I(t) + Q(t)K(t) = ρQ(t)K(t).

(9)

Combining (8) and (9) leads to the following result. Proposition 5. Under OPT, DU, CRS, and LH, a positive welfare difference can be indicated by a positive difference in wealth (Q K − Q K > 0), or by a posi˙  K )) − (Y  + tive difference in real NNP plus anticipated capital gains ((Y  + Q   ˙ Q K ) > 0). The fundamental equation in this result is (9), stating that real NNP + anticipated capital gains = real interest rate · real wealth. Given that DU and LH imply a constant rate of interest equal to the utility discount rate ρ, the anticipated capital gains capture the effects of a nonstationary technology.8 Hence, if we add ST to the list of assumption, then we arrive at real NNP = real interest rate · real wealth. However, the needed assumptions – OPT, DU, ST, CRS, and LH – makes this not a very interesting result as a basis for empirical estimation.

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Under OPT, DU, CRS, and LH, a variant of Pezzey’s (2004) one-sided sustainability test (cf. Proposition3) can be obtained. Proposition 6. Under OPT, DU, CRS and LH, the current level of utility cannot be sustained forever if wealth decreases (d(QK)/dt < 0), or if growth in the sum of real ˙ NNP and anticipated capital gains is negative (d(Y + QK))/dt < 0). The proof is similar as the one for Proposition 3: If OPT, DU, CRS and LH are ˙ is a Hicks–Weitzman stationassumed, then it follows from (8) and (9) that Y + QK ary equivalent of future utility
∞   −ρ(s−t) ˙ Y (t) + Q(t)K(t) e ds = V (K(t)) (10) ∞

t

since t = 1/ρ. Moreover, by OPT, (6) holds, where U¯ is the maximum level of utility that can be sustained forever from time t on, given the initial stocks ˙ it follows from (6) at time t. Since, under LH, U (C) = PC and Y = PC + QK, and (10) that U (C(t)) exceeds the maximum sustainable level if, at t, d(QK))/dt = ˙ + QK ˙ < 0, which is, by (9), is equivalent to (d(Y + QK))/dt ˙ QK < 0. Hence, under OPT, DU, CRS, and LH: e−ρ(s−t) ds

r Welfare improvement can be measured by increasing wealth (or by growth in real NNP plus anticipated capital gains)

r It is a necessary condition for sustainable development that wealth (or, equivalently, real NNP plus anticipated capital gains) does not decline However, the results depend on LH and CRS, which are strong and controversial linearity assumptions. 6. METHODS FOR SATISFYING INFORMATIONAL DEMANDS When doing practical estimation, the assumptions above represent serious informational demands. What techniques can be used to satisfy these informational demands? All results require that one must be able to account for changes in society’s productive capacity. Such changes may be caused by:

r Accumulation of ordinary reproducible capital r Technological change or human capital accumulation r Reduced resource availability r In the case of open economies, changing terms-of-trade The assumption of ST means that all such changes are captured by the vector of investments flow, where the size of these flows can be measured and valued at

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efficiency or accounting prices. What can be done if the assumption of ST is not satisfied, so that it is not the case that all changes in society’s productive capacity correspond to stock changes that can be measured and valued? One – purely formal, but in principle important – method consists of letting time be an additional state variable; i.e., an additional capital component. This reformulates the problem as one of measuring the “value of passage of time.” A first attempt at a practical solution is to assume that the value of passage of time does not change over time. Then time does not contribute to changes in the value of net investments. Hence, time need not be included when calculating growth in real NNP. On the other hand, one must calculate how the real value of consumption changes over time, where the consumption vector must include e.g., environmental amenities. Growth in real NNP will also give a right qualitative result if the value of passage of time is in fixed proportion to total NNP. The Dasgupta–Heal–Solow model of capital accumulation and resource depletion illustrates this possibility: With fixed factor shares, the value of resource depletion is a fixed proportion of total NNP. Therefore, real NNP growth measures welfare improvement even if resource depletion is not included, while the measurable value of net investments can be grossly inaccurate if resource depletion is left out. A model where technological progress is endogenous – in the sense that human capital accumulation equals the fraction of net output that is used for neither consumption nor accumulation of ordinary reproducible capital – illustrates another possibility. Assume that net output (=real NNP) is observable, but that it is not possible to observe how net output not used to augment the stock of ordinary capital is split between consumption, on the one hand, and investment in human capital, on the other. Then, real NNP growth can be used for indicating welfare improvement (and sustainability), while the sign of the measurable value of net investments may not give a correct indication since human capital accumulation cannot be distinguished from consumption. Hence, in spite of the theoretical equivalence between real NNP growth and the value of net investments expressed in Propositions 1 and 2 (as well as Proposition 3), there seems to be cases where the informational requirements are smaller when using real NNP growth as an indicator for welfare improvement (and sustainability). On the other hand, the value of net investments does not need valuation of the components of the extended consumption vector C capturing environmental amenities.9 A second attempt at a practical solution is to try directly to measure the value of passage of time, using forward-looking terms. Such method has been suggested by e.g., by Aronsson et al. (1997), Pezzey (2004), Sefton and Weale (1996), and Vellinga and Withagen (1996). In particular, Sefton and Weale (1996) show how to take into account changing terms-of-trade being faced by a resource exporter. A third attempt at a practical solution is to assume a constant interest rate (being implied by DU and LH), and then combine this assumption with OPT and CRS. As pointed out in Sect. 5, anticipated capital gains will, under this set of assumptions, capture the effects of changes in society’s productive capacity. When using this method, one must, as mentioned in Sect. 2, face the practical problems that arise

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when trying to satisfy the CRS assumption, especially in a world where natural and environmental resources are important. Moreover, unanticipated (“windfall”) capital gains as well as capital gains arising from changing interest rates must be excluded. The method seems to be of practical interest when estimating the sustainable income arising from a raw material exporting country’s resource endowment. Sections 4 and 5 list DU and LH among the assumptions that are sufficient for global comparisons; i.e., for comparisons between two points in time that are not adjacent or between two societies that are not similar. We have seen that the assumption of LH ensures that the value of consumption equals the utility derived from consumption. How can utility be measured if the utility function is not homogeneous of degree 1? To investigate this, allow the utility function introduced in Sect. 2 to be a strictly concave function. Let us also – for the purpose of the analysis of the following paragraphs – denote this utility function by U˜ , so U˜ (C) is the well-being derived from the vector of consumption flows C. A technical way to transform U˜ into a utility function U that is homogeneous of degree 1 is to add an additional consumption good, say good 0, where C0 ≡ 1. Then we can define U by U (C0 , C) := C0 · U˜ (C/C0 ) , implying that U is homogeneous of degree 1. The problem of applying the global welfare comparison analysis of Sect. 3 is now “reduced” to determining the price change of the added consumption good 0, so that ∗ (t) = ˙ one can determine a Divisia consumption price index satisfying P˙0 (t) + P(t)C 0, or, equivalently, ∗ ˙ P˙0 (t) = −P(t)C (t) ,

(11)

entailing that prices are in fixed proportion to utility, and it holds w.l.o.g. that P0 (t) = ∂U (1, C(t))/∂C0 and P(t) = ∇U (1, C(t)) = ∇ U˜ (C(t)) .

(12)

Notice that (12) implies that P0 (t) =

∂U (1, C) = U˜ (C(t)) − ∇ U˜ (C(t))C(t) = U˜ (C(t)) − P(t)C(t) . ∂C0

∗ (t) corresponds to the per time unit loss of “consumers’ sur˙ Hence, by (11), P(t)C plus”. By following Weitzman (2001) one can argue that this change in “consumers’ surplus” is in principle observable in a market economy (see also Li and Löfgren, 2002). Global comparisons across space are not only more difficult because they require utility to be measurable. The assumption of ST also become more demanding when comparing two different societies. Recall that ST requires all variable determinants of productive capacity to be included in the vector of capital stocks. Since two different societies are likely to be less similar than the same society at two different times along the time axis, the vector of capital stocks must include more components when making comparisons across space.

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7. SUMMARY OF RESULTS I have considered five assumptions – OPT, DU, ST, CRS, and LH – which can be made separately of each other. This makes altogether 32 different combinations of assumptions, as illustrated in the two tables. Table 14.1 summarizes results in welfare accounting. For 12 combinations, namely those that satisfy ST and either OPT or DU, it has been shown that a local-in-time welfare improvement can be indicated both by a positive value of net investments and by growth in real NNP. The fundamental equation is that change in real NNP equals the real interest on the real value of net investments. For a subset of four combinations, namely those that satisfy DU, ST, and LH, it has been shown that a positive global welfare difference can be indicated both by a positive real value of stock differences and by a positive difference in real NNP. The fundamental equation is that real NNP equals the real interest on the present value of future consumption. For two combinations, namely those that satisfy OPT, DU, CRS, and LH, it has been shown that a global welfare difference can be indicated both by a positive wealth difference and by a positive difference in real NNP plus anticipated capital gains. Only if all five assumptions are made – namely OPT, DU, ST, CRS, and LH – is it the case that both (1) real NNP is the real interest on real wealth and (2) differences in wealth have welfare significance.

Table 14.1. Overview of Assumptions and Results in Welfare Accounting Not ST

ST

Not CRS

Not CRS

CRS

LH Not – – OPT Not LH DU

– qI > 0, Y˙ > 0 welf. improvem.

– qI > 0, Y˙ > 0 welf. improvem.

 K

QdK > 0, Y  > Y  greater welfare

 K

LH qK = pCe− r dτ dt

qI > 0, Y˙ > 0 welf. improvem.

qI > 0, Y˙ > 0 welf. improvem.

 QK = µU (C)dt – =  qK pCe− r dτ dt

qI > 0, Y˙ > 0 welf. improvem. – qI > 0, Y˙ > 0 welf. improvem.

qI > 0, Y˙ > 0 welf. improvem. – qI > 0, Y˙ > 0 welf. improvem.

CRS

Not LH Not DU

Not LH Not DU LH OPT – – Not LH DU LH

K



Q K > Q K greater welfare

 K

QdK > 0, Y  > Y  greater welfare K

QdK > 0, Y  > Y  greater welfare K

Q K > Q K , Y  > Y  greater welfare

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Table 14.2. Overview of Assumptions and Results in Sustainability Accounting Not ST Not CRS

ST Not CRS

CRS

qI ≥ 0, Y˙ ≥ 0 may ind. sust.

qI ≥ 0, Y˙ ≥ 0 may ind. sust.

qI ≥ 0, Y˙ ≥ 0 may ind. sust. – qI < 0, Y˙ < 0 ⇒ unsustain.

qI ≥ 0, Y˙ ≥ 0 may ind. sust. – qI < 0, Y˙ < 0 ⇒ unsustain.

d(QK)/dt < 0 qI < 0, Y˙ < 0 ⇒ unsustain. ⇒ unsustain.

d(QK)/dt < 0, Y˙ < 0 ⇒ unsustain.

CRS

Not LH Not DU LH Not – – OPT Not LH DU LH

Not LH Not DU LH OPT – – Not LH DU LH

Table 14.2 summarizes results in sustainability accounting. For combinations that satisfy only DU and ST, it appears that the value of net investments (or, equivalently, real NNP growth) is a quite unreliable indicator of whether current utility can potentially be sustained forever. For eight combinations, namely those that satisfy OPT and ST, it has been argued that a non-negative value of net investments (or equivalently, non-negative rate of real NNP growth) may serve as an exact indicator of sustainability, provided that the social preferences take sustainability into account. For four combinations, namely those that satisfy OPT, DU, and ST, it has been shown that a negative value of net investments (or equivalently, negative rate of real NNP growth) implies that development is unsustainable. For two combinations, namely those that satisfy OPT, DU, CRS, and LH, it has been shown that decreasing real wealth implies that development is unsustainable. The analysis of this paper illustrates how stronger results require stronger assumptions, and thereby impose harder informational requirements. Acknowledgments: I thank Kenneth Arrow, Wolfgang Buchholz, Lawrence Goulder, Nick Hanley, Peter Hammond, and John Pezzey for helpful comments. An earlier version, entitled “Assumptions and results in welfare accounting”, was presented at a workshop on “Putting Theory to Work: The Measurement of Genuine Wealth” held at Stanford Institute for Economic Policy Research, 25–26 April 2002. I gratefully

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acknowledge the hospitality of the research initiative on the Environment, the Economy and Sustainable Welfare at Stanford University and financial support from the Hewlett Foundation and CESifo Munich. NOTES 1 The present paper extends my earlier overview, Asheim (2000), by considering multiple consumption

goods and by presenting additional results. However, it leaves out concepts that do not easily generalize to a multiple-consumption-good setting, and does not discuss the relationship between green national accounting and social cost–benefit analysis. 2 It is worthwhile also to analyze the case with a changing population. There are results available (cf., e.g., Hamilton, 2002) under exponential population growth when only per capita consumption matters, provided that one is willing to assume CRS, introduced below. Contributions where population growth need not be exponentital and where instantaneous well-being also depends on population size are emerging (cf., e.g., Arrow et al., 2003b; Asheim, 2004). These result cannot as easily be integrated into the present taxonomy and are not treated here. 3 Formally, “price-supported” means that there is an infinite-dimensional hyperplane, containing the implemented path, that separates all feasible paths from those that are socially preferable. Malinvaud (1953) introduced this mathematical tool to the study of dynamic infinite-horizon economies. 4 With an optimal resource allocation mechanism, λ(t) is the marginal utility of current expenditures. 5 Under his stronger set of assumptions, this equation was used by Weitzman (1976, (14)). 6 This should be borne in mind when, e.g., Arrow et al. (2003a) under ST and DU identify the term “sustainable development at t” with non-negative value of net investments at t, instead of making use of the definition considered here, namely that utility at t can potentially be sustained forever. 7 Cf. Hulten (1987) for a discussion of the properties of a Divisia index. 8 See, e.g., Vincent et al. (1997). 9 There does not seem to be much empirical work that tries to crosscheck QI and Y˙ as measures of welfare and sustainability. See, however, Hanley et al. (1999).

REFERENCES Aronsson, T., Johansson, P.O., and Löfgren, K.-G. (1997), Welfare Measurement, Sustainability and Green National Accounting. Edward Elgar, Cheltenham, UK Arrow, K., Dasgupta, P.S., and Mäler, K.-G. (2003a), Evaluating projects and assessing sustainable development in imperfect economies, Environmental and Resource Economics 26, 647–685 Arrow, K., Dasgupta, P.S., and Mäler, K.-G. (2003b), The genuine savings criterion and the value of population, Economic Theory 21, 217–225 Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Asheim, G.B. (1996), Capital gains and ‘net national product’ in open economies, Journal of Public Economics 59, 419–434 (Chap. 10 of the present volume) Asheim, G.B. (2000), Green national accounting: why and how? Environment and Development Economics 5, 25–48 Asheim, G.B. (2004), Green national accounting with a changing population, Economic Theory 23, 601– 619. (Chap. 19 of the present volume) Asheim, G.B. and Buchholz, W. (2003), The malleability of undiscounted utilitarianism as a criterion of intergenerational justice, Economica 70, 405–422 (Chap. 5 of the present volume) Asheim, G.B. and Weitzman, M. (2001), Does NNP growth indicate welfare improvement?, Economics Letters 73, 233–239 (Chap. 17 of the present volume)

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Dasgupta, P.S. (2001), Valuing objects and evaluating policies in imperfect economies, Economic Journal 111, C1–C29 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Mäler, K.-G. (2000), Net national product, wealth, and social well-being, Environment and Development Economics 5, 69–93 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Hamilton, K. (1994), Green adjustments to GDP, Resources Policy 20, 155–168 Hamilton, K. (2002), Saving effort and population growth: theory and measurement, The World Bank Hanley, N., Moffatt, I., Faichney, R., and Wilson, M. (1999), Measuring sustainability: a time series of alternative indicators for Scotland, Ecological Economics 28, 55–73 Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 67, 972–974 Hartwick, J.M. (1990), National resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291–304 Hicks, J. (1946), Value and capital, 2nd edition. Oxford University Press, Oxford Hulten, C.R. (1987), Divisia index, in Eatwell, J., M. Milgate, M. and Newman, P. (eds.), The New Palgrave: A Dictionary of Economics. Macmillan, London and Basingstoke. Li, C.-C. and Löfgren, K.-G. (2002), On the choice of metrics in dynamic welfare analysis: Utility vs. money measures, Umeå Economic Studies 590. Malinvaud, E. (1953), Capital accumulation and efficient allocation of resources, Econometrica 21, 233–268 Pezzey, J.C.V. (1994), Theoretical Essays on Sustainability and Environmental Policy. Ph.D. thesis, University of Bristol Pezzey, J. (2004), One-sided sustainability tests with amenities, and changes in technology, trade and population, Journal of Environment Economics and Management 48, 613–631 Samuelson, P. (1961), The evaluation of ‘social income’: Capital formation and wealth, in Lutz, F.A. and Hague, D.C. (eds.), The Theory of Capital, St. Martin’s, New York Sefton, J.A. and Weale, M.R. (1996), The net national product and exhaustible resources: The effects of foreign trade, Journal of Public Economics 61, 21–47 Sefton, J.A. and Weale, M.R. (2006), The concept of income in a general equilibrium, Review of Economic Studies 73, 219–249 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Vincent, J.R., Panayotou, T., and Hartwick, J.M. (1997), Resource Depletion and Sustainability in Small Open Economies, Journal of Environmental Economics and Management 33, 274–286 Vellinga, N. and Withagen, C. (1996), On the Concept of Green National Income, Oxford Economic Papers 48, 499–514 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162 Weitzman, M.L. (2001), A contribution to the theory of welfare accounting, Scandinavian Journal of Economics 103, 1–23

CHAPTER 15 NET NATIONAL PRODUCT AS AN INDICATOR OF SUSTAINABILITY

1. INTRODUCTION Net National Product (NNP) can potentially serve several objectives, among others to measure value added and to be an indicator of welfare. In the aftermath of the World Commission on Environment and Development, however, it also seems important to investigate whether the concept of NNP can serve as an indicator of sustainability. My point of departure, following Hicks (1946, Chapter 14), is therefore to require that NNP should measure what can be consumed in the present period without reducing future consumption possibilities1 and, in line with this, to argue that the NNP should equal the maximum per capita consumption level that can be sustained. If sufficiently many facilitating assumptions are made, then a concept of NNP that serves as an exact indicator of sustainability can easily be constructed. For instance, for a closed economy with a constant population, a stationary technology, and with only one capital good, it follows that the present does not reduce future consumption possibilities if and only if it does not decrease the stock of the single capital good. Hence, under the assumption that the one capital good k is identical to the one consumption good x, NNP defined as yt := xt + k˙t measures the maximal sustainable consumption at time t, because then xt
yt is equivalent to k˙t  0. The purpose of this note is to establish that with multiple capital goods, it is not in general possible to construct an exact indicator of sustainability on the basis of current price information as usually suggested. This conclusion holds even if (1) the remaining facilitating assumptions are retained and (2) the existence of an intertemporal competitive equilibrium is assumed; in particular, price information for all goods (including natural capital goods) is assumed to be available in an economy without market imperfections of any kind. The background for the interest in the problem of sustainability is that human economic activity leads to the depletion of natural capital. It therefore becomes an important question whether our accumulation of (real and human) man-made capital is sufficient to make up for the decreased availability of natural capital. Hence, sustainability is interesting only within models of heterogeneous capital.

Originally published in Scandinavian Journal of Economics 96 (1994), 257-265. Reprinted with permission from blackwell.

215 Asheim, Justifying, Characterizing and Indicating Sustainability, 215–223 c 2007 Springer


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Furthermore, even with a constant population and a stationary technology, an efficient path giving rise to constant consumption (or constant utility) will not be a stationary path. In contrast, the stocks of natural capital will tend to be depleted, while the stocks of man-made capital will tend to be accumulated. As argued by Bliss (1975), there will not in general exist one constant rate of interest along a nonstationary path in a heterogeneous capital model. Rather, each capital good will have its own rate of interest that will vary over time. Moreover, maximizing the sum of utilities disocunted at a constant rate will not in general give rise to a constant utility path.2 In contrast to Weitzman (1976), Solow (1986), Hartwick (1990), and Mäler (1991), who lay the foundation for a concept of NNP that is adjusted for the depletion of natural and environmental resources, I find it more appropriate for the discussion of NNP as an indicator of sustainability to build on a framework which allows for a nonconstant rate of utility discounting. Section 2 presents such a framework, which is then used in Section 3 to discuss the seminal analyses of Weitzman (1976) and Hartwick (1977). In Section 4 it is shown through a counter example that the concept of NNP which – on the basis of these contributions – has been suggested in order to take into account the depletion of natural capital is not in general an indicator of sustainability. The interpretation of this result is that – even in an economy without market imperfections of any kind – the competitive prices at a given time cannot, with multiple capital goods, provide information on whether consumption exceeds the sustainable level since, in an intertemporal competitive equilibrium, the relative prices of the multiple capital goods will depend on the entire equilibrium path.

2. THE GENERAL FRAMEWORK The general analysis of a constant population economy with heterogeneous capital builds on Dixit et al. (1980). The indicator of the quality of life u(x) depends on the net output vector x, where u(·) is a stationary and differentiable utility function. ˙ be feasible if and only if (x, k, k) ˙ ∈ Y , where Y is a smooth and convex Let (x, k, k) set and where k is the vector of nonnegative capital stocks. The set Y is assumed ˙ ∈ Y and k˙ 
k, ˙ then to satisfy free disposal of investment flows; i.e., if (x, k, k)  (x, k, k˙ ) ∈ Y . The components of k that represent nonrenewable natural resources can only be depleted, and the corresponding components of k˙ must be nonpositive. Stocks that directly influence the quality of life (e.g., environmental resources) are taken into account by letting these components of k correspond to components of x. For environmental resources, the assumption of free disposal of investment flows means that these positive valued resources can be freely destroyed; hence, negatively valued waste products can be freely generated, not freely disposed of. Accumulated knowledge from learning and research activities may be included as components of k, thus allowing for endogenous technological progress; see Weitzman (1976).

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∗ A feasible path (x∗t , k∗t , k˙ t )∞ t=0 will be called competitive at present value prices ∞ (pt , qt )t=0 and utility discount factors λt > 0 if and only if: ∗ (i) For each t, (x∗t , k∗t , k˙ t ) maximizes instantaneous profit pt x + qt k˙ + q˙ t k sub˙ ∈ Y. ject to (x, k, k)

(ii) For each t, x∗t maximizes λt u(x) − pt x over all x. The imputed rents to the assets are equal to −˙qt , measuring the marginal productivity of the capital stocks. For capital goods that are unproductive as stocks (e.g., nonrenewable natural resources), the corresponding components of −˙qt are zero (i.e., the Hotelling rule). The corresponding components of qt equal the profits or rents that arise when such resources are depleted. Note that (i) combined with the assumption of free disposal of investment flows implies that the vector qt is nonnegative. ∗ A competitive path (x∗t , k∗t , k˙ t )∞ t=0 will be called regular at present value prices ∞ (pt , qt )t=0 and utility discount factors λt > 0 if and only if:
∞ (a) 0 λt u(x∗t ) dt exists and is finite (b)

qt k∗t → 0 to t → ∞.

Condition (b) entails that the value of the capital stocks along a regular path equals the present value of the rents that arise from the future productivity and depletion
∞ ∗ of the stocks: qt k∗t = t [(−˙qs )k∗s + qs (−k˙ s )] ds. A regular path is efficient and
∞ maximizes 0 λt u(xt ) dt over all feasible paths (xt , kt , k˙ t )∞ t=0 with given initial stocks k since, for each t, λt > 0, qt kt  0, and, by (i) and (ii), λt [u(xt − u(x∗t )]

T d[qt k∗t − qt kt ]/dt; i.e., 0 λt [u(xt ) − u(x∗t )] dt
qT k∗T . Hence, a regular path provides price information for the net output vector and capital stocks and can be realized as a competitive equilibrium if the intergenerational altruism of each generation
∞ t is represented by t λs u(xs ) ds. With the price information provided by a regular path, is it possible to construct a concept of NNP that can serve as an indicator of sustainability? Let (vt∗ )∞ t=0 denote ∗ the NNP in terms of utility along the regular path (x∗t , k∗t , k˙ t )∞ . Generalizing the t=0 analysis for the case with one capital good to multiple capital goods, a natural ∗ candidate for a concept of NNP in utility terms is vt∗ := u(x∗t ) + qt k˙ t /λt . If net output is a scalar x, the NNP may alternatively be defined in terms of the single consumption good. For this case, let (yt∗ )∞ t=0 denote the NNP in consumption terms ∗ ∞ ∗ ∗ ˙ along the regular path (xt , kt , kt )t=0 . A natural candidate for this “dollar-value” ∗ NNP, see Hartwick (1990), is yt∗ := xt∗ + qt k˙ t / pt . The question of whether vt∗ ∗ (or yt∗ ) is an indicator of sustainability becomes a question of whether qt k˙ t  0 is equivalent to u(x∗t ) (or xt∗ ) not exceeding the maximum sustainable utility (con∗ sumption) level. Two alternative foundations for the use of vt∗ := u(x∗t ) + qt k˙ t /λt ∗ (or yt∗ := xt∗ + qt k˙ t / pt ) can be found in Weitzman (1976) and Hartwick (1977). Can

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these contributions be used to defend vt∗ (or yt∗ ) as an indicator of sustainability? Is capital management sustainable if and only if the market value of net investments is nonnegative? 3. THE CONTRIBUTIONS OF WEITZMAN (1976) AND HARTWICK (1977) The following is a generalized version of the result reported in Weitzman (1976). ∗

∞ Proposition 1: If a path (x∗t , k∗t , k˙ t )∞ t=0 is regular at prices (pt , qt )t=0
and utility ∗ ∞ discount factors λt = e−r t , r > 0, then for each t, (u(x∗t ) + qt k˙ t /λt ) · t λs ds =
∞ ∗ t λs u(xs ) ds. ∗ Proof. By the Lemma below, d(λt u(x∗t ))/dt = λ˙ t u(x∗t ) − d(qt k˙ t )/dt or ∗ ∗ ∗ ∗ ∗ ∗ ∗ d(λ (u(xt ) + qt k˙ t )/dt = λ˙ t u(xt ). Hence, λt u(xt ) + qt k˙ t − (λT u(xT ) + qT k˙ T ) =

T t T ∗ ∗ ˙ ˙ t (−λs )u(xs ) ds = r t λs u(xs ) ds with r = −λt /λt > 0. Assuming that the growth rate of each capital stock and price is bounded above, it follows from (a) and ∗ (b) that λT u(x∗T ) + qT k˙ T → 0 as T → ∞.3 ∗ ∞ Lemma 1: If a path (x∗t , k∗t , k˙ t )∞ t=0 is competitive at prices (pt , qt )t=0 and utility ∗ ∗ ∗ discount factors λt > 0, then for each t, λt du(xt )/dt = pt x˙ t = −d(qt k˙ t )/dt.

Proof. The result follows from Dixit et al. (1980, proof of Theorem 1). Hence, if the utility discount rate −λ˙ t /λt is constant and equal to r , the present   value of future utility equals the present value for receiving the utility flow u x∗t +
∞  ∞ ∗ qt k˙ t /λt for all s  t. However, if x∗s s=t uniquely maximizes t λs u(xs ) ds over all feasible paths and does not yield constant utility, then Proposition 1 implies that   ∗ it is not feasible to sustain a constant utility level equal to u x∗t + qt k˙ t /λt . Also,  ∗ ∗ qt k˙ t  0 does not imply that u xt is sustainable. This result, which undermines a claim made by Mäler (1991, p. 11) and Hulten (1992, p. 17), will be formally established in Section 4. The result originally stated by Weitzman (1976) follows as a corollary. Corollary 1: (Weitzman 1976) Suppose u(x) = x with net output x being a scalar. If ∗ ∞ a path (xt∗ , k∗t , k˙ t )∞ discount factors λt = t=0 is regular at prices ( pt , qt )t=0 and utility
∞
∞ ∗ −r t ∗ ˙ e , r > 0, then for each t, λt = pt and (xt + qt kt / pt ) · t ps ds = t ps xs∗ ds. Hence, if the own interest rate of consumption good is constant, the present value ∗ of future consumption equals the present value of consuming xt∗ + qt k˙ t / pt for all s  t. Weitzman (1976) did not make the incorrect claim that this implies that it is ∗ feasible to sustain a consumption level equal to xt∗ + qt k˙ t / pt . Moreover, note Svensson’s (1986) observation that Proposition 1 (or Corollary 1) cannot be established without the assumption of a positive and constant utility discount rate (or consumption interest rate); in the model of Solow (1974) and Hartwick (1977), this assumption is inconsistent with sustainable development. This alone undermines the relevance of this result for the study of NNP and sustainability.

NET NATIONAL PRODUCT AS AN INDICATOR OF SUSTAINABILITY

219

Hartwick (1977) finds that, in a closed economy with a constant population and ∗ a stationary technology steering along an efficient path with qt k˙ t = 0 for all t, the utility level is constant and equal to the maximal sustainable level. ∗ Proposition 2: (Hartwick 1977) If a path (x∗t , k∗t , k˙ t )∞ t=0 is regular at prices   ∗ (pt , qt )∞ and utility discount factors λ > 0, and for each t, qt k˙ t = 0, then u x∗t t t=0 is constant.

  Proof. The Lemma implies that along a competitive path, u x∗t is constant if and ∗ ∗ only if qt k˙ t is constant. Having qt k˙ t = 0 for all t is therefore sufficient. Dixit et al. (1980) show the converse under weak assumptions. If the utility level is ∗ ∗ constant along an efficient path, then qt k˙ t = 0 for all t. Hence, qt k˙ t = 0 for all t ∗ becomes equivalent to u(xt ) being equal to the maximum sustainable utility level for   ∗ all t. This supports vt∗ := u x∗t + qt k˙ t /λt as an indicator of sustainability. In the context of a competitive economy, Hartwick’s rule states that a competitive equilibrium leads to a completely egalitarian utility path if and only if, at all times, the values of depleted natural capital measured in competitive prices equals the reinvestment in man-made capital. However, Hartwick’s rule does not claim that a competitive economy which, for the moment, at market value reinvests depleted natural capital in man-made capital, manages its stocks of natural and man-made capital in a sustainable manner. For it is conceivable that such reinvestment is achieved because the competitive prices of natural capital are low. This, in turn, can be caused by the economy not being managed in a sustainable manner. If future generations are poorer than we are, they will be unable to “bid’ highly through the intertemporal competitive equilibrium for the depletable natural capital we manage, leading to low prices of such capital today. Although Hartwick’s rule implies that ∗ qt k˙ t = 0 at any time t if the economy follows an efficient and egalitarian utility   ∗ path, it cannot be concluded that u x∗t is sustainable if qt k˙ t = 0 at some time t. This is shown formally in Section 4. Hence, Hartwick’s rule characterizes a sustainable development; it is not a prescriptive rule for a sustainable development.

4. A COUNTER EXAMPLE Consider a model in which the flow of natural capital (a nonrenewable resource), −k˙n , is combined with a stock of man-made capital, km , in order to produce a consumption good, x. The model, that fits into the framework of Section 2, is described by the following technology and utility function: x + k˙m
(km )a (−k˙n )b , u(xt ) =

b < a < a + b < 1,

−(η−t) −xt ,

where η > 1 denotes the elasticity of marginalutility.

(1) (2)

220

GEIR B. ASHEIM

For positive initial stocks, an efficient path with positive and constant consumption is feasible; see Solow (1974). Still, positive discount rate r , the unique
∞ for−rany ∗ t u(x ) dt over all feasible paths forces path (xt∗ , k∗t , k˙ t )∞ maximizing e t t=0 0 consumption to eventually approach zero; cf. Dasgupta and Heal (1974, 1979, pp. 292–303). Furthermore, (xt∗ )∞ and, as illustrated by Dasgupta and t=0 is single peaked,  ∗ ∞ Heal (1979, Diagram 10.3), for r “large,” xt t=0 starts out with a consumption flow that exceed the maximum sustainable level, while for r “small,” the initial consumption flow falls short of this level. By a continuity argument, there exists a rate of discount r  such that x0∗ equals the maximum sustainable level given the initial  ∞ stocks. Since xt∗ t=0 is efficient, consumption must be increasing in an initial phase, before the eventual phase sets in with consumption decreasing and asymptotically approaching zero. For later reference, let t  denote the time at which consumption reaches its peak.
∞ −r  t ∗ Consider the path (xt∗ , k∗t , k˙ t )∞ e u(xt ) dt over all feat=0 maximizing 0 sible paths. By standard arguments there exist supporting present value prices ∗ ∗ ˙∗ ∞ ( p t , q t )∞ t=0 such that (x t , kt , kt )t=0 is regular at these prices and utility discount  ∗ factors λt = e−r t . Hence, (xt∗ , k∗t , k˙ t )∞ t=0 can be realised as a competitive equiif the intergenerational altruism of each generation t is represented by
librium ∞ ˙∗ t λs u(x s ) ds. What can be said about the behavior of qt kt along this path? ∗ Note that by construction, consuming x0 for all t  0 is feasible and efficient.
∞ Hence, since (x∗t )∞ t=0 uniquely maximizes 0 λt u(x t ) dt over all consumption paths,
∞
∞ it follows that 0 λt u(xt∗ ) dt > 0 λt u(x0∗ ) dt. Moreover, Proposition 1 applies
∞
∞ ∗ and can be used to establish that (u(x0∗ ) + q0 k˙ 0 /λ0 ) 0 λt dt > 0 λt u(x0∗ ) dt. ∗ Hence, q0 k˙ 0 > 0 even though consumption at time 0 equals the maximum sustainable level. For s > t  t  , u(xs∗ ) < u(xt∗ ). Hence, by Proposition 1 it follows that
∞
∞
∞ ∗ ∗ (u(x∗t ) + qt k˙ t /λt ) t λs ds = t λs u(xs∗ ) ds < t λs u(xt∗ ) ds and qt k˙ t < 0 for ∗ all t  t  . Since xt∗ is increasing for t ∈ [0, t  ], by the Lemma, qt k˙ t is differentiable  and decreasing for t ∈ [0, t ]. Consequently, there exists a unique t  ∈ (0, t  ) such ∗ that qt  k˙ t  = 0. From the facts that (a) consuming x0∗ for all t  0 is feasible and efficient, and (b) xt∗ is increasing for t ∈ [0, t  ], it follows that for each t ∈ (0, t  ], xt∗ exceeds the maximum sustainable consumption level given k∗t . These findings are illustrated in Figure 15.1 and can be summarised as follows. Proposition 3: For the model described by (1) and (2), there exist positive initial stocks k, a utility discount rate r  > 0, and some t  > 0 such that the path ∗ ∞ −r  t (xt∗ , k∗t , k˙ t )∞ t=0 is regular at prices ( pt , qt )t=0 and utility discount factors λt = e and satisfies: –

∗ For t = 0, qt k˙ t > 0 and xt∗ equals the maximum sustainable consumption level.

–

For t ∈ (0, t  ), qt k˙ t > 0 and xt∗ exceeds the maximum sustainable consumption level.

∗

221

NET NATIONAL PRODUCT AS AN INDICATOR OF SUSTAINABILITY

xt∗

∗

Max sust. level given k0

x 0∗

Max sust. level given kt∗

xt∗ t"

t'

t'"

t

.

qt ∗ kt∗

t" t'

t

Figure 15.1. Illustration of the counter example.

–

∗

For t = t  , qt k˙ t = 0 and xt∗ exceeds the maximum sustainable consumption level.

∗ ∗ For a discussion of this result, consider t = t  for which qt k˙ t = 0, but with  ∗ ∞xt exceeding the maximum sustainable consumption level. By the structure of xt t=0 there exists some time t  > t  such that xs∗ > xt∗ for s ∈ (t  , t  ) and xs∗ < xt∗ for
t 
∞   s ∈ (t  , ∞). By Proposition 1, t  e−r s [u(xs∗ ) − u(xt∗ )] ds = t  e−r s [u(xt∗ ) − ∗ u(xs )] ds. Hence, if utility could have been invested with a rate of return r  , then xt∗ would have been sustainable. However, the utility flow is skewed toward the present for the precise reason that such an investment would give rise to a lower rate of

222

GEIR B. ASHEIM

return. This again hinges on the decreasing productivity of the technology as manmade capital is accumulated and the flow of resource extraction dwindles. The above discussion explains why the generalized Weitzman result (Proposition 1) cannot be used to support the interpretation, made by e.g., Mäler (1991, p. 11) and Hulten ∗ (1992, p. 17), of u(x∗t ) + qt k˙ t /λt as the maximum sustainable level of utility flow. The behavior of the constructed regular path at t = t  also serves to reject the ∗ claim that qt k˙ t = 0 at a given time t implies that u(xt∗ ) equals the maximum ∗ sustainable level. In relation to Hartwick’s rule (qt k˙ t = 0 for all t implies that u(xt∗ ) equals the maximum sustainable utility level), this can be interpreted as follows. The relative price of man-made in terms of natural capital in an intertemporal competitive equilibrium depends on the entire future equilibrium path. In the present model, a path that distributes utility in favor of generations in the near future increrases this relative price, leading to a higher valuation of the investment in man-made capital relative to the depletion of natural capital. Thus, the insufficient altruism (relative to the requirement of sustainability) that the present generation extends to future generations increases to present NNP above the maximal sustainable level of utility flow. Hence, it would seem impossible to develop the concept of NNP into an indicator of sustainability, even if prices for the valuation of natural and environmental resources were readily available through a perfect intertemporal competitive equilibrium. Acknowledgments: I am grateful for comments by John Hartwick, Leif Sandal, Erling Steigum, Cees Withagen, and a referee. Part of this research was financed by the Norwegian Ministry of Environment and by the Norwegian Research Council for Applied Social Science through the Norwegian Research Centre in Organization and Management. After completing the final version, I have been made aware of Pezzey (1993) who independently reports the result of proposition 3. NOTES 1 “[I]t would seen that we ought to define a man’s income as the maximum value which he can consume

during a week, and still expect to be as well off at the end of the week as he was in the beginning” (Hicks, 1946, p. 172). See Scott (1990) for a recent discussion of Hicksian income, with a reply and a comment in the same issue of Journal of Economic Literature by Robert Eisner and David Bradford. 2 In fact, implementing an efficient constant consumption path in the model analyzed by Solow (1974) and Hartwick (1977) corresponds to maximizing the sum of utilities discounted at a positive and decreasing rate; see the model of Section 4. 3 Proof of this claim is available on request from the author.

REFERENCES Bliss, C. (1975), Capital Theory and the Distribution of Income. Cambridge University Press, Cambridge Dasgupta, P. and Heal, G. (1974), The optimal depletion of exhaustible resources. Review of Economic Studies (Symposium), 3–28 Dasgupta, P. and Heal, G. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge

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Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximum paths of capital accumulation and resource depletion. Review of Economic Studies 47, 551–556 Hartwick, J. (1977), Intergenerational equity and the investing of rents from exhaustible resources. American Economic Review 66, 972–974 Hartwick, J. (1990), Natural resources, national accounting and economic depreciation. Journal of Public Economics 43, 291–304 Hulten, C.R. (1992), Accounting for the wealth of nations: The net versus gross output controversy and its ramifications. Scandinavian Journal of Economics 94 (Supplement), 9–24 Hicks, J. (1946), Value and Capital. Second edition, Oxford University Press, Oxford Mäler, K.-G. (1991), National accounts and environmental resources. Environmental and Resource Economics 1, 1–15 Pezzey, J. (1993), The optimal sustainable depletion of non-renewable resources. I: Capital-resource substitution. University College, London Scott, M. (1990), Extended accounts for national income and product: A comment. Journal of Economic Literature 28, 1172–1186 Solow, R.M. (1974), Intergenerational equity and exhaustible resources. Review of Economic Studies (Symposium), 29–45 Solow, R.M. (1986), On the intergenerational allocation of natural resources. Scandinavian Journal of Economics 88, 141–149 Svensson, L.E.O. (1986), Comment on R. M. Solow. Scandinavian Journal of Economics 88, 153–155 Weitzman, M. (1976), On the welfare significance of national product in a dynamic economy. Quarterly Journal of Economics 90, 156–162

CHAPTER 16 ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

Abstract. Weitzman provides a foundation for NNP as the stationary equivalent of a wealth-maximizing path when there is a constant interest rate and no exogenous technological progress. Here, the implications of Weitzman’s foundation are explored in a case encountered in resource models, i.e., the case of nonconstant interest rates. In a setting that allows for exogenous technological progress, an expression for NNP is obtained that adjusts Green NNP for anticipated capital gains and interest rate effects to produce a measure that indicates sustainability. This result is important when measuring the relative sustainability of resource rich and resource poor countries.

1. INTRODUCTION How should net national product (NNP) be adjusted for the depletion of natural and environmental resources? A prerequisite for examining this problem is to decide on an underlying notion of income. At the personal level, Hicks (1946) defines income as the consumption that, if kept constant would yield the same present value as a person’s actual future receipts.1 Since market prices are not influenced by any single person, this notion of income equals a person’s maximal sustainable consumption. At the national level, national income can likewise be defined as the consumption that, if kept constant, would yield the same present value as the actual future consumption path. Unless the nation is a small open economy faced with given international prices, this notion of national income will exceed sustainable consumption; see Weitzman (1976, pp. 159–160). Hence, it is the hypothetical stationary equivalent of future consumption. Weitzman (1976, p. 160) shows that the conventional measure of NNP— consumption + value of net investments in capital stocks—equals this notion of national income under the assumption of a constant interest rate and no exogenous technical progress. Weitzman (1976, p. 157) emphasizes that stocks of knowledge accumulated through learning and research activities as well as stocks of natural and environmental resources must be contained in the vector of capital stocks for this result to obtain. Moreover, competitive prices for such stocks must be assumed to be available. Hence, Weitzman requires that both endogenous technological progress

Originally published in Scandinavian Journal of Economics 99 (1997), 355–370. Reproduced with permission from Blackwell.

225 Asheim, Justifying, Characterizing and Indicating Sustainability, 225–240 c 2007 Springer


226

GEIR B. ASHEIM

as well as resource depletion be taken into account by the conventional measure. Such an expanded conventional measure has been suggested by Hartwick (1990), Mäler (1991), and others, and is usually referred to as Green NNP. Thus, Weitzman (1976) provides a Hicksian foundation for the concept of Green NNP under given assumptions. Weitzman (1997) shows that Green NNP is a poor measure of the hypothetical stationary equivalent if there is exogenous technological progress, while the example of Asheim (1994) can be used to show that Green NNP does not necessarily equal the hypothetical stationary equivalent if there is not a constant interest rate. Faced with the discrepancy, one possibility is to rely on an alternative foundation for Green NNP; e.g., Green NNP may serve as a measure of welfare changes resulting from small policy changes; see Mäler (1991) and Dasgupta (1996). The route pursued here is instead to apply Weitzman’s (1976) foundation by asking how NNP would need to be defined in terms of current prices and quantities in order to equal the hypothetical stationary equivalent of future consumption. In certain resource models, ethical acceptable outcomes seem to entail that the assumption of a constant interest rate may be violated not only in terms of consumption, but even in terms of utility. It is therefore a major challenge to extend Weitzman’s foundation to the case where interest rates are not constant. By offering such an analysis in a setting that allows for exogenous technological progress, the present analysis holds also (i) for any economy where accumulated knowledge cannot be represented by augmented capital stocks, and (ii) for open economies whose “technology” is changing exogenous due to changing terms of trade. This cases of exogenously accumulating knowledge and/or exogenously changing terms of trade are treated in a number of contributions; see the references listed in Sect. 3. In contrast to most of these contributions, but following Asheim (1986, 1996), I here assume constant returns to scale. The amounts to an assumption that all flows of future earnings can be treated as currently existing capital. Exogenous technological progress contributes to the appreciation of capital; i.e., capital gains. In the deterministic setting of this study, there is no windfall profit or loss, cf., e.g., Hicks (1946, p. 178); rather, capital gains are fully anticipated. An example is the anticipated capital gains on in situ resources when the future development of the resource price is known. If interest rates are not constant, then there is a term structure of interest rates. In order to generalize Weitzman’s (1976) analysis, it turns out that the infinitely longterm interest rate is significant. In particular, I establish here that: NNP satisfying the Weitzman foundation = (long-term interest rate) · (current wealth) . Under the assumption of constant returns to scale, but allowing for exogenous technological progress, this in turn is shown to imply that NNP should include capital

ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

227

gains: NNP satisfying the Weitzman foundation = consumption + value of net investments + anticipated capital gains + (rate of change of long-term interst rate) · (current wealth) . NNP must be adjusted for interest rate changes because, without a constant interest rates, the present value of a constant flow of future earnings will vary. This part of capital gains should not be included when NNP is based on the Weitzman foundation. Kemp and Long (1998) and Hartwick and Long (1999) also analyze the case of nonconstant interest rates, but their analyzes do not appear to be based on the Weitzman foundation. The paper is organized as follows. In Sect. 2, a problem of wealth maximization is used as the point of departure when defining a concept of NNP that satisfies the Weitzman foundation. This definition is used to derive an expression for NNP in Sect. 3; two special cases where the expression coincides with Green NNP are also investigated. The implications of this expression in open economies are illustrated in Sect. 4 by a numerical example of a two-country world, and by discussing the empirical relevance of the results for the measurement of sustainability in open economies. The results are also compared to Sefton and Weale (1996) by showing how their interesting analysis would change in their concept of NNP were based on the Weitzman foundation. Some of the formal analysis is contained in an Appendix. Throughout it is assumed that there exists an intertemporal competitive equilibrium in a constant population economy. The issues of how to define NNP when efficiency prices are not available, or when population is not constant, are not addressed.

2. THE WEITZMAN FOUNDATION IN TERMS OF CONSUMPTION AND UTILITY Weitzman (1976) considers an economy maximizing at time t


∞

e−r (s−t) x(s)ds

(1)

t

over all feasible consumption paths. Here, x(s) denotes consumption at time s, and r is the positive consumption discount rate. Expression (1) measures current wealth. Let (x ∗ (s))∞ s=t be a consumption path maximizing (1) over all feasible consumption paths. Let y(t) denote the consumption NNP at time t. To satisfy the Weitzman foundation of NNP, y(t) has to be the hypothetical stationary equivalent; i.e., the level of consumption that, if sustained indefinitely, would yield the same wealth as

228

GEIR B. ASHEIM

the wealth-maximizing path:


∞

e−r (s−t) y(t)ds =




t

∞

e−r (s−t) x ∗ (s)ds ,

t

or
y(t) = r

∞

e−r (s−t) x ∗ (s)ds .

t

Kemp and Long (1982) generalize Weitzman’s analysis to the case of a concave utility function.2 They consider an economy maximizing at time t


∞

e−ρ(s−t) u(x(s))ds

(2)

t

over all feasible consumption paths, where u(·) is a time-invariant, strictly increasing, concave and differentiable utility function, and ρ is the positive utility discount rate. Expression (2) measures the current discounted utilitarian welfare. Let (x ∗ (s))∞ s=t be a consumption path maximizing (2) over all feasible consumption paths. Let v(t) denote the utility NNP at time t. To satisfy the Weitzman foundation of NNP, v(t) has to be the hypothetical stationary equivalent; i.e., the level of consumption that, if sustained indefinitely, would yield the same welfare as the welfare-maximizing path:
∞
∞ e−ρ(s−t) v(t)ds = e−ρ(s−t) u(x ∗ (s))ds , t

t

or
v(t) = ρ

∞

e−ρ(s−t) u(x ∗ (s))ds .

t

Expression (2) is a controversial welfare criterion since, in certain resource models, it gives rise to ethically unacceptable implications. For instance, in the Dasgupta–Heal–Solow model of capital accumulation and resource depletion, the discounted utilitarian criterion yields a maximizing path in which consumption utilitarian criterion yields a maximizing path in which consumption converges to zero even though paths with positive and nondecreasing consumption are feasible; see Dasgupta and Heal (1974), Solow (1974) and Sect. 4 below. As an alternative, Solow (1974) explores the implications in the Dasgupta–Heal–Solow model of the Rawlsian maximin principle on utility. In the same model I have followed Calvo’s (1978) suggestion that the maximin principle be used on discounted utilitarian welfare in Asheim (1988), as well as imposed Dalton’s (1920) principle of transfers as a side constraint when (2) is maximized in Asheim (1991). Each of these alternative criteria gives rise to an efficient path that can be supported by utility discount factors.

ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

The economy acts as if it maximizes
∞

λ(s)u(x(s))ds

229

(3)

t

over all feasible consumption paths, where λ(s) is the positive utility discount factor applicable at time s. The path (λ(s))∞ s=t yields the present value prices at which marginal utility at one time can be exchanged for marginal utility at some other time. −ρs , there is one Hence, (λ(s))∞ s=t determines utility interest rates. If λ(s) = λ(0)e constant utility interest rate: ρ=−

λ˙ (t) λ(t) = ∞ λ(t) t λ(s)ds

for all t. However, if (2) is not the welfare criterion, λ(s) will not, in general, be an exponential function. Then there is a term structure of utility interest rates. The instantaneous (very short-term) utility interest rates is ρ0 (t) := −

λ˙ (t) . λ(t)

The infinitely long-term utility interest rate is λ(t) ρ∞ (t) :=  ∞ t λ(s)ds

∞ (provided that ρ0 (t) is positive and does not fall “too fall” so that t λ(s)ds exists). Note that, along a maximizing path, 1/ρ∞ (t) is the price at time t, in terms of utility, of a utility annuity from time t on. Note also that ρ˙∞ (t)/ρ∞ (t) = ρ∞ (t) − ρ0 (t). Hence, ρ∞ (t) is decreasing if and only if ρ0 (t) > ρ∞ (t). ∗ ∞ maximize (3) over all feasible consumption paths. In particular, s=t  ∞Let (x (s)) ∗ t λ(s)u(x (s))ds exists. Let v(t) still denote utility NNP. To satisfy the Weitzman foundation, v(t) must be the stationary equivalent of (u(x ∗ (s)))∞ s=t :
∞
∞ λ(s)v(t)ds = λ(s)u(x ∗ (s))ds , t

or

t

∞


∞ λ(s)u(x ∗ (s))ds λ(s) ∞ v(t) = u(x ∗ (s))ds . (4) = ρ∞ (t) λ(t) t t λ(s)ds ∞ Since v(t) ˙ can be shown to equal (λ(t)/ t λ(s)ds)(v(t) − u(x ∗ (t))), v(t) is the solution to t

v(t) ˙ = ρ∞ (t) · (v(t) − u(x ∗ (t))) .

(5)

The differential equation (5) can be rewritten as v(t) − u(x ∗ (t)) = v(t)(1/ρ ˙ ∞ (t)). This yields the following interpretation. The difference between the stock of utility

230

GEIR B. ASHEIM

annuities at time t and the actual utility level at time t equals the rate at which utility annuities can be accumulated times the price of such annuities. It is natural to seek a justification for v(t) in terms of sustainable income; see Hicks (1946, Chap. 14, “Income No. 3) and Nordhaus’s (1995) discussion of Fisher (1906). As noted in the introduction, a constant utility flow equal to v(t) is only attainable if the actual utility path (u(x ∗ (s)))∞ s=t could be changed to a constant utility path without changing the supporting prices. In particular, if (x ∗ (s))∞ s=t is the unique path maximizing (3) and (x ∗ (s))∞ does not yield constant consumption, s=t then a constant utility flow equal to v(t) is not attainable. Hence, the Weitzman foundation gives an upper bound for the level of utility that is actually sustainable. Likewise u −1 (v(t)) is an upper bound for sustainable consumption. With a justification in terms of sustainable income, it is desirable to obtain a measure that is as close as possible to what is actually sustainable. It is shown below that the Weitzman foundation in terms of consumption yields a more accurate measure. Therefore, let us turn again to consumption NNP. Under regularity conditions (see the Appendix) there exists a path of present value prices ( p(s))∞ s=t such that maximizing (3) is equivalent to the maximization of
∞ p(s)x(s)ds (6) t

over all feasible consumption paths. Expression (6) divided through by p(t) measures current wealth. It must hold for all s that x ∗ (s) maximizes λ(s)u(x(s)) − p(s)x(s); hence, if x ∗ (s) is interior, then λ(s)u  (x ∗ (s)) = p(s). In analogy with λ(s), p(s) determines a term structure of consumption interest rates. The instantaneous (very short-term) consumption interest rates is r0 (t) := −

p(t) ˙ . p(t)

The infinitely long-term consumption interest rate is r∞ (t) :=  ∞ t

p(t) . p(s)ds

In analogy, 1/r∞ (t) is the price at time t, in terms of consumption, of a consumption annuity from time t on. Note also that r˙∞ (t)/r∞ (t) = r∞ (t) − r0 (t). Hence, r∞ (t) is decreasing if and only if r0 (t) > r∞ (t). Let y(t) still denote consumption NNP. To satisfy the Weitzman foundation, y(t) must be the stationary equivalent of (x ∗ (s))∞ s=t :
∞
∞ p(s)y(t)ds = p(s)x ∗ (s)ds , t

or

t

∞ y(t) =

t

p(s)x ∗ (s)ds ∞ = r∞ (t) t p(s)ds


t

∞

p(s) ∗ x (s)ds . p(t)

(7)

ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

231

Hence, consumption NNP is the infinitely  ∞ long-term consumption interest rate times current wealth. Since y˙ (t) = ( p(t)/ t p(s)ds)(y(t) − x ∗ (t)), y(t) is the solution to y˙ (t) = r∞ (t) · (y(t) − x ∗ (t)) .

(8)

This differential equation has the same interpretation as (5). Consumption NNP y(t) is also an upper bound for sustainable income. This can be shown by repeating the above argument for v(t). Therefore, to establish that y(t) is a more accurate measure of sustainable income than v(t), it is sufficient to show that v(t) ≥ u(y(t)), or equivalently, u −1 (v(t)) ≥ y(t). By (4) and (7) and the property that λ(s)u  (x ∗ (s)) = p(s) for all s, it follows that
∞
∞ λ(s)(v(t) − u(y(t)))ds = λ(s)[u(x ∗ (s)) − u(y(t)) t

t

+ u  (x ∗ (s))(y(t) − x ∗ (s))]ds , where the term in brackets is nonnegative by the concavity of u. Therefore, since λ(s) > 0 for all s, v(t) ≥ u(y(t)), with strict inequality if (x ∗ (s))∞ s=t is not constant and u is strictly concave. For a small open economy, ( p(s))∞ s=t is exogenously determined by the international capital market, and (x ∗ (s))∞ s=t can be changed into a constant consumption path without changing the supporting prices. Hence, as argued by Brekke (1996, Chap. 4), y(t) is an exact indicator of sustainability for such an economy. 3. EXPRESSIONS FOR CONSUMPTION NNP Through (7), consumption NNP is expressed as the infinitely long-term consumption interest rate times the discounted value of future consumption, the latter term having been interpreted as wealth. This is not the form that national accountants find useful, since wealth —especially as given by (7)—is not easily measured; see, e.g., Usher (1994). Therefore, combine (7) and (8) to yield y˙ (t) r∞ (t) 
∞ 
∞   p(s) ∗ p(s) ∗ d r˙∞ (t) ∗ x (s)ds + x (s)ds = x (t) + dt p(t) r∞ (t) t p(t) t

y(t) = x ∗ (t) +

(9)

Hence, consumption NNP equals consumption plus growth of current wealth plus the rate of change in the infinitely long-term consumption interest rate times current wealth. In this section, I investigate cases where expression (9) can be evaluated using current prices and quantities only. For this purpose, consider a constant population economy as described by Dixit et al. (1980) and reproduced in the Appendix, but where the present analysis allows for exogenous technological progress. Here k∗ (t) is the vector of capital stocks at time t, and Q(t) are the competitive prices of the capital

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stocks in terms of current consumption. Within this framework, three different cases will be analyzed. In the first two cases, Green NNP—i.e., consumption plus the value of net investments: x ∗ (t) + Q(t)k˙ ∗ (t)—is shown to satisfy the Weitzman foundation. In both cases, the assumption of no exogenous technological progress is combined with either a constant consumption interest rate or constant consumption. Case 1: No exogenous technological progress, and a constant consumption interest rate ( p(s) = p(0)e−r s ; i.e., r = r0 (s) = r∞ (s) for all s). This is the case considered by Weitzman (1976). Since r˙∞ (t) = 0, the third term of the r.h.s. of (9) is equal to zero. Also, since p(s + )/ p(t + ) = e−r (s−t) is constant as a function of , ∞ the second term of the r.h.s. of (9) equals t ( p(s)/ p(t))x˙ ∗ (s)ds. Lemma 1 in the Appendix implies that this latter term equals Q(t)k˙ ∗ (t) in the absence of exogenous technological progress; hence, by (9), y(t) = x ∗ (t) + Q(t)k˙ ∗ (t). This amounts to an alternative demonstration of Weitzman’s (1976) original result: with no exogenous technological progress and a constant consumption interest rate. Green NNP satisfies the Weitzman foundation. Case 2: No exogenous technological progress, and a constant consumption interest rate (x ∗ (s) = x ∗ for all s). By (7) it follows that y(t) = x ∗ , while the converse of Hartwick’s (1977) rule, cf. Dixit et al. (1980), reproduced as Lemma 2 in the Appendix, implies that Q(t)k˙ ∗ (t) = 0 in the absence of exogenous technological progress. Hence, y(t) = x ∗ (t) + Q(t)k˙ ∗ (t). Even though the second and third terms of (9) cancel out in this case, there are important resource models in which each of the terms differs from zero. A justly well-known example is the Dasgupta–Heal– Solow model, where Solow’s (1974) shows that the consumption interest rates are decreasing along the constant consumption path. The above assumption of no exogenous technological progress is more appropriate for a closed than for an open economy. The “technology” of an open economy trading in a competitive world economy has to include its trade opportunities. Therefore, the assumption of no exogenous technological progress will be violated if its terms of trade are changing. In resource models, Hotelling’s rule implies that a resourceexporting economy will enjoy improving terms of trade, which in the present context correspond to exogenous technological progress. If such an economy follows a constant consumption path, its consumption level will not equal Green NNP; see Asheim (1986, 1996), Sefton and Weale (1996), Hartwick (1995), Vincent et al. (1997) and Sect. 4 below. Changing terms of trade are also analyzed by Kemp and Long (1998). Even for a closed economy, the assumption of no exogenous technological progress is strong. This has been emphasized and/or analyzed by Aronsson and Löfgren (1993, 1995), Hartwick and Long (1999), Kemp and Long (1982), Nordhaus (1995), Weitzman (1997), Withagen (1996) and others. Therefore, let us turn to the case of exogenous technological progress. If the analysis allows for exogenous technological progress, can current NNP satisfying the Weitzman foundation be expressed in terms of current prices and quantities, and hence be an operational measure of NNP? It can under the

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alternative assumption that the technology exhibits constant returns to scale (CRS) in the capital stocks. The assumption is in the spirit of Lindahl (1933, pp. 401–402) and implies that all factors of production, including labor, are dealt with as capital that is evaluated by the present value of future earnings. It amounts to assuming that all flows of future earnings can be treated as currently existing capital. CRS means that in the hypothetical case where all capital stocks were a given percentage larger, consumption and investments could be increased by the same percentage. This clearly allows for stocks in fixed supply—like “raw labor” and land—that cannot actually be accumulated. The CRS assumption is not restrictive since the existence of an intertemporal competitive equilibrium entails that returns to scale are nonincreasing. Hence, CRS can be obtained by adding an additional fixed capital stock with which returns to scale become constant. Case 3: Constant returns to scale. As shown in Lemma 3 in the Appendix, if the ∗ (s) = ˙ assumption of CRS is imposed, then, for all s, x ∗ (s) + Q(s)k˙ ∗ (s) + Q(s)k d r0 (s)Q(s)k∗ (s) or, equivalently, p(s)x ∗ (s) + ds [ p(s)Q(s)k∗ (s)] = 0. This yields ∞ p(t)Q(t)k∗ (t) = t p(s)x(s)ds or, equivalently, Q(t)k∗ (t) =




∞ t

p(s) x(s)ds . p(t)

Hence, the current value of the capital stocks equal current wealth. It now follows from (9) that r˙∞ (t) ∗ ˙ y(t) = x ∗ (t) + Q(t)k˙ ∗ (t) + Q(t)k Q(t)k∗ (t) . (t) + r∞ (t)

(10)

The first two terms constitute Green NNP. The hypothetical stationary equivalent of ∗ (t) and ˙ future consumption adjusts this measure for anticipated capital gains Q(t)k the rate of change in the infinitely long-term consumption interest rate. Since y(t) = x ∗ (t) + Q(t)k˙ ∗ (t) under the assumptions of Case 1, (10) provides an alternative demonstration of the result shown in Asheim (1996, Proposition 1), namely that in an economy with CRS and no exogenous technological progress, a constant consumption interest rate implies that there are no capital gains. It is now apparent that Weitzman’s original result is somewhat misleading. In the case of no exogenous technological progress and a constant consumption interest rate, Weitzman (1976) shows that NNP should exclude capital gains in order to satisfy the Weitzman foundation. However, as argued above, there are no capital gains in this special case. On the other hand, (10) implies that anticipated capital gains should be fully included in the presence of exogenous technological progress, provided that there is a constant consumption interest rate. Therefore, in the case of a constant consumption interest rate considered by Weitzman (1976), a concept of NNP that includes anticipated capital gains satisfied his foundation independently of whether there is exogenous technological progress.

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4. OPEN ECONOMIES: A NUMERICAL EXAMPLE AND EMPIRICAL EVIDENCE Consider a closed economy with no exogenous technological progress and constant consumption (Case 2 above). Then Green NNP correctly ignores nonzero aggregate capital gains, since the last two terms of (10) cancel out. Let this closed economy be split into open economies with constant subpopulations, each having access to the same CRS technology. Allow for free trade and sufficient factor mobility to ensure overall productive efficiency. If terms of trade are not constant, then the “technology” of each open economy is not stationary. This in turn means that the corresponding terms of (10)—for each open economy—do not cancel out. In the context of resource models, an open economy with relatively large stocks of in situ resources will have large capital gains in relation to its wealth, while an open economy with relatively small such stocks will in comparison have small capital gains in relation to its wealth. The other adjustment term, however, is proportional to wealth. Hence, even when Green NNP needs no adjustment at the aggregate level to satisfy the Weitzman foundation, the necessary adjustments at the disaggregate level will influence the distribution of NNP between different countries. A numerical example is provided prior to discussing the empirical relevance of this observation. The numerical example is based on the Dasgupta–Heal–Solow model of manmade capital accumulation and resource depletion, in which a flow of an nonrenewable resource, −k˙ R , is combined with constant human capital, k H , and manmade capital, kC , in order to produce a consumption good, x. With its technology described by x + k˙C ≤ (k H )1−a−b (kC )a (−k˙ R )b , b < a < a + b < 1, this model fits into the framework of Sect. 3. Choose a = 0.20 and b = 0.16. Solow (1974) shows that positive and constant consumption can be sustained indefinitely by letting accumulated man-made capital substitute for a diminishing resource extraction. Let k∗ (t) = (k ∗H , kC∗ (t), k ∗R (t)) denote the capital vector along such an efficient maximin path, with x ∗ as the corresponding consumption level, where variables without time dependence are constant. The investment in man-made capital is constant along this path, implying that total output is constant, with a fraction 1 − b going to consumption and a fraction b going to investment in man-made capital. By normalizing total output to 1, x ∗ = 1 − b = 0.84 and k˙C∗ = b = 0.16. If the constant consumption path is implemented as a competitive equilibrium, interest rates are decreasing (r0 (t) = a/kC∗ (t) = 0.20/kC∗ (t), r∞ (t) = (a − b)/kC∗ (t) = 0.04/kC∗ (t), and r˙∞ (t)/r∞ (t) = −bkC∗ (t) = −0.16/kC∗ (t)), competitive capital prices and price changes are given by: Q H (t) =

1−a−b a−b

1−a−b Q˙ H = a−b

 

kC∗ (t) k ∗H b k ∗H



 = 16

 =

2.56 , k ∗H

kC∗ (t) k ∗H

 ,

ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

235

Q˙ C = 0 ,  ∗   ∗  kC (t) k (t) b = 4 C∗ , Q R (t) = ∗ a − b k R (t) k R (t)   0.80 a b ˙ Q R (t) = = ∗ , ∗ a − b k R (t) k R (t) QC = 1 ,

while resource extraction −k˙ ∗R (t) equal b/Q R (t) = 0.16/Q R (t). Let this competitive world economy be split into two countries, between which human capital is distributed evenly (k 1H = k 2H = 12 k ∗H ), while only country 1 is endowed with the resource (k 1R (t) = k ∗R (t), k 2R (t) = 0), and only country 2 owns man-made capital (kC1 (t) = 0, kC2 (t) = kC∗ (t)). Productive efficiency is ensured by using half of the capital stock and resource flow in each country. Assume that each country implements a constant consumption path. As I show in Asheim (1996, Table 1, Case 3), each country keeps consumption constant if k˙ 1R (t) = k˙ ∗R (t), k˙ 2R (t) = 0, k˙C1 (t) = 0, and k˙C2 (t) = k˙C∗ (t), with x 1 = 0.48 and x 2 = 0.36. Green NNP then becomes x 1 + Q(t)k˙ 1 (t) = x 1 + Q R (t)k˙ ∗R (t) = 0.48 − 0.16 = 0.32 x 2 + Q(t)k˙ 2 (t) = x 1 + k˙C∗ (t) = 0.36 + 0.16 = 0.52 . Hence, for each country, Green NNP differs at any time from the constant consumption level: 0.32 v. 0.48 for country 1; 0.52 v. 0.36 for country 2. Let us now turn to the adjustments for capital gains and the rate of change in the infinitely long-term consumption interest rate, as specified by (10). No j adjustment is needed for human capital since, for each j = 1, 2, Q˙ H k H + j (˙r∞ (t)/r∞ (t))Q H (t)k H = 1.28 − (0.16/kC∗ (t)) · 8kC∗ (t) = 0. Since country 1 has no stock of man-made capital, r˙∞ (t) r˙∞ (t) 1 ˙ (t) + Q(t)k Q(t)k1 (t) = Q˙ R (t)k 1R (t) + Q R (t)k 1R (t) r∞ (t) r∞ (t) = 0.80 −

0.16 ∗ 4k (t) = 0.16 , kC∗ (t) C

while, since country 2 has no resource stock, r˙∞ (t) r˙∞ (t) 2 ˙ (t) + Q(t)k Q(t)k2 (t) = Q˙ C kC2 (t) + Q C kC2 (t) r∞ (t) r∞ (t) = 0−

0.16 ∗ k (t) = 0.16 . kC∗ (t) C

Therefore, it follows from (10) that y 1 = 0.32 + 0.16 = 0.48 and y 2 = 0.52 − 0.16 = 0.36. Hence, for each country, the NNP based on the Weitzman foundation is at all times equal to the constant consumption level.

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Hence, Green NNP underestimates the sustainability of resource rich countries (like country 1 in the example) and overestimates the sustainability of resource poor countries (like country 2). Vincent et al. (1997) note this problem and point to a study by Pearce and Atkinson (1993), who calculated Green NNP for a number of countries. Japan turns out to consume much less that its Green NNP and is ranked as the most sustainable of the 21 countries presented. Indonesia consumes more than its Green NNP and is regarded as an unsustainable economy. Given that Indonesia is the relatively more resource rich of the two countries, these conclusions may not survive adjustments for capital gains and interest rate effects. Sefton and Weale (1996), hereafter SW, also investigated the concept of NNP in open economies. Instead of considering the solution to the differential equation y˙ (t) = r∞ (t) · (y(t) − x ∗ (t)) as in the present paper, SW determined consumption NNP (or “income”) as the solution to the differential equation Y˙ (t) = r0 (t) · (Y (t) − x ∗ (t)). This yields ∞
∞ ∗ (s)ds (− p(s))x ˙ p(s) ∗ = x (s)ds . r0 (t) (11) Y (t) = t p(t) p(t) t If there is one constant consumption interest rate, then r = r0 (s) = r∞ (s) for all s; hence, y(t) = Y (t). Also if (x ∗ (s))∞ t=s is a constant consumption path, so that x ∗ (s) = x ∗ for all s, then y(t) = Y (t) = x ∗ since
∞
∞ p(s) p(s) ds = ds = 1 . r0 (s) r∞ (t) p(t) p(t) t t However, in general, if there is not a constant consumption interest rate or if consumption is not constant, then Y (t) differs from NNP based on the Weitzman foundation.3 Still, SW’s consumption NNP has the following attractive features: in a world economy with no exogenous technological progress, it equals Green NNP. Furthermore, Green NNP is split into NNPs for the open economies in such a way that, if consumption is kept constant in an open economy, then NNP for this open economy equals its constant consumption level. SW claim that their measure of NNP has the advantage that it can be calculated using observable market prices. In contrast, elsewhere I have measured the maximum sustainable consumption, using the prices that would exist along such an efficient constant consumption path.4 As seen above, however, the measure suggested by SW does not in general satisfy the Weitzman foundation, except in special cases, one of which is the constant consumption case. The same conclusion holds for the measure of the maximum sustainable consumption suggested in Asheim (1996, Proposition 3). However, in contrast to SW’s claim, it turns out that the measure provided in Asheim (1986) does satisfy the Weitzman foundation also when using observable market prices along  ∞ paths where consumption is not constant. To see this, substitute Q(t)k∗ (t) = t ( p(s)/ p(t))x ∗ (s)ds into (7) to yield ∞ y(t) = p(t)Q(t)k∗ (t)/ t ( p(s)ds, the r.h.s. of which is identical to the r.h.s. of (S) in Asheim (1986). Even though this (S) is intended to measure consumption along an efficient constant consumption path, the present analysis has shown that

ADJUSTING GREEN NNP TO MEASURE SUSTAINABILITY

237

this expression in fact measures NNP satisfying the Weitzman foundation even at prices supporting nonconstant consumption paths. APPENDIX Following Dixit et al. (1980), but allowing for exogenous technological progress, ˙ ˙ (x(s), k(s), k(s)) is feasible if and only if (x(s), k(s), k(s)) ∈ F(s), where F(s) is a convex set of feasible triples at time s, and where k(s) is a nonnegative vector of capital stocks at time s. For stocks like “raw labor” and land that are in fixed supply, the corresponding components of k˙ equal 0. For other positive stocks, the F(s) is assumed to satisfy free disposal of investment flows; i.e., for each s, if ˙ ∈ F(s) and k˙  ≤ k˙ with k˙  differing from k˙ for such stocks only, then (x, k, k) (x, k, k˙  ) ∈ F(s). Call a feasible path (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t competitive at present value consump∞ tion and capital prices ( p(s), q(s))∞ s=t (and utility discount factors (λ(s))s=t ) if: (i) For each s, p(s) > 0 and (x ∗ (s), k∗ (s), k˙ ∗ (s)) maximizes instantaneous profit ˙ ∈ F(s). ˙ p(s)x + q(s)k˙ + q(s)k subject to (x, k, k) (ii) For each s, λ(s) > 0, and x ∗ (s) maximizes λ(s)u(x) − p(s)x over all x.) Note that (i) combined with the assumption of free disposal of investment flows imply that the vector q(s) is nonnegative for stocks that are not in fixed supply. Moreover, Q(s) := q(s)/ p(s) are the current value capital prices used in Sects. 3 and 4. ∞ Call a competitive path (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t regular at ( p(s), q(s))s=t (and ∞ utility discount factors (λ(s))s=t ) if: (a) (b)

q(s)k∗ → 0 as s → ∞; ∞ ∞ ∗ ∗ t p(s)x (s)ds exists (and t λ(s)u(x (s))ds exists).

A regular path (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t maximizes  
∞
∞ p(s)x(s)ds and λ(s)u(x(s))ds t

t

∞ with initial stocks k(t) = k∗ (t). ˙ over all feasible paths (x(s), k(s), k(s)) s=t With no exogenous technological progress, the following lemmas are obtained.

Lemma 1: If F(s) is time invariant, and (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t regular at ( p(s), ∞ ∗ ˙ q(s))∞ s=t , then t p(s) x˙ (s)dds = q(t)k(t). Proof. It follows from (i) that, for all s, p(s)x(s) +

d ds (q(s)k(s))

≤ p(s)x ∗ (s) +

∗ d ds (q(s)k (s)) .

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Hence, by (a),
∞


p(s)x(s)ds − q(t)k(t) ≤

t

∞

p(s)x ∗ (s)ds − q(t)k∗ (t) .

t

In particular, since F(s) is time invariant (i.e., no exogenous technological progress),
∞
∞ p(s)x ∗ (s + )ds − p(s)x ∗ (s)ds ≤ q(t)k∗ (t + ) − q(t)k∗ (t) , t

t

which implies that   ∗   ∗
∞ k (t + ) − k∗ (t) x (s + ) − x ∗ (s) ds ≤ q(t) p(s)   t if  > 0 and
∞ t

  ∗  k (t + ) − k∗ (t) x ∗ (s + ) − x ∗ (s) ds ≥ q(t) p(s)   

if  < 0. By taking limits, this establishes the result, provided that x˙ ∗ (s) exists for a.e. s and q(s)k˙ ∗ (s) → 0 as s → ∞. The converse of Hartwick’s (1977) rule is a straightforward consequence of Lemma 1. Lemma 2 (Dixit et al. (1980)): If F(s) is time invariant, (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t ∗ ˙∗ regular at ( p(s), q(s))∞ s=t , and x˙ (s) = 0 for all s, then q(t)k (t) = 0. Returning to the case with exogenous technological progress, but instead assuming that the technology exhibits CRS, then the following lemma can be established. Note that the assumption of CRS is equivalent to assuming that F(s) is a convex cone. Lemma 3: If, for each s, F(s) is a convex cone, and (x ∗ (s), k∗ (s), k˙ ∗ (s))∞ s=t regular  ∗ (t) = ∞ p(s)x ∗ (s)ds. at ( p(s), q(s))∞ , then q(t)k s=t t Proof. If F(s) is a convex cone, then (x ∗ (s), k∗ (s), k˙ ∗ (s)) maximizes p(s)x + ˙ ∈ F(s) only if ˙ q(s)k˙ + q(s)k subject to (x, k, k) ∗ ˙ (s) = 0 . p(s)x ∗ (s) + q(s)k˙ ∗ (s) + q(s)k

Hence, p(s)x ∗ (s) + ∞ ∗ t p(s)x (s)ds.

d ∗ ds (q(s)k (s))

= 0 for all s, such that, by (a), q(t)k∗ (t) =

Acknowledgments: I wish to thank CES, University of Munich for hospitality and support during this research as well as participants at a CES seminar for helpful comments. I also thank John Hartwick, James Sefton and Martin Weitzman for interesting discussions and correspondence, and two referees for valuable suggestions. Earlier versions of this paper were circulated under the title “The Weitzman foundation of NNP with nonconstant interest rates”.

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NOTES 1 After considering the possibilities of changing interest rates and changing prices, Hicks (1946) writes

on p. 174: “Income No. 3 must be defined as the maximum amount of money which the individual can spend this week, and still expect to be able to spend the same amount in real terms in each ensuing week”. Then, on p. 184: “The standard stream corresponding to Income No. 3 is constant in real terms . . . . We ask . . . how much the would be receiving if he were getting a standard stream of the same present value as his actual expected receipts. This amount is his income. 2 Kemp and Long (1982) also generalize Weitzman’s analysis to multiple consumption goods and exogenous technological progress. To focus attention on the central issue of this paper, multiple consumption goods will not be considered here. The single consumption good is assumed to be an indicator of instantaneous well-being derived from the situation in which people live. Hence, it includes more than material consumption. 3 Hartwick and Long (1999) characterize investment behavior along a constant consumption path in the context of nonconstant interest rates and exogenous technological progress. It can be shown that their analysis is related to that of SW. 4 Asheim (1986, 1996).

REFERENCES Aronsson, T. and Löfgren, K.-G. (1993), Welfare consequences of technological and environmental externalities in the Ramsey growth model, Natural Resource Modeling 7, 1–14 Aronsson, T. and Löfgren, K.-G. (1995), National product related welfare measures in the presence of technological change: Externalities and uncertainty, Environmental and Resource Economics 5, 321– 332 Asheim, G.B. (1986), Hartwick’s rule in open economies, Canadian Journal of Economics 19, 395–402 [Erratum 20, (1987) 177] (Chap. 9 of the present volume) Asheim, G.B. (1988), Rawlsian intergenerational justice as a Markov-perfect equilibrium in a resource technology, Review of Economic Studies 55, 469–483 (Chap. 6 of the present volume) Asheim, G.B. (1991), Unjust intergenerational allocations, Journal of Economic Theory 54, 350–371 (Chap. 7 of the present volume) Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Asheim, G.B. (1996), Capital gains and ‘net national product’ in open economies, Journal of Public Economics 59, 419–434 (Chap. 10 of the present volume) Brekke, K.A. (1996), Economic Growth and the Environment. Edward Elgar, Cheltenham, UK Calvo, G. (1978), Some notes on time inconsistency and RawlsŠ maximin criterion, Review of Economic Studies 45, 97–102 Dalton, H. (1920), The measurement of the inequality of incomes, Economic Journal 30, 348–361 Dasgupta, P.S. (1996), The economics of the environment, Environment and Development Economics 1, 387–428 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Fisher, I. (1989), The Nature of Capital and Income. Macmillan, New York Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Hartwick, J.M. (1990), National resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291–304 Hartwick, J.M. (1995), Constant consumption paths in open economies with exhaustible resources, Review of International Economics 3, 275–283 Hartwick, J.M. and Long, N.V. (1999), Constant consumption and the economic depreciation of natural capital: The non-autonomous case, International Economic Review 40, 53–62

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Hicks, J. (1946), Value and capital, 2nd edition. Oxford University Press, Oxford Kemp, M.C. and Long, N.V. (1996), On the evaluation of social income in a dynamic economy: Variations on a Samuelsonian theme, in Fewel, G.R. (ed.), Samuelson and Neoclassical Economics. Kluwer, Dordrecht Kemp, M.C. and Long, N.V. (1998), On the evaluation of national Income in a dynamic Economy: Generalization, in Jäger, K. and Koch, K.-J. (eds.), Trade, Growth, and Economic Policies in Open Economies. Springer, Berlin Heidelberg New York Lindahl, E. (1933), The concept of income, in Bagge, G. (ed.), Economic Essays in Honor of Gustav Cassel. George Allen & Unwin, London Mäler, K.-G. (1991), National accounts and environmental resources, Environmental and Resource Economics 1, 1–5 Nordhaus, W.D. (1995), How should we measure sustainable income? Cowles Foundation DP 1101, Yale University Pearce, D.W. and Atkinson, G.D. (1993), Capital theory and the measurement of sustainable development: An indicator of “weak” sustainability, Ecological Economics 8, 103–108 Sefton, J.A. and Weale, M.R. (1996), The net national product and exhaustible resources: The effects of foreign trade, Journal of Public Economics 61, 21–47 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Usher, D. (1994), Income and the Hamiltonian, Review of Income and Wealth 40, 123–141 Vincent, J.R., Panayotou, T., and Hartwick, J.M. (1997), Resource Depletion and Sustainability in Small Open Economies, Journal of Environmental Economics and Management 33, 274–286 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162 Weitzman, M.L. (1997), Sustainability and technological progress, Scandinavian Journal of Economics 99, 1–13 Withagen, C. (1996), Sustainability and investment rules, Economics Letters 53, 1–6

CHAPTER 17 DOES NNP GROWTH INDICATE WELFARE IMPROVEMENT?

GEIR B. ASHEIM Department of Economics, University of Oslo P.O. Box 1095 Blindern NO-0317 Oslo, Norway Email: [email protected]

MARTIN L. WEITZMAN Department of Economics, Harvard University Littauer Center, Cambridge, MA 02138, USA Email: [email protected]

Abstract. We show that instantaneous increases in real NNP over time are an accurate indicator of true dynamic welfare improvements. This highlights a connection between the theory of green (or comprehensive) national income accounting and the theory of real price indices.

1. INTRODUCTION It has been known for sometime now that the current-value Hamiltonian of an optimal growth problem represents in welfare terms the level of stationary-equivalent future utility. It is also apparent that a current-value Hamiltonian is essentially comprehensive NNP expressed in utility units. Somewhat less apparent is how actually to use the above insights in a world where measurable NNP is expressed in monetary (rather than utility) units. In this paper, we show that welfare is increasing instantaneously over time if and only if real NNP is increasing instantaneously over time. Thus, contrary to some opinions that have been expressed in the literature, time changes in real NNP mirror accurately changes in dynamic welfare, at least locally. Originally published in Economics Letters 73 (2001), 233–239. Reproduced with permission from Elsevier.

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The framework used for establishing the above result is the standard multisector optimal growth model with time-invariant technology. The result may be useful because it shows an intriguing connection between the theory of green (or comprehensive) accounting and the theory of price deflators. In particular, the paper establishes a new conceptual link between the Divisia index of real consumption prices and dynamic welfare evaluation. 2. THE MODEL Let the vector C represent an m-dimensional fully-disaggregated consumption bundle, containing everything that influences current well-being, including environmental amenities and other externalities. (Supplied labor corresponds to negative components.) Current consumption is presumed to be fully observable, along with its associated m-vector of efficiency prices. For any consumption-flow {C(t)}, overall intertemporal welfare is measured by
∞ W ({C(t)}) := e−ρt U (C(t))dt , (1) 0

where U is a given concave and nondecreasing utility function with continuous second derivatives, while ρ is a given utility discount rate. There are n capital goods, including stocks of natural resources, environmental assets, human capital (like education and knowledge capital accumulated from R&Dlike activities), and other nonorthodox forms. The stock of capital of type j (1 ≤ j ≤ n) at time t is denoted K j (t), and its corresponding net investment flow is I j (t) = K˙ j (t). The n-vector K = {K j } denotes all capital stocks, while I = {I j } stands for the corresponding n-vector of net investments. The net investment flow of a natural capital asset is negative if the overall extraction rate exceeds the replacement rate. We are imagining an idealized world where the coverage of capital goods is so comprehensive, and the national accounting system so complete, that there remain no unaccounted-for residual growth factors. Thus, all sources of future growth are fully “accounted-for” as investments that are valued at their efficiency prices and included in national product. Formally, the (m + n)-dimensional attainable-possibilities set at any time t is a function S only of the capital stocks K(t) at that time. Therefore, the consumption–investment pair (C(t), I(t)) is attainable at time t if and only if (C(t), I(t)) ∈ S(K(t)) .

(2)

The set of attainable possibilities S(K) is presumed to be convex. 3. OPTIMAL GROWTH Consider the standard optimal growth problem: Maximize (1) subject to constraints ˙ (2) and K(t) = I(t), and obeying the initial condition K(0) = K0 , where K0 is given. In what follows, it is assumed for simplicity that an optimal solution exists

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and is unique. Let {C∗ (t)}, {I∗ (t)}, and {K∗ (t)} represent the optimal trajectories of consumption, investment and capital, and write
∞ e−ρ(s−t) U (C∗ (s))ds W ∗ (t) := t

for maximized welfare at time t. Let {
(t)} represent the trajectory of the dual vector of shadow investment prices, relative to utility being the numeraire. Applying the maximum principle of control theory to the above optimization problem, and letting the current-value Hamiltonian be given by H (C, I;
) = U (C) +
I , we obtain that, at each t, (C∗ (t), I∗ (t)) maximizes H (C, I;
(t)) subject to (C, I) ∈ S(K∗ (t)): H ∗ (t) = H(K∗ (t),
(t)) :=

max

(C,I)∈S(K∗ (t))

H (C, I;
(t))

= U (C∗ (t)) +
(t)I∗ (t) .

(3)

Refer to
(t)I∗ (t) as the value of net investments. Furthermore, we have as co-state differential equations that ˙ ∇HK (K∗ (t),
(t)) = ρ
(t) −
(t) ,

(4)

where ∇ denotes a vector of partial derivatives. Finally, the relevant transversality conditions are e−ρT
(T )K∗ (T ) → 0 and e−ρT H(K∗ (T ),
(T )) → 0 as T → ∞, implying that e−ρT
(T )I∗ (T ) → 0 as T → ∞ (cf. Michel, 1982). There is a basic result – cf. Weitzman (1976) and Dixit et al. (1980) – that is of fundamental importance for the analysis that follows. Lemma 1: Under the given assumptions, ˙ ∗ (t) + d(
(t)I∗ (t))/dt = ρ
(t)I∗ (t) ∇U (C∗ (t))C holds at any t. Proof. By (3), (4), and the envelope theorem, it follows that: ˙ = (ρ
−
)I ˙ ∗ +
I ˙ ∗ = ρ
I∗ . H˙ ∗ = ∇HK I∗ + ∇H



(5)

However, (3) also directly implies that ˙ ∗ + d(
I∗ )/dt . H˙ ∗ = ∇U (C∗ )C

(6)

The lemma is obtained by combining (5) and (6). Lemma 1 implies the following result, noted by e.g., Dasgupta and Mäler (2000) and Pemberton and Ulph (2001).

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Proposition 2: Under the given assumptions, W˙ ∗ (t) =
(t)I∗ (t) holds at any t. Proof. The proposition is obtained through integration since
∞ ∗ ∗ ˙ e−ρ(s−t) U (C∗ )ds W (t) = −U (C (t)) + ρ
=

t ∞

˙ ∗ ds = − e−ρ(s−t) ∇U (C∗ )C

t




∞

 d(e−ρ(s−t)
I∗ )/ds ds ,

t

where the second equality follows by integrating by parts, and the third equality follows from Lemma 1. This result means that the value of net investments has the following welfare significance: Maximized welfare is increasing if and only if
I∗ is positive. Measurable comprehensive net national product (NNP) is frequently identified in the literature with the “linearized” Hamiltonian (cf. e.g., Hartwick, 1990), being the sum of the “value of consumption” and the value of net investments, measured in monetary units. While Proposition 2 implies that welfare is increasing if and only if measurable NNP exceeds the value of consumption, this is a different kind of welfare significance than the one sought by Weitzman (1976), where higher welfare is indicated by higher NNP. The latter interpretation would translate here into a result that welfare is increasing along the time axis if and only if measurable NNP is also increasing. Can such a result be established? 4. NNP IN NOMINAL PRICES If the optimal growth trajectory is realized through an intertemporal competitive equilibrium, market prices will be expressed in monetary units. Neither the vector of marginal utilities, ∇U (C∗ ), nor the vector of investment prices in utility units,
, are directly observable. Rather, what may be observed directly are nominal prices at time t for consumption goods and investment flows, given respectively by p(t) = ∇U (C∗ (t))/λ(t), q(t) =
(t)/λ(t) , and a nominal interest rate at time t, r (t), given by r (t) = ρ −

˙ λ(t) , λ(t)

where λ(t) > 0 is the not-directly-observable marginal utility of current expenditures, which may depend on the “quantity of money” at time t.

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At any time t, consumers maximize utility and producers maximize profit: C∗ (t) maximizes U (C) − λ(t)p(t)C , ∗

∗

(7)

∗

˙ (C (t), I (t), K (t)) maximizes p(t)C + q(t)I − (r (t)q(t) − q(t))K over all (C, I, K) satisfying (C, I) ∈ S(K) ,

(8)

where r (t)q j (t) − q˙ j (t) is the cost of holding one unit of capital good j. We have that (7) follows from the concavity of U , while (8) follows from the convexity of S(K) ˙ = ρλq − for any K, the maximum principle, and the property that ∇HK = ρ
−
˙λq − λq˙ = λ(r q − q). ˙ This latter property also means that Lemma 1, expressed in nominal prices, yields ˙ ∗ (t) + d(q(t)I∗ (t))/dt = r (t)q(t)I∗ (t) . p(t)C

(9)

Define comprehensive NNP in nominal prices, y(t), as the sum of the nominal value of consumption and the nominal value of net investments y(t) := p(t)C∗ (t) + q(t)I∗ (t) . It follows from Proposition 2 that maximized welfare is increasing if and only if NNP exceeds the value of consumption: W˙ ∗ (t) > 0 ⇔ y(t) − p(t)C∗ (t) = q(t)I∗ (t) > 0 . However, since the level of NNP in nominal prices at t depends on λ(t), and λ(t) is arbitrary, the condition that y˙ (t) > 0 cannot signify welfare improvement. For a change in NNP (as opposed to a comparison of NNP with the value of consumption) to indicate a change in welfare, NNP must be measured in real prices. How then should NNP in real prices be determined? 5. NNP IN REAL PRICES AND LOCAL COMPARISONS In this section, we build upon a finding by Sefton and Weale (2006) that a Divisia consumption price index is of essential importance when expressing comprehensive NNP in real prices. By using such a price index, we show that a claim made by Dasgupta and Mäler (2000, Sect. 7.1) – namely that comprehensive real NNP cannot be used for intertemporal welfare comparisons – is incorrect. The application of a price index {π(t)} turns nominal prices {p(t), q(t)} into real prices {P(t), Q(t)}, P(t) = p(t)/π(t), Q(t) = q(t)/π(t) , implying that the real interest rate, R(t), at time t is given by R(t) = r (t) −

π˙ (t) . π(t)

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A Divisia price index satisfies ∗ (t) ˙ p(t)C π˙ (t) = , π(t) p(t)C∗ (t)

˙ ∗ = 0 at each t: implying that PC   ∗ ˙ ∗ ˙ ∗ = d p C∗ = π pC − π˙ pC = 0 . PC 2 dt π π Define comprehensive NNP in real Divisia prices, Y (t), as the sum of the real value of consumption and the real value of net investments: Y (t) := P(t)C∗ (t) + Q(t)I∗ (t) . Proposition 3: Under the given assumptions,   Y˙ (t) = R(t) Y (t) − P(t)C∗ (t) holds at any t. Proof. It follows from the definition of Y (t) that     ˙ ∗ + d(QI∗ )/dt = RQI∗ = R Y − PC∗ , Y˙ = d PC∗ + QI∗ /dt = PC ˙ ∗ = 0, and the third equality is obtained where the second equality follows since PC since (9) holds also for {P(t), Q(t)} and {R(t)}. Combining Propositions 2 and 3, we have the main result of this paper. Proposition 4: Provided that the real interest rate is positive, growth in real NNP means that welfare is increasing. 6. DISCUSSION The case of a so-called “cake-eating” economy – where no production takes place and consumption at time t equals the extraction at time t of a nonrenewable and finite natural resource – might seem to imply that a theorem like Proposition 3 cannot be established. A cake-eating economy’s comprehensive NNP is identical to zero, since consumption at each point in time equals extraction, and thus, the value of consumption and the value of net investment add up to zero. How can this result be reconciled with Proposition 3? The key issue here is that the real interest rate, R, in a “cake-eating” economy is identical to zero. Hence, even though a negative value of net investment in the resource (by Proposition 2) means that welfare is decreasing, by Proposition 3 comprehensive NNP is constant and equal to zero. Note that the paradox vanishes for any economy where R > 0. The real prices used in Proposition 3 are derived through a Divisia consumption price index. Hence, although welfare improvement is indicated by real growth in NNP – comprising the value of both consumption and net investments – only the consumption goods (including supplied labor as negative components) should be

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˙ ∗= used as quantity weights in the price index. In fact, since ∇U (C∗ ) = λπ P and PC 0, real Divisia prices satisfy the condition that increased instantaneous well-being is indicated by growth in real consumption expenditures:     ˙ ∗ + PC ˙ ∗ > 0 ⇔ d PC∗ /dt = PC ˙∗ > 0. d U (C∗ ) /dt = ∇U (C∗ )C We have shown in this paper that welfare stock improvements can be indicated by real NNP flow changes locally in time. However, unless Y (t) is monotone between t and t , it does not necessarily follow that Y (t ) < Y (t ) indicates that welfare stock is higher at time t when compared to an earlier point in time, t . The underlying reason is that the consumption bundle used as weights in a Divisia price index changes continuously over time. Even though an increase in P(t)C∗ (t) means that instantaneous well-being – i.e. utility – increases at time t, such a local result does not translate easily or directly into a general statement for making global welfare stock comparisons.

Acknowledgments: We thank Robert Cairns for comments. Asheim is grateful for the hospitality of Harvard University and the financial support of the Research Council of Norway. REFERENCES Dasgupta, P.S. and Mäler, K.-G. (2000), Net national product, wealth, and social well-being, Environment and Development Economics 5, 69–93 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Hartwick, J.M. (1990), National resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291–304 Michel, P. (1982), On the transversality condition in infinite horizon optimal control problems, Econometrica 50, 975–985 Pemberton, M. and Ulph, D. (2001), Measuring income and measuring sustainability, Scandinavian Journal of Economics 103, 25–40 Sefton, J.A. and Weale, M.R. (2006), The concept of income in a general equilibrium, Review of Economic Studies 73, 219–249 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162

CHAPTER 18 A GENERAL APPROACH TO WELFARE MEASUREMENT THROUGH NATIONAL INCOME ACCOUNTING

GEIR B. ASHEIM Department of Economics University of Oslo P.O. Box 1095 Blindern NO-0317 Oslo, Norway Email: [email protected]

WOLFGANG BUCHHOLZ Department of Economics University of Regensburg DE-93040 Regensburg Germany Email: [email protected]

Abstract. We develop a framework for analyzing national income accounting using a revealed welfare approach that is sufficiently general to cover, e.g., both the standard discounted utilitarian and maximin criteria as special cases. We show that the basic welfare properties of comprehensive national income accounting, previously ascribed only to the discounted utilitarian case, extend to this more general framework. In particular, it holds under a wider range of circumstances that real NNP growth (or equivalently, a positive value of net investments) indicates welfare improvement. We illustrate the applicability of our approach in the Dasgupta–Heal–Solow model of capital accumulation and resource depletion.

1. INTRODUCTION Net national product (NNP) represents the maximized value of the flow of goods and services that are produced by the productive assets of a society. If NNP increases, then society’s capacity to produce has increased, and – one might think – society is better Originally published in Scandinavian Journal of Economics 106 (2004), 361–384. Reproduced with permission from Blackwell.

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off. Although such an interpretation is often made in public debate, the assertion has been subject to controversy in the economic literature. While Samuelson (1961, p. 51) writes that “[o]ur rigorous search for a meaningful welfare concept has led to a rejection of all current income concepts . . . ”, Weitzman (1976), in his seminal contribution, shows that greater NNP indicates higher welfare if: (a) Dynamic welfare equals the sum of utilities discounted at a constant rate. (b)

Current utility equals the market value of goods and services consumed.

Weitzman’s result is truly remarkable – as it means that changes in the stock of forward looking welfare can be picked up by changes in the flow of the value of current production – but, unfortunately, very strong assumptions are invoked. Recently, Asheim and Weitzman (2001) have established that assumption (b) can be relaxed when concerned with whether welfare is increasing locally in time: real NNP growth corresponds to welfare improvement even when current utility does not equal the market value of current consumption, as long as NNP is deflated by a Divisia consumption price index. It is the purpose of the present paper to show how also Weitzman’s assumption (a) can be relaxed and a “snapshot” of the change in society’s current performance still indicates change in dynamic welfare. Why relax the assumption of discounted utilitarianism? First of all, such an assumption restricts the use of NNP comparisons for indicating welfare changes to situations where it can readily be determined that society maximizes the sum of utilities discounted at a constant rate. Moreover, there is a contradiction between having welfare correspond to discounted utilitarianism, on the one hand, and being concerned with welfare improvement, on the other hand. For example in the Dasgupta–Heal–Solow model (Dasgupta and Heal, 1974, 1979; Solow, 1974) of capital accumulation and resource depletion, eventually society’s welfare is optimally decreasing along the discounted utilitarian path. Increasing welfare over time does not have independent interest when society implements a path that maximizes the sum of discounted utilities. In contrast, real-world societies care about whether welfare is improving, both in terms of what proponents of economic growth may refer to as “progress” and in terms of what environmentalists call “sustainability.” To incorporate concerns for progress and sustainability we here extend Weitzman’s (1976) remarkable result by developing a framework for national accounting that is sufficiently general to include, in addition to discounted utilitarianism, also cases like (i) maximin, (ii) undiscounted utilitarianism, and (iii) discounted utilitarianism with a sustainability constraint. In these cases, nondecreasing current welfare does entail that current utility can be sustained indefinitely (as opposed to the case of unconstrained discounted utilitarianism; cf. Asheim, 1994; Pezzey, 1994). Within this general framework we demonstrate how changes in dynamic welfare are revealed by current NNP and hence by observable current prices and quantities. It is a prerequisite for the positive results on the relationship between welfare improvement and change in current NNP – from Weitzman (1976) to the current paper – that the list of goods and services included in NNP is comprehensive. The national accounts are “comprehensive” if all variable determinants of current

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productive capacity are included in the vector of capital stocks, and if all variable determinants of current well-being are included in the vector of consumption flows. For example one must “green” national accounts by (i) including depletion and degradation of natural capital as negative components to the vector of investment goods, and (ii) adding flows of environmental amenities to the vector of consumption goods. In this paper we first show that, even outside the realm of discounted utilitarianism, the value of net investments has the following welfare significance: Welfare is increasing if and only if the value of net investments is positive.1 Thus, in an economy with natural capital, welfare is increasing if and only if the accumulation of manmade capital (including stocks of knowledge) in value more than compensates for natural resource depletion and environmental degradation. Then, using the analysis of Asheim and Weitzman (2001), we establish that a positive value of net investments corresponds to real NNP growth. Hence, under quite general assumptions, welfare improvement is indicated by increasing real NNP or by the value of consumption falling short of NNP so that the value of net investments is positive. The empirical challenges of making national accounts comprehensive are the same in this more general framework. Forward-looking terms to capture technological progress and changing terms-of-trade (cf. Aronsson and Löfgren, 1993; Sefton and Weale, 1996) can also be incorporated here. Moreover, Asheim (2004) shows how the framework can be extended to accommodate exogenous population growth. We present the basic model in Sect. 2, before we in Sect. 3 take a look at national income accounting in the special cases of discounted utilitarianism and maximin. Then we turn in Sect. 4 to our general framework for revealed welfare analysis, and show in Sect. 5 how this framework means that real NNP growth can indicate welfare improvement in a general setting. Finally, we illustrate the applicability of our framework in Sect. 6 by considering progress and sustainability in the Dasgupta– Heal–Solow model, and conclude in Sect. 7.

2. THE MODEL Consider the model used by Weitzman (1970, 2001) and Asheim and Weitzman (2001), who generalize Weitzman (1976) by allowing multiple consumption goods. Let C represent a m-dimensional consumption vector that includes also environmental amenities and other externalities. (Supplied labor corresponds to negative components.) Let U be a given concave and nondecreasing utility function with continuous partial derivatives that assigns instantaneous utility U (C) to any consumption vector C. Assume an idealized world where C contains all variable determinants of current instantaneous well-being, implying that society’s instantaneous well-being is increased by moving from C to C if and only if U (C ) < U (C ). Let K denote a n-dimensional capital vector that includes not only the usual kinds of man-made capital stocks, but also stocks of natural resources, environmental assets, human capital (like education and knowledge capital accumulated from R&D˙ stand for like activities), and other durable productive assets. Moreover, let I (= K)

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the corresponding n-vector of net investments. The net investment flow of a natural capital asset is negative if the overall extraction rate exceeds the replacement rate. Assume again an idealized world where K contains all variable determinants of current productive capacity, implying that the set of attainable (m + n)-dimensional consumption-investment pairs is a function S only of the available capital stocks K, not of time. Hence, (C, I) is attainable given K if and only if (C, I) ∈ S(K), where S(K) is a convex and smooth set that constitutes current productive capacity. It holds that NNP is the maximized market value of current productive capacity in a perfect market economy where C and I are included in NNP and valued at market prices (cf. Sect. 5). As time passes, NNP changes both because K, and thus productive capacity S(K), change due to a nonzero vector of net investments, and because market prices of consumption and investment flows change. Since NNP is used for (a) consumption now and (b) accumulation of capital goods yielding increased future consumption, relating NNP growth to welfare improvement requires a notion of dynamic welfare: The welfare judgements must not only take into account the utility derived from current consumption, but must also reflect the utility possibilities that future consumption will give rise to. For this purpose, we assume that society’s welfare judgements are described by complete and transitive social preferences on the set of utility paths. However, these underlying social preferences are assumed not to be directly observable by the national accountant. What the national accountant can observe at any point in time is how the agents in society make decisions according to a resource allocation mechanism that assigns an attainable consumption-investment pair (C(K), I(K)) to any vector of capital stocks K.2 We assume that the functions C and I are continuous everywhere and differentiable almost everywhere, and that there exists a unique solution {K∗ (t)} ˙ ∗ (t) = I(K∗ (t)) that satisfies the initial condition to the differential equations K K∗ (0) = K0 , where K0 is given. Hence, {K∗ (t)} is the capital path that the resource allocation mechanism implements. Write C∗ (t) := C(K∗ (t)) and I∗ (t) := I(K∗ (t)). Since the resource allocation mechanism in this manner implements a utility path {U (C∗ (t))} for any vector of initial capital stocks K0 , the social preferences yield a complete and transitive binary relation on the set of capital vectors, under the presumption that paths are implemented by the resource allocation mechanism. Assume that, for given social preferences and resource allocation mechanism, this binary relation can be represented by an ordinal welfare index, W , that is unique up to a monotone transformation, signifying that society’s dynamic welfare is increased by moving from K to K if and only if W (K ) < W (K ). Also, assume that W is continuous and differentiable everywhere. To retain focus, we refer to optimal control theory and the maximum principle throughout the next sections under standard assumptions, without explicitly stating what these assumption are. 3. DISCOUNTED UTILITARIANISM AND MAXIMIN A main motivation for the analysis of the present paper is that it applies to a variety of methods for aggregating the interests of different generations in social evaluation. Discounted utilitarianism is the conventional example of social preferences

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in the intertemporal context. A prime example of an alternative welfare criterion is maximin – i.e., the ranking of paths according to the utility of the worst-off generation – as proposed by Rawls (1971) and Solow (1974). Analyzing these often applied kinds of social preferences and the corresponding resource allocation mechanisms will point to properties that will ensure welfare significance of national income accounting also for a wider class of social preferences and resource allocation mechanisms. In the special cases of discounted utilitarianism and maximin we will – in line with a standard constructive technique for preference representation in consumer theory (see, e.g., Mas-Colell et al., 1995, pp. 47–8, and Varian, 1992, p. 97) and inspired by Hicks (1946, Chap. 14) and Weitzman (1970, 1976) in the present context – identify the level of the welfare index, W (K), with the utility level that if held constant is equally as good as the implemented utility path given K as the vector of initial stocks. This makes W (K) a stationary equivalent of future utility. Discounted utilitarianism. Social preferences are represented by
∞ e−ρt U (C(t))dt ,

(1)

0

where ρ is a positive and constant utility discount rate. Assume that the resource allocation mechanism, for any vector of initial capital stocks K0 , implements a path {C∗ (t), I∗ (t), K∗ (t)} that maximizes (1) over all feasible consumption paths. By the maximum principle there exists a path {
(t)} of investment prices in terms of utility such that (C∗ (t), I∗ (t)) maximizes U (C) +
(t)I subject to (C, I) ∈ S(K∗ (t)) at each t. Associate welfare W (K0 ) with the utility level that if held constant is equally as good as the implemented path:  ∞ −ρt
∞ e U (C∗ (t))dt 0 W (K ) = 0  ∞ −ρt e−ρt U (C∗ (t))dt . =ρ e dt 0 0 It is a main result of Weitzman (1970, (5) and (16)) (reported in Weitzman, 1976, in the case where C is one-dimensional and U (C) = C) that
∞ U (C∗ (0)) +
(0)I∗ (0) = ρ e−ρt U (C∗ (t))dt (2) 0

Hence, W (K0 ) = U (C∗ (0)) +
(0)I∗ (0) under discounted  ∞ utilitarianism. Since
(0) is the vector of partial derivatives of 0 e−ρt U (C∗ (t))dt w.r.t. the initial stocks, we obtain that the vector of partial derivatives of W , ∇W (K0 ), equals ρ
(0). By the maximum principle it now follows that (C∗ (0), I∗ (0)) maximizes ρU (C) + ∇W (K0 )I subject to (C, I) ∈ S(K0 ) ,   since ρU (C) + ∇W (K0 )I = ρ · U (C) +
(0)I and ρ > 0. Maximin. Social preferences are represented by inft U (C(t)). Assume that the resource allocation mechanism implements maximin and that this leads to an

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efficient path with constant utility; formally, what Burmeister and Hammond (1977) and Dixit et al. (1980) call a regular maximin path. Then with K∗ (0) = K0 as the initial condition there exists a path of utility discount factors  ∞ {µ(t)} such that it is as if the implemented path {C∗ (t), I∗ (t), K∗ (t)} maximizes 0 µ(t)U (C(t))dt over all ∞ feasible consumption paths. Since U (C∗ (t)) is constant, it follows that 0 µ(t)dt is finite. This holds if the supporting utility discount rates, −µ(t)/µ(t), ˙ are positive and do not decrease too fast. Again, by the maximum principle there exists a path {
(t)} of investment prices in terms of utility such that (C∗ (t), I∗ (t)) maximizes U (C) +
(t)I subject to (C, I) ∈ S(K∗ (t)) at each t. Associate welfare W (K0 ) also in this case with the utility level that if held constant is equally as good as the implemented path: ∗

W (K ) = U (C (t)) = 0

∞ 0

µ(t)U (C∗ (t))dt ∞ . 0 µ(t)dt

By the converse of Hartwick’s rule (cf. Hartwick, 1977; Dixit et al., 1980; Withagen and Asheim, 1998; Mitra, 2002), we have that
(t)I∗ (t) = 0 at each t. Hence, W (K0 ) = U (C∗ (0)) +
(0)I∗ (0) even under maximin. ∞  Since
(0) is the vector of partial derivatives of 0 µ(t)/µ(0) U (C∗ (t))dt w.r.t. 0 the initial stocks, we obtain by invoking the  envelope theorem that ∇W (K ) equals ∞ ρ ∗
(0), where ρ ∗ := µ(0) µ(t)dt is the infinitely long-term supporting util0 ity discount rate at time 0 (i.e., the discounted average of the instantaneous discount rates, −µ(t)/µ(t), ˙ from time 0 on). By the maximum principle it follows that (C∗ (0), I∗ (0)) maximizes ρ ∗ U (C) + ∇W (K0 )I subject to (C, I) ∈ S(K0 ) ,   since ρ ∗ U (C) + ∇W (K0 )I = ρ ∗ · U (C) +
(0)I and ρ ∗ > 0.3 Two observations follow from the cases of discounted utilitarianism and maximin: 1.

Refer to U (C∗ (0)) +
(0)I∗ (0) as net national product in terms of utility or “utility NNP.” For both cases, utility NNP represents dynamic welfare globally; i.e., welfare is greater if and only if utility NNP is greater.

2.

Interpret ρ and ρ ∗ as Lagrangian multipliers for the constraint that U (C) ≥ U (C∗ (0)). For both cases, welfare improvement at time 0, ∇W (K0 )I, is maximized subject to (a) (C, I) ∈ S(K0 ), and (b) U (C) ≥ U (C∗ (0)).

The analysis of the next sections will show how observation 2 can be used as the basis for revealed welfare analysis in which welfare significance of national accounting aggregates are obtained. In contrast, we show that observation 1 cannot be generalized; it does for example not apply to the case of undiscounted utilitarianism (as will be shown in Sect. 6). We now turn to the general analysis.

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4. RESOURCE ALLOCATION AND WELFARE IMPROVEMENT Fix the underlying, but unobservable, social preferences used to rank utility paths, and consider a resource allocation mechanism. What properties on the resource allocation mechanism are both (i) strong enough for the underlying welfare concerns to be revealed through national income accounting and (ii) weak enough to hold for a wide range of circumstances? In this section, we answer this question by imposing two properties that hold if the most preferred paths under discounted utilitarianism and maximin are implemented, but, as illustrated in Sect. 6, have broader application. The first of these properties is the following. Property 1 (Implementation of an efficient path): Let {C∗ (t), I∗ (t), K∗ (t)} be the path implemented by the resource allocation mechanism with K∗ (0) = K0 as initial condition. There exists a continuous path of positive supporting utility discount factors {µ(t)}, with corresponding discount rates −µ(t)/µ(t) ˙ positive at almost  being ∞ every t, such that it is as if {C∗ (t), I∗ (t), K∗ (t)} maximizes 0 µ(t)U (C(t))ddt over all feasible consumption paths with K∗ (0) = K0 as initial condition. This property is clearly satisfied when discounted utilitarianism is implemented, and also for maximin when implementation of this criterion  ∞leads to a regular maximin path (cf. Sect. 3). The maximization is as if since 0 µ(t)U (C(t))dt is not necessarily the primitive objective of the society. For example in the maximin case, {µ(t)} simply characterizes the implemented path without having any intrinsic welfare significance. If Property 1 holds, then the maximum principle yields efficiency prices supporting the efficient path. In particular, there exists a continuous path {
(t)} of investment prices in terms of utility such that, at each t, (C∗ (t), I∗ (t)) maximizes U (C) +
(t)I subject to (C, I) ∈ S(K∗ (t)) . This yields the maximized current-value Hamiltonian: H ∗ (t) = H (K∗ (t),
(t)) :=

max

(C,I)∈S(K∗ (t))

U (C) +
(t)I = U (C∗ (t)) +
(t)I∗ (t) .

Refer to
I∗ as the value of net investments. Furthermore, µ(t) ˙ ˙
(t) −
(t) , ∇K H (K∗ (t),
(t)) = − µ(t)

where ∇ denotes a vector of partial derivatives. The following basic result – which is at the heart of the analyzes of, e.g., Weitzman (1976, cf. (14)) and Dixit et al. (1980, cf. Theorem 1) – can now be established. Lemma 1: If Property 1 holds, then U (C∗ (t)) +
(t)I∗ (t) is continuous and   ˙ ∗ ˙ ∗ (t) + d
(t)I∗ (t) /dt = − µ(t) ∇U (C∗ (t))C µ(t)
(t)I (t) holds at almost every t.

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Proof. H ∗ = U (C∗ ) +
I∗ is continuous since K∗ and
are continuous. Otherwise, adapt the proof of Asheim and Weitzman (2001, Lemma 1). This result says that change in utility NNP equals the supporting utility discount rate times the value of net investments. Let, as in Sect. 2, the binary relation over vectors of stocks, induced from the social preferences for given resource allocation mechanism, be represented by a welfare index, W , that is unique up to a monotone transformation. To ensure that the underlying welfare concerns can be revealed through national income accounting, we make, in addition to Property 1, the following assumption: The resource allocation mechanism and the accompanying welfare index satisfy that welfare improvement is maximized subject to (C, I) being attainable and utility being at least U (C(K)). This is stated by the following property, where ρ(K) is formally a Lagrangian multiplier on the lower bound for utility. Property 2 (No waste of welfare improvement): ρ(K) > 0 such that:

For every K, there exists

(C(K), I(K)) maximizes ρ(K)U (C) + ∇W (K)I subject to (C, I) ∈ S(K) . We have observed in Sect. 3 that Property 2 holds when discounted utilitarianism and (under regularity conditions) maximin are implemented; ρ(K) can be interpreted as a supporting utility discount rate in these cases. In all our examples, we show that Properties 1 and 2 hold for resource allocation mechanisms that are optimal in the sense that they, for any initial stocks, implement paths that are weakly preferred to any feasible path according to the social preferences. We conjecture that Properties 1 and 2 are necessary for optimal resource allocation if the social preferences and the technological environment satisfy the following condition: There does not exist an alternative path that, compared to an optimal path, has higher utility in an initial period, at the end of which the alternative path is deemed as good as the optimal path.4 The investigation of such a primitive condition on preferences and technology seems, however, to require a discrete time framework and, thus, falls outside the scope of the present paper. By writing W ∗ (t) := W (K∗ (t)) for the welfare level along {C∗ (t), I∗ (t), K∗ (t)}, so that W˙ ∗ (t) = ∇W (K∗ (t))I∗ (t), we obtain the following result. Lemma 2: If Properties 1 and 2 hold, then at every t, W˙ ∗ (t) = ρ(K∗ (t))
(t)I∗ (t) . Proof. Since U is concave and S(K) is convex and smooth, there is a unique ndimensional hyperplane that supports the set of feasible (n + 1)-dimensional utilityinvestment vectors. By comparing the maximum principle with Property 2, we can conclude that ∇W (K∗ (t)) = ρ(K∗ (t))
(t) and, thus, W˙ ∗ (t) = ∇W (K∗ (t))I∗ (t) = ρ(K∗ (t))
(t)I∗ (t) hold at every t.

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Lemma 2 shows that the sign of the value of net investments along the implemented path indicates whether welfare is increasing.5 The main result of the present section follows from Lemmas 1 and 2. Proposition 1: If Properties 1 and 2 hold, then dynamic welfare is increasing if and only if there is growth in U (C∗ (t)) +
(t)I∗ (t). Thus, changes in dynamic welfare according to the unspecified aggregation of the interests of different generations are revealed through changes in utility NNP. Proposition 1 is a result for local-in-time comparisons along the implemented path. It does not imply that utility NNP represents dynamic welfare globally (i.e., welfare is greater if and only if utility NNP is greater), which would entail that W (K∗ (t)) = U (C∗ (t)) +
(t)I∗ (t) holds at each t. To shed on theproblems involved, note that W˙ ∗ = ρ(K∗ )
I∗  light ∗ ∗ (by Lemma 1). Hence, the ˙ (by Lemma 2), and d U (C ) +
I∗ /dt = (−µ/µ)
I ∗ ∗ combination of ρ(K ) = −µ/µ ˙ and
I = 0 precludes that utility NNP can represent welfare globally. As shown in Sect. 3, it works for discounted utilitarianism because ρ(K∗ ) = −µ/µ, ˙ and it works for maximin because
I∗ = 0. However, in general we must allow for cases where ρ(K∗ ) = −µ/µ ˙ is combined with
I∗ = 0 ∗ ∗ ∗ and, thus, W (K ) = U (C ) +
I cannot hold at each t. Indeed, Prop. 3 of Sect. 6 presents a case where global representation of welfare by means utility NNP is not possible. 5. REAL NNP GROWTH AND LOCAL COMPARISONS Until now we have considered NNP and the value of net investments in utility terms. Utility, however, is not observable directly, while market prices in principle are. Comprehensive NNP that is measurable by market prices is often identified in the literature with the “linearized” Hamiltonian (cf. Hartwick, 1990), being the sum of the value of consumption and the value of net investments, measured in monetary units. To demonstrate that welfare is increasing locally in time along the implemented path if and only if such measurable NNP is also increasing, we now adapt the analysis of Asheim and Weitzman (2001) to the present more general setting. What may be observed directly at each time t are nominal prices for consumption goods and investment flows, p(t) = ∇U (C∗ (t))/λ(t) and q(t) =
(t)/λ(t), where λ(t) > 0 is the continuous not-directly-observable marginal utility of current expenditures. Comprehensive NNP in nominal prices, y(t), is then defined by: y(t) := p(t)C∗ (t) + q(t)I∗ (t) . If Property 1 holds, then y(t) = max(C,I)∈S(K∗ (t)) p(t)C + q(t)I. By Lemma 2, dynamic welfare is increasing if and only if NNP in nominal prices exceeds the value of consumption. However, since NNP in nominal prices at t depends on λ(t), and λ(t) is arbitrary, y˙ (t) > 0 need not signify welfare improvement. For a change in NNP to indicate a change in welfare, NNP must be measured in

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real prices. The application of a price index {π(t)} turns nominal prices {p(t), q(t)} into real prices {P(t), Q(t)} by imposing P(t) = p(t)/π(t) and Q(t) = q(t)/π(t) at each t. Following Asheim and Weitzman (2001), we use a Divisia consumption price index, ∗ (t) ˙ p(t)C π˙ (t) = , π(t) p(t)C∗ (t)

˙ ∗ = 0 holds. Comprehensive NNP in real implying that π(t) is continuous and PC Divisia prices, Y (t), is then defined by: Y (t) := P(t)C∗ (t) + Q(t)I∗ (t) . Lemma 3: If Property 1 holds, then Y (t) is continuous and   Y˙ (t) = R(t) Y (t) − P(t)C∗ (t) holds at almost every t, where the real interest rate, R(t), at time t is given by R(t) = −µ(t)/µ(t) ˙ − λ˙ (t)/λ(t) − π(t)/π(t). ˙ Proof. That Y is continuous follows from the continuity of U (C∗ ) +
I∗ (cf. Lemma 1) since U has continuous partial derivatives and both λ and π are continuous. Otherwise, adapt the proof of Asheim and Weitzman (2001, Prop. 3). Since Lemma 3 entails that change in real NNP = real interest rate · value of net investments and, by Lemma 2, a positive value of net investments indicates welfare improvement, we obtain the main result of Asheim and Weitzman (2001) in our generalized setting. Proposition 2: Provided that Properties 1 and 2 hold and the real interest rate is positive, dynamic welfare is increasing if and only if there is growth in measurable NNP in real Divisia prices. Proof. The result follow from Lemmas 2 and 3 since

  ρ(K∗ )
I∗ = ρ(K∗ )λπ QI∗ = ρ(K∗ )λπ Q Y − PC∗ ,

where ρ(K∗ ), λ, and π are all positive. As noted by Asheim and Weitzman (2001), real NNP growth indicates welfare improvements locally in time. Unless real NNP grows in a monotone manner between t  and t  , it does not necessarily follow that a higher real NNP at t  than t  indicates that welfare is higher at t  compared to t  . 6. PROGRESS AND SUSTAINABILITY IN A RESOURCE MODEL We finally use the Dasgupta–Heal–Solow (DHS) model of capital accumulation and resource depletion to illustrate the applicability of our framework. In the DHS model, a stock of man-made capital (K M ) is combined with extracted raw material

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259

from a stock of a natural resource (K N ) to produce output that can be split between consumption and investment. For tractability, we assume that the production function is Cobb-Douglas and exhibits CRS, implying that the consumption-investment pair (C, IM , IN ) is attainable given (K M , K N ) if and only if a C + IM ≤ K M · (−IN )b , b < a < a + b = 1 ,

where C ≥ 0, IN ≤ 0, K M ≥ 0, and K N ≥ 0, The assumption that b < a is required to ensure that progress and sustainability are feasible in the present setting. Consider paths for which C, −IN , K M , and K N remain positive throughout so that smoothness of the attainable set is satisfied. Let the ratio between man-made capital and output be denoted by κ:  K b KM M κ= a = . K M · (−IN )b −IN If the implemented path satisfies Property 1, then the real interest rate along the path measures the marginal productivity of K M and is given by R(t) = a/κ ∗ (t), where κ ∗ (t) is the capital-output ratio along the implemented path at time t. Moreover, the real investment prices are given by: a

Q M (t) = 1 and Q N (t) = b · κ ∗ (t) b ,

(3)

since, with output as numeraire, Q N (t) measures the marginal productivity of −IN . The Hotelling rule for short-run efficiency yields Q˙ N (t)/Q N (t) = R(t), implying κ˙ ∗ (t) = b .

(4)

If, in addition to (4), the following transversality conditions are satisfied, lim

t→∞

∗ (t) KM a

b · κ ∗ (t) b

= 0 and

lim K N∗ (t) = 0 ,

t→∞

(5)

then routine calculations show that, by setting U(C) = C, Property 1 holds: The a ∞  implemented path maximizes 0 1/ b · κ ∗ (t) b · C(t)dt over all feasible paths, 0 , K 0 )  0. for any initial stocks (K M N 6.1. Progress Paths Combine efficiency (Property 1) with an exogenous investment rule, a IM = β K M (−IN )b , b < β < a ,

(6)

leading to sustained progress (cf. Hamilton, 2002, for an independent and similar investigation). We now apply our revealed welfare analysis (by invoking Property 2) and show how real NNP growth picks up the welfare improvement that the sustained consumption growth entails. We confirm this welfare result by establishing that the investment rule is optimal under undiscounted utilitarianism. We observe that – even though growth in measurable real NNP measures welfare improvement locally in time – utility NNP cannot represent welfare globally.

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a (−I )b equals resource rents; i.e., the share of output that is Note that bK M N attributable to extraction of raw material. Hartwick’s rule (cf. Hartwick, 1977; Dixit et al., 1980) entails that reinvesting resource rents forever leads to constant consumption. Since β > b, more than resource rents are reinvested by following (6), leading to progress in the sense that consumption increases in a sustained manner. Since β < a, feasibility of the implemented path is ensured. The efficiency conditions (4) and (5) and the investment rule (6) determines a resource allocation mechanism specifying C, IM , and IN as functions of capital stocks, (K M , K N ). It follows from the definition of κ and the investment rule (6) that these functions can, for positive capital stocks, be described by:

KM C(K M , K N ) = (1 − β) ·  K  κ KMN

(7)

KM IM (K M , K N ) = β ·  K  κ KMN

(8)

KM IN (K M , K N ) = −   1 , κ KKMN b

(9)

where, by imposing (4) and K N part of (5) as efficiency conditions, we can calculate the capital-output ratio as an explicit function of K M /K N , 

KM κ KN



− ab

= (a − β)



KM KN

b a

,

(10)

0 , K 0 )  0 at and check that K M part of (5) is satisfied. For given initial stocks (K M N ∗ (t), K ∗ (t)}, time 0, (8)–(10) determine the implemented path of capital stocks, {K M N which in turn yields the implemented paths of consumption and investment flows: ∗ (t), K ∗ (t)), I ∗ (t) = I (K ∗ (t), K ∗ (t)), and I ∗ (t) = I (K ∗ (t), C ∗ (t) = C(K M M N N M M N N M ∗ K N (t)). By combining (7) and (8) with (4), we can establish that consumption grows at a positive (but decreasing) rate since β > b:

β −b C˙ ∗ (t) = ∗ > 0. ∗ C (t) κ (t) Moreover, by combining (3) with (10), it follows that the relative price of natural capital in terms of man-made capital is positively related to β, the parameter that indicates society’s emphasis on progress: K ∗ (t) Q N (t) b = · M . Q M (t) a − β K N∗ (t)

(11)

By assuming that the implemented path does not waste opportunity for welfare improvement – i.e., by adding Property 2– Prop. 2 implies that welfare is increasing

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261

if and only if there is real NNP growth, where real NNP can be written as ∗ C ∗ (t) + Q M (t)IM (t) + Q N (t)IN∗ (t) = a ·

∗ (t) KM ∗ κ (t)

due to the constant factor shares. It follows from (4) and (8) that the growth rate of NNP equals that of consumption. Thus, the revealed welfare analysis picks up that consumption increases in a sustained manner. By Lemma 3 increased welfare can also be indicated by a positive value of net ∗ (t) + Q (t)I ∗ (t) > 0. Since (8)–(10) imply investments: Q M (t)IM N N IN (K M , K N ) a − β KN =− · , IM (K M , K N ) β KM

(12)

and β > b, (11) implies that welfare is improving along the implemented path: ∗ ∗ Q M (t)IM (t) + Q N (t)IN∗ (t) = IM (t) (1 − b/β) > 0 .

Property 2 (cf. the proof of Lemma 2) entails that any welfare index W (K M , K N ) satisfies ∂ W (K M ,K N ) ∂ KN ∂ W (K M ,K N ) ∂ KM

=

N QN b KM = = . · M QM a − β KN

(13)

It is a direct consequence of (13) that welfare can be represented by a−β

W (K M , K N ) = K M

K Nb .

(14)

Moreover, it follows from (12) that the implemented path in (K M , K N )-space is a−β β described by K M K N being constant, with K˙ M = IM > 0 and K˙ N = IN < 0. That welfare is improving along the implemented path can now alternatively be seen by comparing the iso-welfare contours given by (14) with the contour that describes the implemented path in (K M , K N )-space. Are there explicitly specified social preferences such this resource allocation mechanism for any vector of initial stocks implements a most preferred path? This question is answered by observing that the resource allocation mechanism can be derived from the utilitarian problem of maximizing, without discounting,
∞ − 1−β −C(t) β−b dt (15) 0

over all feasible paths. It follows from Dasgupta and Heal (1979, pp. 303–308) that an undiscounted utilitarian optimum exists if (1 − β)/(β − b) > (1 − a)/(a − b), which implies b < β < a. This, of course, is the assumption we have made. Now, rank positive initial stocks, (K M , K N ), by the maximum value of the integral in (15) they give rise to. We find that this ranking can be represented by (14). This confirms the result that we have already derived through our revealed welfare analysis, namely that welfare is increasing along the implemented path.

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Under discounted utilitarianism utility NNP represents dynamic welfare globally (cf. (2) in Sect. 3 as well as Weitzman, 1970, 1976). Moreover, under maximin, the converse of Hartwick’s rule implies that utility NNP is equal to the constant utility level and therefore represents welfare globally. For the progress paths that we analyze here in the context of the DHS model, however, it turns out that utility NNP cannot represent welfare globally. Proposition 3: Consider the resource allocation mechanism determined by (7)–(10) in the context of the DHS model. There exists no utility function such that net national product in terms of utility represents dynamic welfare globally. ∗ (t) + Proof. It follows from (6) and the constant factor shares that Q M (t)IM Q N (t)IN∗ (t) = (β − b)C ∗ (t)/(1 − β). Since M (t) = U  (C ∗ (t)) · Q M (t) and N (t) = U  (C ∗ (t)) · Q N (t), this implies

Utility NNP = U (C(K M , K N )) + U  (C(K M , K N ))

β −b C(K M , K N ) 1−β

for an arbitrary U function, when the pair of capital stocks is (K M , K N ). Assume that utility NNP represents welfare globally. Then utility NNP must be invariant when moving along any iso-welfare contour defined by (14):   β −b β − b  ∂C ∂C U  (C) + U  (C) + U  (C) C dKM + dKN = 0 . 1−β 1−β ∂ KM ∂ KN Since C(K M , K N ) increases when moving along an iso-welfare contour by increasing K M and decreasing K N , the second parenthesis is nonzero and U  (C)a/(β − b) + U  (C)C = 0 must hold. Hence, U is in the class of affine transformations of U (C) = −C

1−β − β−b

.

(16)

Since the implemented path maximizes (15), the supporting utility discount rate is zero throughout for any utility function in this class. Under these circumstances, Lemma 1 – extrapolated to the case with a zero utility discount rate – implies that utility NNP does not change as a consequence of nonzero value of net investments. Indeed, it follows that utility NNP is zero for any pair of initial stocks if the utility function is given by (16). Hence, utility NNP has no welfare significance within the class of utility functions that are affine transformations of (16). Property 1 does not hold for any utility function in the class considered in the proof of Prop. 3, i.e., that is an affine transformation of (16). If we instead use a utility function so that Property 1 holds for a path of supporting utility discount factors {µ(t)} with discount rates −µ(t)/µ(t) ˙ that are positive (U (C) = C is an example), it follows from the analysis of Sefton and Weale (1996) that
∞ µ(t) µ(t) ˙ Utility NNP = U (C ∗ (t))dt . − µ(t) µ(0) 0 Hence, utility NNP is a weighted average of future utility. However, by following a given iso-welfare contour defined by (14) as K M → ∞ and K N → 0, and

GENERAL APPROACH TO WELFARE MEASUREMENT

263

considering the consumption paths that would be implemented for these initial conditions, it can be shown that the minimal consumption (which occurs at time 0) along these paths goes to infinity. This means that the constant welfare along such an iso-welfare contour cannot be expressed as a weighted average of future utility. However, even though utility NNP cannot represent dynamic welfare globally, Prop. 2 implies that growth in measurable NNP in real prices measures welfare improvement locally in time, even in the current setting. This illustrates the generality of the positive result that we report in Proposition 2.

6.2. Sustainability as a Constraint Consider a society that deems unsustainable development unacceptable, and which adopts a resource allocation mechanism that among the acceptable sustainable paths implements the path that maximizes the sum of discounted utilities


∞

e−ρt U (C(t))dt ,

(17)

0

where ρ is a positive and constant utility discount rate.6 We confirm that Properties 1 and 2 hold and show how measurement of welfare improvement through real NNP growth can be useful for the management of society’s assets: Real NNP growth approaching zero indicates that unconstrained development is no longer sustainable. A consumption path is said to be sustainable if, at all times, current consumption does not exceed the maximum sustainable consumption level given the current capital stocks. Since unconstrained maximization of (17) in the DHS model leads to consumption converging to zero as time goes to infinity, the sustainability constraint imposed on the implemented path is binding. Since by (4) the real interest rate R(t) = a/κ ∗ (t) is decreasing along any efficient path, the sustainability constraint binds in an eventual phase with constant consumption, which can possibly be preceded by an unconstrained utilitarian phase with increas0 , K 0 )  0, the implemented path ing consumption. For given initial stocks (K M N ∗ (t), I ∗ (t), K ∗ (t), K ∗ (t)} can be determined by maximizing (17) subject {C ∗ (t), IM N M N to the constraint that consumption is nondecreasing; standard arguments imply that such a path exists. Therefore, if, in this example, social preferences over paths are represented by (17) on the set of nondecreasing consumption paths, while paths that are not nondecreasing are strictly less preferred, then it follows that the resource allocation mechanism defined above implements a most preferred path also in this case. Since implemented 0 , K 0 ) can be associated with the utility level paths are nondecreasing, welfare W (K M N that if held constant is equally as good when evaluated by (17): ∞ 0 W (K M , K N0 )

=

0

e−ρt U (C ∗ (t))dt ∞ =ρ −ρt dt 0 e


0

∞

e−ρt U (C ∗ (t))dt .

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For simplicity, assume constant elasticity of marginal utility; i.e., for all   C > 0, − U  (C) · C /U  (C) = η > 0. Then the resource allocation mechanism becomes homogenous of degree 1 since production exhibits CRS. In particular, the capital-output ratio κ is a function of K M /K N , and the dividing line between the sustainability unconstrained and constrained regimes is a ray in (K M , K N )space. Consumption increases if and only if the infinitely long-term real interest rate exceeds ρ. This rate equals the inverse of the value of a perpetual bond; hence, it is the discounted average of the instantaneous real interest rate, a/κ ∗ (s), from time t on: ∞  a  1/ b · κ ∗ (s) b · (a/κ ∗ (s))ds a−b t . = ∗ ∞  a ∗ κ (t) b t 1/ b · κ (s) ds Note that the infinitely long-term interest rate (a − b)/κ ∗ (t) is smaller than the instantaneous rate a/κ ∗ (t) since the latter is decreasing throughout. In the eventual sustainability constrained phase, the resource allocation mechanism implements efficient paths with constant consumption, implying that the resource allocation mechanism is described by (7)–(10) with β = b. Since this phase is entered when the infinitely long-term interest rate (a − b)/κ ∗ (t) equals ρ, it follows from (10) that paths are in the unconstrained discounted utilitarian phase if 1  KM a−b b <ρ , KN ρ and in the eventual sustainability constrained phase otherwise. For tractability, assume also that U (C) = C b so that η = a. Then the unconstrained discounted utilitarian phase is characterized by (7)–(9) with β = β(κ) = 1 − a

ρκ a 2 − b2 · e

ρκ−(a−b) ab

,

where (4) implies that κ is an increasing function of K M /K N for K M /K N < ρ((a − b)/ρ)1/b since K˙ M = IM > 0 and K˙ N = IN < 0, and continuous at K M /K N = ρ((a − b)/ρ)1/b since κ ∗ (t) is differentiable w.r.t. time. Note that β is a decreasing function of κ (and by (4) of time) and converges to b as κ approaches the value (a − b)/ρ at which time paths enter into the sustainability constrained phase. Hence, output, C, IM , and IN are throughout continuous functions of (K M , K N ) and time. 0 , K 0 )  0 and consider the implemented path deterFix the initial stocks (K M N mined by the resource allocation mechanism described above. Let τ denote the time at which the implemented path enters the eventual sustainability constrained phase. 0 /K 0 ≥ ρ((a − b)/ρ)1/b . We now Set τ = 0 if the path starts in this phase, i.e., if K M N verify that this path does indeed satisfy Properties 1 and 2 and maximize (17) subject to consumption being nondecreasing.

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Property 1 holds for the continuous path of supporting utility discount factors {µ(t)} determined (up to the choice of numeraire) by µ(t) µ(t) a−b ∞ for t ∈ [τ, ∞) , (18) = ρ for t ∈ [0, τ ) and  ∞ = ∗ κ (t) t µ(s)ds t µ(s)ds implying that utility discount rates are positive: −µ(t)/µ(t) ˙ = ρ for t ∈ (0, τ ) and −µ(t)/µ(t) ˙ = a/κ ∗ (t) for t ∈ (τ, ∞). This can be seen by choosing a path of investment prices in terms of utility {M (t), N (t)} so that the current-value Hamiltonian is maximized at any point in time: M (t) = U  (C ∗ (t)) · Q M (t) = bC ∗ (t)−a a

N (t) = U  (C ∗ (t)) · Q N (t) = bC ∗ (t)−a · bκ ∗ (t) b . Then the co-state differential equations hold, a κ ∗ (t)

M (t) = −

µ(t) ˙ ˙ M (t) M (t) −  µ(t)

0=−

µ(t) ˙ ˙ N (t) , N (t) −  µ(t)

and the consumption path satisfies Ramsey’s rule: C˙ ∗ (t) µ(t) ˙ a = − + a . κ ∗ (t) µ(t) C ∗ (t)

(19)

Since −µ(t)/µ(t) ˙ jumps from ρ = (a − b)/κ ∗ (τ ) to a/κ ∗ (τ ) when the sustainability constrained phase is entered, it follows from (19) that the rate of consumption growth decreases abruptly from b/(aκ ∗ (τ )) to 0 at that time. The following result, the proof of which is available from the authors, establishes formally that (17) is maximized subject to consumption being nondecreasing. 0 , K 0 )  0, the path implemented by the Lemma 4: For any initial stocks (K M N ∞ resource allocation mechanism described above maximizes 0 e−ρt C(t)b dt over all feasible nondecreasing consumption paths.

To apply the revealed welfare analysis of Sects. 4 and 5, we must show that Property 2 holds. It follows from (18) that ∞
∞ µ(t) τ µ(s)ds −ρt ∞ for t ∈ [0, τ ) and  ∞ e−ρs ds . = ρe =ρ µ(s)ds µ(s)ds τ 0 0 Hence, since consumption is constant in the eventual sustainability constrained 0 , K 0 ) can be rewritten as follows: phase, we have that W (K M N ∞
∞ µ(t)C ∗ (t)b dt 0 0 −ρt ∗ b e C (t) dt = 0  ∞ . W (K M , K N ) = ρ 0 0 µ(t)dt

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∞ Since (M (0), N (0)) is the vector of partial derivatives of 0 µ(t)C ∗ (t)b dt w.r.t. the initial stocks, we obtain by invoking the envelope theorem that 0 , K0) 0 , K0) ∂ W (K M ∂ W (K M µ(0) µ(0) N N = ∞ = ∞ M (0) and N (0) . ∂ KM ∂ KN 0 µ(t)dt 0 µ(t)dt   ∞  0 , K 0 ) = µ(0) Hence, Property 2 holds since, by setting ρ(K M N 0 µ(t)dt , the ∗ (0), I ∗ (0)) maximizes maximum principle implies that (C ∗ (0), IM N 0 , K N0 )C b + ρ(K M

0 , K0) 0 , K0) ∂ W (K M ∂ W (K M N N IM + IN ∂ KM ∂ KN

over all attainable consumption-investment pairs. Hence, Prop. 2 is applicable and welfare is increasing if and only if there is growth ∗ (t)/κ ∗ (t). Since (4) implies that the growth rate of real NNP equals in real NNP, a K M (β(κ) − b)/κ, welfare is increasing as long as the path remains in the unconstrained utilitarian phase, during which β(κ) > b. Since β(κ) reaches b at the point in time at which the sustainability constraint becomes binding, the observation that the growth rate of real NNP decreases toward zero indicates that unconstrained development is no longer sustainable. Hence, the information on welfare changes offered by the growth rate of real NNP is useful for the management of society’s assets, given that unsustainable paths are deemed socially unacceptable. Note that consumption yields no such indication, since the rate of consumption growth falls discontinuously to zero at the time the path enters the sustainability constrained phase. By Lemma 3 increased welfare can also be indicated by the value of net invest∗ (t) + Q (t)I ∗ (t) = I ∗ (t)(1 − b/β(κ ∗ (t))), being positive. Again ments, Q M (t)IM N N M β(κ) > b during the unconstrained utilitarian phase implies that welfare is increasing, while the observation that the value of net investments decreases toward zero as β(κ) approaches b indicates that unconstrained development is no longer sustainable. Thus, also the sign of the value of net investments is useful for asset management. NNP growth (and the value of net investments) indicate when the sustainability constraint becomes binding precisely because policies implementing sustainable development are expected and, hence, reflected in the ratio of investment prices. Sustainability cannot be indicated in this way if instead an unconstrained utilitarian path is expected to be followed throughout (cf. Asheim, 1994; Pezzey, 1994). 7. CONCLUDING REMARKS We have established that real NNP growth – or equivalently, a positive value of net investments – can be used to indicate welfare improvement, independently of the welfare criterion adopted, in a constant population society with comprehensive national accounting. Provided it holds that the implemented policies lead to an efficient path that does not waste opportunity for welfare improvement, the underlying – but unspecified and unobservable – welfare judgements are revealed through prices and quantities that are available in a perfect market economy. In such a revealed

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welfare approach we need not know the actual social preferences when drawing welfare conclusions on the basis of national accounting aggregates. We have thus shown that the result of Asheim and Weitzman (2001) – namely that increasing measurable NNP in real Divisia prices indicates welfare improvement in a multiple consumption good setting even when utility itself is not measurable – holds even in situations where society does not subscribe to discounted utilitarianism. The present analysis covers also circumstances where, for example, progress and sustainability are important concerns. We have exemplified this in Sect. 3 by showing that, in general, maximin is encompassed by the present approach, and in Sect. 6 by considering two resource allocation mechanisms in the Dasgupta–Heal– Solow model of capital accumulation and resource depletion; one that implements undiscounted utilitarianism and one that maximizes the sum of discounted utilities within the subset of sustainable paths. In the latter case, real NNP growth approaching zero indicates that unconstrained development is no longer sustainable.

Acknowledgments: We have benefited from discussions with Kenneth Arrow and Martin Weitzman. We also thank Finn Førsund, Lawrence Goulder, Peter Hammond, Geoffrey Heal, and David Miller as well as three anonymous referees for helpful comments. An earlier version was circulated under the title “Progress, sustainability, and comprehensive national accounting.” Asheim appreciates the hospitality of the research initiative on the Environment, the Economy and Sustainable Welfare at Stanford University, where much of this work was done. We gratefully acknowledge financial support from the Hewlett Foundation through the above mentioned research initiative (Asheim), CESifo Munich (Asheim), and the Research Council of Norway (Ruhrgas grant, both authors). NOTES 1 Under discounted utilitarianism, this is proven by Weitzman (1976, (14)), and reported by, e.g., Hamil-

ton and Clemens (1999), Dasgupta and Mäler (2000), and Pemberton and Ulph (2001). 2 This is inspired by Dasgupta (2001, p. C20) and Dasgupta and Mäler (2000). 3 The observation that ∇W (K∗ (t)) is proportional to (t) constitutes

a simple proof of the converse of Hartwick’s rule: Constant utility implies that 0 = dW (K∗ (t))/ dt = ∇W (K∗ (t))I∗ (t), which due to proportionality of ∇W (K∗ (t)) and (t) yields (t)I∗ (t) = 0. See also Cairns (2000). 4 In the case of maximin, this condition holds if maximin paths are regular, while it can fail otherwise, e.g., in a one-sector model with the initial capital stock exceeding the golden rule level. The maximin criterion illustrates that Property 2 does not necessarily mean that there is a linear tradeoff between current utility and welfare improvement. Properties 1 and 2 can hold even if the resource allocation mechanism does not implement an optimal path. For example suppose that society adheres to discounted utilitarianism in a technology where implementation of discounted utilitarianism would have lead to nonconstant utility, but, in fact, a regular maximin path is implemented. Both Properties 1 and 2 hold under such circumstances. 5 Onuma (2003) has independently, through his Prop. 3, derived a result that is similar to our Lemma 2. His result is based on a property called “Generation rationality under a bequest constraint,” which corresponds to our Property 2.

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6 Asheim et al. (2001) present ethical axioms under which only sustainable paths are acceptable in the

DHS model. Discounted utilitarian paths under a sustainability constraint in the DHS model are analyzed in continuous time by Asheim (1986) and Pezzey (1994).

REFERENCES Aronsson, T. and Löfgren, K.-G. (1993), Welfare consequences of technological and environmental externalities in the Ramsey growth model, Natural Resource Modeling 7, 1–14 Asheim, G.B. (1986), Rawlsian intergenerational justice as a Markov-perfect equilibrium in a resource technology, Discussion Paper 6/1986, Norwegian School of Economics and Business Administration, Bergen (A version with descrete time analysis is published in the Review of Economic Studies 55, 469–483, Chap. 6 of the present volume) Asheim, G.B. (1994), Net national product as an indicator of sustainability, Scandinavian Journal of Economics 96, 257–265 (Chap. 15 of the present volume) Asheim, G.B. (2004), Green national accounting with a changing population, Economic Theory 23, 601–619 (Chap. 19 of the present volume) Asheim, G.B., Buchholz, W. and Tungodden, B. (2001), Justifying sustainability, Journal of Environmental Economics and Management 41, 252–268 (Chap. 3 of the present volume) Asheim, G.B. and Weitzman, M. (2001), Does NNP growth indicate welfare improvement?, Economics Letters 73, 233–239 (Chap. 17 of the present volume) Burmeister, E. and Hammond, P.J. (1977), Maximin paths of heterogeneous capital accumulation and the instability of paradoxical steady states, Econometrica 45, 853–870 Cairns, R.D. (2000), Sustainability accounting and green accounting, Environment and Development Economics 5, 49–54 Dasgupta, P.S. (2001), Valuing objects and evaluating policies in imperfect economies, Economic Journal 111, C1–C29 Dasgupta, P.S. and Heal, G.M. (1974), The optimal depletion of exhaustible resources, Review of Economic Studies (Symposium), 3–28 Dasgupta, P.S. and Heal, G.M. (1979), Economic Theory and Exhaustible Resources. Cambridge University Press, Cambridge, UK Dasgupta, P.S. and Mäler, K.-G. (2000), Net national product, wealth, and social well-being, Environment and Development Economics 5, 69–93 Dixit, A., Hammond, P. and Hoel, M. (1980), On Hartwick’s rule for regular maximin paths of capital accumulation and resource depletion, Review of Economic Studies 47, 551–556 Hamilton, K. (2002), Savings, welfare, and rules for sustainability, The World Bank Hamilton, K. and Clemens, M. (1999), Genuine savings in developing countries, World Bank Economic Review 13, 333–356 Hartwick, J.M. (1977), Intergenerational equity and investing rents from exhaustible resources, American Economic Review 66, 972–974 Hartwick, J.M. (1990), National resources, national accounting, and economic depreciation, Journal of Public Economics 43, 291–304 Hicks, J. (1946), Value and capital, 2nd edition. Oxford University Press, Oxford Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995), Microeconomic Theory. Oxford University Press, Oxford Mitra, T. (2002), Intertemporal equity and efficient allocation of resources, Journal of Economic Theory 107, 356–376 Onuma, A. (2003), The broad stock of society’s capital and sustainability, Keio Economic Society DP No. 03-04 Pemberton, M. and Ulph, D. (2001), Measuring income and measuring sustainability, Scandinavian Journal of Economics 103, 25–40 Pezzey, J. (1994), Theoretical Essays on Sustainability and Environmental Policy, Ph.D. thesis, University of Bristol Rawls, J. (1971), A Theory of Justice. Harvard University Press, Cambridge, MA Samuelson, P. (1961), The evaluation of ‘social income’: Capital formation and wealth, in Lutz, F.A. and Hague, D.C. (eds.), The Theory of Capital, St. Martin’s Press, New York

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Sefton, J.A. and Weale, M.R. (1996), The net national product and exhaustible resources: The effects of foreign trade, Journal of Public Economics 61, 21–47 Solow, R.M. (1974), Intergenerational equity and exhaustible resources, Review of Economic Studies (Symposium), 29–45 Varian, H.R. (1992), Microeconomic Analysis. Norton, New York Weitzman, M.L. (1970), Aggregation and disaggregation in the pure theory of capital and growth: A new parable, Cowles Foundation Discussion Paper 292 Weitzman, M.L. (1976), On the welfare significance of national product in a dynamic economy, Quarterly Journal of Economics 90, 156–162 Weitzman, M.L. (2001), A contribution to the theory of welfare accounting, Scandinavian Journal of Economics 103, 1–23 Withagen, C. and Asheim, G.B. (1998), Characterizing sustainability: The converse of Hartwick’s rule, Journal of Economic Dynamics and Control 23, 159–165 (Chap. 11 of the present volume)

CHAPTER 19 GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

Abstract. Following Arrow et al. (2003), this paper considers green national accounting when population is changing and instantaneous well-being depends both on per capita consumption and population size. Welfare improvement is shown to be indicated by an expanded “genuine savings indicator,” taking into account the value of population growth, or by an expanded measure of real NNP growth. Under CRS, the measures can be related to the value of per capita stock changes and per capita NNP growth, using a result due to Arrow et al. (2003). The results are compared to those arising when instantaneous well-being depends only on per capita consumption.

1. INTRODUCTION How can welfare improvement be measured by national accounting aggregates when population is changing? The answer depends on whether a bigger future population for a given flow of per capita consumption leads to a higher welfare weights for people living at that time, or, alternatively, only per capita consumption mattes. When applying discounted utilitarianism to a situation where population changes exogenously through time, it seems reasonable to represent the instantaneous wellbeing of each generation by the product of population size and the utility derived from per capita consumption. This is the position of “total utilitarianism.” which has been endorsed to by, e.g., Meade (1955) and Mirrlees (1967), and which is the basic assumption in Arrow et al.’s (2003) study of savings criteria with a changing population. Within a utilitarian framework, the alternative position of “average utilitarianism,” where the instantaneous well-being of each generation depends only on per capita consumption, have been shown to yield implications that are not ethically defensible.1 However, if maximin is applied as a dynamic welfare criterion, then the instantaneous well-being of each generation will be represented by the utility derived from per capita consumption. Moreover, if the welfare criterion cares about sustainability (in the sense that current per capita utility should not exceed what is potentially sustainable), then it becomes important to compare the level of individual utility for different generations, irrespectively of how population size develops. Therefore,

Originally published in Economic Theory 23 (2004), 601–619. Reproduced with permission from Springer.

271 Asheim, Justifying, Characterizing and Indicating Sustainability, 271–289 c 2007 Springer


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utility derived from per capita consumption seems more relevant in a discussion of sustainability. Following a suggestion by Samuelson (1961, p. 52), the welfare analysis in the present paper does not presuppose a utilitarian framework, and allows for the possibility that a requirement of sustainability is imposed. At this level of generality, one cannot give a definite answer to the question of whether only per capita consumption matters. In Sects. 3–6 of this paper, I follow the basic assumption of Arrow et al. (2003) by letting – in the tradition of “total utilitarianism” – instantaneous well-being depend not only on per capita consumption, but also population size. Within a model with multiple consumption and capital goods, I derive four ways for indicating welfare improvement, which are generalized or novel results. In Sect. 7, I compare the results to those arising when instantaneous well-being depends only on per capita consumption, and not on population size. In line with Arrow et al. (2003), I treat population as a form of capital. Section 2 introduces the model, while Sect. 8 concludes. 2. MODEL Following Arrow et al. (2003), I assume that population N develops exogenously over time. The population trajectory {N (t)}∞ t=0 is determined by the growth function N˙ = φ(N ) and the initial condition N (0) = N 0 . Two special cases are exponential growth, φ(N ) = ν N , where ν denotes the constant growth rate, and logistic growth,
 φ(N ) = ν¯ N 1 − NN∗ , where ν¯ denotes the maximum growth rate, and N ∗ denotes the population size that is asymptotically approached. As mentioned by Arrow et al. (2003), the latter seems like the more acceptable formulation in a finite world. In general, denote by ν(N ) the rate of growth of population as a function of N , where ν(N ) = φ(N )/N . Denote by C = (C1 , . . . , Cm ) the nonnegative vector of commodities that are consumed. To concentrate on the issue of intertemporal distribution, I assume that goods and services consumed at any time are distributed equally among the population at that time. Thereby the instantaneous well-being for each individual may be associated with the utility u(c) that is derived from the per capita vector of consumption flows, c := C/N . Assume that u is a time-invariant, increasing, concave, and differentiable function. That u is time-invariant means that all variable determinants

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of current well-being are included in the vector of consumption flows. At any time, labor supply is assumed to be exogenously given and equal to the population size at that time. Denote by K = (K 1 , . . . , K n ) the nonnegative vector of capital goods. This vector includes not only the usual kinds of man-made capital stocks, but also stocks of natural resources, environmental assets, human capital, and other durable productive assets. Corresponding to the stock of capital of type j, K j , there is a net ˙ denotes the vector of net investment flow: I j := K˙ j . Hence, I = (I1 , . . . , In ) = K investments. The quadruple (C, I, K, N ) is attainable if (C, I, K, N ) ∈ C, where C is a convex and smooth set, with free disposal of consumption and investment flows. The set of attainable quadruples does not depend directly on time. I thus make an assumption of “green” or comprehensive accounting, meaning that current productive capacity depends solely on the vector of capital stocks and the population size. If C is a cone, then the technology exhibits constant returns to scale. An assumption of constant returns to scale will be imposed only in Sects. 6 and 7. Society makes decisions according to a resource allocation mechanism that assigns to any vector of capital stocks K and any population size N a consumptioninvestment pair (C(K, N ), I(K, N )) satisfying that (C(K, N ), I(K, N ), K, N ) is attainable.2 I assume that there exists a unique solution {K∗ (t)}∞ t=0 to the differ˙ ∗ (t) = I(K∗ (t), N (t)) that satisfies the initial condition K∗ (0) = ential equations K K0 , where K0 is given. Hence, {K∗ (t)} is the capital path that the resource allocation mechanism implements. Write C∗ (t) := C(K∗ (t), N (t)) and I∗ (t) := I(K∗ (t), N (t)). Say that the program {C∗ (t), I∗ (t), K∗ (t)}∞ t=0 is competitive if, at each t: 1.

(C∗ (t), I∗ (t), K∗ (t), N (t)) is attainable

2.

There exist present value prices of the flows of utility, consumption, labor input, and investment, (µ(t), p(t), w(t), q(t)), with µ(t) > 0 and q(t) ≥ 0, such that C1 C∗ (t) maximizes µ(t)u(C/N (t)) − p(t)C/N (t) over all C, ˙ C2 (C∗ (t), I∗ (t), K∗ (t), N (t)) maximizes p(t)C − w(t)N + q(t)I + q(t)K over all (C, I, K, N ) ∈ C.

Here C1 corresponds to utility maximization, while C2 corresponds to intertemporal profit maximization. Assume that the implemented program {C∗ (t), I∗ (t), K∗ (t)}∞ t=0 is competitive with finite utility and consumption values,  ∞  ∞ µ(t)N (t)u(C∗ (t)/N (t))dt and p(t)C∗ (t)dt exist, 0

0

and that it satisfies a capital value transversality condition, lim q(t)K∗ (t) = 0.

t→∞

(1)

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It follows that the implemented program {C∗ (t), I∗ (t), K∗ (t)}∞ t=0 maximizes  ∞ µ(t)N (t)u(C/N (t))dt 0

over all programs that are attainable at all times and satisfies the initial condition. Moreover, writing c∗ (t) := C∗ (t)/N (t), it follows from C1 and C2 that p(t) = µ(t)∇c u(c∗ (t)), ∂C(K∗ (t),

N (t))

(2) N (t))

, ∂N ∂N ˙ −q(t) = p(t)∇K C(K∗ (t), N (t)) + q(t)∇K I(K∗ (t), N (t)). w(t) = p(t)

+ q(t)

∂I(K∗ (t),

(3) (4)

3. WELFARE ANALYSIS Write U (K, N ) := N u(C(K, N )/N ) and U ∗ (t) := U (K∗ (t), N (t)) for the flow of total utility. In line with the basic analysis of Arrow et al. (2003), I assume for the next four sections that U ∗ (t) measures the social level of instantaneous well-being at time t. Assume that, at time t, society’s dynamic welfare is given by a Samuelson-Bergson welfare function defined over paths of total utility from time t to infinity, and that this welfare function does not depend on t. Moreover, assume that, for a given initial condition, the optimal path is time-consistent, and that society’s resource allocation mechanism implements the optimal path. If the welfare indifference surfaces in infinite-dimensional utility space are smooth, then, at time t, {µ(s}∞ s=t are local welfare weights on total utility flows at different times.3 Following a standard argument in welfare economics, as suggested by Samuelson (1961, p. 52) in the current setting, one can conclude that dynamic welfare is increasing at time t if and only if  ∞ µ(s)U˙ ∗ (s)ds > 0. (5) t

To show that this welfare analysis includes discounted total utilitarianism, assume for the rest of this paragraph only that society through its implemented program maximizes the sum of total utilities discounted at a constant rate ρ. Hence, the dynamic welfare of the implemented program at time t is  ∞ e−ρ(s−t) U ∗ (s)ds. t

Then the change in dynamic welfare is given by   ∞  d  ∞ −ρ(s−t) ∗ e U (s)ds = −U ∗ (t) + ρ e−ρ(s−t) U ∗ (s)ds dt t t  ∞ ρt =e e−ρs U˙ ∗ (s)ds, t

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where the second equality follows by integrating by parts. Hence, (5) follows by −ρt }∞ . setting {µ(t)}∞ t=0 = {e t=0 Turn now to the question of how to determine (5) by means of current prices and quantities. Since u is concave and differentiable, and C is a convex and smooth set, with free disposal of consumption flows, it follows that, at each t, U(t) := {(U, I, K)| U = N (t)u(C/N (t)) and (C, I, K, N (t)) ∈ C} is a convex and smooth set. Furthermore, it follows from C1 and C2 that, at each t, (U ∗ (t), I∗ (t), K∗ (t)) maximizes ˙ µ(t)U + q(t)I + q(t)K over all (U, I, K) ∈ U(t). In particular, ˙ −q(t) = µ(t)∇K U (K∗ (t), N (t)) + q(t)∇K I(K∗ (t), N (t)).

(6)

Denote by ψ(t) the marginal value of population growth, measured in present value terms. Since ψ(t) is measured in present value terms, the decrease of the value ˙ of population growth, −ψ(t), equals the marginal productivity of the population stock: ∂U (K∗ (t), N (t)) ∂I(K∗ (t), N (t)) ˙ −ψ(t) = µ(t) + q(t) + ψ(t)φ  (N (t)). (7) ∂N ∂N It follows from (2) and the definition of U (K , N ) that ∂C(K∗ (t), N (t)) ∂U (K∗ (t), N (t)) = v(t) + p(t) , ∂N ∂N
 where v(t) := µ(t) u(c∗ (t)) − ∇c u(c∗ (t))c∗ (t) denotes the marginal value of consumption spread, measured in present value terms. Hence, using (3), (7) can be rewritten as µ(t)

˙ −ψ(t) = v(t) + w(t) + ψ(t)φ  (N (t)),

(8)

and increasing N leads to three different kinds of marginal contributions: 1.

Consumption is spread on more people: v(t).

2.

Output increases: w(t).

3.

Population growth increases: ψ(t)φ  (N (t)).

By combining (6) and (7), one obtains
 µU˙ ∗ = µ ∇K U · I∗ + ∂∂UN · φ(N )
∗

 ˙ ˙ + qI˙∗ + ψφ(N = − qI ) + ψ dtd φ(N ) = − dtd qI∗ + ψφ(N ) . Assuming that


 lim q(t)I∗ (t) + ψ(t)φ(N (t)) = 0

t→∞

(9)

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holds as an investment value/population growth value transversality condition, one arrives at the following result by integrating (9) and using (5) as an indicator of welfare improvement. Proposition 1: Dynamic welfare is increasing at time t if and only if q(t)I∗ (t) + ψ(t)φ(N (t)) > 0. This formally generalizes the “genuine savings indicator” to a case with population change by indicating welfare improvement by means of a positive value of net investments and population growth. However, while q can in principle be observed as market prices in a perfect market economy or calculated as efficiency prices provided the resource allocation mechanism implements an efficient program, one needs to consider how to calculate ψ. This question is posed in Sect. 4. 4. VALUE OF POPULATION GROWTH How can the marginal value of population growth, ψ(t), be calculated? Solving (8) and imposing lim ψ(t) = 0

t→∞

as a terminal condition yields  ψ(t) = t

∞

 φ(N (s))
φ(N (t)) v(s) + w(s) ds.

(10)

Hence, the value of population growth is the integral of an expression that consists of two factors, v(s) + w(s) and φ(N (s))/φ(N (t)). Let us investigate (10) by discussing these factors.

4.1. The Sign of v(s) + w(s) If one assumes that the value of consumption, pC∗ , exceeds the total functional share of labor, wN , then it follows from (2) and the definition of v that u(c∗ ) ≤ 0 is a sufficient condition for v + w to be negative. That u(c∗ ) is negative, means that instantaneous well-being is reduced if an additional person is brought into society and offered the existing per capita consumption flows.4 Note that, since u is increasing, u(c) is negative for vectors with small consumption flows. For example if c is one-dimensional and u(c) = ln c, then u(c) ≤ 0 if and only if c ≤ 1. Since per capita consumption and, thus, utility derived from per capita consumption increases with development, one obtains the conclusion that v + w < 0 is more likely to hold for less developed societies. Since w > 0, it is sufficient for v + w to be positive that v and, thus, u(c∗ ) − ∇c u(c∗ )c∗

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are nonnegative. That u(c∗ ) − ∇c u(c∗ )c∗ is positive, means that instantaneous wellbeing is increased if an additional person is brought into society even when the total consumption flows are kept fixed and must be spread on an additional person.5 Note that it follows from the concavity of u that u(c) − ∇c u(c)c is nondecreasing as c increases along a ray where different commodities are consumed
in fixed proportions.  For example if c is one-dimensional and u(c) = ln c, then d u(c) − u  (c)c /dc = 1/c, and u(c) − u  (c)c ≥ 0 if and only if c ≥ e. Since it is reasonable to assume that per capita consumption as well as the marginal productivity of labor increases with development, one obtains the conclusion that v + w > 0 is more likely to hold for more developed societies. Note that v and, thus, ψ are not invariant under an additive shift in the utility function (cf. Arrow et al., 2003, p. 224). 4.2. The Development of φ(N (s))/φ(N (t)) If there is constant absolute population growth at all future times, then φ(N (s))/φ(N (t)) equals 1 throughout and (10) simplifies to  ∞
 ψ(t) = v(s) + w(s) ds. t

If future absolute population growth is lower than the present – which occurs on the decreasing part of a logistic growth function – then φ(N (s))/ φ(N (t)) is smaller than 1 throughout and it holds that  ∞
 ψ(t) < v(s) + w(s) ds, t

provided that v(s) + w(s) > 0. If future absolute population growth is higher than the present – which occurs with exponential growth, entailing that the rate of growth of population, ν(N ) = φ(N )/N , is constant – then φ(N (s))/φ(N (t)) is greater than 1 throughout and it holds that  ∞
 ψ(t) > v(s) + w(s) ds, t

provided that v(s) + w(s) > 0. 4.3. Measuring the Value of Population Growth by the Present Value of Future Wages If it holds that the total value of the current population N (t) valued by ψ(t) is approximated the present value of future wages, then the total value of population growth, ψ(t)φ(t), can be approximated through multiplying the present value of future wages by the population growth rate, ν(N ) = φ(N )/N .

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To investigate the merits of such an approximation, note that −


 d(ψ N ) = − ψ˙ N + ψ N˙ dt
 = v + w + ψφ  (N ) N − ψφ(N )
 = wN + µ u(c∗ ) − ∇c u(c∗ )c∗ N + ν  (N )N ψ N ,

(11)

where φ  (N ) = d(ν(N )N )/d N = ν  (N )N + ν(N ) has been used to establish the last equality. Hence, the total value of the current population N (t) valued by ψ(t) can be expressed as follows:  ∞ ψ(t)N (t) = w(s)N (s)ds 

t



t

∞

+

∞

+


 µ(s) u(c∗ (s)) − ∇c u(c∗ (s))c∗ (s) N (s)ds

(12)

ν  (N (s))N (s)ψ(s)N (s)ds,

t

provided that the following population value transversality condition holds: lim ψ(t)N (t) = 0.

t→∞

(13)

In the special case where u is homogeneous of degree 1, it follows that u(c∗ ) − ∇c u(c∗ )c∗ = 0 throughout, so that in (12) the second term on the rhs. is equal to zero. In the special case where growth is exponential so that the growth rate ν(N ) is constant, it follows that ν  (N ) = 0 throughout, so that in (12) the third term on the rhs. is equal to zero. Hence, a linearly homogeneous u combined with exponential population growth are sufficient for the total value of population growth, ψ(t)φ(t), to be equal to  ∞ w(s)N (s)ds. (14) ν(N ) · t

In a developed society one would expect that  ∞
 µ(s) u(c∗ (s)) − ∇c u(c∗ (s))c∗ (s) N (s)ds > 0, t

while



∞

ν  (N (s))N (s)ψ(s)N (s)ds < 0,

t

since exponential growth cannot be maintained indefinitely. Hence, in a developed society, the two additional terms in (12) go in different directions, implying that it cannot easily be determined whether (14) over- or underestimates the total value of population growth, ψ(t)φ(t).

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5. REAL NNP GROWTH AS A WELFARE INDICATOR To investigate to what extent real NNP growth indicates welfare improvement in the presence of a changing population, I follow Asheim and Weitzman (2001) and Sefton and Weale (2006) by using a Divisia consumption price index when expressing comprehensive NNP in real prices. The application of a price index {π(t)} turns the present value prices {p(t), q(t)} into real prices {P(t), Q(t)}, P(t) = p(t)/π(t) Q(t) = q(t)/π(t), implying that the real interest rate, R(t), at time t is given by ˙ (t) R(t) = − ππ(t) .

A Divisia consumption price index satisfies ∗ (t) ˙ π˙ (t) p(t)C = , π(t) p(t)C∗ (t)

˙ ∗ = 0: implying that PC ˙ ∗= PC

d dt

p π

C∗ =

˙ ∗ − π˙ pC∗ π pC = 0. π2

Define comprehensive NNP in real Divisia prices, Y (t), as the sum of the real value of consumption and the real value of net investments: Y (t) := P(t)C∗ (t) + Q(t)I∗ (t). Define likewise real prices for utility, consumption spread, and population growth: M(t) = µ(t)/π(t) V (t) = v(t)/π(t) (t) = ψ(t)/π(t). Since ˙ ˙ Q(t) = q(t)/π(t) + R(t)Q(t) ˙ ˙ (t) = ψ(t)/π(t) + R(t)(t), it follows from (9) that M U˙ ∗ +

d dt


∗ 
 QI + φ(N ) = R QI∗ + φ(N ) .

(15)

280

GEIR B. ASHEIM


Moreover, keeping in mind that U ∗ = N u(c∗ ), V = M u(c∗ ) − ∇c u(c∗ )c∗ , P = ˙ ∗ = 0, one obtains M∇c u(c∗ ), and PC  
 ˙∗ M U˙ ∗ = M u(c∗ ) − ∇c u(c∗ )c∗ φ(N ) + ∇c u(c∗ )C (16)
∗ d = V φ(N ) + dt PC . Hence, by combining (15) and (16), it follows that

 Y˙ + V φ(N ) + dtd φ(N ) = R QI∗ + φ(N ) . In view of Proposition 1, this leads to the following result. Proposition 2: Dynamic welfare is increasing at time t if and only if
 Y˙ (t) + V (t)φ(N (t)) + dtd (t)φ(N (t)) > 0, provided that the real interest rate, R(t), is positive. I end this section by discussing the following question: If national accountants can estimate the “genuine savings indicator,” QI∗ , and real growth in comprehensive NNP, Y˙ , but not the terms that capture the welfare effects of population change, which of QI∗ and Y˙ is the better indicator of welfare improvement? If one assumes that, in a more developed society:

r Absolute population growth, φ(N ), is positive but decreasing toward zero r The marginal value of consumption spreading, V , is positive, entailing that also the marginal value of population growth, , is positive

then it follows that both V φ(N ) and φ(N ) are positive, but eventually decreasing. As the society is getting near to having population saturated at N ∗ (if a logistic growth function is followed),
then V φ(N ) > 0 due to a positive value of consumption spread, while dtd φ(N ) < 0 since population growth is decreasing ˙ toward zero.
 Hence, when using Proposition 2 and approximating Y + V φ(N ) + d ˙ dt φ(N ) by Y , one would be missing two terms with opposite signs. On the other hand, when using Proposition 1 and approximating QI∗ + φ(N ) by QI∗ , one would be missing one term with a positive sign. Thus, if it is impractical or impossible to calculate terms involving the value of population change, then real NNP growth, Y˙ , may be an interesting alternative to the “genuine savings indicator,” QI∗ , as an approximate indicator of welfare improvement. In the present section I have considered real growth in total NNP, not real growth in per capita NNP. To be able to analyze per capita measures – as I will do in section the next – one needs to impose an assumption of constant returns to scale. One can, however, use (16) to make the following observation:
 ∗ )/dt M U˙ ∗ = Mu(c∗ )φ(N ) + PC∗ · d(PC − ν(N ) . PC∗

GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

281

Hence, if u(c∗ ) is nonnegative – so that instantaneous well-being is not decreased if an additional person is brought into society and offered the existing per capita consumption flows – and real per capita consumption increases throughout, then it follows from (5) that dynamic welfare is improving. 6. PER CAPITA MEASURES In this section I consider two per capita measures: “value of net changes in per capita stocks” and “real growth in per capita NNP.” These will be considered in turn in each of the two following sections. In both sections I impose the additional assumption of constant returns to scale. 6.1. Value of Net Changes In Per Capita Stocks Denote by k∗ (t) := K∗ (t)/N (t) the vector of per capita capital stocks. Since k˙ ∗ =

˙∗ K N

−

N˙ K∗ N N

=

I∗ N

− ν(N )k∗

and φ(N ) = ν(N )N , it follows that
 qI∗ + ψφ(N ) = qk˙ ∗ + ν(N ) qk∗ + ψ N (cf. Arrow et al., 2003, p. 222).
Hence, by Proposition 1, welfare is increasing at time t if and only if qk˙ ∗ + ν(N ) qk∗ + ψ > 0. This means that dynamic welfare can be improving even if the
value netchanges in per capita stocks, qk˙ ∗ , is negative, provided that the term ν(N ) qk∗ + ψ is sufficiently positive. How can qk∗ + ψ be calculated? To allow analysis of this question – in particular, to derive a generalized version of Arrow et al.’s (2003) Theorem 2 – one must impose constant returns to scale by assuming that C is a convex cone. Then it follows directly from C2 that, at each t, ∗ ˙ p(t)C∗ (t) − w(t)N (t) + q(t)I∗ (t) + q(t)K (t) = 0,

(17)

or, equivalently, 
d q(t)K∗ (t) = p(t)C∗ (t) − w(t)N (t). − dt

(18)

This means that the value of capital equals the present value of the difference between the value of consumption and the functional share of labor,  ∞
 q(t)K∗ (t) = p(s)C∗ (s) − w(s)N (s) ds, t

provided that (1) holds as a capital value transversality condition.

282

GEIR B. ASHEIM

It now follows from (2), (11), and (18) that
 d(ψ N ) = − pC∗ − wN + µu(c∗ )N + ν  (N )N ψ N t
 d(qK∗ ) + N µu(c∗ ) + ν  (N )ψ N . = dt or, equivalently,

 d qK∗ + ψ N − = N µu(c∗ ) + ν  (N )ψ N . dt If (1) and (13) hold as transversality conditions, then  ∞
 q(t)K∗ (t) + ψ(t)N (t) = N (s) µ(s)u(c∗ (s)) + ν  (N (s))ψ(s)N (s) ds, −

t

and, by dividing by N (t), ∗



∞

q(t)k (t) + ψ(t) = t

 N (s)
∗  N (t) µ(s)u(c (s)) + ν (N (s))ψ(s)N (s) ds.

(19)

By differentiating both sides of (19) w.r.t. time, one obtains 
d qk∗ + ψ = µu(c∗ ) + ν  (N )ψ N + ν(N )qk∗ + ν(N )ψ − dt
 = µu(c∗ ) − ν  (N )qK∗ + φ  (N ) qk∗ + ψ , where I have followed Arrow et al. (2003) by using φ  (N ) = ν  (N )N + ν(N ) to establish the last equality. Integrating this yields  ∞  φ(N (s))
∗  ∗ q(t)k∗ (t) + ψ(t) = (20) φ(N (t)) µ(s)u(c (s)) − ν (N (s))q(s)K (s) ds. t

In light of Proposition 1, (19) and (20) lead to the following result: Proposition 3 (Arrow et al., 2003, Theorem 2): Assuming constant returns to scale, dynamic welfare is increasing at time t if and only if
 q(t)k˙ ∗ (t) + ν(N (t)) q(t)k∗ (t) + ψ(t) > 0, where



∗

q(t)k (t) + ψ(t) =

∞

 N (s)
∗  N (t) µ(s)u(c (s)) + ν (N (s))ψ(s)N (s) ds

∞

 φ(N (s))
∗  ∗ φ(N (t)) µ(s)u(c (s)) − ν (N (s))q(s)K (s) ds.

t

 =

t

u(c∗ )

is nonnegative – so that instantaneous well-being is not decreased if Hence, if an additional person is brought into society and offered the existing per capita consumption flows – and ν  (N ) < 0 – so that the rate of growth of population decreases

GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

283

as population increases, then qk∗ + ψ > 0, meaning that welfare improvement is possible even if the value of net changes in per capita stocks is negative. In the case of exponential population growth – so that the rate of growth of population is constant – it follows that ν  (N ) = 0, N (s)/N (t) = φ(N (s))/ φ(N (t)), and  ∞ N (s) ∗ q(t)k∗ (t) + ψ(t) = N (t) µ(s)u(c (s))ds. t

6.2. Real Growth in Per Capita NNP In the previous section I have translated the result on the generalized “genuine savings indicator” in a setting where there is population growth, Proposition 1, into a finding, Proposition 3, stated in per capita terms. In this section I do the same for Proposition 2 by translating a result on real growth in total NNP into a finding stated in terms of real growth in per capita NNP. The obtained result will be reported below as Proposition 4. Also in this section I impose the additional assumption of constant returns to scale. Write y(t) := Y (t)/N (t) for real per capita NNP and i∗ (t) := I∗ (t)/N (t) for the vector of per capita investment flows. It follows from (17) that
 ˙ k∗ y = Pc∗ + Qi∗ = W + RQ − Q ˙ with W (t) = w(t)/π(t) denoting the real wage rate. More˙ since −q/π = RQ − Q, over, y˙ =

Y˙ N

−

N˙ Y N N

=

Y˙ N

− ν(N )y

Hence, by combining these two observations it follows that
 Y˙ ˙ ∗ N = y˙ + ν(N )W + ν(N ) RQ − Q k .

(21)

˙ ˙ one obtains ˙ = R − , Likewise, since −ψ/π = V + W + φ  (N ) and −ψ/π ˙ = R. (22) V + W + φ  (N ) + 
 The stage is now set for expressing Y˙ + V φ(N ) + dtd φ(N ) in per capita terms and thus, derive a fourth expression that indicates welfare improvement: Using (21) and (22) it follows through tedious but straightforward calculations that

∗  1 ˙ d ˙ ∗ N Y + V φ(N ) + dt (φ(N )) = y˙ − ν(N )Qk + ν(N )R Qk +  . On the basis of Props. 2 and 3, this leads to the following result. Proposition 4: Assuming constant returns to scale, dynamic welfare is increasing at time t if and only if
 ∗ ˙ (t) + ν(N (t))R(t) Q(t)k∗ (t) + (t) > 0, y˙ (t) − ν(N (t))Q(t)k

284

GEIR B. ASHEIM

provided that the real interest rate, R(t), is positive, where Q(t)k∗ (t) + (t) =

 

∞

 N (s)
∗  N (t) M(s)u(c (s)) + ν (N (s))(s)N (s) ds

∞

 φ(N (s))
∗  ∗ φ(N (t)) M(s)u(c (s)) − ν (N (s))Q(s)K (s) ds.

t

= t

In the setting of a one-sector model like the one considered by Arrow et al. (2003) – where consumption, investment, and capital are all one-dimensional, and output is split between consumption and investment – the anticipated capital gains are zero: Q˙ = 0. Under this simplifying assumption one can draw the following conclusion from Proposition 4: If a positive interest R, a nonnegative u(c∗ ),
rate ∗ and a positive and decreasing ν(N ) lead to ν(N )R Qk +  being positive, then welfare improvement is possible even if real per capita NNP is decreasing.

7. WELFARE WHEN ONLY PER CAPITA CONSUMPTION MATTERS In this section I investigate how the welfare analysis will change if I – instead of letting total utility, N (t)u(c∗ (t)), constitute the instantaneous well-being at time t – let instantaneous well-being at time t depend only on the utility derived from the vector of per capita consumption flows, u(c∗ (t)), but not on the size of the population, N (t). This is the underlying assumption made by, e.g., Hamilton (2002). The following is a straightforward adaptation of the analysis of Sect. 3. Write U˜ (K, N ) := u(C(K, N )/N ) and U˜ ∗ (t) := U˜ (K∗ (t), N (t)) for the flow of per capita utility. In the alternative welfare analysis, U˜ ∗ (t) measures the social level of instantaneous well-being at time t, and dynamic welfare is increasing at time t if and only if  ∞ µ(s) ˜ U˙˜ ∗ (s)ds > 0, (23) t

where, for each t, µ(t) ˜ = µ(t)N (t). In analogy to the demonstration in Sect. 3, it can be shown that this welfare analysis includes discounted utilitarianism, where society through its implemented program maximizes the sum of per capita utilities discounted at a constant rate ρ. However, as pointed out by Dasgupta (2001b, Sect. 6.4), it is hard to offer an ethical defense for such discounted average utilitarianism. The framework of Asheim and Buchholz (2004) can, however, be used show how the welfare analysis of Sect. 3 applies also to cases like maximin, where society through its implemented program maximizes infimum of per capita utilities, and welfare criteria that impose sustainability as a constraint. In such cases, it seems appropriate to adopt the assumption of the present section and let instantaneous wellbeing depend only on per capita consumption, and not on population size (see also Pezzey, 2004, Sect. 4).

GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

285

Since u is concave and differentiable, and C is a convex and smooth set, with free disposal of consumption flows, it follows that, at each t, ˜ U(t) := {(U˜ , I, K)| U˜ = u(C/N (t)) and (C, I, K, N (t)) ∈ C} is a convex and smooth set. Furthermore, it follows from C1 and C2 that, at each t, (U˜ ∗ (t), I∗ (t), K∗ (t)) maximizes ˙ µ(t) ˜ U˜ + q(t)I + q(t)K ˜ over all (U˜ , I, K) ∈ U(t). In particular, ˜ (K∗ (t), N (t)) + q(t)∇K I(K∗ (t), N (t)). ˙ −q(t) = µ(t)∇ ˜ KU

(24)

˜ Denote by ψ(t) is the marginal value of population growth, measured in present value terms, under the alternative welfare analysis. It follows that ∂ U˜ (K∗ (t), N (t)) ∂I(K∗ (t), N (t))  ˜ + q(t) + ψ(t)φ (N (t)) ∂N ∂N  ˜ (N (t)), = v(t) ˜ + w(t) + ψ(t)φ

˙˜ −ψ(t) = µ(t) ˜

(25)

where the second equality follows from (2), (3), and the definition of U˜ (K , N ), with v(t) ˜ := − p(t)c∗ (t) denoting the marginal value of consumption spread, measured in present value terms, under the alternative welfare analysis. Hence, increasing N leads to three different kinds of marginal contributions: 1.

Consumption is spread on more people: v(t) ˜

2.

Output increases: w(t)

3.

 (N (t)) ˜ Population growth increases: ψ(t)φ

By combining (24) and (25), one obtains
 ˜ µ˜ U˜˙ ∗ = µ˜ ∇K U˜ · I∗ + ∂∂ UN · φ(N )
∗ 

˙˜ ˜ ˙ + qI˙∗ + ψφ(N = − qI ) + ψ˜ dtd φ(N ) = − dtd qI∗ + ψφ(N ) . Assuming that

(26)


 ˜ lim q(t)I∗ (t) + ψ(t)φ(N (t)) = 0

t→∞

holds as an investment value/population growth value transversality condition, one arrives at the following result by integrating (26), and using (23) as an indicator of welfare improvement. Proposition 5: Dynamic welfare is increasing at time t if and only if ˜ q(t)I∗ (t) + ψ(t)φ(N (t)) > 0.

286

GEIR B. ASHEIM

Solving (25) and imposing ˜ =0 lim ψ(t)

t→∞

as a terminal condition yields ˜ ψ(t) =−



∞

t

 φ(N (s))
∗ φ(N (t)) p(s)c (s) − w(s) ds.

It follows that the marginal value of population growth is negative, provided that the value of consumption, pC∗ , exceeds the total functional share of labor, wN . This reflects that population growth means that society’s assets must be shared among more people, while – under this alternative welfare analysis – there is no countervailing effect. Note that ψ˜ is invariant under an additive shift in the utility function. By repeating the analysis of Sect. 5, it follows that dynamic welfare is increasing at time t if and only if
 ˜ Y˙ (t) + V˜ (t)φ(N (t)) + dtd (t)φ(N (t)) > 0, provided that the real interest rate, R(t), is positive, where, under the alternative welfare analysis, V˜ (t) = v(t)/π(t) ˜ is the marginal value of consumption spread, ˜ ˜ and (t) = ψ(t)/π(t) is the marginal value of population growth, measured in real terms. Adapting the analysis of Sect. 6.1 to the welfare criterion considered in the present section, yields the following result: Proposition 6: Assuming constant returns to scale, dynamic welfare is increasing at time t if and only if
 ˜ q(t)k˙ ∗ (t) + ν(N (t)) q(t)k∗ (t) + ψ(t) > 0, where ˜ q(t)k∗ (t) + ψ(t) =



∞ t



=− t

N (s)  ˜ N (t) ν (N (s))ψ(s)N (s)ds ∞

φ(N (s))  ∗ φ(N (t)) ν (N (s))q(s)K (s)ds.


If a positive and decreasing ν(N ) leads to ν(N ) qk∗ + ψ˜ being positive (since ν  (N ) < 0, and keeping in mind that ψ˜ < 0), then welfare improvement is possible even if the value of net changes in per capita stocks is negative. This result has a clear intuitive interpretation: When the rate of population growth is decreasing, it is not necessary for the current generation to compensate fully for current population growth in order for dynamic welfare to be nondecreasing. Since ν  (N ) = 0 if population growth is exponential, one obtains as a corollary the following result, shown by Hamilton (2002) under discounted average utilitarianism and by Dasgupta (2001b, p. 258) under “dynamic average utilitarianism.”6

GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

287

Proposition 7: Assuming constant returns to scale and exponential population growth, dynamic welfare is increasing at time t if and only if q(t)k˙ ∗ (t) > 0. Note that this result does not entail that dynamic welfare is increasing if and only if real per capita wealth is increasing, since time-differentiating real per capita wealth, QK∗ /N , yields d(QK∗ /N ) qk˙ ∗ ˙ ∗, = + Qk dt π ˙ ∗ = 0. However, since where it does not follow from our assumptions that Qk  ˙∗ qk˙ ∗ QK QK∗
QI∗ N˙ QK∗ π = N − N N = N QK∗ − ν , it follows that qk˙ ∗ can be signed by comparing the ratio of the value of net investments and wealth with the rate of growth of population.7 By repeating the analysis of Sect. 6.2, it follows that

 ˙ ∗ + ν(N )R Qk∗ +  ˜ = R Qk˙ ∗ + ν(N )(Qk∗ + ) ˜ . y˙ − ν(N )Qk (27) Hence, Proposition 6 implies that dynamic welfare is increasing at time t if and only if
 ∗ ˙ ˜ y˙ (t) − ν(N (t))Q(t)k (t) + ν(N (t))R(t) Q(t)k∗ (t) + (t) > 0, provided that the real interest rate, R(t), is positive. Define z(t) by z(t) := P(t)c∗ (t) + Q(t)k˙ ∗ (t). Since y = Pc∗ + Qi∗ and k˙ ∗ = i∗ − ν(N )k∗ , it follows that
 ˙ ∗ − ν(N )Qk˙ ∗ − ν  (N )φ(N )Qk∗ . z˙ = dtd y − ν(N )Qk∗ = y˙ − ν(N )Qk

(28)

By combining (27) and (28) with an assumption of exponential population growth ˜ = 0), it follows that (so that ν  (N ) = 0 and Proposition 6 implies Qk∗ + 
 z˙ (t) = R(t) − ν(N (t)) Q(t)k˙ ∗ (t), which by Proposition 7 means that the following result is obtained. Proposition 8: Assuming constant returns to scale and exponential population growth, dynamic welfare is increasing at time t if and only if z˙ (t) > 0, provided that the real interest rate net of population growth, R(t) − ν(N (t)), is positive. It is important to observe that z˙ is not real growth in per capita NNP; rather, z˙ is real growth in the sum of the value of per capita consumption and the value of net changes in per capita stocks.

288

GEIR B. ASHEIM

8. CONCLUDING REMARKS In this paper, I have followed a standard argument in welfare economics – which was suggested by Samuelson (1961, p. 52) in the current setting – and identified welfare improvement with  ∞
 d N (s)u(c∗ (s)) ds > 0, µ(s) ds t

if both population size and per capita consumption contribute to instantaneous wellbeing, or,  ∞ 
∗ d µ(s) ˜ ds u(c (s)) ds > 0. t

if only per capita consumption matters. In each case, the analysis encompasses discounted utilitarianism (which, however, seems more defendable in the former case). Through Props. 1–8 I have established eight ways to indicate welfare improvement, depending on which welfare criterion is adopted, on whether population growth is exponential, and on whether the technology exhibits constant returns to scale. Thereby, this paper offers a substantial generalization and extension of Arrow et al.’s (2003) analysis and results. These two cases are of interest for different reasons: Following Arrow et al. (2003, p. 221), there are strong arguments in favor of associating instantaneous well-being with total utility, N u(c∗ ), when investigating whether utilitarian welfare is increasing over time. However, in line with Pezzey (2004), one might argue that per capita utility, u(c∗ ), is more relevant in a discussion of sustainability. This would mean that development is said to be sustainable at the current time, if the current level of individual utility derived from per capita consumption flows, u(c∗ ), can potentially be sustained indefinitely. If per capita utility, u(c∗ ), is sustained throughout, so that d(u(c∗ ))/dt ≥ 0 at all times, then it follows directly from the criterion for welfare improvement that welfare is nondecreasing, in the case where only per capita consumption matters for instantaneous well-being. However, the converse implication does not hold: Nondecreasing welfare does not imply that current instantaneous well-being can potentially be maintained forever. In fact, it can be shown (cf. Pezzey, 2004, Sect. 4) – under the ∞ is an exponentially decreasing function – that nondecreasing provision that {µ(t)} ˜ t=0 welfare is a necessary, but not sufficient, condition for sustainable per capita utility. Acknowledgments: This paper is inspired by the recent investigation of the genuine savings criterion and the value of population by Arrow et al. (2003). I thank Kenneth Arrow, Partha Dasgupta, Lawrence Goulder, and a referee for helpful discussions and comments. I gratefully acknowledge the hospitality of the Stanford University research initiative on the Environment, the Economy and Sustainable Welfare, and financial support from the Hewlett Foundation through this research initiative.

GREEN NATIONAL ACCOUNTING WITH A CHANGING POPULATION

289

NOTES 1 See Dasgupta (2001b, Sect. 6.4) for a discussion of the deficiency of “average utilitarianism.” Dasgupta

(2001b, p. 100) suggests “dynamic average utilitarianism” as an alternative, where discounting seems to arise due to an exogenous and constant per-period probability of extinction. Since the present paper abstracts from any kind of uncertainty, this alternative criterion will not be discussed here. 2 This is inspired by Dasgupta (2001a, p. C20) and Dasgupta and Mäler (2000). 3 By identifying the social level of instantaneous well-being at time t with U ∗ (t), I assume that there are stable welfare indifference surfaces in infinite-dimensional space when the well-being of each generation is measured by total utility, irrespectively of how consumption flows and population size develop. Discounted total utilitarianism leads to linear indifference surfaces in this space. 4 Observe that u has not been normalized to satisfy u(0) = 0. The analysis allows for the possibility that there are per capita consumption flows, c ≥ 0, such that u(c) < 0. Cf. the concepts of “well-being subsistence,” as discussed by Dasgupta (2001b, Chapter 14) in the tradition of Meade (1955) and Dasgupta (1969), and “critical-level utilitarianism,” as proposed by Blackorby and Donaldson (1984). 5 As discussed by Dasgupta (2001b, Chapter 14), the condition u(c) − ∇ u(c)c = 0 is important in c classical utilitarian theories of optimal population; see also Meade (1955) and Dasgupta (1969). 6 Cf. Note 1. “Dynamic average utilitarianism” (as introduced by Dasgupta, 2001b, pp. 100 and 258) coincides with discounted average utilitarianism when population growth is exponential. 7 I am grateful to Partha Dasgupta for making this observation.

REFERENCES Arrow, K., Dasgupta, P.S., and Mäler, K.-G. (2003), The genuine savings criterion and the value of population, Economic Theory 21, 217–225 Asheim, G.B. and Buchholz, W. (2004), A general approach to welfare measurement through national income accounting, Scandinavian Journal of Economics 106, 361–384 (Chap. 18 of the present volume) Asheim, G.B. and Weitzman, M. (2001), Does NNP growth indicate welfare improvement?, Economics Letters 73, 233–239 (Chap. 17 of the present volume) Blackorby, C. and Donaldson, D. (1984), Social criteria for evaluating population change, Journal of Public Economics 25, 13–33 Dasgupta, P.S. (1969), On the concept of optimum population, Review of Economic Studies 36, 294–318 Dasgupta, P.S. (2001a), Valuing objects and evaluating policies in imperfect economies, Economic Journal 111, C1–C29 Dasgupta, P.S. (2001b), Human Well-Being and the Natural Environment. Oxford University Press, Oxford Dasgupta, P.S. and Mäler, K.-G. (2000), Net national product, wealth, and social well-being, Environment and Development Economics 5, 69–93 Hamilton, K. (2002), Saving effort and population growth: theory and measurement, The World Bank Meade, J.E. (1955), Trade and Welfare. Oxford University Press, Oxford Mirrlees, J.A. (1967), Optimal growth when the technology is changing, Review of Economic Studies (Symposium) 34, 95–124 Pezzey, J. (2004), One-sided sustainability tests with amenities, and changes in technology, trade and population, Journal of Environment Economics and Management 48, 613–631 Samuelson, P. (1961), The evaluation of ‘social income’: Capital formation and wealth, in Lutz, F.A. and Hague, D.C. (eds.), The Theory of Capital, St. Martin’s Press, New York Sefton, J.A. and Weale, M.R. (2006), The concept of income in a general equilibrium, Review of Economic Studies 73, 219–249

INDEX

Cake-eating economy/technology, 108, 109, 119, 126, 246 Capital gains, 8, 11, 155, 156–162, 164, 206–210, 226, 227, 233–236, 284 Characterizing sustainability, 3, 7 Classical utilitarianism, 21–24, 27, 30 Closed economy(ies), 147, 149, 151, 155, 156, 215, 219, 232, 234 Competitive equilibrium, 10, 11, 143, 153, 157, 158, 217, 219, 220, 234 Competitive path, 128, 132, 134, 139–142, 193, 217, 219, 237 Constant population, 7–10, 71, 105, 127, 130–133, 135, 137–140, 142, 147, 148, 155–158, 164, 215, 216, 219, 227, 231, 266 Dasgupta-Heal-Solow (DHS) model/technology, 6, 20, 41, 42, 44, 47, 64, 70, 71, 73–76, 79, 80, 126, 127, 133–135, 138–140, 143, 144, 203, 206, 208, 228, 232, 234, 250, 251, 258, 267 Discounted utilitarianism, 5–7, 9, 12, 13, 14, 20–24, 27–31, 35, 37, 47, 48, 63, 65, 66–70, 72, 74–76, 180, 199, 250–257, 259, 262, 267, 271, 284, 288 Distributional conflicts (between generations), 29, 46, 48, 53–55 Dynamic welfare, 6, 7, 11, 12, 197, 199, 201, 202, 204, 206, 241, 242, 250, 252, 254, 257, 258, 262, 263, 271, 274, 276, 280–287 Efficiency condition, 4, 25, 28, 30, 260 Environmental resources, 2, 3, 7, 8, 10, 11, 127, 129, 155, 162, 164, 165, 173, 200, 204, 209, 216, 222, 225 Equal treatment (of generations), 4, 5, 21, 25, 34–36, 53, 65 Equity condition, 4, 5, 25, 54, 55, 56 Ethical preferences, 34, 36, 46, 47, 49 Generalized Hartwick rule, 132, 139, 140, 143, 148, 166

Genuine savings indicator, 201, 276, 280, 283 Hartwick rule, 125–129, 131–133, 139–143, 147–151, 156, 159, 166 Hotelling rule, 125, 126, 134, 135, 137, 138, 148, 151, 165, 217, 259 Indicating sustainability, 3, 205 Indicator of sustainability, 10, 11, 139, 155, 162, 164, 203, 211, 215–219, 222, 231 Intergenerational equity, 9, 126, 133, 180 Intergenerational justice, 1, 4, 5–7, 19–21, 24–26, 30, 35, 48, 54, 57, 58, 74, 83, 103, 104, 133 Intragenerational justice, 1 Justifying sustainability, 3, 48 Leximin, 5–7, 26, 27, 30, 31, 46, 47, 53, 54, 56–58 Man-made capital, 2, 3, 6, 7, 10, 38, 40–42, 47, 64, 70, 126, 127, 133–135, 137–142, 172, 173, 180, 182, 198, 215, 216, 219, 222, 234, 235, 251, 258–260, 273 Maximin, 5, 22, 23, 26, 30, 83, 84, 86–88, 91–93, 114, 119, 121, 148, 151, 159, 162, 180–184, 188, 192, 193, 194, 200, 228, 250, 253, 254, 257, 271, 284 National accounting, 11, 162, 164, 197, 198, 200, 202, 212, 242, 250, 254, 266, 267, 271 Net national product (NNP), 155, 215, 225, 244, 249 Natural resources, 2, 3, 7, 8, 10, 11, 21, 34, 42, 71, 126, 165, 198, 216, 217, 242, 251, 273 Natural capital, 7, 9, 10, 40, 42, 126, 127, 133, 139, 140–144, 160, 198, 200, 204, 215, 216, 219, 222, 242, 251, 252, 260

291

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INDEX

Open economy(ies), 8, 143, 147, 151, 152, 156, 158, 164, 225, 231, 232, 234, 236 Population growth, 12, 31, 71, 212, 251, 275–280, 283, 285–289 Productive technologies, 5, 66, 103, 111–114, 119, 120 Ramsey model/technology, 72, 75, 79, 81, 131–133, 139–141, 172 Real prices, 201, 202, 205, 206, 245, 246, 258, 263, 279 Resource allocation mechanism, 199, 201, 202, 212, 252, 253, 255, 256, 260–265, 267, 273, 274, 276 Social choice theory, 4, 24, 34, 49 Social preferences, 24–26, 28–30, 34–37, 42–45, 47–49, 54, 55, 65, 66, 74, 76, 199, 203, 211, 252, 253, 255, 256, 261, 263, 267 Stationary equivalent, 203, 226, 229, 230, 241, 253 Stationary technology, 7–10, 74, 85, 114, 117, 147, 148, 151, 165, 172–174, 176, 215, 216, 219

Subjective preferences, 6, 7, 105, 113, 114, 121 Sustainability accounting, 13, 197, 199, 211 Sustainable development, 1–7, 20, 29, 64, 65, 131, 143, 155, 203, 207, 218, 219, 266 Sustainable income, 8, 209, 230, 231 Sustainable paths/streams, 34, 40, 43, 48, 263, 267 Sustained development, 2, 49 Technological progress, 2, 12, 21, 71, 129, 144, 156, 164, 165, 174, 176, 200, 226, 231–232, 236–239, 251 Terms-of-trade, 8, 143, 156, 162, 164, 207, 208, 251 Undiscounted utilitarianism, 5–7, 23, 35, 37, 65–70, 74, 76, 259, 267 Value of net investments, 7–12, 126, 127, 129, 130 Welfare improvement, 11, 12, 202, 207, 210, 245, 250–252, 254, 256, 258, 263, 276, 279, 283, 286, 288 Welfare accounting, 210, 211

Sustainability, Economics, and Natural Resources 1. Kant, Shashi, and Berry, R. Albert (Eds.). 2005. Economics, Sustainability, and Natural Resources: Economics of Sustainable Forest Management. ISBN 978-1-4020-3465-7 2. Kant, Shashi, and Berry, R. Albert (Eds.). 2005. Institutions, Sustainability, and Natural Resources: Institutions for Sustainable Forest Management. ISBN 978-1-4020-3479-4 3. Asheim, Geir B. 2007. Justifying, Characterizing, and Indicating Sustainability. ISBN 978-1-4020-6199-8

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